Evo-SETI: Life Evolution Statistics on Earth and Exoplanets 3030519309, 9783030519308

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Claudio Maccone

Evo-SETI Life Evolution Statistics on Earth and Exoplanets

Evo-SETI

Claudio Maccone

Evo-SETI Life Evolution Statistics on Earth and Exoplanets

123

Claudio Maccone Istituto Nazionale di Astrofisica (INAF) Rome, Italy International Academy of Astronautics (IAA) Paris, France

ISBN 978-3-030-51930-8 ISBN 978-3-030-51931-5 https://doi.org/10.1007/978-3-030-51931-5

(eBook)

© Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

This book is about the evolution of life on Earth as well as on exoplanets, where Alien Civilizations might live. SETI, (Search for Extraterrestrial Intelligence), has been trying to detect Alien Civilizations scientifically since 1959. Evo-SETI (Evolution and SETI) is thus the appropriate title for this book, merging Evolution and SETI in a strong mathematical framework. But fear not if this book is “too mathematical”! We will take you step-by-step to understand it. And it all started back in the 1980s, when this author watched for the first time the TV series “Life on Earth: A Natural History” by David Attenborough (https://en.wikipedia.org/wiki/ Life_on_Earth_(TV_series)). At that time, this author was a mathematical physicist in his forties, and he then decided to cast the Evolution of Life on Earth into mathematical equations. It took him over thirty years to create the mathematical book that you now see here.

1 Overcome Theorem (Peak-Locus Theorem): Growth of Life on Earth over the last 3.5 Billion Years We shall overcome…we shall overcome… https://www.youtube.com/watch?v= RkNsEH1GD7Q. This famous pacifist song by Joan Baez inspired the author (now, in November 2019) to rename “Overcome Theorem” the following mathematical result, that was his most important mathematical discovery made during the ten years 2010–2020. The author used to call it “Peak-Locus Theorem” in all his papers and books published in between 2010 and 2020. The older name “Peak-Locus Theorem” is mathematically correct, since it refers to the geometric locus of the peaks, as shown by the red solid curve in Fig. 1 hereafter. However, the name “Peak-Locus Theorem” also somehow hides the intuitive ideas that we now explain in a popular way for the reader’s benefit. The significance of this theorem may be immediately understood easily. Please have a look at Fig. 1.

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Fig. 1 OVERCOME Theorem (previously called PEAK-LOCUS Theorem (PLT) in all papers, 2010–2020)

On the horizontal axis is the time, ranging, say, from 0 to 10 in some arbitrary time units. On the vertical axis, the solid red curve is an exponential, ranging from 1 (since econstant*0 = 1) to a value of about 7.6. In between the horizontal time axis and the increasing red exponential are three different curves, named b1_lognormal (the blue curve), b2_lognormal (the green curve) and b3_lognormal (the fuchsia curve). Please worry not if you do not understand the meaning of the “b-lognormal” name: you will do so later. Just look at their shape: they start each at a different instant in time: b1 = 0 (meaning that 0 is the birth time for the b1_lognormal), b2 = 2 (meaning birth time 2 for the b2_lognormal) and b3 = 5 (meaning birth time 5 for the b3_lognormal). Then, going left to right, each curve elongates itself in order to have its maximum exactly on the red exponential curve. But… while doing so, the peak of each curve becomes narrower and narrower, and higher and higher, in such a way that the area under the curve is exactly the same for all curves and, conventionally, we assume that the common value of this area equals one. So the b-lognormals both stretch up and slim up at the same time, because the area under them is assumed to be the same for all of them (That is called the “normalization condition” fulfilled by the b-lognormals). How does all this apply to reality ? Well, think of evolution of life on Earth over the last 3.5 billion years: (1) For simplicity, suppose that life started exactly 3.5 billion years ago with the first molecule capable of reproducing itself: RNA (i.e. the “RNA world”, then leading to the “DNA world” still dominating life today). (2) Then suppose that each b-lognormal corresponds to a certain living species, born during the course of evolution just at the instant where that b-lognormal starts.

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(3) Clearly, the number of living species grew up enormously in the 3.5 billion years of evolution: from 1 (RNA, just 3.5 billion years ago) to 50 million nowadays, as most biologists say. In other words, we now assume that the vertical axis in Fig. 1 represents the number of species living on Earth while the time elapsed over 3.5. billion years. (4) Is this realistic? In other words, is it realistic to assume that the red upper curve is exactly an exponential? Clearly not so, since we know that many species died in the course of evolution, like dinosaurs, killed by an asteroid that hit the Earth about 65 million years ago. So, the “exact” exponential must be replaced by something else: a curve that we do not know well, since it actually may have oscillated up and down around the exponential in an unpredictable way. (5) No problem: mathematicians have a way to get around these difficulties. They replace the exponential curve by a “stochastic process”, i.e. a fluctuating curve just like the two different curves shown in Fig. 2. So, mathematicians assume that the mean value of the GBM is just the exponential red curve in Fig. 1. That “marries” the exponentially increasing number of living species on Earth over 3.5 billion years with the need to take into account the mass extinctions that we know did take place in the past: like the “two main ones” occurred about 250 million years ago (primary to secondary era transition) and 65 million years ago (dinosaurs’ demise and transition between secondary and tertiary eras, i.e. reptiles to mammals transition), plus several “smaller” extinctions.

Fig. 2 Two particular realizations of the stochastic process called geometric Brownian motion (GBM) taken from the Wikipedia site http://en.wikipedia.org/wiki/Geometric_Brownian_motion. Their mean values correspond to the exponential shown in red in Fig. 1 (with other numeric values, for simplicity)

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(6) Please stop a moment to think about what our OVERCOME Theorem (Peak-Locus Theorem) means, as we now explain. (7) Each new species overcomes the older species. Meaning that each new species is a little more complex than the older species from which it derives, but each new species also is more apt than the older species to face the difficulties of being alive. That is what Darwin called “natural selection”, and that is the way we have cast Darwin’s ideas into maths! Please note that we do not care about the specific DNA changes enabling the transition from an older species to a newer one: that is the task of molecular biologists and related scientists. We just care about the overall mathematical picture of the evolution of life on Earth over the last 3.5 billion years since… we later want to extrapolate that into the future and find out if Extraterrestrial Civilizations are living around us somewhere in the universe. That is SETI, the search for extraterrestrial intelligence https://en.wikipedia.org/wiki/Search_for_extraterrestrial_intelligence. (8) So, what type of physical units shall we adopt for what is on the vertical axis in Figs. 1 and 2? Just “the number of living species” (as we just said) or something else “more profound”? Well, we might think of complexity since each newer species is more complex than the previous ones. Moreover, complexity theory has been studied by scholars over the past 50 years, and so we might refer to this book as “a book about the mathematics of complexity”. But this author is primarily a mathematical physicist, and so he prefers to resort to the information unit that even children understand nowadays: bits, bytes, megabytes, gigabytes, terabytes and even petabytes, just like 1 petabyte is the number of raw data that SETI scientists had just now collected thanks to the “Breakthrough Listen” Project, to which this author is affiliated https://en.wikipedia.org/wiki/ Breakthrough_Listen. As you may have understood already, we want to see information measured in bits on the vertical axis in Figs. 1 and 2. (9) Information theory is the wonderful achievement of Claude Elwood Shannon (1916–2001); please see https://en.wikipedia.org/wiki/Claude_Shannon. In particular, Shannon introduced in 1948 the notion of entropy of a probability density, that we will largely use in this book: we simply call it “Shannon entropy”. Then, we discovered that the Shannon entropy is the natural measure of the evolution of life on Earth and on exoplanets. But… Shannon entropy… of which probability densities? Of the b-lognormals appearing in Fig. 1, of course. Fortunately, the Shannon entropy of a b-lognormal is given by a simple formula expressing it in terms of the two free b-lognormal parameters, mu and sigma, and so it becomes possible to compute the Shannon entropy of the GBM having as mean value the red solid exponential in Fig. 1. (10) Evo-Entropy. By Evo-Entropy, we mean the Shannon entropy of the evolution of life on Earth (and on exoplanets). Then, little by little in the years between 2010 and 2015, this author came to make an unexpected discovery: if one assumes that the growth of the number of species is exponential (as in Fig. 1), then the corresponding Evo-Entropy is just LINEAR, that is JUST A STRAIGHT LINE !!! In other words, we discovered that the Evo-Entropy of life forms grew just like a straight line, ranging from zero bits at the

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beginning of life (just 3.5 billion years ago), to 25.575 bits nowadays, as we now show in a diagram. (11) Figure 3 shows the time evolution of the entropy of life forms (Evo-Entropy) as a straight line ranging between 0 at the time of the origin of life on Earth (3.5 billion years ago) and 25.575 bit nowadays (zero time). We will later discover that this straight line is not exactly the Evo-entropy: actually we had to drop the minus sign in front of the traditional Shannon entropy definition (someone calls “negentropy” this “signed-reversed Shannon entropy), and we had to add a constant (i.e. time-independent term) to the traditional Shannon entropy definition in order to get the neat straight line growing from zero to 25.575 bits as shown in Fig. 3. But these are “marginal details” good for nerds. (12) So, Fig. 3 shows the time evolution of the entropy of life forms (Evo-Entropy) as a straight line ranging between 0 at the time of the origin of life on Earth (3.5 billion years ago) and 25.575 bit nowadays (zero time). This innocent-looking Fig. 3, however, hides a much more profound meaning that we now wish to point out: Figure 3 is our “mathematical proof” that the Molecular Clock indeed

Fig. 3 Evo-Entropy (in bits per individual) of the latest species appeared on Earth over the last 3.5 billion years. This shows that a man (nowadays) is 25.575 bits more evolved than the first form of life, i.e. “something capable of reproducing itself”, i.e. RNA, in our assumption, about 3.5 billion years ago (it might be 3.8, but worry not!)

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is a fundamental law of nature. All biologists nowadays know about the molecular clock https://en.wikipedia.org/wiki/Molecular_clock, discovered in 1962 by Émile Zuckerkandl (1922–2013) and Linus Pauling (1901–1994). Their discovery paved the way to the neutral theory of molecular evolution (https://en.wikipedia.org/wiki/Neutral_theory_of_molecular_evolution) by Moto-o Kimura (1924–1994) and others, basically saying that “molecular evolution obeys the laws of quantum physics and not the natural selection of Darwin”. Thus, molecular evolution on Earth must be the same on exoplanets too: not a small feat, confirming the idea that life must be present “everywhere in the universe”. (13) And so we now reach the most profound consequence of the straight line in Fig. 3. This is our Evo-SETI SCALE for evolution of life in the universe. The unit of the scale is called EE (Earth Evolution) and equals 25.575 bits. For a planet like Mars, the value on this scale is much smaller than 1 EE (if life actually appeared on Mars) and zero if no life ever existed on Mars. But for an exoplanet hosting a civilization more advanced than human’s on Earth nowadays, the value of Evo-Entropy must be higher or much higher than 1 EE. This is our Scale of Life in the Universe.

2 NASA Versus SETI: 1992–2020 We have a picture for you (this author is the second person to the left of the screen, “toasting” with his coffee cup). The present time is not the first time NASA takes an interest into SETI. The NASA SETI Program actually started on 12 October 1992, exactly 500 years after Christopher Columbus landed in America for the first time. The person writing these lines was then aged 44 and was already deeply involved with SETI: he had the privilege to embark on one of the twelve buses that left Pasadena, California (the town where Caltech and NASA-JPL are located) on 11 October and get to Barstow by the night. There, Carl Sagan gave the dinner speech. On the following morning, we reached the site of NASA’s Deep Space Network antenna at Goldstone, in the Mojave Desert. This antenna is shown in Fig. 4 of the JBIS 1999 paper written by Steven J. Garber of the NASA History office and now available at the site https:// history.nasa.gov/garber.pdf. There Carl Sagan officially opened NASA’s All-Sky-Survey SETI search, while, at the same time, at the Arecibo radio telescope in Puerto Rico (at the time the world’s largest) Bernard “Barney” Oliver, John Billingham, John Rummel, Jill Tarter, Seth Shostak and other SETI professionals officially opened the NASA Targeted Search, a SETI search on 778 stars of solar type nearby the Sun in the Galaxy. Good old days, that NASA-SETI Opening on 12 October 1992. But in less than a year’s time the NASA SETI Program was over, terminated by Congress, as per the description given by Steven J. Garber in the JBIS 1999 paper mentioned above.

Fig. 4 This picture was shot on 27 September 2018, at the Lunar and Planetary Institute (LPI) in Houston, Texas, and shows the about 50 participants invited by NASA to attend the NASA Technosignatures Workshop described in detail at the website https://www.hou.usra.edu/meetings/technosignatures2018/. The word “Technosignatures” seems rather cumbersome, but it really means SETI, the search for extraterrestrial intelligence, advocated since 1959 by great minds like Giuseppe Cocconi, Phil Morrison, Frank Drake, Carl Sagan, Jill Tarter, Nikolay Kardashev and many more

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Evolution as INCREASING NUMBER OF SPECIES

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Number of LIVING SPECIES on Earth

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Time in billions of years Fig. 5 Biological evolution as the increasing number of living species on Earth between 3.5 billion years ago and now. The red solid curve is the exponential mean value mGBM(t) of the geometric Brownian motion (GBM) stochastic process LGBM(t), while the blue dot–dot curves above and below the exponential mean value are the two standard deviation upper and lower curves, respectively. The “Cambrian Explosion” of life, that started around 542 million years ago, is evident in the plot just before the value of –0.5 billion years in time, where all three curves “start leaving the time axis and climbing up”. Notice also that the starting value of living species 3.5 billion years ago is the number ONE by definition and this FIRST SPECIES is RNA, in our understanding, but it “looks like” zero in this plot since the vertical scale (which is the true scale here, not a log scale) does not show it. Notice finally that nowadays (i.e. at time t = 0) the two standard deviation curves have exactly the same distance from the middle mean value red solid curve, i.e. 30 million living species more or less the mean current value of 50 million species. These are assumed values that we used just to exemplify the GBM mathematics: biologists might assume other numeric values. But our describing equations are going to remain the same, apart from all assumed numbers

What happened next? Well, it happened that “people of good will” kept SETI alive in spite of all difficulties. And not just in the USA only. In Russia, France, Italy, Australia and Argentina were a few “heroic radio astronomers” willing to “risk their good reputation” by doing SETI searches rather than just ordinary astrophysical work. This situation lasted approximately until 2010, when Britain’s Astronomer Royal, Lord Martin Rees, declared that “absence of evidence is not evidence of absence”. In the meantime, the discovery of more and more planets orbiting around other stars (exoplanets), and the success of space missions looking for such exoplanets, like NASA’s “Kepler”, convinced various scientific establishments of leading countries that SETI was “respectable”. In 2015 a new SETI revolution occurred: Russian-born American billionaire Yuri Milner gathered a meeting at the Royal Society in London and declared that he

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was willing to pay $100 million in the next 10 years for SETI searches to be conducted at an unprecedented scale by his Breakthrough Listen (BL) Program (https://breakthroughinitiatives.org/initiative/1) taking place at the Department of Astronomy of the University of California at Berkeley (U. C. Berkeley). Then, an explosion of SETI searches started. Several American Universities followed U. C. Berkeley’s example and so did some other countries, like Italy, where this author lives, and the UK, France, the Netherlands, Russia and especially China, that, in 2017, opened up her brand-new, 500 m large fast radio telescope https://drive.google.com/file/d/0B3YGjLNLQSmtZEowOE1ISHAxejg/view, now the world’s largest. So much for radio SETI, traditionally based on the assumption that Aliens would transmit radio messages around the hydrogen line of having 21 cm in wavelength, or, equivalently, 1420 MHz in frequency https://en.wikipedia.org/wiki/Hydrogen_ line. But what if Aliens transmit at other frequencies? Well, if they send laser beams around, then optical SETI might possibly detect these, as per the popular description at the site https://www.centauri-dreams.org/2017/07/17/detectionpossibilities-for-optical-seti/. So much for ongoing SETI searches worldwide as of 2020.

3 Physics, Chemistry and Mathematics are the Same all Over the Universe Is humankind ready for such a contact with Aliens ? Most probably not so yet, in this author’s view. We actually know “nothing” about them. We do not know how far they live from us. Most important, we do not know how much more technologically advanced than us they might possibly be. And that is scary. Knowing nothing about Aliens, this author tried to resort to mathematics. This is because “two and two makes four everywhere in the universe”, i.e. for both us and extraterrestrials (abbreviated ETs, in the sequel). In fact, we know for sure that fundamental physics and chemistry are the same all over the universe, and we know so because the spectral lines in the light reaching us from the stars are the same as ours. But there is more. The frequencies of these spectral lines are computed on Earth by virtue of quantum mechanics. Thus, also quantum mechanics must be the same for both us and Aliens. In turn, quantum mechanics is part of high-level mathematics (eigenvalues, eigenvectors and the Hilbert space https://en.wikipedia. org/wiki/Hilbert_space) and so the equations we scientists work with every day with must be the same for both us and Aliens. In other words still, not only physics and chemistry are the same for both us and ETs, but mathematics too has got to be the same. And some “philosophers” denying so are just (sorry!) too ignorant about modern science, and we will not pay attention to them any longer. So, let us use mathematics, the only product of the human mind that may enable us to make further discoveries even before the relevant experimental counterpart is

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found. Our motto is “To measure is to understand” https://www.goodreads.com/ quotes/632992-measurement-is-the-first-step-that-leads-to-control-and.

4 Classical Drake Equation (1961) Versus Statistical Drake Equation (2008) The foundational equation of SETI is the Drake equation (1961) https://en. wikipedia.org/wiki/Drake_equation. It was generalized by this author into the statistical Drake equation in 2008, as described in Chap. OVERCOME Theorem, that is PEAK-LOCUS Theorem, Proven by Maxima of this book as well as at the Wikipedia site https://link.springer.com/chapter/10.1007/978-3-642-27437-4_1. Both the classical equation of Frank Drake (1961) and the statistical equation of Claudio Maccone (2008) estimate the number N of communicating civilizations now existing in our galaxy, the Milky Way, but they do so in different ways: for Drake, N is the product of the seven positive numbers listed at the Drake equation site given earlier. For Maccone, N is the product of a very large number of factors (theoretically speaking, an “infinite” number of factors), each of which is a probability density function (abbreviated “pdf”) having its own mean value and its own standard deviation, both known to scientists by virtue of experimental measurements. Then, whatever the probability density functions might possibly be, the so-called central limit theorem of statistics shows that the probability density function of N is a lognormal. More details about lognormals and b-lognormals are in the next section.

5 The b-Lognormal Equation (i.e. Probability Density Function = pdf), Easily Proven as Follows This book is based on the fundamental notion of a lognormal pdf (in the time), with the consequent b-lognormal pdf in the time, that is just a lognormal starting at the initial instant b=birth other than zero. To let even the preface to this book be self-contained in this regard, we now provide an easy proof of the b-lognormal equation as a probability density function (pdf). Just start from the well-known Gaussian or normal pdf ðxlÞ2

e  2 r2 Gaussian or normal pdf ðx; l; rÞ ¼ pffiffiffiffiffiffi with 2p r 8 > <  1\x\ þ 1; independent variable:  1\l\ þ 1; a real parameter ¼ where the peak is: > : r [ 0; a positive parameter = the standard deviation: This pdf has two parameters

ð1Þ

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(1) l turns out to be the mean value of the Gaussian and the abscissa of its peak. Since the independent variable x may take up any value between – ∞ and + ∞, i.e. it is a real variable, so l must be real too. (2) r turns out to be the standard deviation of the Gaussian and so it must be a positive variable. (3) Since the Gaussian is a pdf, it must fulfil the normalization condition.

Z1 1

ðxlÞ2

e 2 r 2 pffiffiffiffiffiffi dx ¼ 1 2pr

ð2Þ

and this is the equation we need in order to “discover” the b-lognormal. Just perform in the integral Eq. (2) the substitution x ¼ lnðtÞ (where ln is the natural logarithm, i.e. the one to base e = 2.718281828459045…). Then Eq. (2) is turned into the new integral Z1 0

ðln ðtÞlÞ2

e 2 r2 pffiffiffiffiffiffi dt ¼ 1: 2p r t

ð3Þ

But this Eq. (3) may be regarded as the normalization condition of another random variable, ranging “just” between zero and þ 1, and this new random variable we call “lognormal” since it “looks like” a normal one except that x is now replaced by x ¼ lnðtÞ and t now also appears at the denominator of the fraction. In other words, the lognormal pdf is ðlnðtÞlÞ2

e  2 r2 lognormal pdf ðt; l; rÞ ¼ pffiffiffiffiffiffi with 2p  r  t 8 > < 0  t\ þ 1; independent variable: > :

ð4Þ

 1\l\ þ 1; a real parameter ðno special name): r [ 0; a positive parameter ðno special nameÞ:

At the site https://en.wikipedia.org/wiki/Log-normal_distribution, various plots of Eq. (4) are shown. Just one more step is required to jump from the “ordinary lognormal” Eq. (4) (i.e. the lognormal starting at t = 0) to the b-lognormal, that is the lognormal starting at any real instant b (“b” stands for “birth”). Since this simply is a shifting along the time axis from 0 to the new real time origin b, in mathematical terms it means that we have to replace t by (t–b) everywhere in the pdf Eq. (4). Thus, the b-lognormal pdf must have the equation

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e  2 r2 with b-lognormal pdf ðt; l; r; bÞ ¼ pffiffiffiffiffiffi 2p  r  ðt  bÞ 8 b\t\ þ 1; time = real independent variable starting at b : > > <  1\l\ þ 1; a real parameter ðno special name): r [ 0; a positive parameter ðno special nameÞ: > > : b ¼ birth time, the (real) time when the b-lognormal starts:

ð5Þ

Sometime, the b-lognormal Eq. (5) is called “three-parameter lognormal” by statisticians. That is correct since the three parameters appearing in Eq. (5) are ðl; r; bÞ. However, we prefer to call it b-lognormal to stress its biological meaning as the probability density representing the lifetime of any living being, born at the instant b. We will later use b-lognormals also to represent the lifetime of Historic Civilizations too, like the nine Historic Civilizations that will be studied in the sequel of this preface.

6 Assume Life on Earth Started 3.5 Billion Years Ago: Then ts = –3.5 109 Years, and Zero Time is Now Question: is it possible to measure mathematically the biological evolution (also called Darwinian evolution) that paleontologists tell us occurred on Earth over (about) the last 3.5 billion years? Our answer is “yes” as we show that in this book. Darwin (1859, “The Origin of Species”) had the great merit to dispel all religious assumptions from the biological sciences by realizing that animals and plants evolved from primitive forms up to humans over that long amount of time. How long exactly? Radioactivity gave the relevant answer starting about 1910, after Darwin had passed away in 1882. Here again mathematics plays a key role in radioactivity, as given by the Harry Bateman’s analytical solution to the experimental differential equations discovered by physicists Pierre and Marie Curie and others; see https://en.wikipedia.org/wiki/Bateman_equation. So, the “long time” of evolution that Darwin did not know about was estimated over the last 100 years by paleontologists, biologists, geneticists and other scientists to be over 3.5 billion years, possibly 3.8, or even 4 (as someone claims), given that the Earth originated from the Sun about 4.5 billion years ago (like all planets in the Solar System). In all chapters of this book, we shall assume for simplicity that life on Earth started exactly 3.5 billion years ago, just to fix the ideas. This number will be denoted by ts (= time of start [of life on Earth]) in all equations. So, ts = –3.5 109 years, assuming that the zero time is nowadays, i.e. that times prior to nowadays are denoted by negative numbers (in years), while future times will be positive numbers for us. The first form of life, biologists say nowadays, was RNA and DNA then followed. We all know that DNA is “the molecule of life”, meaning that every baby is

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born out of the merging of the father’s and mother’s DNA. But DNA is the same for all living members of a certain living species. Darwin could hardly understand what is happening when a new species is originated by an older species: the answer was found after 1900 by the British school of mathematical geneticists (Francis Galton, Karl Pearson, Godfrey Hardy, Ronald Fisher, J. B. S. Haldane and Francis Crick, just to name a few). Many American geneticists also contributed after about 1900. In this book, we assume that the number of living species on Earth over the last 3.5 billion years is well represented, mathematically speaking, by a geomeric Brownian motion (GBM). In fact: (1) We know that the first living species (meaning the first molecule capable of reproducing itself) was RNA 3.5 billion years ago. In the language of mathematics, this is the same as saying that the initial condition of our GBM is one, that is Ns = 1 at ts = –3.5*10^9 years. (2) Nowadays, i.e. at the time t = 0 in our time convention, the supposed number of species living on Earth is about 50 million (so many biologists guess, including insect species in the count), but this number could be quite higher if we include bacteria in the count. At the moment, we shall assume Ne = 50 million, where Ne means the number of living species at the end of the observed timespan, i.e. nowadays. (3) Then the geometric Brownian motion (GBM, site https://en.wikipedia.org/wiki/ Geometric_Brownian_motion) is our way to cast biological evolution mathematically, i.e. by virtue of just a few, simple statistical equations. (4) Figures 1, 2, 3 and 5 summarize all that just in four plots! (5) In the English-speaking world, this plots also goes under the name of “Malthusian growth”, since Thomas Malthus (1766–1834) used a similar exponential curve in 1798 to describe the human population growth (https://en. wikipedia.org/wiki/Malthusian_growth_model). But we prefer to use the term “exponential growth”. In this book, we assume that the number of living species on Earth over the last 3.5 billion years is well represented, mathematically speaking, by a GBM.

7 How to Take Extinct Species into Account by Virtue of the Stochastic Process Called (Incorrectly) “Geometric Brownian Motion” (GBM) Mathematically, naïve folks might object that many species went extinct during the 3.5 billion years of biological evolution, and so representing evolution by virtue of a simple, increasing exponential curve in the time is an oversimplified view. Well, this objection is easily answered by mathematicians: biological evolution is not exactly the exponential shown above in red, but is rather a stochastic process (i.e. a

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random function of the time) the mean value of which is indeed the above exponential. Wall Street economists have studied that stochastic process since about fifty years ago to represent mathematically the fair price or theoretical value for a call or a put option based on six variables such as volatility, type of option, underlying stock price, time, strike price and risk-free rate. These mathematical studies are called Black-Scholes models (https://en.wikipedia.org/wiki/Black%E2%80% 93Scholes_model), and the Nobel prize in Economics was awarded in 1997 to Merton and Scholes (William Black had unfortunately passed away already). So, what this author did, in the practice, was to pick up the GMB used in the Black-Scholes model and reuse it to describe biological evolution over the last 3.5 billion years. With a caveat: this GBM is not what physicists call a Brownian motion, so, please, kindly tune up your words according to the person you are talking to: in physics, a Brownian motion is a Gaussian stochastic process, while a Wall Street’s GBM actually is a lognormal stochastic process, i.e. a process of the type e = 2.71828… raised to the Gaussian process. Hard to explain by words, but easy by equations, as we shall see in this book.

8 Our Earlier Mathematical Definition of “Lifetime” of a Living Being: It is a “b-Lognormal” in the Time The first significant innovation put forward by this author in his Evo-SETI Theory was his definition of lifetime of a living being as a b-lognormal in the time. As per Eq. (5), a b-lognormal is a lognormal in between the starting point (b = birth) and the “senility” point s = the time of the lognormal’s descending inflexion, as shown in Fig. 6. After the instant s, the curve is a descending straight line until it reaches the time axis again at the “death time” d.

9 Our First Innovative Discovery made in 2010: The Two b-Lognormal History Formulae The most important mathematical consequences of our definition of life are the following two b-lognormal history formulae: they express the two parameters r and l of the b-lognormal pdf in terms of the three parameters ðb; s; d Þ of birth, senility and death, respectively. They were the first important step ahead in Evo-SETI Theory made by this author and were discovered by him at a date prior to 10 March 2011 (the full proof is given at the pages 172–181 in Chap. 6, Appendix 6.B, of this author’s 2012 book Mathematical SETI):

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Fig. 6 Lifetime of every living being as a b-lognormal between birth b and death d, with a junction at senility s. More definitions like childhood, youth, maturity, decline, vitality and fertility are just obvious consequences

(

ds ffi pffiffiffiffiffiffi r ¼ pffiffiffiffiffiffi db sb

l ¼ lnðs  bÞ þ

ðdsÞðd þ b2sÞ ðdbÞðsbÞ

ð6Þ

In other words, if one assigns the birth, senility and death time of a living being, then the corresponding b-lognormal plot is found immediately. The proof is so easy that even a good freshman should be able to find it!

10

History of 9 “Western” Historic Civilizations as b-Lognormals: Ancient Egypt to USA

The unexpected bonanza of the history formulae Eq. (6) is that they apply not only to the lifetime of any living being: they also apply to the lifetime of Historic human Civilizations. Just look at the following Fig. 7, please. So, let us review our history inputs given by Table 1. There are 3  9 ¼ 27 input numbers (27 real numbers).

Fig. 7 Lifetime of nine Western Civilizations as b-lognormals. From Ancient Egypt, Greece and Rome, to the Italian Renaissance and then the colonial empires of Portugal, Spain, France, Britain and the USA, the history formulae (Eq. 6) are just all that is required to represent history mathematically: you just give the numbers for b, s, and d for each civilization, as given in Table 1, and the nine above b-lognormal plots come out immediately! Note that, along the vertical axis, are just real positive numbers and nothing else. In fact, the b-lognormals are just probability densities, i.e. curves staying only above the time axis and having the area below the curve (the integral) equal to one. As easy as that!

xx Preface

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Table 1 Birth, peak, decline and death times of nine Historic Western Civilizations (3100 BC– 2035 AD), plus the relevant peak heights. They are shown in Fig. 2 as nine b-lognormal probability density functions (pdfs) b = Birth time

p = Peak time

s = Decline = senility time

d = Death time

P= Peak ordinate

Ancient Egypt

3100 BC Lower and Upper Egypt unified. First Dynasty

689 BC Assyrians invade Egypt in 671 BC. leave 669 BC

30 BC Cleopatra’s death: last Hellenistic queen

8.313  10–4

Ancient Greece

776 BC First Olympic Games, from which Greeks compute years

1154 BC Luxor and Karnak temples edified by Ramses II by 1260 BC 434 BC Pericles’ Age. Democracy peak. Arts and Science peak. Aristotle

30 BC Cleopatra’s death: last Hellenistic queen

2.488  10–3

Ancient Rome

753 BC Rome founded. Italy seized by Romans by 270 BC, Carthage and Greece by 146 BC, Egypt by 30 BC. Christ 0

323 BC Alexander the Great dies. Hellenism starts in Near East 273 AD Aurelian builds new walls around Rome after Military Anarchy, 235–270 AD

476 AD Western Roman Empire ends. Dark Ages start in West. Not in East

2.193  10–3

Renaissance Italy

1250 Frederick II dies. Middle Ages end. Free Italian towns start Renaissance

1564 Council of Trent ends. Catholic and Spanish rule.

1660 Cimento shut. Bruno burned 1600. Galileo died 1642

5 749  10–3

Portuguese Empire

1419 Madeira Island discovered. African coastline explored by 1498

1822 Brazil independent, other colonics retained

1999 Last colony, Macau, lost to Republic of China

3.431  10–3

Spanish Empire

1402 Canary islands are conquered by 1496. In 1492, Columbus

1805 Spanish fleet lost at Trafalgar

1898 Last colonies lost to the USA

5 938  10–3

117 AD Rome at peak: Trajan in Mesopotamia. Christianity preached in Rome by Saints Peter, Paul against slavery by 69 AD 1497 Renaissance art and architecture. Birth of Science. Copernican Revolution (1543) 1716 Black slave trade to Brazil at its peak. Millions of blacks enslaved or killed 1798 Largest extent of Spanish colonies in America: California

(continued)

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Table 1 (continued)

French Empire

British Empire

b = Birth time

p = Peak time

discovers America 1524 Verrazano first in New York bay. Cartier in Canada 1534

settled since 1769 1812 Napoleon 1 dominates continental Europe and reaches Moscow

1588 Spanish Armada defeated. British Empire’s expansion starts

s = Decline = senility time

d = Death time

P= Peak ordinate

1870 Napoleon III defeated. Third Republic starts. World Wars

1962 Algeria lost as most colonies. Fifth Republic starts in 1958 1974 Britain joins the EEC and loses most of her colonies

4 279  10–3

1904 1947 8.447 British Empire After World  10–3 peak. Top Wars One British Science: and Two, Faraday. India gets Maxwell, independent Darwin, Rutherford USA Empire 1898 1972 2001 2035 0 013 Philippines, Moon Landings, 9/11 terrorist Singularity? Cuba, Puerto 1959–72: attacks: Will the Rico seized from America leads decline. USA yield Spain the world Obama 2009 to China? Table 1. You just input the three numbers b, s and d for each civilization (i.e. b-lognormal), and the nine plots shown in Fig. 7 come out automatically. Isn’t this surprising? Well, even more surprising it would be if the SETI scientists would discover an Alien Civilization in space, and then we could figure out mathematically how much “more advanced than us” they are. This is precisely the goal of this mathematical book, i.e. the goal of our Evo-SETI Theory, an acronym standing for “Evolution and SETI”.

11

Molecular Clock Predicted by our Evo-SETI Theory as the Linear Evo-Entropy of the GBMs

What is a clock ? Mathematically speaking, a clock is a straight line in a diagram with time on the horizontal axis and whatever changes linearly in time on the vertical axis, like sand in an hourglass. Then, Fig. 3 is a clock. Let us now consider the “molecular clock”, i.e. the greatest experimental discovery of molecular biology if not of all of astrobiology (site: https://en.wikipedia. org/wiki/Molecular_clock). We report here just the same the description of the molecular clock provided by Wikipedia: the molecular clock is a figurative term for a technique that uses the mutation rate of biomolecules to deduce the time in prehistory when two or more life forms diverged. The biomolecular data used for such calculations are usually nucleotide sequences for DNA or amino acid

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sequences for proteins. The benchmarks for determining the mutation rate are often fossil or archaeological dates. The molecular clock was first tested in 1962 on the haemoglobin protein variants of various animals and is commonly used in molecular evolution to estimate times of speciation or radiation. So what? So, this author thinks that his Evo-SETI Theory embodies the existence of the molecular clock over 3.5 billion years since the appearance of life on Earth: the molecular clock is nothing but the Evo-Entropy (Shannon entropy with a reversed sign in front, and starting at zero 3.5 billion years ago) of the geometric Brownian motion, i.e. of the exponentially increasing number of species over the same 3.5 billion years. Wow! That is a discovery! Let us make an historical comparison with two great scientists of the past: Johannes Kepler (1571–1630) and Isaac Newton (1642–1527). Kepler discovered the laws of planetary motion by translating mathematically all the careful observations of Mars made by his teacher Tycho Brahe (1546–1601) into three simple equations: https://en. wikipedia.org/wiki/Kepler%27s_laws_of_planetary_motion. But Newton did better still: he proved mathematically that the three Kepler’s laws were just mathematical consequences of his single law, the law of universal gravitation: https://en.wikipedia. org/wiki/Newton%27s_law_of_universal_gravitation. By doing so, Newton established a new branch of science, called celestial mechanics up to about 1950 and astrodynamics after the advent of space missions and nowadays (2020) we may compute in advance the orbits of all spacecrafts to any requested precision. Similarly, we claim that our Evo-SETI Theory has mathematized the molecular clock. We claim that this mathematical discovery of ours will have consequences in the future of atrobiology that we can hardly envisage nowadays: it is just too early yet.

12

Information Gaps Among Different Civilizations as INFORMATION Entropy Differences Among Them

One of the possibilities of Evo-SETI Theory is to describe mathematically how much a civilization is more advanced than another one by computing the difference among the two relevant Evo-Entropies. Please look at Table 2. This is an antisymmetric matrix (also called a skew-symmetric matrix) expressing the Evo-Entropy GAPS (in bits/individual) among the nine Historic Western Civilizations described in Fig. 3 and Table 1. We are now able to understand the #1 success of Evo-SETI Theory: that is the hability to quantify the GAP among different civilizations by virtue of just one number, rather than using a mountain of words (as Historians do). For instance, using data similar to Table 2, we could prove that the Evo-Entropy gap between the Aztecs and the Spaniards when they clashed in 1519 was about 3.85 bits/individual. In other words, back in 1519, each Spaniard was, on the average, 3.85 bits more knowledgeable than any Aztecs. That is why the Aztecs had a psychological

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Table 2 Information gaps = (Shannon) entropy differences in bits/individual among the nine Historic Western Civilizations (3100 BC–2035 AD) shown in Fig. 2 and Table 1. INFORMATION CAP in bits/individual

Egypt

Greece

Rome

Italy

Portugal

Spain

France

Britain

USA

Egypt

0

1.467

1.622

2.795

2.008

3.425

4.624

4.335

3.864

Greece

–1.467

0

0.154

1.328

0.541

1.958

3.157

2.868

2.397

Rome

–1.622

–0.154

0

1.173

0.386

1.804

3.002

2.713

2.242

Italy

–2.795

–1.328

–1.173

0

–0.7872

0.630

1.829

1.540

1.069

Portugal

–2.008

–0.541

–0.386

0.7872

0

1.418

2.616

2.327

1.856

Spain

–3.425

–1.958

–1.804

–0.630

–1.418

0

1.198

0.909

0.438

France

–4.624

–3.157

–3.002

–1.829

–2.616

–1.198

0

–0.289

–0.759

Britain

–4.335

–2.868

–2.713

–1.540

–2.327

–0.909

0.289

0

–0.471

USA

–3.864

–2.397

–2.242

–1.069

–1.856

–0.438

0.759

0.471

0

breakdown. So, Evo-SETI Theory could even be used in mathematical psychology, making that research field more quantitative than it has been up to now.

13

Introducing Energy into Evo-SETI Theory

Evo-SETI Theory, as described up to now, dealt only with (the Shannon) entropy, i.e. the entropy of Shannon’s Information Theory, and not with energy at all. That was the situation that this author faced prior to the year 2015. However, both entropy and energy are the two pillars of classical thermodynamics, and so we can hardly expect to create a good model for the evolution of life on Earth and exoplanets if we just use entropy only: we must insert energy to Evo-SETI Theory too. How can we do so? The answer to this question was found by this author on 22 November 2015, when he discovered his new “Logpar History Formulae”, as described in the next and following sections.

14

Our 2015 New Mathematical Definition of “Lifetime of a Living Being”: A “Logpar” Power Curve in the Time

On 22 November 2015, this author made one more mathematical discovery. He replaced the b-lognormal described so far by virtue of another curve that he had never considered earlier: the “logpar”. The logpar acronym stands for “LOGnormal plus PARabola” and means a curve that is a b-lognormal only between birth and peak time, but actually just a parabola between peak time and death time. Let us see this LOGPAR plot in the case of the Civilization of Ancient Rome (753 BC–476 AD). Evidently, the adoption of a logpar instead of a b-lognormal abolishes the senility instant s and replaces its role by the role of the peak time p. In the practice, it is much easier to estimate someone’s peak time p than senility time s, and so

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abandoning b-lognormals in favour of logpars greatly simplify things for the applications of Evo-SETI Theory to real cases. In other words, we will only have to specify the logpar triplet (b, p, d) in order to get an easily understandable logpar curve for all further calculations based on it.

15

Our Two Logpar History Formulae

Thus, the real discovery that this author made on 22 November 2015 was the discovery of the logpar history formulae, expressing the logpar’s r and l in terms of the “easy triplet” b, p, d and not requesting s any more : 8 < :

r¼

pffiffi pffiffiffiffiffiffiffi 2 dp pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi

2 dðb þ pÞ

l ¼ lnðp  bÞ þ r2

ð7Þ

But that is just the most evident change. Other more subtle mathematical changes, that become evident only by doing the calculations, are (1) We have dropped the hypothesis that life is a probability density as it was the case for the b-lognormals. In other words, the logpar curves are positive curves not fulfilling the normalization condition requesting that the area under the b-lognormal (i.e. the area under the curve in between b and þ 1) must equal one. (2) Logpars are now positive power curves measured in Watts. With this interpretation, the area under each logpar, i.e. the integral (with respect to the time, of course) of the logpar in between the birth b and any instant z before death d, is the energy necessary to that living being to live up to the instant z (we call this integral of the logpar the “progressive energy” of the alive being). (3) Consequently, the lifetime energy of that living being is the “final” progressive energy up to death time d. (4) A further key result provided by logpars (and not by b-lognormals) is the “Principle of Minimum Energy”, saying that the derivation of the logpar history formulae Eq. (7) is possible only under the assumption that lifetime energy described at point (3) is a minimum with respect to the positive parameter r. In words, this mathematical discovery very much resembles the “principle of least action” of paramount importance in physics. But we confess that we still have to dig more profoundly about this principle in future mathematical papers. The potential consequences of this minimum energy principle might be enormous for the applications of Evo-SETI Theory to the study of all living beings, but at the moment (December 2020) all this still is premature.

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16

Approaching Immortality, That is Letting the Death Time d Approach Infinity

One of the classical themes of science fiction is “Living Beings Approaching Immortality”. Actually, the life expectancy of humans has already increased considerably during the last few decades, so one may wonder whether, in the far future, humans will become “nearly immortals” or so. Evo-SETI Theory offers a mathematical clue about immortality: just let the death time approach infinity, that is let d ! 1 in the logpar history formulae Eq. (7), and see what happens. Well, it happens that r ! 1 and this is a striking novelty with respect to b-lognormals, where 0\r\1 still in accordance with the meaning of r as standard deviation of the Gaussian = normal distribution. In other words, having abandoned the normalization condition in the transition from b-lognormals to logpars, now that pays off for further improvements. For instance, what happens if we let d ! 1 in the Ancient Rome logpar plot in Fig. 8? The result is most surprising, and it is shown in Fig. 9, as we now describe.

History of the ROMAN CIVILIZATION as a LOGPAR finite curve 0.0006 0.0005 0.0004 0.0003 0.0002 0.0001 0

900

800

700

600

500

400

300

200

100

0

100

200

300

400

500

600

Time in years: negative years = B.C., 0 = Christ born, positive years = A.D. Fig. 8 Representation of the history of the Roman civilization as a logpar finite curve: the ascent of Rome is shown in red (753 BC–117 AD) and the decline in blue (117 AD–476 AD). Rome was funded in 753 BC, i.e. in the year –753 in our notation, or b ¼ 753. Then, the Roman republic and empire (the latter since the first emperor, Augustus, roughly after 27 BC) kept growing in conquered territory until it reached its peak (maximum extension, up to Susa in current Iran) in the year 117 AD, i.e. p ¼ 117, under emperor Trajan. Afterwards, it started to decline and loose territory until the final collapse in 476 AD (d ¼ 476, Romulus Augustulus, last emperor). Thus, just three points in time are necessary to summarize the history of Rome: b ¼ 753; p ¼ 117; d ¼ 476: No further intermediate point, like senility in between peak and death, is necessary since we now used a logpar rather than a b-lognormal (b-lognormals only had been used by this author in the years 2009–2017)

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Logpar Energy’s Approach to Immortality “in the Long Run”

17

Figure 9 shows not only “The Decline and Fall of the Roman Empire” (117–378 AD, just to use Edward Gibbon’s great book’s title), but also the Dark Ages (Middle Ages), say, in between 476 AD and the Italian Renaissance (after roughly 1300 AD): it took 1000 years to the European Western Civilization to recover from the Barbaric Invasions within the Roman Empire firstly, and then within the Romance Countries secondly. The solid red line in Fig. 9 is the energy of the once Roman lands, normalized to the ½ value at the peak time energy of Trajan in 117 AD. ENERGY of Roman Empire and (later) Western Countries: 117-2000 AD ARBITRARY ENERGY SCALE (logarithmic?)

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000

Years since Christ's birth (zero year) Fig. 9 Total energy, i.e. total work produced each year by the Roman Empire, starting after its peak, that occurred in the year 117 under Emperor Trajan. This is the solid red curve given by Eq. (56) of the paper “ENERGY OF EXTRA-TERRESTRIAL CIVILIZATIONS ACCORDING TO EVO-SETI THEORY” (Acta Astronautica, 144 (2018), 202–213). We see that, after Trajan, the empire started to decline, producing less and less total energy and reaching its minimum in the year 385.844 AD  386 AD. These were the years (and actually decades, or even a few centuries) of the Barbarian Invasions inside the Western Roman Empire, after the Visigoths had inflicted the first severe defeat to the Romans at the battle of Hadrianople in 378 AD. Then, the “Dark Ages of the Western Civilizations”, or “Middle Ages”, lasted about ten centuries, and it was not until about 1300 AD that Western Europe started flourishing again, reaching about the same total energy level that the former Roman Empire had had under Trajan. This level is shown in the graph by the thin solid blue horizontal line. After roughly 1300 AD, the Italian Renaissance developed and then expanded into the whole of Western Europe in the following centuries. In addition, the dot–dot red line is the oblique asymptote to the total energy. Finally, while the horizotal time scale is in agreement with the historic facts, the vertical scale of this graph is completely arbitrary, and we had to re-scale it to the correct energy value (measured in Joules) in our above-mentioned 2018 paper

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But Fig. 9 also shows the oblique asymptote (dot–dot straight line) that the above energy approaches at higher and higher values of the time: the same Trajan normalized value of ½ was reached again during the Italian Renaissance around 1523 AD. After that, energy increased like a straight line (the asymptote).

18

Our Two Logell History Formulae (Discovered in 2018)

In 2019, this author submitted one more paper to the International Journal of Big History, by the title of “TWO POWER CURVES YIELDING THE ENERGY OF A LIFETIME IN EVO-SETI THEORY”: the paper was accepted for publication. This paper introduces the new concept of a LOGELL Power Curve, where the part of the curve between peak and death is the descending quarter of an ellipse, rather than a descending parabola, as in the logpar. The name LOGELL actually means “LOGnormal in between birth and peak, and ELLipse in between peak and death”. The difference between logpar and logell for the case of Ancient Rome is shown in Fig. 10. Once again it was possible to derive the logell history formulae, given by: 8 pffiffipffiffiffiffiffiffiffi p dp < pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r¼ pðdbÞ þ ðp4ÞðpbÞ ð8Þ : l ¼ lnðp  bÞ þ r2 And once again r ! 1 if we let d ! 1 into the upper Eq. (8), i.e. “in the long run”, as we now explain. Rome's LOGPAR (red) and LOGELL (red before peak, blue after peak) 0.0006 0.0005 0.0004 0.0003 0.0002 0.0001 0

900

800

700

600

500

400

300

200

100

0

100

200

300

400

500

600

Fig. 10 The difference between logell and logpar is only in their behaviour between peak p and death d, i.e. an ellipse for the logell (in blue) and a parabola for the logpar (in red). The common part of the curve prior to the peak (shown in red here) is a b-lognormal, that is a lognormal probability density function (pdf) in the time starting at the birth time b and reaching its peak at time p. In this way, the finite lifetime of any living being or civilization is a power curve (power means measured in Watts, as in physics) and the area under this power curve is the total energy that the living being or civilization needs in order to cope for its own existence. In fact, the energy is the just integral of the power in the time, or, if you prefer, the power is just the derivative of the energy with respect to the time, by definition

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xxix

Logell Energy Approach to Immortality “in the Long Run”

Just as the logpars approach immortality “in the long run”, and their respective energies do so by virtue of their own oblique asymptote, so do the logells. Figure 11 hereafter shows both the logpar (in red) and the logell (in blue) asymptotes for Ancient Rome’s decline, fall and recovery after the Barbarian Invasions. Basically, the difference between logpars and logells is that logells allow for a larger amount of energy (area under each power curve) than the logpars towards the end of life. So logells also allow for a faster recovery than logpars do. This feature shows neatly up in the two recovery times shown in Fig. 11: (1) Rome’s total energy as a function of D (i.e. d, the death time regarded now as the independent variable): The solid curve in red is the logpar energy curve. The dot–dot straight line in red is its oblique asymptote. (2) The solid curve in blue is the logell energy curve. The dot–dot straight line in blue is its oblique asymptote. (3) As we see from Fig. 11, the recovery of the Western Civilization after the fall of the Western Roman Empire (476 AD) happened at two different times given by the blue and the red solid curve, respectively. The blue curve (logell) intercepts the 0.5 axis near the year 800 AD, that is the time when Charlemagne was appointed “Holy Roman Emperor” by Pope Leo III and the Carolingian Renaissance started: see the Wikipedia website https://en.wikipedia.org/wiki/ Carolingian_Renaissance. The red solid curve (LOGPAR) is the Italian Renaissance https://en.wikipedia.org/wiki/Italian_Renaissance, crossing the 0.5 axis around 1400 AD.

ARBITRARY ENERGY SCALE (logarithmic?)

Rome's LOGELL energy (blue) and LOGPAR energy (red) with asymptotes 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000

Years since Christ's birth (zero year)

Fig. 11 Logell versus logpar for the recovery of the Western European Civilization: 800 1400 AD

AD

or

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20

Big History: A New Popular Way of Describing the Evolution of the Universe Over About 13.8 Billion Years: Big Bang to Humans (And Possibly Beyond, into the Future, Including ETs)

BIG HISTORY (https://en.wikipedia.org/wiki/Big_History) is a new research discipline that, in this author’s view, is best described by the very same words of the mentioned Wikipedia relevant site. We now report these words just the same. “Big history is an academic discipline which examines history from the big bang to the present. Big history resists specialization and searches for universal patterns or trends. It examines long time frames using a multidisciplinary approach based on combining numerous disciplines from science and the humanities and explores human existence in the context of this bigger picture. It integrates studies of the cosmos, Earth, life and humanity using empirical evidence to explore cause-and-effect relations and is taught at universities and primary and secondary schools often using web-based interactive presentations.” This author became enamoured with Big History in 2013, while he was already working to his mathematical Evo-SETI Theory. He then joined the International Big History Association (IBHA) and attended the relevant conferences in San Rafael (California, 2014), Amsterdam (The Netherlands, 2016) and at Villanova University (USA, 2018). Then, he felt confident enough to run his own “Big History and SETI” Conference in Milan, Italy, on 15–16 July 2019, with about 30 speakers from many countries worldwide. All talks were tape-recorded and may be found at the International Big History Association (IBHA) website https://bighistory.org/setiand-big-history/. In particular, this author’s own presentation of his Evo-SETI Theory is found at the website: https://www.youtube.com/watch?v=Mcsj_IGuUQU&list=PL2DlytCDr RCnSglJ9_nCbEFWhFgTgYmR-&index=7&t=182s. That video describes all the most important Evo-SETI results that we have mentioned so far.

21

Our “E Pluribus Unum” Theorem: i.e. The Connection Between the Lifetime of Many Single Individuals and the Lifetime of Their Civilization

We also discovered the mathematical connection (called by us “E Pluribus Unum” Theorem, i.e. “One [Civilization] out of many [Individuals])” between the lifetimes of many single individuals (actually an infinity of single individuals) and the lifetime of the whole civilization. This connection is the Evo-SETI translation of the central limit theorem (CLT) of statistics described at the site https://en.wikipedia. org/wiki/Central_limit_theorem.

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These results are mathematically rather difficult and were published in the first part of the paper entitled “New Evo-SETI results about civilizations and molecular clock”, International Journal of Astrobiology, 16 (1): 40–59 (2017).

22

Important: The Lifetime in Between Birth and Peak Always is a b-LOGNORMAL, No Matter Whether We Use a Straight Line In Between Senility and Death (i.e. a b-lognormal), or a Parabola (i.e. a Logpar) or an Ellipse (i.e. a Logell) Power Curve for the Decline in One’s Lifetime

The following “obvious” remark is in reality quite important: whether we use a b-lognormal, or a logpar, or a logell curve to represent the lifetime of a living being or a civilization, the part between birth and peak is THE SAME for all curves, and that is just a b-lognormal climbing from birth to peak. In other words still, only the second part of the curve, i.e. the one between peak and death, changes according to the assumed decline profile, and is: (1) b-lognormal: from senility s followed by a straight line at senility s, or (2) logpar: descending parabola from peak p to death d, or (3) logell: descending quarter-of-ellipse from peak p to death d. On the contrary, the area under the curve in between birth and peak is always the same, i.e. the area under the same b-lognormal between birth and peak. We discovered that this area is given by Zp Birth to Peak ENERGY ¼ 2 erf ð xÞ ¼ pffiffiffi p

Zx

b-lognormalðt; l; r; bÞ dt ¼

1  erf 2

  prffiffi 2

with

b

et dt: 2

0

ð9Þ In Eq. (9) erf ð xÞ is the well-known “error function” of statistics and probability, i.e. the “integral of the Gaussian”. Note that Eq. (9) is independent of both b and p and depends on r only, i.e. we may shift the b-lognormal forth and back along the time axis as we please, but the only variable “that matters” is r, i.e. the standard deviation of the original Gaussian now reverberated into the b-lognormal since lognormal = exp(Gaussian).

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23

Preface

Ontogeny (Ontogenesis) and Other Applications of Evo-SETI Theory to the Bio-Medical Sciences

Ontogeny (until recently also called Ontogenesis, website https://en.wikipedia.org/ wiki/Ontogeny) is the part of a baby’s life in between birth and puberty. Thus, ontogeny is the “building up” period in any living creature’s life: the body “automatically piles up” all necessary and sufficient “tools” to be later transmitted to any offspring. Around 2018–19, this author discovered an equation giving the energy of ontogeny, but he did not have the time to publish it, constrained between his duties in other fields of science (astronautics). So, the result that we now show is unpublished and is mentioned here in public for the first time and without any mathematical proof with apologies for the lack of proof. First of all, consider man as an example. Suppose that the average life of a man lasts 80 years (we do not make any distinction between men and women at this point). Suppose also that the peak age in a man’s lifetime is 40. Then, upon setting b ¼ 0, p ¼ 40 and d ¼ 80 into the logpar history formulae Eq. (7), they produce the logpar power curve shown in Fig. 12. Three different segments of the logpar power curve exist: (1) Ontogeny (i.e. ontogenesis) in between birth and puberty at age around 12 or 13 (black solid curve). Puberty is assumed by us to coincide with the ascending inflexion of the b-lognormal. (2) Youth between puberty at 12 or 13 and peak at 40 (RED solid curve). (3) Decline between the peak at age 40 and death at 80 (decline’s parabola). The energy of ontogeny is clearly given by the area under the black segment of this logpar power curve. In other words, the energy of ontogeny is given by the integral, that we were able to compute analytically MAN's LIFETIME as a LOGPAR power curve: black = ONTOGENY 0.01 0.009 0.008 0.007 0.006 0.005 0.004 0.003 0.002 0.001 0

0

5

10

15

20

25

30

35

40

45

50

55

60

65

70

75

80

Years since the MAN's birth Fig. 12 Logpar power curve of a man’s lifetime, assuming peak at age 40 and death at age 80 years

Preface

xxxiii puberty Z

b-lognormalðt; l; r; bÞ dt

ENERGY of ONTOGENY ¼ birth

¼

1  erf

pffiffiffiffiffiffiffiffiffi

r2 þ 4 þ 3 r 3 22

 :

2

ð10Þ

Please spend a minute to realize the importance of this result: we are talking about the energy requested to build up a new living being from its birth up to its puberty (that we identify with the b-lognormal ascending inflexion time, called a = adolescence in previous papers by this author). A lot of further biomedical research could and should be done, based on equation Eq. (10). Unfortunately, at age 71, this author is unable to pursue further research along these lines, but young researchers should think about equation Eq. (10). Actually, this author was able to do better than just Eq. (10): he was able to prove that the progressive energy of a living being between birth and any instant z before the peak is given by the following equation Eq. (11): Birth_to_any_instant_z_before_Peak__PROGRESSIVE_ENERGY__for_any_ Living_Being =  Zz ¼

1  erf b-lognormalðt; l; r; bÞ dt ¼

lnðpb pzb ffiffi Þ 2r

 þ prffiffi2

2

with

b  z  p:

b

ð11Þ We leave to the reader to prove that Eq. (10) is the particular case of Eq. (11) when the upper limit of the integral is the abscissa of the increasing inflexion (puberty). Also, just a glance at Eq. (11) shows that Eq. (9) is the particular case z ¼ p of Eq. (11). Final delight: when you kiss your baby about 4.3 years old, and see that he/she is so lively, well… it is all fault of the third derivative of the b-lognormal Eq. (5) equalled to zero! In fact, such an equation has three roots: ½t ¼ %er

pffiffiffiffiffiffiffiffiffiffi r2 þ 3 2 r2 þ l

þ b; t ¼ %er

pffiffiffiffiffiffiffiffiffiffi r2 þ 3 2 r2 þ l

þ b; t ¼ %e l2 r þ b 2

ð12Þ That, with reference to Fig. 12, have the three numerical values (in years since Man’s birth): ½t ¼ 4:300541166719219; t ¼ 98:07031362681045; t ¼ 20:53668476130368 ð13Þ

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Preface

Well, to understand the meaning of Eq. (13), just remember that the curvature radius of any curve b-lognormal(t) is given by radius of curvature of b-lognormalðtÞ ¼  1þ

d 2 b-lognormalðtÞ dt2



2 32

ð14Þ

d b-lognormalðtÞ dt

Now we introduce an “unjustified approximation”, consisting in supposing that one has d b-lognormalðtÞ ¼0 dt

ð15Þ

during the ontogeny, i.e. during the time of the black solid curve in Fig. 12. While the usage of this approximation is uncertain here, yet it might be “numerically not too wrong”, as “company engineers” usually do when applying Eq. (14) to their own “practical company problems”. In this event, Eq. (14) reduces to the much easier equation typical of engineers: radius of curvature of b-lognormalðtÞ 

d 2 b-lognormalðtÞ dt2

ð16Þ

In this supposition, it follows that the instants of the maxima and minima of the radius of curvature are approximately given by the zeros of the first derivative of Eq. (16), that is, the instants of the maxima and minima of the radius of curvature are approximately given by the ZEROS OF THE THIRD DERIVATIVE of the b-lognormal: d 3 b-lognormalðtÞ 0 dt3

ð17Þ

The zeros of this equation Eq. (17) are analytically exactly given by Eq. (12) and, numerically for the man input triplet ½b ¼ 0; p ¼ 40; d ¼ 80, these three instants (in years since the Man’s birth) are given by Eq. (13). So, when you caress your baby aged approximately 4.3 years, please realize that he/she is living the maximum time of his/hers ontogeny rate! Wow! In other words, actually more scientific words than the previous poetic words referring to babies aged 4.3 years, our Evo-SETI Theory shows that (approximately only, because of the “engineers’ assumption” Eq. (15)) the maximum ontogeny activity occurs at the time tMax

Ontogeny Activity

¼ er

pffiffiffiffiffiffiffiffiffi r2 þ 32 r2 þ l

þ b ¼ ðp  bÞe

ffi ffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffi

p p p 2 dp p þ 5d6b þ 2 ðdpÞ p þ d2b

þb ð18Þ

Preface

xxxv

In Eq. (18), l and r were of course expressed in terms of the input triplet ðb; p; d Þ by virtue of the logpar history formulae Eq. (7). Next: what about the second root in Eq. (12), numerically corresponding to 98.070 years? Answer: this root must be discarded since it goes beyond the assumed death age of 80, and so it has no biological meaning at all. And what about the third root in Eq. (12), that is tAge

REASON

2 ðdpÞ

¼ el2 r þ b ¼ ðp  bÞe p þ d2b þ b 2

of

ð19Þ

Well, this is no less than the “Age of Reason”, i.e. the time in one’s life when he/she decides what to do in the rest of his/her life! and for man it equals 20.536 years, as Eq. (13) shows. In conclusion, this author regrets that he was unable to publish a great paper about ONTOGENY mathematically investigated by the zeros of the third derivative of the b-lognormal, as quickly described above. There is a lot to discover in this field, and we leave that to youg folks.

24

Letting Maxima (NASA Symbolic Manipulator of the Apollo Flights to the Moon) do the Calculations

No mathematician, however good, can conduct lengthy calculations by hand any more. Since the 1950s, however, excellent “symbolic manipulators” have been created to help mathematicians. As of 2020, Mathematica, Maple and a few other codes can do algebra, the calculus, differential and integral equations and even tensors without making mistakes that are not just silly programming mistakes. This author loves Maxima (formerly called Macsyma, see the website http:// maxima.sourceforge.net/). In fact, Maxima come to you for free, and that is what the “poor” students want. Also, the Maxima files (written in Lisp, so Maxima is actually an Articifial Intelligence = AI code) were created over 50 years ago at the AI Lab of MIT with NASA funds (to check the orbits to the Moon). Therefore, after 50 years of use, Maxima is pratically bug-free. Chapter OVERCOME Theorem, that is PEAK-LOCUS Theorem, Proven by Maxima of this book is thus the Maxima code proving that molecular clock straight line in Fig. 3 is actually the Evo-Entropy (=Shannon entropy with a reversed sign and starting at zero just 3.5 billion years ago) of the running b-lognormal (RbL) of the lognormal process L(t) starting at t = ts and having an arbitrarily assigned mean value m(t). This is the best result of Evo-SETI Theory, since it may be extended to exoplanets and SETI, as we shall see in this book.

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A note about REPETITIONS in this book. Then, Charge!

Dear Readers, This book is a collection of LEGALLY AUTHORIZED REPRINTS of PAPERS published by this author during the years 2010–2020. As such, this book contains many REPETITIONS of the text, equations and theorems. In fact, whenever submitting a new paper to a journal, it was necessary for this author to SUMMARIZE all previous results so that the new Reviewers would NOT reject the submitted paper just because these Reviewers were “stranger” to the topics. In other words, when you are opening NEW PATHS to Research, you must FIGHT HARD against old-fashioned Reviewers. Are you convinced that Evo-SETI Theory leads to profound insights touching not just astrophysics, history, big history and biology, but also sociology, psychology, medicine and other fields of science? This author hopes so, especially since just today, February 6th, 2021, he is 73 years old, and it is high time for him to pass the testimony on to new generations of mathematically trained scientists of all kinds. Best wishes!

Links to this Author’s Online Scientific Activity During the COVID-19 Pandemics Lockdown The sudden Covid-19 pandemics forced this author to stay locked at his apartment in Turin ever since 9 March 2020. Nevertheless, he was able to continue online the following activities: (1) On 25 March 2020, he was able to chair a 3 h “Moon Farside Negotiations” Symposium of the International Academy of Astronautics (IAA) that you might wish to download here: https://drive.google.com/drive/folders/1SOh6um FMUiNArgL_BggjCdzYbjtNenah?usp=sharing This activity is related to what described in the Chapter about SETI Space Missions of the present book. (2) A consequence was the Scientific American article by Leonard David that you find here: https://www.scientificamerican.com/article/astronomers-battle-spaceexplorers-for-access-to-moons-far-side/ and the other website http://www. leonarddavid.com/radio-silence-on-the-lunar-farside-a-frequency-freak-out-asgovernments-private-groups-eye-moon-exploration-goals/. (3) About the same time, two Event Horizon interviews to this author were put online at this website: https://www.eventhorizonshow.com/watch/ about the KLT for mathematical SETI and the searches for ET civilizations living around the Centre of the Milky Way Galaxy, respectively. Thanks if you would like to watch them. Turin, Italy February 6 2021

Claudio Maccone

Contents

OVERCOME Theorem, that is PEAK-LOCUS Theorem, Proven by Maxima . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 OVERCOME Theorem, that is PEAK-LOCUS Theorem . . . . . . . . . 2 Evo-Entropy(p): Measuring “How Much Evolution” Occurred . . . . . 3 Perfectly LINEAR Evo-Entropy When the Mean Value Is Perfectly Exponential (A GBM): This Is just the Molecular Clock . . . . . . . . . 4 Introducing EE, the Evo-SETI Unit: An Amount of Information Equal to the Evo-Entropy Reached by Life Evolution on Earth Nowadays . 5 Conclusions About Evo-Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Evo-SETI Mathematics: Part 1: Entropy of Information. Part 2: Energy of Living Forms. Part 3: The Singularity . . . . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Part 1: Entropy of Information as the Measure of Evolution, Peak-Locus Theorem, and Scale of Biological Evolution (Evo-SETI Scale) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Purpose of This Chapter . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 A Simple Proof of the b-Lognormal Probability Density Function (PDF) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Biological Evolution as the Exponential Increase of the Number of Living Species . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Biological Evolution on Earth Was just a Particular Realization of Geometric Brownian Motion in the Number of Living Species . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 During the Last 3.5 Billion Years Life Forms Increased like a Lognormal Stochastic Process . . . . . . . . . . . . . . . . . . . . . 2.6 Mean Value of the Lognormal Process L(t) . . . . . . . . . . . . 2.7 LðtÞ Initial Conditions at ts . . . . . . . . . . . . . . . . . . . . . . . .

.. .. ..

1 2 3

..

6

. . . .

8 9 10 25

... ...

27 27

... ...

30 30

...

31

...

32

...

34

... ... ...

36 36 38

. . . .

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2.8 LðtÞ Final Conditions at te > ts . . . . . . . . . . . . . . . . . . . . . . 2.9 Important Special Cases of mL ðtÞ . . . . . . . . . . . . . . . . . . . . . 2.10 Boundary Conditions When mL ðtÞ Is a First, Second or Third Degree Polynomial in the Time (t − ts) . . . . . . . . . . . . . . . . 2.11 Peak-Locus Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.12 Evo-Entropy(p): Measuring “How Much Evolution” Occurred . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.13 Perfectly Linear Evo-Entropy When the Mean Value Is Perfectly Exponential (a GBM): This Is just the Molecular Clock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.14 Introducing EE, the Evo-SETI Unit: An Amount of Information Equal to the Evo-Entropy Reached by Life Evolution on Earth Nowadays . . . . . . . . . . . . . . . . . . . . . . . 2.15 Markov-Korotayev Alternative to Exponential: A Cubic Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.16 Evo-Entropy of the Markov-Korotayev Cubic Growth . . . . . . 2.17 Comparing the Evo-Entropy of the Markov-Korotayev Cubic Growth to a Hypothetical (1) Linear and (2) Parabolic Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.18 Conclusions About Evo-Entropy . . . . . . . . . . . . . . . . . . . . . 2.19 Life as a Finite b-Lognormal as Assumed by This Author Prior to 2017 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.20 b-Lognormal History Formulae and Their Applications to Past History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Part 2: Energy of Living Forms by “Logpar” Power Curves . . . . . . 3.1 Introduction to Logpar Power Curves . . . . . . . . . . . . . . . . . . 3.2 Finding the Parabola Equation of the Right Part of the Logpar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Finding the b-Lognormal Equation of the Left Part of the Logpar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Area Under the Parabola on the Right Part of the Logpar Between Peak and Death . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Area Under the Full Logpar Curve Between Birth and Death . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 The Area Under the Logpar Curve Depends on Sigma Only, and Here Is the Area Derivative W.R.T. r . . . . . . . . . . . . . . 3.7 Exact “History Equations” for Each Logpar Curve . . . . . . . . 3.8 Considerations on the Logpar History Equations . . . . . . . . . . 3.9 Logpar Peak Coordinates Expressed in Terms of ðb; p; d Þ Only . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.10 History of Ancient Rome as an Example of How to Use the Logpar History Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.11 Area Under Rome’s Logpar and Its Meaning as “Overall Energy” of the Roman Civilization . . . . . . . . . . . . . . . . . . . .

Contents

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3.12 The Energy Function of d Regarded as a Function of the Death Instant d, Hereafter Renamed D . . . . . . . . . . . . . . . . . . . . . . 3.13 Discovering an Oblique Asymptote of the Energy Function, Energy(D), While the Death Instant D Is Increasing Indefinitely . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.14 The Oblique Asymptote for the “History of Rome” Case . . . 3.15 What if Hadn’t Rome Fallen? Discovering the Straight Line Parallel to the Asymptote but Starting at the Rome Power Peak . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.16 Energy Output of the Sun as a G2 Star Over the About 10 Billion Years of Its Lifetime . . . . . . . . . . . . . . . . . . . . . . 3.17 Energy Output of an M Star Over 45 Billion Years of Lifetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.18 Mean Power in a Lifetime . . . . . . . . . . . . . . . . . . . . . . . . . . 3.19 Lifetime Mean Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.20 Logpar Power Curves Versus b-Lognormal Probability Densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.21 Conclusions About Logpars . . . . . . . . . . . . . . . . . . . . . . . . . Part 3: Before and After the Singularity According to Evo-SETI Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Every Exponential in Time Has just a Single Knee: The Instant at Which Its Curvature Is Highest . . . . . . . . . . . . . . . . . . . . 4.2 GBM Exponential as Mean Value of the Increasing Number of Species Since the Origin of Life on Earth . . . . . . . . . . . . . 4.3 Deriving the Knee Time for GBMs . . . . . . . . . . . . . . . . . . . 4.4 Knee-Centered Form of the GBM Exponential . . . . . . . . . . . 4.5 Finding WHEN the GBM Knee Will Occur According to the Author’s Conventional Values for ts and B . . . . . . . . . . . . . . 4.6 Ray Kurzweil’s 2006 Book “the Singularity Is Near” . . . . . . 4.7 Kurzweil’s Singularity Is the Same as Our GBM’s Knee in Our Evo-SETI Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 Measuring the Pace of Evolution B by the Average Number m0 of Species Living on Earth NOW . . . . . . . . . . . . . . . . . . . . 4.9 An Unexpected Discovery: The “Origin-to-Now” (“OTN”) Equation Relating the Time of the Origin of Life on Earth (ts) to m0 (the Average Number of Species Living on Earth Right Now) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.10 Solving the “Origin-to-Now” Equation NUMERICALLY for the Two Cases of −3.5 and −3.8 Billion Years of Life on Earth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.11 But… Biologists Are UNABLE to Measure m0 Experimentally! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.12 Lognormal pdf of the GBM . . . . . . . . . . . . . . . . . . . . . . . . .

xxxix

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4.13 Finding the GBM Parameter r . . . . . . . . . . . . . . . . . . . . . . . 4.14 Numerical Standard Deviation Nowadays for the Two Cases of Life Starting 3.5 and 3.8 Billion Years Ago . . . . . . . . . . . 4.15 Numerical r for the Two Cases of Life Starting 3.5 and 3.8 Billion Years Ago . . . . . . . . . . . . . . . . . . . . . . . . . . 4.16 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . 108

SETI, Evolution and Human History Merged into a Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 SETI and Darwinian Evolution Merged Mathematically . . . . . . . . . 1.1 Introduction: The Drake Equation (1961) as the Foundation of SETI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Statistical Drake Equation (2008) . . . . . . . . . . . . . . . . . . . . . 1.3 The Statistical Distribution of N Is Lognormal . . . . . . . . . . . 1.4 Darwinian Evolution as Exponential Increase of the Number of Living Species . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Introducing the ‘Darwin’ (D) Unit, Measuring the Amount of Evolution that a Given Species Reached . . . . . . . . . . . . . . 1.6 Darwinian Evolution Is just a Particular Realization of Geometric Brownian Motion in the Number of Living Species . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 GBM as the Key to Stochastic Evolution of All Kinds . . . . . . . . . . 2.1 The N(t) GBM as Stochastic Evolution . . . . . . . . . . . . . . . . . 2.2 Our Statistical Drake Equation Is the Static Special Case of N(t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 GBM as the Key to Mathematics of Finance . . . . . . . . . . . . 3 Darwinian Evolution Re-defined as a GBM in the Number of Living Species . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 A Concise Introduction to Cladistics and Cladograms . . . . . . 3.2 Cladistics: Namely the GBM Mean Exponential as the Locus of the Peaks of b-Lognormals Representing Each a Different Species Started by Evolution at Time t = b . . . . . . . . . . . . . 3.3 Cladogram Branches Are Increasing, Decreasing or Stable (Horizontal) Exponential Arches as Functions of Time . . . . . 3.4 KLT-Filtering in Hilbert Space and Darwinian Selection Are “the Same Thing” in Our Theory… . . . . . . . . . . . . . . . . 3.5 Conclusions About Our Statistical Model for Evolution and Cladistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Lifespans of Living Beings as b-Lognormals . . . . . . . . . . . . . . . . . 4.1 Further Extending b-Lognormals as Our Model for All Lifespans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Infinite b-Lognormals . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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. . 120 . . 121 . . 121 . . 123 . . 125 . . 126 . . 126

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From Infinite to Finite b-Lognormals: Defining the Death Time, d, as the Time Axis Intercept of the b-Lognormal Tangent Line at Senility s . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Terminology About Various Time Instants Related to a Lifetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Terminology About Various Time Spans Related to a Lifetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Normalizing to One All the Finite b-Lognormals . . . . . . . . . 4.7 Finding the b-Lognormals Given b and Two Out of the Four a, p, s, d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Golden Ratios and Golden b-Lognormals . . . . . . . . . . . . . . . . . . . . 5.1 Is r Always Smaller Than 1? . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Golden Ratios and Golden b-Lognormals . . . . . . . . . . . . . . . 6 Mathematical History of Civilizations . . . . . . . . . . . . . . . . . . . . . . 6.1 Civilizations Unfolding in Time as b-Lognormals . . . . . . . . . 6.2 Eight Examples of Western Historic Civilizations as Finite b-Lognormals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Plotting All b-Lognormals Together and Finding the Trends . 6.4 b-Lognormals of Alien Civilizations . . . . . . . . . . . . . . . . . . . 7 Extrapolating History into the Past: Aztecs . . . . . . . . . . . . . . . . . . . 7.1 Aztecs–Spaniards as an Example of Two Suddenly Clashing Civilizations with Large Technology Gap . . . . . . . . . . . . . . . 7.2 ‘Virtual Aztecs’ Method to Find the ‘True Aztecs’ b-Lognormal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 b-Lognormal Entropy as ‘Civilization Amount’ . . . . . . . . . . . . . . . 8.1 Introduction: Invoking Entropy and Information Theory . . . . 8.2 Exponential Curve in Time Determined by Two Points Only . 8.3 Assuming that the Exponential Curve in Time Is the GBM Mean Value Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 The ‘No-Evolution’ Stationary Stochastic Process . . . . . . . . . 8.5 Entropy of the ‘Running b-Lognormal’ Peaked at the GBM Exponential Mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 Decreasing Entropy for an Exponentially Increasing Evolution: Progress! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7 Six Examples: Entropy Changes in Darwinian Evolution, Human History Between Ancient Greece and Now, and Aztecs and Incas Versus Spaniards . . . . . . . . . . . . . . . . . . . . . . . . . 8.8 b-Lognormals of Alien Civilizations . . . . . . . . . . . . . . . . . . . 9 Conclusion: Summary of Technical Concepts Described . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Evolution and Mass Extinctions as Lognormal Stochastic Processes . . . 171 1 Introduction: Mathematics and Science . . . . . . . . . . . . . . . . . . . . . . . . 172 2 A Summary of the ‘Evo-SETI’ Model of Evolution and SETI . . . . . . . 173

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Important Special Cases of mL(t) . . . . . . . . . . . . . . . . . . . . . . . . . . Introducing b-lognormals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Peak-Locus Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Entropy as the Evolution Measure . . . . . . . . . . . . . . . . . . . . . . . . . Evo-SETI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mass Extinctions of Darwinian Evolution Described by a Decreasing GBM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 GBMs to Understand Mass Extinctions of the Past . . . . . . . . 8.2 Example: The K–Pg Mass Extinction Extending Ten Centuries After Impact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Mass Extinctions Described by an Adjusted Parabola Branch . . . . . 9.1 Adjusting the Parabola to the Mass Extinctions of the Past . . 9.2 Example: The Parabola of the K–Pg Mass Extinction Extending Ten Centuries After Impact . . . . . . . . . . . . . . . . . 10 Cubic as the Mean Value of a Lognormal Stochastic Process . . . . . 10.1 Finding the Cubic When Its Maximum and Minimum Times Are Given, in Addition to the Five Conditions to Find the Parabola . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Markov–Korotayev Biodiversity Regarded as a Lognormal Stochastic Process Having a Cubic Mean Value . . . . . . . . . . . . . . . . . . . . . . . 11.1 Markov–Korotayev’s Work on Evolution . . . . . . . . . . . . . . . 12 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Supplementary Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

New Evo-SETI Results About Civilizations and Molecular Clock . . . 1 Part I: New Results About Civilizations in Evo-SETI Theory . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 A Simple Proof of the b-Lognormal’s pdf . . . . . . . . . . . . . . 1.3 Defining ‘Life’ in the Evo-SETI Theory . . . . . . . . . . . . . . . . 1.4 History Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Death Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Birth–Peak–Death (BPD) Theorem . . . . . . . . . . . . . . . . . . . . 1.7 Mathematical History of Nine Key Civilizations Since 3100 BC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 b-Scalene (Triangular) Probability Density . . . . . . . . . . . . . . 1.9 Uniform Distribution Between Birth and Death . . . . . . . . . . . 1.10 Entropy Difference Between Uniform and b-Scalene Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.11 ‘Equivalence’ Between Uniform and b-Lognormal Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.12 b-Lognormal of a Civilization’s History as CLT of the Lives of Its Citizens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1.13 The Very Important Special Case of Ci Uniform Random Variables: E-Pluribus-Unum Theorem . . . . . . . . . . . . . . . . . 2 Part 2: New Results About Molecular Clock in Evo-SETI Theory . . 2.1 Darwinian Evolution as a Geometric Brownian Motion (GBM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 A Leap Forward: For Any Assigned Mean Value mL(t) We Construct Its Lognormal Stochastic Process . . . . . . . . . . 2.3 Completing [3]: Letting ML(t) There Be Replaced Everywhere by mL(t), the Assigned Trend . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Peak-Locus Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 tsGBM and GBM Sub-cases of the Peak-Locus Theorem . . . 2.6 Shannon Entropy of the Running b-Lognormal . . . . . . . . . . . 2.7 Introducing Our… Evo-Entropy(p) Measuring How Much a Life Form Has Evolved . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 The Evo-Entropy(p) of tsGBM Increases Exactly Linearly in Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Supplementary Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Life Expectancy and Life Energy According to Evo-SETI Theory . . . 1 Part 1: Logpar Curves and Their History Equations . . . . . . . . . . . . 1.1 Introduction to Logpar “Finite Lifetime” Curves . . . . . . . . . . 1.2 Finding the Parabola Equation of the Right Part of the Logpar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Finding the b-Lognormal Equation of the Left Part of the Logpar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Area Under the Parabola on the Right Part of the Logpar Between Peak and Death . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Area Under the Full Logpar Curve Between Birth and Death . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 The Area Under the Logpar Curve Depends on Sigma Only, and Here Is the Area Derivative w.r.t. Sigma . . . . . . . . . . . . 1.7 Exact “History Equations” for Each Logpar Curve . . . . . . . . 1.8 Considerations on the Logpar Least-Energy History Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9 Logpar Peak Coordinates Expressed in Terms of ðb; p; d Þ Only . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Part 2: Energy as the Area Under All Logpar Power Curves . . . . . . 2.1 The Area Under a Logpar and Its Meaning as “Lifetime Energy” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Part 3: Mean Energy in a Lifetime and Lifetime Mean Value . . . . . 3.1 Mean Energy in a Lifetime . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Lifetime Mean Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Part 4: Adolescence Formulae (Or Puberty Formulae) . . . . . . . . . . . 4.1 Logpar’s Increasing Inflexion Time as Adolescence Time (Or Puberty Time for Living Beings) . . . . . . . . . . . . . . . . . . 5 Part 5: Life Expectancy and Fertility in Logpars . . . . . . . . . . . . . . . 5.1 Reconsidering the Death Time d as a Living Being’s Life Expectancy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Introducing the Living Being’s End-Of-Fertility (EOF) Time . 5.3 Life Expectancy of Living Beings . . . . . . . . . . . . . . . . . . . . 5.4 Fertility Span of Living Beings . . . . . . . . . . . . . . . . . . . . . . 5.5 A Numerical Example About the Most Important Case: Man . 5.6 Checking Numerically the (Small) Difference Between History Formulae and Adolescence Formulae for Man . . . . . . . . . . . 6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Evo-SETI: A Mathematical Tool for Cladistics, Evolution, and SETI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Purpose of This Chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 During the Last 3.5 Billion Years, Life Forms Increased as in a (Lognormal) Stochastic Process . . . . . . . . . . . . . . . . . . . . . 3 Mean Value of the Lognormal Process L(t) . . . . . . . . . . . . . . . . . . 4 L(t) Initial Conditions at ts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 L(t) Final Conditions at te > ts . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Important Special Cases of m(t) . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Boundary Conditions When m(t) Is a First, Second, or Third Degree Polynomial in the Time (t − ts) . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Peak-Locus Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 EvoEntropy(p) as a Measure of Evolution . . . . . . . . . . . . . . . . . . . 10 Perfectly Linear EvoEntropy When the Mean Value Is Perfectly Exponential (GBM): This Is just the Molecular Clock . . . . . . . . . . . 11 Markov-Korotayev Alternative to Exponential: A Cubic Growth . . . 12 EvoEntropy of the Markov-Korotayev Cubic Growth . . . . . . . . . . . 13 Comparing the EvoEntropy of the Markov-Korotayev Cubic Growth, to the Hypothetical (1) Linear and (2) Parabolic Growth . . . . . . . . . 14 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Proof of the CUBIC MEAN VALUE Equation . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Statistical Drake Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Step 1: Letting Each Factor Become a Random Variable . . . . . . . . 2.1 Step 2: Introducing Logs to Change the Product into a Sum 2.2 Step 3: The Transformation Law of Random Variables . . . .

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Step 4: Assuming the Easiest Input Distribution for Each Di: The Uniform Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Step 5: A Numerical Example of the Statistical Drake Equation with Uniform Distributions for the Drake Random Variables Di . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Step 6: Computing the Logs of the Seven Uniformly Distributed Drake Random Variables Di . . . . . . . . . . . . . . . . 3.3 Step 7: Finding the Probability Density Function of N, but Only Numerically, Not Analytically . . . . . . . . . . . . . . . . 4 The Central Limit Theorem (CLT) of Statistics . . . . . . . . . . . . . . . . 5 The Lognormal Distribution Is the Distribution of the Number N of Extraterrestrial Civilizations in the Galaxy . . . . . . . . . . . . . . . . . 6 Comparing the CLT Results with the Non-CLT Results . . . . . . . . . 7 Distance of the Nearest Extraterrestrial Civilization as a Probability Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Classical, Non-probabilistic Derivation of the Distance of the Nearest ET Civilization . . . . . . . . . . . . . . . . . . . . . . . 7.2 Probabilistic Derivation of the Probability Density Function for ET_Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Statistical Properties of This Distribution . . . . . . . . . . . . . . . 7.4 Numerical Example of the ET_Distance Distribution . . . . . . . 8 The “DATA ENRICHMENT PRINCIPLE” as the Best CLT Consequence upon the Statistical Drake Equation (Any Number of Factors Allowed) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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SETI and SEH (Statistical Equation for Habitables) . . . . . . . . . . . . . 1 Introduction to SETI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 The Key Question: How Far Are They? . . . . . . . . . . . . . . . . . . . . . 3 Computing N by Virtue of the Drake Equation (1961) . . . . . . . . . . 4 The Drake Equation Is Over-Simplified . . . . . . . . . . . . . . . . . . . . . 5 The Statistical Drake Equation by Maccone (2008) . . . . . . . . . . . . . 6 Solving the Statistical Drake Equation by Virtue of the Central Limit Theorem (CLT) of Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 An Example Explaining the Statistical Drake Equation . . . . . . . . . . 8 Finding the Probability Distribution of the ET-Distance by Virtue of the Statistical Drake Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 9 The “Data Enrichment Principle” as the Best CLT Consequence upon the Statistical Drake Equation (Any Number of Factors Allowed) . . 10 Habitable Planets for Man . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 The Statistical Dole Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 The Number of Habitable Planets for Man in the Galaxy Follows the Lognormal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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13 The Distance Between Any Two Nearby Habitable Planets Follows the Maccone Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 A Numerical Example: A Some Hundred Million Habitable Planets Exist in the Galaxy! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Distance (Maccone) Distribution of the Nearest Habitable Planet to Us According to the Previous Numerical Inputs . . . . . . . . . . . . . 16 Comparing the Statistical Dole and Drake Equations: Number of Habitable Planets Versus Number of ET Civilizations in This Galaxy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 SEH, the “Statistical Equation for Habitables” Is just the Statistical Dole Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Societal Statistics by Virtue of the Statistical Drake Equation . . . . . . 1 Introducing the Drake Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 The Drake Equation is Over-Simplified . . . . . . . . . . . . . . . . . . . . . 3 The Statistical Drake Equation by Maccone [3] . . . . . . . . . . . . . . . 4 Solving the Statistical Drake Equation by Virtue of the Central Limit Theorem (CLT) of Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 An Example Explaining the Statistical Drake Equation . . . . . . . . . . 6 The “Data Enrichment Principle” as the Best CLT Consequence upon the Statistical Drake Equation (Any Number of Factors Allowed) . . 7 Habitable Planets for Man . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 The Statistical Dole Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 The Number of Habitable Planets for Man in the Galaxy Follows the Lognormal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 An Example Explaining the Statistical Dole Equation: Some Hundred Million Habitable Planets Exist in the Galaxy! . . . . . . . . . . . . . . . . 11 Comparing the Statistical Dole and Drake Equations: Number of Habitable Planets Versus Number of ET Civilizations in This Galaxy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 The Probability Distribution of the Ratio of Two Lognormally Distributed Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Breaking the Drake Equation up into the Dole Equation Times the Drake Equation’s Societal Part . . . . . . . . . . . . . . . . . . . . . . . . . 14 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Evolution and History in a New “Mathematical SETI” Model . . . . . . . . 401 1 Introduction: Interstellar Flight and SETI Are the Two Sides of the Same Coin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402 1.1 Astronautics and Interstellar Flight Since the Moon Landings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402

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The “FOCAL” Space Missions Enabling Us to Use the Sun as a Gravitational Lens (1992) . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 A “RADIO BRIDGE” Among Two Stars to Enable the Radio Link Among Their Civilizations (2011) . . . . . . . . . . . . . . . . 1.4 A Galactic Internet Might Be in Use Already, but by Aliens, Not by Humans yet (2021) . . . . . . . . . . . . . . . . . . . . . . . . . SETI and Darwinian Evolution Merged Mathematically . . . . . . . . . 2.1 Introduction: The Drake Equation (1961) as the Foundation of SETI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Statistical Drake Equation (2008) . . . . . . . . . . . . . . . . . . . . . 2.3 The Statistical Distribution of N Is Lognormal . . . . . . . . . . . 2.4 Darwinian Evolution as (Overall) Exponential Increase in the Number of Living Species . . . . . . . . . . . . . . . . . . . . . 2.5 Darwinian Evolution Is Just a Particular Realization of Geometric Brownian Motion in the Number of Living Species . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Geometric Brownian Motion (GBM) Is the Key to Stochastic Evolution of All Kinds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 The N(t) GBM as Stochastic Evolution . . . . . . . . . . . . . . . . . 3.2 Our Statistical Drake Equation Is the Static Special Case of N(t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 The N(t) Stochastic Process Is a Geometric Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Darwinian Evolution Re-defined as a GBM in the Number of Living Species . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introducing the “DARWIN” (D) Unit, Measuring the Amount of Evolution that a Given Species Reached . . . . . . . . . . . . . . 4.2 Cladistics, Namely the GBM Mean Exponential as the Locus of the Peaks of b-Lognormals Representing Each a Different Species Started by Evolution at the Time t = b . . . . . . . . . 4.3 Cladogram Branches Are Made up by Increasing, Decreasing or Stable (Horizontal) Exponential Arches . . . . . . . . . . . . . . 4.4 Conclusions About Our Statistical Model for Evolution and Cladistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Life-Spans of Living Beings as b-Lognormals . . . . . . . . . . . . . . . . 5.1 Further Extending b-Lognormals as Our Model for All Life-Spans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Infinite b-Lognormals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 From Infinite to Finite b-Lognormals: Defining the Death Time, d, as the Time Axis Intercept of the Tangent at Senility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Terminology of Various Time Instants Related to a Lifetime . 5.5 Terminology of Various Time Spans Related to a Lifetime . .

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5.6 5.7

Normalizing to One All Finite b-Lognormals . . . . . . . . . . . . Finding the b-Lognormals Given b and Two Out of the Four a, p, s, d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Golden Ratios and Golden b-Lognormals . . . . . . . . . . . . . . . . . . . . 6.1 Is r Always Smaller Than 1? . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Golden Ratios and Golden b-Lognormals . . . . . . . . . . . . . . . 7 Mathematical History of Human Civilizations . . . . . . . . . . . . . . . . . 7.1 Civilizations Unfolding in Time as b-Lognormals . . . . . . . . . 7.2 Eight Examples of Western Civilizations as Finite b-Lognormals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Plotting All b-Lognormals Together and Finding Exponentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Extrapolating History into the Past: Aztecs . . . . . . . . . . . . . . . . . . . 8.1 Aztec–Spaniards as an Example of Two Suddenly Clashing Civilizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 “Virtual Aztecs” Method to Find the “True Aztecs” B-Lognormal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 b-Lognormal Entropy as “Civilization Amount” . . . . . . . . . . . . . . . 9.1 Introduction: Invoking Entropy and Information Theory . . . . 9.2 Exponential Curve in Time Determined by Two Points Only . 9.3 Assuming that the Exponential Curve in Time Is the GBM Mean Value Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 The “No-Evolution” Stationary Stochastic Process . . . . . . . . 9.5 Entropy of the “Running B-Lognormal” Peaked at the GBM Exponential Mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6 Decreasing Entropy for an Exponentially Increasing Evolution: Progress! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.7 Six Examples: Entropy Changes in Darwinian Evolution, Human History Between Ancient Greece and Now, and Aztecs and Incas Versus Spaniards . . . . . . . . . . . . . . . . . . . . . . . . . 9.8 b-Lognormals of Alien Civilizations . . . . . . . . . . . . . . . . . . . 10 Aliens: How Much More Advanced Than US? . . . . . . . . . . . . . . . . 10.1 Extrapolating the Human Past 8000 Years into the Future . . . 10.2 Extrapolating 100,000 Years into the Future . . . . . . . . . . . . . 10.3 Extrapolating a Million Years into the Future . . . . . . . . . . . . 10.4 Fermi Paradox: Extrapolating Ten Million Years into the Future . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Spaceflight, SETI and the Future of Humankind . . . . . . . . . . . . . . . 11.1 Spaceflight as of 2013 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Big History: One More Step Ahead . . . . . . . . . . . . . . . . . . . 11.3 Conclusion: The Grand Vision of Universal Evolution . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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SETI as a Part of Big History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction: New Statistical Mechanisms . . . . . . . . . . . . . . . . . . . . 2 Merging SETI and Darwinian Evolution Statistically . . . . . . . . . . . . 2.1 The Drake Equation (1961) as the Foundation of SETI . . . . . 2.2 Statistical Drake Equation (2008) . . . . . . . . . . . . . . . . . . . . . 2.3 The Statistical Distribution of N is Lognormal . . . . . . . . . . . 2.4 Darwinian Evolution as the Exponential Increase of the Number of Living Species . . . . . . . . . . . . . . . . . . . . . 2.5 Introducing the “Darwin” (d) Unit, Measuring the Amount of Evolution that a Given Species Reached . . . . . . . . . . . . . . 2.6 Darwinian Evolution Is Just a Particular Realization of Geometric Brownian Motion in the Number of Living Species . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Geometric Brownian Motion (GBM) as the Key to Stochastic Evolution of All Kinds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 The N(t) GBM as Stochastic Evolution . . . . . . . . . . . . . . . . . 3.2 Statistical Drake Equation as the Static Special Case of N(t) . 3.3 GBM as the Key to the Mathematics of Finance . . . . . . . . . . 3.4 Adjusting the GBM: Letting It Take the Value of One at Its Start (Time t = tSTART) and Deriving Its Current Mean Value and Standard Deviation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Example: Darwinian Evolution as a GBM Taking the Value of One at Its Start (Time t = tSTART) with Known Current Mean Value and Standard Deviation . . . . . . . . . . . . . . . . . . . . . . . 4 Big History as the Statistical Drake Equation Extended by Adding the “Missing Initial Part” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Big Bang to Current Stars: The “Missing Initial Part” of the Drake Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Mass Extinctions in the Course of Darwinian Evolution Understood by Virtue of a Decreasing GBM . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 A Brand-New Discovery: GBMs to Understand Mass Extinctions of the Past . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Example: The K–Pg Mass Extinction Extending Ten Centuries After Impact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix: Cyclic Phenomena as Lognormal Stochastic Processes . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lognormals for SETI, Evolution and Mass Extinctions . . . . . . . . . . . 1 Introduction: New Statistical Mechanisms . . . . . . . . . . . . . . . . . . . . 2 Mass Extinctions of Darwinian Evolution Described by a Decreasing Geometric Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 GBMs to Understand Mass Extinctions of the Past . . . . . . . . 2.2 Example: The K–Pg Mass Extinction Extending Ten Centuries After Impact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Mass Extinctions Described by an Adjusted Parabola Branch . . . 3.1 Adjusting the Parabola to the Mass Extinctions of the Past 3.2 Example: The Parabola of the K–Pg Mass Extinction Extending Ten Centuries After Impact . . . . . . . . . . . . . . . 4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Statistical Drake–Seager Equation for Exoplanet and SETI Searches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 The Classical Drake Equation (1961) . . . . . . . . . . . . . . . . . . . . . . 3 Transition from the Classical to the Statistical Drake Equation . . . 4 Step 1: Letting Each Factor Become a Random Variable . . . . . . . . 5 Step 2: Introducing Logs to Change the Product into a Sum . . . . . 6 Step 3: The Transformation Law of Random Variables . . . . . . . . . 7 Step 4: Assuming the Easiest Input Distribution for Each Di: The Uniform Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Step 5: Computing the Logs of the 7 Uniformly Distributed Drake Random Variables Di . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 The Central Limit Theorem (CLT) of Statistics . . . . . . . . . . . . . . . 10 The Lognormal Distribution is the Distribution of the Number N of Extraterrestrial Civilizations in the Galaxy . . . . . . . . . . . . . . . . 11 Data Enrichment Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 The Statistical Seager Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 13 The Extremely Important Particular Case When the Input Random Variables Are Uniform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Evo-SETI Entropy Identifies with Molecular Clock . . . . . . . . . 1 Purpose of This Chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 During the Last 3.5 Billion Years Life Forms Increased Like a (Lognormal) Stochastic Process . . . . . . . . . . . . . . . . . . . . . 3 Important Special Cases of m(t) . . . . . . . . . . . . . . . . . . . . . . 4 Peak-Locus Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Entropy as Measure of Evolution . . . . . . . . . . . . . . . . . . . . . 6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Evo-SETI SCALE to Measure Life on Exoplanets . . . . . . . . . . 1 Purpose of This Chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 During the Last 3.5 Billion Years Life Forms Increased Like a (Lognormal) Stochastic Process . . . . . . . . . . . . . . . . . . . . . 3 Mean Value of the Lognormal Process L(t) . . . . . . . . . . . . .

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L(t) Initial Conditions at ts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . L(t) Final Conditions at te > ts . . . . . . . . . . . . . . . . . . . . . . . . . . . Important Special Cases of mL(t) . . . . . . . . . . . . . . . . . . . . . . . . . . Boundary Conditions When mL(t) is a First, Second or Third Degree Polynomial in the Time (t–ts) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Peak-Locus Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Entropy as Measure of Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Kullback–Leibler Divergence (or, Better, “Distance”) Among Any Two Living Species . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Evo-SETI Theory and Information Gap Among Civilizations . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 A Simple Proof of the b-Lognormal’s Pdf . . . . . . . . . . . . . . . . . 3 History Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Death Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Birth-Peak-Death (BPD) Theorem . . . . . . . . . . . . . . . . . . . . . . . 6 Information Entropy (SHANNON ENTROPY) as the Measure of a Civilization’s Advancement . . . . . . . . . . . . . . . . . . . . . . . . 7 Information Gaps, Namely Entropy Differences Among the Nine Historic Western Civlizations . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Kurzweil’s Singularity as a Part of Evo-SETI Theory . . . . . . . . . . . . 1 Geometric Brownian Motion (GBM) Is Key to Exponential Stochastic Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Darwinian Evolution as the Exponential Increase of the Number of Living Species . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Darwinian Evolution Was Just a Particular Realization of Geometric Brownian Motion in the Number of Living Species . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Knee of Any Exponential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Every Exponential in Time Has Just a Single Knee: The Instant at Which Its Curvature Is Highest . . . . . . . . . . . . . . . . . . . . 2.2 (GBM) Exponential as Mean Value of the Increasing Number of Species Since the Origin of Life . . . . . . . . . . . . . . . . . . . 2.3 Deriving the Knee Time for GBMs . . . . . . . . . . . . . . . . . . . 2.4 Knee-Centered Form of the GBM Exponential . . . . . . . . . . . 2.5 Finding When the GBM Knee Will Occur According to the Author’s Conventional Values for ts and B . . . . . . . . . 3 Kurzweil’s “the Singularity Is Near” (Is Nowadays) . . . . . . . . . . . . 3.1 Ray Kurzweil’s 2006 Book “the Singularity Is Near” . . . . . .

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Kurzweil’s Singularity Is the GBM’s Knee in Our Evo-SETI Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Measuring the Pace of Evolution B by Measuring the Average Number m0 of Species Living on Earth Right Now . . . . . . . 3.4 An Unexpected Discovery: The “Origin-to-Now” (“OTN”) Equation Relating the Time of the Origin of Life on Earth ts to m0 the Average Number of Species Living on Earth Right Now . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Solving the “Origin-to-Now” Equation Numerically for the Two Cases of −3.5 and −3.8 Billion Years of Life Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 No Way for the Biologists to Measure m0 Experimentally! . . 4 Upper and Lower Standard Deviation Curves . . . . . . . . . . . . . . . . . 4.1 Lognormal PDF of the GBM . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Finding the GBM Parameter r . . . . . . . . . . . . . . . . . . . . . . . 4.3 Numerical Standard Deviation Nowadays for the Two Cases of Life Starting 3.5 and 3.8 Billion Years Ago . . . . . . . . . . . 4.4 Numerical r for the Two Cases of Life Starting 3.5 and 3.8 Billion Years Ago . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Evo-entropy of the b-Lognormals Having Their Peaks on the GBM Exponential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Peak-Locus Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Entropy as Measure of Evolution . . . . . . . . . . . . . . . . . . . . . 6 The Korotayev-Markov Alternative Evolution Theory with a Cubic-like Mean Value in Time . . . . . . . . . . . . . . . . . . . . . 6.1 Peak-Locus Theorem When the Mean Value Is a Polynomial in the Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Finding the Cubic When Its Maximum and Minimum Times Are Given, in Addition to the Five Conditions to Find the Parabola . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Markov-Korotayev Biodiversity Regarded as a Lognormal Stochastic Process Having a Cubic Mean Value . . . . . . . . . . 7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . APPENDIX, i.e. our: Evo-SETI SINGULARITY THEOREM. A NEW RESULT connecting three quite different facts like: (1) The time ts of the origin of life on Earth (RNA); (2) The number m0 of Species living NOW * 140 million; (3) The SINGULARITY is just (a few years from) NOW! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Energy of Extra-Terrestrial Civilizations According to Evo-SETI Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Part 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Logpar Curves and Their History Equations . . . . . . . . . . 2 Part 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Energy as the Area Under Logpar Curves . . . . . . . . . . . . 3 Part 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Mean Energy in a Lifetime and Lifetime Mean Value . . . 4 Part 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Conclusions: The Advantages of Logpar Power Curves . . Appendix A: Supplementary Data . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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The Evo-SETI Unit of Evolution is EE = Earth Evolution = 25.575 bit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Purpose of This Chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 During the Last 3.5 Billion Years Life Forms Increased Like a (Lognormal) Stochastic Process . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Mean Value of the Lognormal Process L(t) . . . . . . . . . . . . . . . . . . 4 L(t) Initial Conditions at ts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 L(t) Final Conditions at te > ts . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Important Special Cases of m(t) . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Boundary Conditions When m(t) Is a First, Second or Third Degree Polynomial in the Time (t − ts) . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Peak-Locus Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 EvoEntropy(p) as Measure of Evolution . . . . . . . . . . . . . . . . . . . . . 10 Perfectly Linear EvoEntropy When the Mean Value Is Perfectly Exponential (GBM): This Is Just the Molecular Clock! . . . . . . . . . . 11 Introducing the EE Evo-SETI Unit: Information Equal to the EvoEntropy Reached by the Evolution of Life on Earth Nowadays . 12 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Appendix A: Supplementary Data . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Evo-SETI Quartics Yielding ET Civilizations’ Energy . . . . . . . . . . . . 1 Introduction to Evo-SETI Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Quartic Curve in the Time t Representing the Power (Measured in Watts) of a Civilization Between Its Inception (b = Birth) and Its End (d = Death) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Conditions Imposed on Our Quartic Power Curve . . . . . . . . . . . . . . 4 Separating Our Quartic Curve into the Product of the Two Above Boundary Conditions Times a Quadratic Polynomial in the Time . . 5 Expressing the Separated Quartic in Terms of b, d and the Derivatives Db and Dd of this Quartic at the Initial and End Times, Respectively . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

643 645 645 655 655 665 665 669 669 672 673

. . 675 . . 676 . . . . .

. . . . .

676 677 678 680 681

. . 683 . . 684 . . 685 . . 688 . . . .

. . . .

690 690 691 691

. . 693 . . 693

. . 694 . . 694 . . 695

. . 696

liv

The General Quartic and Its Five Coefficients Expressed in Terms of b, d and the Derivatives Db and Dd of the Quartic at the Initial and End Times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Energy of the General Quartic, i.e. Area Under the General Quartic and the Time Axis, i.e. Integral of the Quartic Between b and d . . . 8 Defining the Smooth-Start Quartic, That Is the Quartic with Zero Derivative at b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Energy of the Smooth-Start Quartic . . . . . . . . . . . . . . . . . . . . . . . . 10 Defining the Symmetric Quartic, i.e. When the Derivatives Db and Dd of the Quartic at the Initial and End Times are Equal to Each Other . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Energy of the Symmetric Quartic, i.e. Area Under the Symmetric Quartic, i.e. Integral of the Symmetric Quartic Between b and d . . . 12 The Special Case of the Symmetric Quartic with Zero Derivatives at Both b and d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Energy of the Symmetric Quartic with Zero Derivatives at Both b and d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Determining the Last Unknown, A, If One Knows the Energy that a Civilization or a Living Being Used (or Produced) During Its Own Lifetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Contents

6

KLT for an Expanding Universe with SETI Applications . . . . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Claudio Maccone’s 1981 Exact Analytical Solution Yielding the KLT for All Time-Rescaled Gaussian Stochastic Processes (that is Gaussian Noises) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 The Machinery of Maccone’s 1981 Exact KLT for Relativistic Frames in Arbitrary Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 The 1981 Exact KLT Machinery in Case of Relativistic Frames in Uniform Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Friedman-Lemaître-Robertson-Walker (FLRW) Metric and the Friedman Equations with K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 KLT Time-Rescaling Function f ðtÞ for Motion of Spaceships, Particles and Light Inside Any Expanding Universe Given Its aðtÞ Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Motion of a “Star Trek Spaceship” with “Flight Law” Equal to mðtÞ Inside an Expanding Universe Having the Expansion Law aðtÞ . . . . 8 Motion of a Constant-Speed mconst Beam of Particles Inside an Expanding Universe Having the Expansion Law aðtÞ . . . . . . . . . . .

. . 697 . . 698 . . 698 . . 700

. . 700 . . 701 . . 701 . . 702

. . . .

. . . .

703 703 704 713

. . 715 . . 715

. . 716 . . 716 . . 720 . . 723

. . 724 . . 727 . . 728

Contents

Motion of c-Speed Electromagnetic Waves, Neutrinos and Gravitational Waves in an Expanding Universe Having the Expansion Law aðtÞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Applications of the KLT to SETI . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

lv

9

SETI Space Missions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 A SETI Space Mission that Never Was: Quasat . . . . . . . . . . . . . . . 3 This Author’s Activity About Exploiting for SETI the Moon Farside Radio Quietness in the Fifteen Years 1995–2010 . . . . . . . . . . . . . . 4 This Author’s First Presentation Ever at the United Nations COPUOS About Legally Protecting the Central Part of the Moon Farside Against Man-Made Radio Pollution . . . . . . . . . . . . . . . . . . . . . . . . 5 Defining PAC, the “Protected Antipode Circle” . . . . . . . . . . . . . . . 6 Need for RFI-Free Radio Astronomy, as Pointed Since 1974 by Both ITU and Jean Heidmann (1923–2000) . . . . . . . . . . . . . . . . 7 A Short Review About the Five Lagrangian Points of the Earth-Moon System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 The Quiet Cone Above the Moon Farside Depends on the Orbits of Telecommunication Satellites Orbiting the Earth . . . . . . . . . . . . . 9 Selecting Crater Daedalus Near the Farside Center, i.e. the Near the Earth Antipode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Our 2010 Vision of the Moon Farside for RFI-free Science, Likely not Valid After 2018 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 The Further Two Lagrangian Points L1 and L2 of the Sun-Earth System: Their “Polluting” Action on the Farside of the Moon . . . . . 12 Attenuation of Man-Made RFI on the Moon Farside . . . . . . . . . . . . 13 A New Mathematical Contribution of Ours: The Blocking Equation for Electromagnetic (em) Waves Emitted at Height H Above the Earth and Reaching the Earth-Moon Axis at a Distance X Above Moon Farside . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 The Years 2018–2019 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 The Farside Spectrum Still Is not Polluted (in August 2019) Except in the S, X and UHF Bands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Queqiao and Chang’e 4 Communications Bands Above the Moon Farside . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 COSMOLOGY: Need for Ultra-Low Frequency Radio Astronomy in Space Within the Quiet Cone Above the Moon Farside, I.E. just at the Lagrangian Point L2, Where Queqiao Is . . . . . . . . . . . . . . . . . . . . . 18 SETI (or “Technosignatures”, According to NASA’s 2018 “New Jargon”) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . .

. . . .

728 728 729 729

. . 733 . . 734 . . 734 . . 735

. . 735 . . 737 . . 738 . . 739 . . 740 . . 741 . . 744 . . 746 . . 747

. . 748 . . 751 . . 752 . . 753

. . 753 . . 754

lvi

Contents

19 Summary About This Author’s Work to Legally Protecting Radio Astronomy on the Farside and Within the Quiet Cone in the Space Above the Farside . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755 20 CONCLUSIONS: The Moon Village Should Be Located Outside the PAC and Along the 180 Degrees Meridian, Possibly Close to the South Pole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 756 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 756 Power and Energy of Civilizations by Logpar and Logell Power Curves (with Ancient Rome Example) . . . . . . . . . . . . . . . . . . . . . . . . 1 Part 1: Logell Curves and Their History Equations . . . . . . . . . . . . . 1.1 Introduction to Logell “Finite Lifetime” Curves . . . . . . . . . . 1.2 Finding the Ellipse Equation of the Right Part of the Logell . 1.3 Finding the b-Lognormal Equation of the Left Part of the Logell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Area Under the Ellipse on the Right Part of the Logell Between Peak and Death . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Area Under the Full Logell Curve Between Birth and Death . 1.6 The Area Under the Logell Curve Depends on Sigma Only, and Here Is the Area Derivative w.r.t. Sigma . . . . . . . . . . . . 1.7 Exact “History Equations” for Each Logell Curve . . . . . . . . . 1.8 Considerations on the Logell History Equations . . . . . . . . . . 1.9 Logell Peak Coordinates Expressed in Terms of ðb; p; d Þ Only . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.10 History of Rome as an Example of How to Use the Logell History Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Part 2: Energy as the Area Under Logell Power Curves . . . . . . . . . 2.1 Area Under Any Logell Power Curve and Its Meaning as “Lifetime Energy” of that Living Being . . . . . . . . . . . . . . 2.2 Discovering an Oblique Asymptote of the Energy Function Energy (D) While the Death Instant D Is Increasing Indefinitely . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Part 3: Mean Power in a Logell Lifetime . . . . . . . . . . . . . . . . . . . . 3.1 Mean Power in a Logell Lifetime . . . . . . . . . . . . . . . . . . . . . 4 Part 4: Logell Lifetime Mean Value . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Lifetime Mean Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Part 5: Conclusions: Which One Is Better? Logell or LOGPAR? . . . 5.1 Conclusions About Rome’s Civilization . . . . . . . . . . . . . . . . 5.2 Conclusions About Evo-Seti Theory as of 2018 . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . .

. . . .

757 758 758 759

. . 761 . . 763 . . 763 . . 764 . . 766 . . 767 . . 768 . . 768 . . 769 . . 769

. . . . . . . . .

. . . . . . . . .

771 773 773 774 774 777 777 778 779

Contents

Logpars and Energy of Nine Historically Important Civilizations . . 1 Summary of Chapter “Energy of Extra-Terrestrial Civilizations According to Evo-SETI Theory” Results that Will Be Used in This Chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Finding the Logpars of Nine Civilizations that Made the History of the World . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Peak Height of Each of the Nine Civilizations . . . . . . . . . . . . . . . 4 Total Energy of Each Civilization . . . . . . . . . . . . . . . . . . . . . . . . 5 After-Peak Energy and Longterm History, that Is, Letting D Approach Infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 SURPRISE: The MINIMUM AFTER-ENERGY Value Is THE SAME for All Nine Civilizations! . . . . . . . . . . . . . . . . . . . . . . . . 7 AFTER-PEAK-ENERGY Curves for All Nine Civilizations . . . . . 8 Central Star, Longterm Energy and the Evolution of a Civilization There . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 LONGTERM POWER for All Nine Civilizations . . . . . . . . . . . . . 10 MAXIMA CODE for All Calculations Described in This Chapter . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

lvii

. . . 781

. . . 782 . . . 785 . . . 785 . . . 787 . . . 787 . . . 788 . . . 789 . . . .

MOLECULAR CLOCK as a Stochastic Process: Evo-Entropy (Shannon Entropy of Evolution) of a Geometric Brownian Motion (GBM) with a LINEAR MEAN VALUE . . . . . . . . . . . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Summary of the Mathematical Appendix Found in Chapter “OVERCOME Theorem, that is PEAK-LOCUS Theorem, Proven by Maxima” of This Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 A Strong Upper Bound upon the Standard Deviation in the Number of Species Living Today: Delta Ne 0,B>0); (%o83) [ A > 0 , B > 0 ] (%i84) tsGBM_case:[m(p)=m(ts)*%e^(B*(p-ts)),sL=sqrt(2*B)]; (%o84) [ m( p ) = m( ts ) %e ( p - ts ) B , sL = 2

B]

(%i85) towards_final_tsGBM_sigma_and_mu_of_p: final_sigma_and_mu_of_m_of_p,tsGBM_case; (%o85) [ σ =

%e p B - ( p - ts ) B 2

π m( ts )

,μ=

%e 2 p B - 2 ( p - ts ) B 4 π m( ts ) 2

-p B]

(%i86) final_tsGBM_sigma_and_mu_of_p: distrib(radcan(towards_final_tsGBM_sigma_and_mu_of_p)); (%o86) [ σ =

%e ts B 2

π m( ts )

,μ=

%e 2 ts B - 4 π p m( ts ) 2 B 4 π m( ts ) 2

Finding the numeric value of B. (%i87) tsGBM_case_at_te:subst(te,p,tsGBM_case); (%o87) [ m( te ) = m( ts ) %e ( te - ts ) B , sL = 2

B]

]

Appendix

23 (%i88) B_of_tsGBM:first(solve(first(tsGBM_case_at_te),B)); log

(%o88) B = -

m( te ) m( ts )

ts - te

(%i89) tsGBM_numeric_values: [ts=-3.5*10^9*yr,m(ts)=1,te=0,m(te)=50*10^6]; (%o89) [ ts = - 3.5 10 9 yr , m( ts ) = 1 , te = 0 , m( te ) = 50000000 ] (%i90) B_of_tsGBM,tsGBM_numeric_values; (%o90) B =

2.8571428571428571 10 -10 log ( 50000000 ) yr

(%i91) B_in_yr:ev(%,numer); (%o91) B =

5.0650095895406915 10 -9 yr

(%i92) yr_vs_sec:yr=365.25*24*60*60*sec; (%o92) yr = 3.15576 10 7 sec (%i93) B_in_sec:B=rhs(B_in_yr),yr_vs_sec; (%o93) B =

1.6050046865226414 10 -16 sec

GBM case. (%i94) GBM_case:[m(p)=A*%e^(B*p),sL=sqrt(2*B)]; (%o94) [ m( p ) = A %e p B , sL = 2 B ] (%i95) final_GBM_sigma_and_mu_of_p: final_sigma_and_mu_of_m_of_p,GBM_case; (%o95) [ σ =

1 2

πA

,μ=

1 4 π A2

-p B]

This completes the study of the Peak-Locus Theorem.

ENTROPY of the Running b-lognormal Shannon ENTROPY of the Running b-Lognormal in bits.

24

OVERCOME Theorem, that is PEAK-LOCUS Theorem, Proven by Maxima (%i96) def_H:H=(log(sqrt(2*%pi)*sigma)+mu+1/2)/log(2); π σ) + μ +

log( 2

(%o96) H =

1 2

log ( 2)

(%i97) towards_def_H_of_m:def_H,final_sigma_and_mu_of_m_of_p; p sL 2

log

%e

2

+

m( p )

(%o97) H =

%e p

sL 2

4 π m( p ) 2

-

p sL 2 2

+

1 2

log ( 2)

(%i98) H_of_m:distrib(radcan(towards_def_H_of_m)); (%o98) H =

2 %e p sL

4 π log ( 2) m ( p ) 2

-

l og ( m ( p ) ) log ( 2)

+

1 2 log ( 2)

(%i99) def_H_of_ts:H_of_ts=subst(ts,p,rhs(H_of_m)); (%o99) H_of_ts =

%e ts sL

2

4 π log ( 2) m ( ts ) 2

-

log ( m ( ts ) ) log ( 2)

+

1 2 log ( 2)

NON-LINEAR Evo-ENTROPY (in bits) of the Running b-Lognormal (%i100) def_NonLinearEvoEntropy_of_p: NonLinearEvoEntropy_of_p=(-rhs(H_of_m)+rhs(def_H_of_ts)); (%o100) NonLinearEvoEntropy_of_p =

%e ts sL

2

4 π log ( 2) m ( ts )

2

-

2 %e p sL

4 π log ( 2) m ( p ) 2

(%i101) NonLinearEvoEntropy(p):=(-H(p)+H(ts)); (%o101) NonLinearEvoEntropy( p ) := - H( p ) + H( ts ) (%i102) 'NonLinearEvoEntropy(ts)=NonLinearEvoEntropy(ts); (%o102) NonLinearEvoEntropy( ts ) = 0

-

log ( m ( ts ) ) log ( 2)

+

l og ( m ( p ) ) log ( 2)

Appendix

25

EXACTLY LINEAR EvoEntropy for Geometric Brownian Motion starting at ts. This is the EXACTLY LINEAR MOLECULAR CLOCK. LINEAR EvoEntropy for the tsGBM case. (%i103) def_linearEvoEntropy_of_p_for_tsGBM: def_NonLinearEvoEntropy_of_p,tsGBM_case,def_Ns_vs_m_of_ts; (%o103) NonLinearEvoEntropy_of_p =

log( Ns %e ( p - ts ) B ) %e 2 p B - 2 ( p - ts ) B log ( 2)

-

4 π log ( 2) Ns 2

+

%e 2 ts B

-

log( Ns )

4 π log ( 2) Ns 2 log ( 2)

(%i104) EvoEntropy_of_p_for_tsGBM:linearEvoEntropy_of_p_for_tsGBM= factor(radcan(rhs(def_linearEvoEntropy_of_p_for_tsGBM))); (%o104) linearEvoEntropy_of_p_for_tsGBM = -

( ts - p ) B log ( 2)

References 1. Maccone, SETI, evolution and human history merged into a mathematical model. Int. J. Astrobiol. 12(3), 218–245 (2013). (That is Chapter “SETI, Evolution and Human History Merged into a Mathematical Model” of the present book) 2. Maccone, Evolution and history in a new “mathematical SETI” model. Acta Astronaut. 93, 317– 344 (2014). (That is Chapter “Evolution and History in a New “Mathematical SETI” Model” of the present book)

Evo-SETI Mathematics: Part 1: Entropy of Information. Part 2: Energy of Living Forms. Part 3: The Singularity

1 Introduction Did you ever wonder if some quantitative relationship exists between: 1. The expected number of living Species on Earth today, that we call m0, and 2. The billions of years (− ts) elapsed since the first Life form (RNA) appeared on Earth about (or more than) 3.5 billion years ago? Well, this author claims that he derived such an unexpected relationship mathematically and it looks simple (ln is the natural log, i.e. the log to base e = 2.71828 . . .): − ts m0 ln(m0) = √ that is 2 − ts − ts − ts ≈ = − ts = m0 ln2 (m0) = √ 0.980 1 2 ln(2)

and finally

m0 ln2 (m0) ≈ − ts In words: the Shannon Entropy, i.e. the amount of information given to us by knowing how many Species (m0) are living on Earth nowadays, is approximately the same number as the amount of billion years elapsed between the Origin of Life on Earth and nowadays (− ts). This is Eq. (211) in this Chapter “Evo-SETI Mathematics: Part 1: Entropy of Information. Part 2: Energy of Living Forms. Part 3: The Singularity”. But its proof requires a long maturation of ideas. Actually, it took ten years (2009–2019) to this author to discover the above equation, plus a good knowledge of the calculus, basic probability and statistics, information theory and the theory of Stochastic Processes (Geometric Brownian Motion in particular), plus a sufficient knowledge of Darwinian Evolution in general.

© Springer Nature Switzerland AG 2020 C. Maccone, Evo-SETI, https://doi.org/10.1007/978-3-030-51931-5_2

27

28

Evo-SETI Mathematics: Part 1: Entropy of Information. …

It all started when this author first watched the TV series “Life on Earth” (1979) by David Attenborough: he then he conceived the idea of “casting the Darwinian evolution of life into mathematical equations”, actually “into some statistical equations”. The result is this Chapter “Evo-SETI Mathematics: Part 1: Entropy of Information. Part 2: Energy of Living Forms. Part 3: The Singularity” and the “Mathematical EvoSETI Theory” associated with it. SETI is the Search for Extra-Terrestrial Intelligence that started in 1959 and is in full swing nowadays. In fact, just as Life is developing on Earth at the pace that we know, the same or something similar might have happened, is happening or will happen on some other exoplanet, as we now discover more and more of them at an exponential growth. So, the discovery of a larger and larger number of Exoplanets raises a question: where does a newly-discovered Exoplanet stand in its capability to develop Life as we know it on Earth? Our tentative answer to this question is our Evo-SETI Theory, a mathematical model aiming at casting Cladistics and the Evolution of Life on Earth over the last 3.5 billion years in terms of a few simple statistical equations based on lognormal probability distributions in the time, rather than in the amounts of something else. The first new notion is that of a b-lognormal, i.e. a lognormal probability density function (pdf) starting at time b > 0 rather than at time zero. The Lifetime of any living form may then be expressed as a b-lognormal starting at b, reaching puberty at the ascending inflexion point a (“adolescence or puberty”), raising up to the peak time p, then starting to decline at the descending inflexion point s (“senility”) and finally going down along a straight line to the intercept with the time axis, that is the “death” of the individual. Based on all this, the author was able to derive several mathematical consequences like the Central Limit Theorem of Statistics re-cast in the language of Evo-SETI theory: from the Lifetime of each individual to the Lifetime of the “big b-lognormal” of the whole population itself to which the individual belongs (“E-Pluribus-Unum Theorem”). In addition, this author discovered the “Peak-Locus Theorem” translating Cladistics in term of Evo-SETI: each SPECIES created by Evolution over 3.5 billion years is a b-lognormal whose peak lies on the exponential in the number of living Species. More correctly still, this exponential is not the exact curve telling us exactly how many Species were on Earth at a given time in the past: on the contrary the exponential is the mean value of a stochastic process called “Geometric Brownian Motion” (GBM) in the mathematics of Finances, so that also the Mass Extinctions of the past are incorporated in Evo-SETI Theory as all-lows of the GBM. But then: what is the Shannon Entropy of each b-lognormal representing a Species? Answer: the Shannon Entropy (with a reversed sign) is the measure of how evolved that Species was, or is now, compared to other Species of the past and of the future. That means measuring evolution: just a single number in bits, typical of Shannon’s Information Theory, rather than a mountain of words! One more key point: what is the equivalent of the molecular clock in evo-SETI theory? Answer: it is the straight line behavior in time of the Shannon Entropy if (and

1 Introduction

29

only if) the exponential is the enveloping curve of all the b-lognormals representing the various Species (called “Evo-Entropy” in our papers). Concluding remark: this author was able to generalize his peak-locus theorem from the simple exponential case to the general case when the mean-value-envelope is not just an exponential, but rather an arbitrary curve that you may chose at will: for instance it as a polynomial of the third degree in the time in the Markov-Korotayev [11, 12] model of evolution, leading then to a non-linear entropy. What a neat mathematical tool for future biologists willing to understand our statistically simple Evo-SETI Theory! Ray Kurzweil’s famous 2006 book “The Singularity Is Near” predicted that the Singularity (i.e. computers taking over humans) would occur around the year 2045. In this chapter we prove that Kurzweil’s prediction is in agreement with the “EvoSETI” (Evolution and SETI)” mathematical model that this author has developed since 2008 in a series of mathematical papers published in both Acta Astronautica and the International Journal of Astrobiology. The key ideas of Evo-SETI are: 1. Evolution of Life on Earth over the last 3.5 billion years was just a realization of a stochastic process in the number of living Species called Geometric Brownian Motion (GBM), well-known in Financial Mathematics (it is the underlying process of the Black-Sholes model for stock prices, for which a Nobel prize in Economics was awarded in 1997, see https://en.wikipedia.org/wiki/Geometric_ Brownian_Motion). It increases exponentially in time and is in agreement with the Statistical Drake Equation of SETI. 2. The level of advancement of each living Species is the (Shannon) ENTROPY of the b-lognormal probability density (i.e. a lognormal starting at the positive time b (birth)) corresponding to that Species. (Peak-Locus Theorem of Evo-SETI theory). 3. Humanity is now very close to the point of minimum radius of curvature of the GBM exponential mean value, called “GBM knee”. We claim that this knee is precisely Kurzweil’s SINGULARITY, in that before the Singularity the exponential growth was very slow (these are animal and human Species made of meat and reproducing sexually over millions of years), whereas, after the Singularity, the exponential growth will be extremely rapid (computers reproducing themselves technologically faster and faster in time). But how is this Singularity structured in detail? Well, first of all we describe what the GBM is, and why it reflects the stochastic exponential increase that occurred in Biological Evolution for over 3.5 billion years. Please notice that the denomination “Geometric Brownian Motion” (taken from Financial Mathematics) is incorrect since the GBM is NOT a Brownian Motion as understood by physicists (i.e. a stochastic process whose probability density function (pdf) is a Gaussian). On the contrary, the GBM is a lognormal process, i.e. a process whose pdf is a lognormal pdf. Next, we compute the time when the GBM knee occurs (i.e. the time of minimum radius of curvature) and find what we call the knee equation, i.e. the relationship between t_knee, ts (the time of the origin of Life on Earth) and B (the rate of growth

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Evo-SETI Mathematics: Part 1: Entropy of Information. …

of the GBM exponential. This equation holds good for any time assumed to be the Singularity time, either in the past, or now, or in the future. Then Ray Kurzweil’s claim that the Singularity is near becomes part of our Evo-SETI Theory in that t_knee is set to zero (i.e. approximately nowadays, when compared to the 3.5 billions of years of past Biological Evolution of Life on Earth). This leads to a very easy form of the GBM exponential as well as to the discovery of a pair of important new equations: 1. The inverse proportionality between the average number of Species living NOW on Earth and B, the pace of evolution. In other words, it would be possible to find B were the biologists able to tell us “fairly precisely” how many Species live on Earth nowadays. Unfortunately, this is not the case since, when it comes to insects and so on, the number of Species is so huge that it is not even known if it ranges in the millions or even in the billions. 2. More promising appears to be another new equation, that we discovered, relating the time of the origin of Life on Earth, ts (that is known fairly precisely to range between 3.5 and 3.8 billion years ago) and the average number of living Species NOW. Finally, the mathematical machinery typical of the Evo-SETI theory is called into action: 1. The Peak-Locus Theorem stating that the GBM exponential is where ALL PEAKS of the b-lognormals running left-to-right are located, so that the blognormals become higher and higher and narrower and narrower (with area = 1 as the normalization). 2. The Shannon entropy as evolution measure of the b-lognormals, more correctly with the sign reversed and starting at the time of the origin of Life on Earth, that is rather called Evo-Entropy. 3. After this point, one more paper should be written to describe… how the blognormal’s “width” would correctly describe the “average duration in time” of each Species (before the Singularity) and of each COMPUTER Species (after the Singularity)… “too much to be done now”.

2 Part 1: Entropy of Information as the Measure of Evolution, Peak-Locus Theorem, and Scale of Biological Evolution (Evo-SETI Scale) 2.1 Purpose of This Chapter This Chapter describes recent developments in a new statistical theory describing Life, Evolution and SETI by mathematical equations. This we call the Evo-SETI mathematical model of Evolution and SETI.

2 Part 1: Entropy of Information as the Measure of Evolution …

31

Our main question is: whenever a new exoplanet is discovered, where does that exoplanet stand in its evolution towards Life as we have it on Earth nowadays, or beyond? This paper is divided into three parts: 1. Evo-Entropy (i.e. the Shannon Information Entropy of the b-lognormals describing each one a different Species), then our discovery called Peak-Locus Theorem, and finally the scale of biological evolution, modelled on what happened on Earth but capable of being extended to other Exoplanets. 2. Energy of every living being as the time integral of our newly-discovered LOGPAR (LOGnormal plus PARabola) power curves. Ancient Rome example. 3. The application of all that to the Singularity, assuming that the Singularity is practically NOW (the knee). After the Singularity, computers will reproduce themselves technologically at an ever increasing exponential pace still given by the Geometric Brownian Motion (GBM) after the knee. But the Evo-Entropy will always grow LINEARY, reaching huge values (Fermi paradox of SETI: we are too backward to match with Alien Computers).

2.2 A Simple Proof of the b-Lognormal Probability Density Function (PDF) This book is based on the notion of a b-lognormal, just as are Refs. [7–9]. To let this Chapter be self-contained in this regard, we now provide an easy proof of the b-lognormal equation as a probability density function (pdf). Just start from the well-known Gaussian or normal pdf 2

_ (x−μ) 2 σ2

e Gaussian_or_normal_pdf(x; μ, σ ) = √

2πσ

⎧ ⎪ ⎨ − ∞ ≤ x ≤ + ∞, − ∞ ≤ μ ≤ + ∞, (1) with ⎪ ⎩ σ ≥ 0.

This pdf has two parameters: 1. μ turns out to be the mean value of the Gaussian and the abscissa of its peak. Since the independent variable x may take up any value between −∞ and +∞, i.e. it is a real variable, so μ must be real too. 2. σ turns out to be the standard deviation of the Gaussian and so it must be a positive variable. 3. Since the Gaussian is a pdf, it must fulfill the normalization condition

∞ −∞

(x−μ)2

e− 2 σ 2 dx = 1 √ 2π σ

(2)

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Evo-SETI Mathematics: Part 1: Entropy of Information. …

and this is the equation we need in order to “discover” the b-lognormal. Just perform in the integral (2) the substitution x = ln(t) (where ln is the natural log). Then (2) is turned into the new integral ∞ 0

(ln (t)−μ)2

e_ 2 σ 2 dt = 1. √ 2π σ t

(3)

But this (3) may be regarded as the normalization condition of another random variable, ranging “just” between zero and + ∞, and this new random variable we call “lognormal” since it “looks like” a normal one except that x is now replaced by x = ln(t) and t now also appears at the denominator of the fraction. In other words, the lognormal pdf is ⎧ ⎨

_

(ln(t)−μ)2 2

lognormal_pdf(t; μ, σ ) = e √2π2 σσ ·t ⎩ holding for 0 ≤ t ≤ ∞, −∞ ≤ μ ≤ + ∞, σ ≥ 0.

(4)

Just one more step is required to jump from the “ordinary lognormal” (4) (i.e. the lognormal starting at t = 0) to the b-lognormal, that is the lognormal starting at any real instant b (“b” stands for “birth”). Since this simply is a shifting along the time axis from 0 to the new real time origin b, in mathematical terms it means that we have to replace t by (t − b) everywhere in the pdf (4). Thus the b-lognormal pdf must have the equation ⎧ ⎨ ⎩

b_lognormal_ pd f (t; μ, σ, b) =

_

(ln(t−b)−μ)2

e√ 2 σ 2 2π σ ·(t−b)

holding for t ≥ b, and up to t = ∞.

(5)

Sometimes, the b-lognormal (5) is called “three-parameter lognormal” by statisticians. This is correct since the three parameters appearing in (5) are (b, μ, σ ). However, we prefer to call it b-lognormal to stress its biological meaning as the probability density representing the Lifetime of any living being, born at the instant b. We will later use b-lognormals also to represent the Lifetime of historic Civilizations, like the nine historic Civilizations that will be studied in Sect. 2.19 hereafter.

2.3 Biological Evolution as the Exponential Increase of the Number of Living Species Consider the Biological Evolution over the last 3.5 billion years, or so. Assuming that the number of Species increased just exponentially over the 3.5 billion years of evolutionary time span is certainly a gross oversimplification of the real situation.

2 Part 1: Entropy of Information as the Measure of Evolution …

33

However, we will assume this exponential increase of the number of living Species in time just temporarily in this section while in Sect. 2.4 Brownian Motions (GBMs). In other words, we assume that 3.5 billion years ago there was on Earth only one living Species, whereas now there may be (say) 50 million living Species or more. Note that the actual number of Species currently living on Earth does not really matter as a number for us at the moment: we just want to stress the exponential character of the growth of the number of Species. Thus, we shall assume that the number of living Species on Earth increases in time as E(t) (standing for “exponential in time”): E(t) = A e B t

(6)

where A and B are two positive constants that we will determine numerically now. This assumption is obviously in agreement with the classical Malthusian theory of population growth. But it also is in line with the more recent “Big History” viewpoint about the whole evolution of the Universe, from the Big Bang up to now, requesting that progress in evolution occurs faster and faster, so that only an exponential growth is compatible with the requirements that (6) approaches infinity for t → ∞, and all its time derivatives are exponentials too, apart from constant multiplicative factors. Let us now adopt the convention that the current epoch corresponds to the origin of the time axis, i.e. to the instant t = 0. This means that all the past epochs of Biological Evolution correspond to negative times, whereas the future ahead of us (including finding ETs) corresponds to positive times. Setting t = 0 in (6), we immediately find E(0) = A

(7)

proving that the constant A equals the number of living Species on Earth right now. We shall assume A = 50 million species = 5 × 107 species.

(8)

To also determine the constant B numerically, consider the two values of the exponential (6) at two different instants t1 and t2 , with t1 < t2 , that is 

E(t1 ) = A e B t1 E(t2 ) = A e B t2 .

(9)

Dividing the lower equation by the upper one, A disappears and we are left with an equation in B only: E(t2 ) = e B (t2 −t1 ) . E(t1 ) Solving this for B yields

(10)

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Evo-SETI Mathematics: Part 1: Entropy of Information. …

B=

ln(E(t2 )) − ln(E(t1 )) . t2 − t1

(11)

We may now impose the initial condition stating that 3.5 billion year ago there was just one Species on Earth, the first one (whether this was RNA is unimportant in the present simple mathematical formulation): 

t1 = −3.5 × 109 years E(t1 ) = 1 whence ln(E(t1 )) = ln(1) = 0.

(12)

The final condition is of course that today (t2 = 0) the number of Species equals A given by (8). Upon replacing both (8) and (12) into (11), the latter yields:  ln 5 × 107 1.605 × 10−16 ln(E(t2 )) = . =− B=− t1 −3.5 109 year sec

(13)

It is important to remark that (13) is the true pace of biological evolution on Earth, i.e. the basic constant of Evo-SETI Theory, like G in Newtonian Gravitation, or just like the speed of light c in physics, and so on. Of course, (13) applies to the evolution of Life on Earth only. In the case of the evolution of Life on other exoplanets, the numerical value of (13) might be quite different, i.e. either smaller or larger, because of the different physical conditions of that exoplanet compared to Earth. We wait for SETI to find those. Having thus determined the numerical values of both A and B, the exponential in (6) is thus fully specified. This curve is plot in Fig. 1 just over the last billion years, rather than over the full range between −3.5 billion years and now, for simplicity.

2.4 Biological Evolution on Earth Was just a Particular Realization of Geometric Brownian Motion in the Number of Living Species Consider again the exponential curve described in the previous section. The most frequent question that non-mathematically minded persons ask this author is: “then you do not take the mass extinctions into account”. Our answer to this objection is that our exponential curve is just THE MEAN VALUE of a certain stochastic process that may run above and below that exponential in an unpredictable way. Such a stochastic process is called Geometric Brownian Motion (abbreviated GBM) and is described, for instance, at the web site: http://en.wikipedia.org/wiki/Geomet ric_Brownian_Motion, from which the following Fig. 2 is taken. In other words, mass extinctions that occurred in the past are indeed taken into account as unpredictable fluctuations in the number of living Species that occurred in the particular realization of the GBM between −3.5 billion years and now. So,

2 Part 1: Entropy of Information as the Measure of Evolution …

35

Fig. 1 Exponential curve representing the growing number of Species on Earth up to now, without taking the mass extinctions into any consideration at all. This simple representation is, of course, oversimplified, inasmuch as we know that several mass extinctions did indeed take place in the past evolution of Life on Earth. But mathematicians do know how to cope for that: we must replace the simple exponential (6) by virtue of a stochastic process such that its mean value is indeed an exponential somehow like (6), but the ordinate may unpredictably oscillate above and below the mean value. This is done in the next section

Fig. 2 Geometric Brownian Motion (GBM). Figure 1 Two particular realizations of the stochastic process called Geometric Brownian Motion (GBM) taken from the relevant Wikipedia site. Their mean values are two exponentials with different values of A and B, that is of μ and σ (i.e. of μ = 1, σ = 0.2 and μ = 0.5, σ = 0.5) in the relevant lognormal probability density functions that we will not write here since irrelevant to the goals of this section now.

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Evo-SETI Mathematics: Part 1: Entropy of Information. …

extinctions are “unpredictable vertical downfalls” in that GBM plot that may indeed happen from time to time, but we don’t know when. Please also notice that: 1. Any particular realization of the GBM occurred over the last 3.5 billion years is very much unknown to us in its numeric details, but… 2. We won’t care either, inasmuch as the theory of stochastic processes only cares about such statistical quantities like the mean value and the standard deviation curves that are deterministic curves in time with well-known equations, as we will see in a moment.

2.5 During the Last 3.5 Billion Years Life Forms Increased like a Lognormal Stochastic Process Let us look at Fig. 3: on the horizontal axis is the time t, with the convention that negative values of t are past times, zero is now, and positive times are future times. Let ts denote the “time of start” of Life on Earth. Then the starting point on the time axis is ts = −3.5 × 109 years i.e. 3.5 billion years ago, the time of the origin of Life on Earth that we assume to be correct. If the origin of Life started earlier than that, say 3.8 billion years ago, the coming equations would still be the same and their numerical values will only be slightly changed. On the vertical axis is the number of Species living on Earth at time t, denoted L(t). This “function of the time” we don’t know in detail, and so it must be regarded as a random function, or stochastic process, with the notation L(t) standing for “Life at time t”. In this paper we adopt the convention that capital letters represent random variables, i.e. stochastic processes if they depend on the time, while lower-case letters mean ordinary variables or functions.

2.6 Mean Value of the Lognormal Process L(t) The most important ordinary, continuous function of the time associated with a stochastic process like L(t) is its mean value, denoted by m L (t) ≡ L(t) .

(14)

The probability density function (pdf) of a stochastic process like L(t) is assumed in Evo-SETI Theory to be a b-lognormal starting at ts. In other words, the relevant pdf equation reads, in its most general form including an arbitrary time function M L (t):

2 Part 1: Entropy of Information as the Measure of Evolution …

37

Fig. 3 Biological Evolution as the increasing number of living species on Earth between 3.5 billion years ago and now. The red solid curve is the mean value of the GBM stochastic process L GBM (t) given by (35), while the blue dot-dot curves above and below the mean value are the two standard deviation upper and lower curves, given by (24) and (25), respectively, with m GBM (t) given by (35). The “Cambrian Explosion” of Life, that on Earth started around 542 million years ago, is evident in the above plot just before the value of −0.5 billion years in time, where all three curves “start leaving the time axis and climbing up”. Notice also that the starting value of living Species 3.5 billion years ago is ONE by definition, but it “looks like” zero in this plot since the vertical scale (which is the true scale here, not a log scale) does not show it. Notice finally that nowadays (i.e. at time t = 0) the two standard deviation curves have exactly the same distance from the middle mean value curve, i.e. 30 million living Species more or less the mean value of 50 million Species. These are assumed values that we used just to exemplify the GBM mathematics: biologists might wish assume other numeric values that we don’t care about in this mathematical paper

L(t)_ pd f (n; M L (t), σ, t) = √  with



e

ln(n) −M L (t)]2 − [ 2 2 σ L (t−ts)

√ 2π σ L t − ts n

n ≥ 0, σ L ≥ 0, and M L (t) = arbitrary function of t. t ≥ ts,

(15)

This assumption is in line with the extension in time of the statistical Drake equation, namely foundational and statistical equation of SETI, as shown in Ref. [4]. The mean value (14) is of course related to the pdf (15) by the defining integral in the number n of living Species on Earth at time t, that is ∞ m L (t) ≡ 0

e

ln(n) −M L (t)]2 − [ 2 2 σ L (t−ts)

n· √ dn . √ 2π σ L t − ts n

(16)

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Evo-SETI Mathematics: Part 1: Entropy of Information. …

The “surprise” is that this integral (16) may be computed exactly with the key result that the mean value m L (t) is given by m L (t) = e M L (t) e

σ L2 2

(t−ts)

.

(17)

In turn, the last equation has the “surprising” property that it may be inverted exactly, i.e. solved for M L (t): M L (t) = ln(m L (t)) −

σ L2 (t − ts) . 2

(18)

2.7 L(t) Initial Conditions at ts Now about the initial conditions of the stochastic process L(t), namely about the value L(ts). We shall assume that the positive number L(ts) = N s > 0

(19)

is always exactly known, i.e. with probability one: Pr{L(ts) = N s} = 1.

(20)

In the practice, N s will be equal to 1 in the theories of evolution of Life on Earth or on an exoplanet (i.e., there must a been a time ts in the past when the number of living Species was just one, let it be RNA or something else), and it will be equal to the number of living Species just before the asteroid/comet impact in the theories of mass extinction of Life on a planet. The mean value m L (t) of L(t) also must equal the initial number N s at the initial time ts, that is m L (ts) = N s .

(21)

Replacing t by ts in (17), one then finds m L (ts) = e M(ts)

(22)

that, checked against (21), immediately yields N s = e M L (ts) that is M L (ts) = ln(N s) . These are the initial conditions for the mean value.

(23)

2 Part 1: Entropy of Information as the Measure of Evolution …

39

After the initial instant ts, the stochastic process L(t) unfolds oscillating above or below the mean value in an unpredictable way. Statistically speaking, however, we expect L(t) “not to depart too much” from m(t) and this fact is graphically shown in Fig. 3 by the two dot-dot blue curves above and below the mean value solid red curve m L (t). These two curves are the upper standard deviation curve

 2 σ (t−ts) L −1 upper_st_dev_curve(t) = m L (t) 1 + e

(24)

and the lower standard deviation curve

 2 σ (t−ts) L −1 lower_st_dev_curve(t) = m L (t) 1 − e

(25)

respectively (Proof: see Table 2 of Ref. [7]). Notice that both (24) and (25), at the initial time t = ts, equal the mean value m(ts) = N s, that is, with probability one again, the initial value N s is the same for all the three curves shown in Fig. 3. The function of the time  2 (26) variation_coefficient(t) = eσL (t−ts) − 1 is called “variation coefficient” by statisticians since the standard deviation of L(t) (be careful: this is just the standard deviation  L (t) of L(t) and not either of the above two “upper” and “lower” standard deviation curves given by (24) and (25), respectively) is  st_dev_curve(t) ≡  L (t) = m L (t)

eσL (t−ts) − 1 2

(27)

Indeed, (27) shows that the variation coefficient (26) is the ratio of (t) to m L (t), i.e. it expresses how much the standard deviation “varies” with respect to the mean value. Having understood this fact, it is then obvious that the two curves (24) and (25) are obtained as  2 m L (t) ±  L (t) = m L (t) ± m L (t) eσL (t−ts) − 1

(28)

as shown by (24) and (25), respectively.

2.8 L(t) Final Conditions at te > ts Now about the final conditions for the mean value curve as well as for the two standard deviation curves.

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Evo-SETI Mathematics: Part 1: Entropy of Information. …

Let us call te the ending time of our mathematical analysis, namely the time beyond which we don’t care any longer about the values assumed by the stochastic process L(t). In the practice, this te is zero (i.e. now) in the theories of evolution of Life on Earth or on an exoplanet, or the time when the mass extinction ends (and Life starts growing up again) in the theories of mass extinction of Life on a planet. First of all, it is clear that, in full analogy to the initial condition (21) for the mean value, also the final condition has the form m L (te) = N e > 0

(29)

where N e is a positive number denoting the number of Species alive at the end time te. But we don’t know what random value will L(te) take. We only know that its standard deviation curve (27) will take at time te a certain positive value that will differ by a certain amount δ N e from the mean value (29). In other words, we only know from (27) that one has δ N e =  L (te) = m L (te)

 2 eσL (te−ts) − 1

(30)

Dividing (30) by (29) the common factor m(te) disappears, and one is left with δNe = Ne

 eσL (te−ts) − 1. 2

(31)

Solving this for σ L finally yields

  2  ln 1 + δNNee σL = . √ te − ts

(32)

This equation expresses the so far unknown numerical parameter σ L in terms of the initial time ts plus the three final-time parameters (te, N e, δ N e). Thus, in conclusion, we have shown that, once the five parameters (ts, N s, te, N e, δ N e) are assigned numerically, the lognormal stochastic process L(t) is completely specified. Finally notice that the square of (32) may be rewritten in the following different form:  ⎧ ⎫  2  1 2  te−ts  ⎨ ⎬ ln 1 + δNNee δNe = ln 1 + σ L2 = (33) ⎩ ⎭ te − ts Ne from which we infer the formula 

e

σ L2

=e

ln

  1 2 te−ts 1+( δNNee )



 = 1+



δNe Ne

1 2  te−ts

.

(34)

2 Part 1: Entropy of Information as the Measure of Evolution …

41

This Eq. (34) enables us to get rid of eσL replacing it by virtue of the four boundary parameters supposed to be known: (ts, te, N e, δ N e). It will be later used in Sect. 2.11 in order to rewrite the Peak-Locus Theorem in terms of the boundary 2 conditions, rather than in terms of eσL . 2

2.9 Important Special Cases of m L (t) The particular case of (14) when the mean value m L (t) is given by the generic exponential m GBM (t) = N0 eμG B M (t−ts)

(35)

is called Geometric Brownian Motion (GBM), and is widely used in financial mathematics, where it represents the “underlying process” of the stock values (Black-Sholes models). This author used the GBM in his previous models of Evolution and SETI Refs. [4–7], since it was assumed that the growth of the number of ET civilizations in the Galaxy, or, alternatively, the number of living Species on Earth over the last 3.5 billion years, grew exponentially (Malthusian growth). Notice that, upon equating the two right-hand-sides of (17) and (35), we find e MGBM (t) e

2 σG BM 2

(t−ts)

= N0 eμG B M (t−ts) .

(36)

Solving this equation for MGBM (t) yields   σG2 B M MGBM (t) = ln N0 + μG B M − (t − ts) . 2

(37)

This is (with ts = 0) just the “mean value showing at the exponent” of the well-known GBM pdf, i.e.

GBM(t)_ pd f (n; N0 , μ, σ, t) =

e−

    2 2 ln(n)− ln N0 + μ− σ2 t 2 σ2 t

√ √ 2π σ t n

, (n ≥ 0).

(38)

We conclude this short description of the GBM as the exponential sub-case of the general lognormal process (15) by warning our readers that “GBM” is a misleading name, since GBM is a lognormal process and not Gaussian one, as the Brownian Motion is indeed in the language of physics. 1. As we mentioned already, another interesting particular case of the mean value function m(t) in (15) is when it equals a generic polynomial in t starting at ts, namely

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Evo-SETI Mathematics: Part 1: Entropy of Information. …



polynomial_degree

m polynomial (t) =

ck (t − ts)k .

(39)

k=0

with ck being the coefficient of the k-th power of the time t in the polynomial (39). We just confine ourselves to mentioning that the case where (39) is a second-degree polynomial (i.e. a parabola in t) may be used to describe the Mass Extinctions on Earth over the last 3.5 billion years (see Ref. [8]). 2. As we did already, we must also introduce the notion of b-lognormal

[ln(t−b)−μ]2

e− 2 σ 2 b-lognormal_pdf(t; μ, σ, b) = √ 2π σ (t − b)

(40)

holding for t ≥ b = birth, and meaning the Lifetime of a living being, let it be a cell, a plant, a human, a civilization of humans, or even an ET civilization (Ref. [7], in particular pages 227–245).

2.10 Boundary Conditions When m L (t) Is a First, Second or Third Degree Polynomial in the Time (t − ts) In Ref. [8] the reader may find a mathematical model of Biological Evolution different from the GBM model described in terms of GBMs. That is the Markov-Korotayev model, for which this author proved the mean value (14) to be a Cubic(t) i.e. a third degree polynomial in t. We summarize hereafter the key formulae proven in Ref. [8] about the case when the assigned mean value m L (t) is a polynomial in t starting at ts, that is: m L (t) =

3 

ck (t − ts)k .

(41)

k=0

1. The mean value is a straight line. Then this straight line simply is the line through the two points (ts, N s) and (te, N e), that, after a few rearrangements, turns out to be: m straight_line (t) = (N e − N s)

t − ts + N s. te − ts

(42)

2. The mean value is a parabola, i.e. a quadratic polynomial in t. Then, the equation of such a parabola reads as below. Equation (43) was actually firstly derived by

2 Part 1: Entropy of Information as the Measure of Evolution …

43

this author in Ref. [8], pages 299–301, in relationship to Mass Extinctions (i.e. it is a decreasing function of the time).

t − ts t − ts 2− + N s. m parabola (t) = (N e − N s) te − ts te − ts

(43)

3. The mean value is a cubic. Then, in Ref. [8], pages 304–307 this author proved, in relation to the Markov-Korotayev model of Evolution, that the cubic mean value of the L(t) lognormal stochastic process is given by the cubic equation in t m cubic (t) = (N e − N s)·     (t − ts) 2(t − ts)2 − 3 tMax + tmin − 2 ts (t − ts) + 6 (tMax − ts) tmin − ts    + N s.  (te − ts) 2(te − ts)2 − 3 tMax + tmin − 2 ts (te − ts) + 6 (tMax − ts) tmin − ts

(44)

Notice that, in (44) one has, in addition to the usual initial and final conditions N s = m L (ts) and N e = m L (te), two more “middle conditions” referring to the two instants (t M , tm ) of at which the Maximum and the minimum of the cubic Cubic(t) occur, respectively: 

tmin = time_of_the_Cubic_minimum tMax = time_of_the_Cubic_Maximum.

(45)

2.11 Peak-Locus Theorem The Peak-Locus theorem is a new mathematical discovery of ours playing a central role in Evo-SETI Theory. In its most general formulation, it holds good for any lognormal process L(t) and any arbitrary mean value m L (t). In the GBM case, it is shown in Fig. 4. The Peak-Locus theorem states that the family of b-lognormals each having its peak exactly located upon the mean value curve (14), is given by the following three equations, specifying the parameters μ( p), σ ( p) and b( p), appearing in (40) as three functions of the independent variable p, the b-lognormal’s peak: that is, if rewritten directly in terms of m L ( p): ⎧ 2 σ2 eσ L p ⎪ μ( p) = − p 2L ⎪ ⎪ 4 π [m L ( p)]2 ⎪ ⎪ σ L2 ⎪ ⎨ e2 p σ ( p) = √ ⎪ 2 π m L ( p) ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 2 b( p) = p − eμ( p) − [σ ( p)] .

(46)

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Evo-SETI Mathematics: Part 1: Entropy of Information. …

Fig. 4 Biological Exponential as the Geometric LOCUS OF THE PEAKS of b-lognormals for the GBM case. Each b-lognormal is a lognormal starting at a time (t = b = birth time) and represents a different SPECIES that originated at time b of the Biological Evolution. This is what biologists call CLADISTICS, just seen now through the glasses of our Evo-SETI model. It is evident that, when the generic “Running b-lognormal” moves to the right, its peak becomes higher and higher and narrower and narrower, since the area under the b-lognormal always equals 1. Then, the (Shannon) ENTROPY of the running b-lognormal is the DEGREE OF EVOLUTION reached by the corresponding SPECIES (or living being, or a civilization, or an ET civilization) in the course of Evolution (see, for instance, Refs. [9, 10])

The proof of (46) is lengthy and was given as a special pdf file (written in the language of the NASA Maxima symbolic manipulator) that the reader may freely download in the web site of Ref. [8]. But we now present an important follow-up result: the Peak-Locus Theorem (46) rewritten not in terms of σ L anymore, but rather in terms of the four boundary parameters supposed to be known: (ts, te, N e, δ N e). To this end, we must insert (34) and (33) into (46), with the result ⎧   p   p 2 te−ts  δ N e 2  2(t−ts) 1+( δNNee ) ⎪ ⎪ ⎪ μ( p) = − ln 1 + ⎪ Ne 4 π [m L ( p)]2 ⎪ ⎪ ⎪ p   ⎪  ⎨ 2 2(t−ts) 1 + δNNee σ ( p) = √ ⎪ ⎪ ⎪ 2 π m L ( p) ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 2 b( p) = p − eμ( p) − [σ ( p)] . In the particular GBM case, the mean value is (35) with μG B M = B, σ L = and N0 = N s = A. Then, the Peak-Locus theorem (46) with ts = 0 yields: ⎧ 1 ⎪ ⎨ μ( p) = 4π A2 − B p, 1 σ = √2π A , ⎪ ⎩ b( p) = p − eμ( p)−σ 2 .

(47)

√

2B

(48)

In this simpler form, the Peak-Locus theorem was already published by the author in Refs. [5–7], while its most general form is (46) and (47).

2 Part 1: Entropy of Information as the Measure of Evolution …

45

2.12 Evo-Entropy( p): Measuring “How Much Evolution” Occurred The (Shannon) Entropy of the b-lognormal (40) is H ( p) =

  √ 1 1 ln 2π σ ( p) + μ( p) + . ln(2) 2

(49)

This is an extremely important result that we are not going to prove now for the sake of brevity, but is proven in Chapter 30 of the author’s book “Mathematical SETI”. This (49) is a function of the peak abscissa p and is measured in bits, as customary in Shannon’s Information Theory. By virtue of the Peak-Locus Theorem (46), it becomes   2 eσ L p 1 1 − ln(m L ( p)) + . H ( p) = ln(2) 4π [m L ( p)]2 2

(50)

One may also rewrite (50) directly in terms of the four boundary parameters (ts, te, N e, δ N e) upon inserting (34) into (50), with the result: ⎧ p  δ N e 2  te−ts ⎪ ⎨ 1 + Ne 1 − ln(m L ( p)) + H ( p) = 2 ln(2) ⎪ 4π [m L ( p)] ⎩

⎫ ⎪ 1⎬ . 2⎪ ⎭

(51)

Thus, (50) or (51) yield the Entropy of each member of the family of ∞1 blognormals (the family’s parameter is p) peaked upon the mean value curve (14). The b-lognormal Entropy (50) is thus the Measure of the Amount of Evolution of that b-lognormal: it measures “the decreasing disorganization in time of what that b-lognormal represents”, let it be a cell, a plant, a human or even a civilization, either historic human or ET. Entropy is thus disorganization decreasing in time. However, that one would prefer to use a measure of the “increasing organization” in time. The Evo-Entropy of p EvoEntropy( p) = −[H ( p) − H (ts)]

(52)

(Entropy of Evolution) is a function of p that has a minus sign in front, thus changing the decreasing trend of the (Shannon) Entropy (28) into the increasing trend of our Evo-Entropy (30). In addition, our Evo-Entropy starts at zero at the initial time ts, as expected: EvoEntropy(ts) = 0.

(53)

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Evo-SETI Mathematics: Part 1: Entropy of Information. …

By some authors, like Léon Brillouin, the reversed-sign Shannon Entropy is called “negentropy” (negative entropy) Negentropy( p) = −H ( p).

(54)

Of course, (54) does not start at zero at the time of the origin of Life on Earth (since conceived for other goals) Negentropy(ts) = 0

(55)

so we still prefer our neat Evo-SETI definition (52). Also, by virtue of (50), and keeping (21) in mind, the Evo-Entropy (54) becomes EvoEntropy( p)_of_the_Lognormal_Process_L(t)    2 2 eσL ts eσ L p m L ( p) 1 − + ln . = ln(2) 4π N s 2 Ns 4π [m L ( p)]2

(56)

Alternatively, we may rewrite (56) directly in terms of the five boundary parameters (ts, N s, te, N e, δ N e) upon inserting (34) into (56), thus finding: EvoEntropy( p)_of_the_Lognormal_Process_L(t) ⎧ ⎫ p ts   δ N e 2  te−ts  δ N e 2  te−ts ⎪ ⎪  ⎨ 1 + 1 + Ne Ne m L ( p) ⎬ 1 − + ln . = 2 2 ⎪ ln(2) ⎪ 4π N s Ns 4π [m L ( p)] ⎩ ⎭

(57)

Let us now remark that the standard deviation at the end time, δ N e, really is irrelevant to compute the Evo-Entropy (57). In fact, the Evo-Entropy (57) is just a continuous curve, and not a stochastic process. So, we may assign δ N e any numeric arbitrary value, and the Evo-Entropy curve must not change. Keeping this in mind, we see that the “true” Evo-Entropy curve (57) is obtained by “squashing down” (57) into the mean value curve m L (t), and that only happens if we let δ N e = 0.

(58)

Inserting (58) into (57), the latter simplifies dramatically into EvoEntropy( p)_of_the_Lognormal_Process_L(t)    1 1 m L ( p) 1 − + ln = ln(2) 4π N s 2 Ns 4π [m L ( p)]2

(59)

which is the final form of the Evo-Entropy (56) and (57) that we will use in the sequel. We may now see very neatly that the final Evo-Entropy (59) is made up by

2 Part 1: Entropy of Information as the Measure of Evolution …

47

three terms: The constant term 1 4π N s 2

(60)

whose numeric value in the particularly important case N s = 1 boils down to 1 = 0.079577471545948 4π

(61)

that is “almost zero”. The “inverse square” term −

1 4π [m L ( p)]2

(62)

that rapidly falls down to zero for m L (t) approaching infinity. In other words, this inverse-square term may become “almost negligible” for large values of the time p. And finally the “dominant term”, i.e. the logarithmic term  ln

m L ( p) Ns

 (63)

that actually is the leading term (“dominant term”) in the Evo-Entropy (59) for large values of the time p. In conclusion, the Evo-Entropy (59) in essence depends basically upon its logarithmic term (63) and so its shape in time must be similar to the shape of a logarithm i.e. nearly vertical at the beginning of the curve and then progressively approaching the horizontal shape, though never reaching it. This curve has no maxima nor minima, nor inflexions.

2.13 Perfectly Linear Evo-Entropy When the Mean Value Is Perfectly Exponential (a GBM): This Is just the Molecular Clock We now prove our crucial discovery that, in the GBM case (35), that is when the mean value is given by the exponential m G B M (t) = N s e

σ L2 2

(t−ts)

= N s e B (t−ts)

(64)

the Evo-Entropy (56) becomes just an exact linear function of the time p since the first two terms inside the braces in (56) just cancel against each other. Proof: just insert (64) into (56) and then simplify, as shown hereafter. EvoEntropy( p)_of_GBM

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Evo-SETI Mathematics: Part 1: Entropy of Information. …

=

= = = = =

⎧ ⎫ ⎪ ⎛ ⎞⎪ ⎪ ⎪ 2 σ ⎪ ⎪ 2 2 L ⎬ 2 ( p−ts) N s e 1 ⎨ eσL ts eσ L p ⎝ ⎠ − + ln

2 ⎪ 4π N s 2 ⎪ ln(2) ⎪ Ns σ L2 ⎪ ⎪ ⎪ ⎩ ⎭ 4π N s e 2 ( p−ts)     2 2 2 σL 1 eσ L p eσL ts ( p−ts) 2 − + ln e 2 ln(2) 4π N s 2 4π N s 2 eσL ( p−ts)   2 eσL ts 1 σ L2 1 − + ( p − ts) 2 ln(2) 4π N s 2 2 4π N s 2 eσL (−ts)   2 2 eσL ts eσL ·(ts) σ L2 1 − + ( p − ts) ln(2) 4π N s 2 4π N s 2 2  2  σL 1 ( p − ts) ln(2) 2 B · ( p − ts). ln(2)

(65)

In other words, the GBM Evo-Entropy, is the function of the variable p given by GBM_EvoEntropy( p) =

B · ( p − ts). ln(2)

(66)

This is, of course, a straight line in the time p starting at the time ts of the Origin of Life on Earth and increasing linearly thereafter. It is measured in bits/individual and is shown in Fig. 5. But… this is the same linear behaviour in time as the molecular clock, that is the technique in molecular evolution that uses fossil constraints and rates of molecular change to deduce the time in geologic history when two Species or other taxa diverged. The molecular data used for such calculations are usually nucleotide sequences for DNA or amino acid sequences for proteins (see Refs. [1, 14]). So, we have discovered that the Evo-Entropy in our Evo-SETI model and the Molecular Clock are the same linear time function, apart for multiplicative constants (depending on the adopted units, like bits, seconds, etc.). This conclusion appears to be of key importance to understand “where a newly discovered exoplanet stands on its way to develop LIFE”.

2 Part 1: Entropy of Information as the Measure of Evolution …

49

Fig. 5 Evo-Entropy (in bits per individual) of the latest Species appeared on Earth during the last 3.5 billion years if the mean value is an increasing exponential, i.e. if our lognormal stochastic process L(t) is a GBM. This straight line shows that a Man (nowadays) is 25.575 bits more evolved than the first form of Life (RNA) 3.5 billion years ago

2.14 Introducing EE, the Evo-SETI Unit: An Amount of Information Equal to the Evo-Entropy Reached by Life Evolution on Earth Nowadays We now propose our new Evo-SETI UNIT of evolution. From the above discussion it follows that the numeric value of the Evo-SETI unit is about 25.575 bits if Life on Earth started 3.5 billion years ago. This unit we propose to call EE (Earth EvoEntropy), so that we define EE as E E = Evo_SETI_unit_of_Evolution_as_the_ difference_in_Information_Content_ between_RNA_and_Humans = 25.575 bits.

(67)

Thus, a planet like Mars will have an Evo-SETI evolution much less than 1 EE in case it hosted any primitive form of Life, even in the past. And just 0 EE in case it did not host any form of Life at all.

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Evo-SETI Mathematics: Part 1: Entropy of Information. …

On the contrary, an exoplanet hosting an ExtraTerrestrial Civilization much more advanced than the Human one will have a EE larger or much larger than 1 according to the higher degree of Evolution reached by that Civilization when compared to current Humans. This is our Evo-SETI SCALE quantifying the Evolution of Life in the Universe and, at the same time, proving once again that the Molecular Clock discovered in 1962 by Emil Zuckerkandl (1922–2013) and Linus Pauling (1901–1994) is indeed a correct and fundamental law of nature.

2.15 Markov-Korotayev Alternative to Exponential: A Cubic Growth The Molecular Clock, shown in Fig. 5 as the linear growth of the Evo-Entropy over the last 3.5 billion years of evolution of Life on Earth, is the key fact in molecular evolution and allows for an immediate quantitative estimate of “how much” (in bits per individuals) any two Species “differ” from each other. This is the key to Cladistics, of course. But, after 2007, this “exponential vision” was challenged by the alternative “cubic vision” (our words) that we now describe. Let us now refer to the important mathematical paper [2, 3] in that new research field nowadays called Big History. Also interesting is the Wikipedia site http://en. wikipedia.org/wiki/Andrey_Korotayev, whose words we now report almost just the same. According to this site, in 2007–2008 the Russian scientist Andrey Korotayev, in collaboration with Alexander V. Markov, showed that a “hyperbolic” mathematical model can be developed to describe the macrotrends of biological evolution. These authors demonstrated that changes in biodiversity through the Phanerozoic correlate much better with the hyperbolic model (widely used in demography and macrosociology) than with the exponential and logistic models (traditionally used in population biology and extensively applied to fossil biodiversity as well). The latter models imply that changes in diversity are guided by a first-order positive feedback (more ancestors, more descendants) and/or a negative feedback arising from resource limitation. Hyperbolic model implies a second-order positive feedback. The hyperbolic pattern of the world population growth has been demonstrated by Korotayev to arise from a second-order positive feedback between the population size and the rate of technological growth. According to Korotayev and Markov, the hyperbolic character of biodiversity growth can be similarly accounted for by a feedback between the diversity and community structure complexity. They suggest that the similarity between the curves of biodiversity and human population probably comes from the fact that both are derived from the interference of the hyperbolic trend with cyclical and stochastic dynamics (Refs. [2, 3] trough [13]). This author was struck by the following Fig. 6 (taken from the above-mentioned Wikipedia site) showing the increase, but not monotonic increase, of the number of Genera (in thousands) during the last 542 million years of Life on Earth making

2 Part 1: Entropy of Information as the Measure of Evolution …

51

Fig. 6 According to Korotayev and Markov, during the Phanerozoic the biodiversity shows a steady but not monotonic increase from near zero to several thousands of genera

up for the Phanerozoic. Thus, this author came to wonder whether the red curve in Fig. 6 could be regarded as the Cubic mean value curve of a lognormal stochastic process, just as the exponential mean value curve is typical of Geometric Brownian Motions. This author’s answer to the above question is “yes”: we may indeed use our Cubic (44) to represent the red line in Fig. 6, thus reconciling the Markov-Korotayev theory with our Evo-SETI theory requiring that the profile curve of Evolution must be the Cubic mean value curve of a certain lognormal stochastic process (and certainly not a GBM in this case). Let us thus consider the following numerical inputs to the Cubic (44) that we derive “by a glance at Fig. 6” (the precision of these numerical inputs is really unimportant at this early stage of “matching” the two theories (ours and the Markov-Korotayev’s) since we are just looking for the “proof of concept”, and better numeric approximations might follow in the future): ⎧ ts = −530 ⎪ ⎪ ⎨ Ns = 1 ⎪ te =0 ⎪ ⎩ N e = 4000.

(68)

In words, the first two Eq. (68) mean that the first of the Genera appeared on Earth about 530 million years ago, i.e. the number of Genera on Earth was zero before 530 million years ago. Also in words, the last two Eq. (53) mean that, at the present time t = 0, the number of Genera on Earth is 4000 on the average. Now, “on the average” means that, nowadays, a standard deviation of about 1000 (plus or minus) affects the average value of 4000. This is shown in Fig. 7 by the grey stochastic process called “all genera”. And this is re-phrased mathematically by assigning the fifth numeric input

52

Evo-SETI Mathematics: Part 1: Entropy of Information. …

Fig. 7 Our Cubic mean value curve (thick red solid curve) plus and minus the two standard deviation curves (thin solid blue and green curve, respectively) give more mathematical information than just the previous Fig. 6. In fact, we now have the two standard deviation curves of the Lognormal stochastic process (24) and (25) that are completely missing in the Markov-Korotayev theory and in their plot shown in Fig. 7. We claim that our Cubic mathematical theory of the Lognormal stochastic process L(t) is a “more profound mathematization” than the Markov-Korotayev Theory of Evolution since it is stochastic, rather than just deterministic, as Korotayev and Markov published it in 2007–2008

δ N e = 1000.

(69)

Then, as a consequence of the five numeric boundary inputs (ts, N s, te, N e, δ N e) plus the standard deviation σ on the current value of Genera, Eq. (32) yields the numeric value of the positive parameter σ

σ =

#   $ $ ln 1 +  δ N e 2 % Ne te − ts

= 0.011.

(70)

Having thus assigned numeric values to the first five conditions, only conditions on the two abscissae of the Cubic Maximum and minimum, respectively, are still to be assigned. A glance to Fig. 6, then, makes us establish them as (of course in millions of years ago): 

tMax = − 400 tmin = − 220.

(71)

2 Part 1: Entropy of Information as the Measure of Evolution …

53

Finally, inserting these seven numeric inputs into the Cubic (44) as well as into both Eq. (27) of the upper and lower standard deviation curves, the final plot shown in Fig. 7 is produced.

2.16 Evo-Entropy of the Markov-Korotayev Cubic Growth What is the Evo-Entropy (59) of the Markov-Korotayev Cubic growth (44)? To answer this question we must obviously insert (44) into (59) and then plot the resulting equation Cubic_EvoEntropy(t)    1 1 1 m Cubic (t) · . = − + ln ln(2) 4π N s 2 Ns 4π [m Cubic (t)]2

(72)

The plot of this function of t is shown in Fig. 8.

Fig. 8 Evo-Entropy (59) of the Markov-Korotayev Cubic mean value (44) of our lognormal stochastic process L(t) representing the growing number of Genera during the Phanerozoic. Starting with the left part of the curve, one immediately notices that, in a few million years around the Cambrian Explosion of 542 million years ago, the Evo-Entropy had an almost vertical growth from the initial value of zero to about the value of 10 bits per individual. These were the few million years where the bilateral symmetry became the dominant trait of all the primitive creatures inhabiting the Earth during the Cambrian Explosion. After that, for about the next 300 million years the Evo-Entropy did not change much. These were the times when bilaterally-symmetric living creatures underwent little or no change in their body structure, like reptiles, birds and very early mammals (roughly after 310 million years ago). But then, after the “mother of all mass extinctions at the end of the Paleozoic” (about 250 million years ago), the Evo-Entropy started growing again in the mammals. Today, according the Markov-Korotayev model, the Evo-Entropy is about 12.074 bits/individual for Humans, i.e. much less than the 25.575 bits/individual predicted by the GBM exponential growth shown in Fig. 5. So the question is: which one model is right? The GBM one, or the Markov-Korotayev one?

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Evo-SETI Mathematics: Part 1: Entropy of Information. …

2.17 Comparing the Evo-Entropy of the Markov-Korotayev Cubic Growth to a Hypothetical (1) Linear and (2) Parabolic Growth Just “for fun”, let us now consider two more types of growth in the Phanerozoic: 1. The LINEAR (=straight line) growth given by the mean value (42) and 2. The PARABOLIC (=quadratic) growth given by the mean value (43). 3. And, of course, we compare these with the CUBIC growth (44) typical for the Markov-Korotayev model. The results of this comparison are shown in the two diagrams (upper one and lower one) in Fig. 7. For the sake of simplicity, we omit all detailed mathematical calculations and just confine ourselves to writing down the equation of the: 1. STRAIGHT LINE EVO-ENTROPY: STRAIGHT_LINE_EvoEntropy(t)    m straight_line (t) 1 1 1 · − . =  2 + ln ln(2) 4π N s 2 Ns 4π m straight_line (t) (73) 2. PARABOLIC (QUADRATIC) EVO-ENTROPY: PARABOLA_EvoEntropy(t)    m parabola (t) 1 1 1 · − . =  2 + ln ln(2) 4π N s 2 Ns 4π m parabola (t)

(74)

3. CUBIC (MARKOV-KOROTAYEV) EVO-ENTROPY: that is just (72) (Fig. 9).

2.18 Conclusions About Evo-Entropy The Biological Evolution of Life on Earth over the last 3.5 to roughly 4 billion years has hardly been cast into any “profound” mathematical form. The molecular clock is an exception in that Zuckerkandl and Pauling cast it in a “straight line” form, i.e. in the easiest possible geometrical form, as early as 1962. Since 2012 this author has tried to do profound mathematics about the evolution of Life on Earth by resorting to lognormal probability distributions in the time, starting each at a different time instant b (birth) and called b-lognormals [7–9]. His discovery (in the years 2010– 2015) of the Peak-Locus Theorem valid for any enveloping mean value (and not just the exponential one (GBM), see the Appendix 1 to Ref. [9]) has made it possible the use of the Shannon Entropy of Information Theory as the correct mathematical tool measuring the Evolution of Life in bits/individual.

2 Part 1: Entropy of Information as the Measure of Evolution …

55

Fig. 9 Comparing the mean value m L (t) (shown in the upper diagram) and the EvoEntropy(t) (shown in the lower diagram) of the Markov-Korotayev numeric inputs (68), (69), (70) and (71) in case the growth was the Cubic mean value (44) (blue solid curve) or the straight line (42) (dashdash orange curve) or the parabola (43) (dash-dot red curve). We see that, for all these three curves, starting with the left part of the curve, in a few million years around the Cambrian Explosion of 542 million years ago, the Evo-Entropy had an almost vertical growth from the initial value of zero to about the value of 10 bits per individual. Again as in Fig. 8, these were the few million years where the bilateral symmetry became the dominant trait of all the primitive creatures inhabiting the Earth during the Cambrian Explosion. After that, for about the next 300 million years the Evo-Entropy did not change much. These were the times when bilaterally-symmetric living creatures underwent little or no change in their body structure, like reptiles, birds and very early mammals (roughly after 310 million years ago). At the time of the “mother of all mass extinctions at the end of the Paleozoic” (about 250 million years ago), the two Evo-Entropy curves of the linear and parabolic growth were about 5 bits/individual higher than curve the Cubic Markov-Korotayev growth. Again: which model is closer to the actual reality?

In conclusion, what happened on Earth over the last 4 billion years is now summarized by a few simple statistical equations, but is just about the evolution of Life on Earth, and not on other Exoplanets. The extension of our Evo-SETI Theory to Life on other Exoplanets will be possible only when SETI, the current scientific Search

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Evo-SETI Mathematics: Part 1: Entropy of Information. …

for ExtraTerrestrial Intelligence, will achieve the first Contact between Humans and an Alien Civilization.

2.19 Life as a Finite b-Lognormal as Assumed by This Author Prior to 2017 Prior to 2017, i.e. prior to his discovery of the LOGPAR power curves described in Part 2 of this paper, this author defined mathematically the LIFTIME of every living being (and the Lifetime of a Civilization too) as shown in the following Fig. 10: “Life” in that Evo-SETI Theory was a ‘finite b-lognormal’ made up by a lognormal pdf between birth b and senility (descending inflexion point) s, plus the straight tangent line at s leading to death d.

Fig. 10 “Life”, defined prior to 2017 in the Evo-SETI Theory as a “finite b-lognormal” made up by a lognormal pdf between birth b and senility (descending inflexion point) s, plus the straight tangent line starting at s leading to death d

2 Part 1: Entropy of Information as the Measure of Evolution …

57

Fig. 11 The b-lognormals of nine Historic Western Civilizations computed by virtue of the History Formulae with the three numeric inputs for b, p and d of each Civilization given by the corresponding line in Table 1

2.20 b-Lognormal History Formulae and Their Applications to Past History Having so defined “Life” as a finite b-lognormal, this author was able to show that, given one’s birth b, death d and (somewhere in between) one’s senility s, then the two parameters μ (a real number) and σ (a positive number) of the b-lognormal (5) are given by the two equations 

d−s √ σ = √d−b s−b μ = ln(s − b) +

(d−s)(b+d−2s) . (d−b)(s−b)

(75)

These were called “History Formulae” by this author for their use in Mathematical History. The mathematical proof of (75) is found in Ref. [9], pages 227–231 and follows directly from the definition of s (as descending inflexion point) and d (as interception between the descending tangent straight line at s and the time axis). In previous versions of his Evo-SETI Theory, the author gave an apparently nonfactorized version of the History Formulae (75) reading 

σ =

√ √ d−s d−b s−b

μ = ln(s − b) +

2s 2 −(3d+b)s+d 2 +bd . (d−b)s−bd+b2

(76)

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Evo-SETI Mathematics: Part 1: Entropy of Information. …

The corresponding s is derived from b, p and d by virtue of a second-order approximation provided by the solution of the quadratic equation in the birth-peak-death theorem, not described in the present paper. Plus the Shannon ENTROPY of the relevant b-lognormal probability density function as the civilization level measure (Tables 2 and 3).

3 Part 2: Energy of Living Forms by “Logpar” Power Curves 3.1 Introduction to Logpar Power Curves The idea is easy behind the notion of a FINITE LIFETIME as the new Logpar curve: we seek to represent any Lifetime by virtue of just three points in time: birth, peak, death (BPD). No other point in between. That is, no other “senility point” s as those appearing in all b-lognormals that this author had published in his Evo-SETI Theory prior to 2017. In fact, it is easier and more natural to describe someone’s Lifetime just in terms of birth, peak and death, than in terms of birth, senility and death, because when the senility arrives is rather uncertain and so hard to define in the practice for any individual. Please look at Fig. 12. The first part, the one on the left, i.e. prior to the peak time p, is just a b-lognormal: it starts at birth time b, climbs up to the adolescence time a (ascending inflexion point of the b-lognormal) (in reality the adolescence time should more properly be called “puberty time” since it marks the beginning of the reproduction capacity for that individual) and finally reaches the peak time at p (maximum, i.e. the point of zero first derivative of the b-lognormal). All this is just ordinary b-lognormal stuff, as we have been “preaching” since about 2012. But now the novelty comes, i.e. the second part, the one on the right: that is just a parabola having its vertex exactly at the peak time p. Notice that this definition automatically implies that the tangent line at the peak is horizontal, i.e. the same for both the b-lognormal and the parabola. Notice also that, after the peak, the parabola plunges down until it reaches the time axis at the death time d. Therefore this new definition of death time d is different from the old definition of d applying to blognormals alone, as we did prior to 2017. And this is our LOGPAR (LOGnormal plus PARabola), a NEW CURVE FINITE IN TIME, namely ranging in time just between birth and death.

1250 Frederick II dies. Middle Ages end. Free Italian towns start Renaissance

1419; Madeira island discovered. African coastline explored by 1498

1402; Canary islands are conquered by 1496. In 1492 Columbus discovers America

Renaissance Italy

Portuguese Empire

Spanish Empire

1798; Largest extent of Spanish colonies in America: California settled since 1769

1716; Black slave trade to Brazil at peak. Millions of blacks enslaved

1497; Renaissance art and architecture. Birth of Science. Copernican revolution (1543)

117 ad; Rome at peak: Trajan in Mesopotamia. Christianity preached in Rome by Saints Peter, Paul against slavery by 69 ad

753 bc Rome founded; Italy seized by Romans by 270 bc, Carthage and Greece by 146 bc, Egypt by 30 bc. Christ born at 0

Ancient Rome

1805; Spanish fleet lost at Trafalgar

1822; Brazil independent, other colonies retained

1564; Council of Trent ends Catholic and Spanish rule

273 ad; Aurelian builds new walls around Rome after Military Anarchy, 235–270 ad

323 bc; Alexander the Great dies. Hellenism starts in Near East

776 bc; First Olympic 434 bc; Pericles’ Age. Games, from which Greeks Democracy peak. Arts and compute years Science peak Aristotle

Ancient Greece

s = Decline = senility time

3100 bc; Lower and Upper 1154 bc; Luxor and Karnak 689 bc Assyrians invade Egypt unified. First temples edified by Ramses Egypt in 671 bc, leave Dynasty II by 1260 bc 669 bc

p = Peak time

Ancient Egypt

b = Birth time

1898; Last colonies lost to the USA

1999; Last colony, Macau, lost to Republic of China

1660; Cimento shut Bruno burned 1600. Galileo died 1642

476 ad; Western Roman Empire ends. Dark Ages start in West Not in East

30 bc Cleopatra’s death: last Hellenistic queen

30 bc Cleopatra’s death: last Hellenistic queen

d = Death time

Table 1 Birth, peak, decline and death times of nine Historic Western Civilizations (3100 bc–2035 ad), plus the relevant peak heights

(continued)

5:938 × 10−3

3:431 × 10−3

5:749 × 10–3

2:193 × 10−3

2:488 × 10−3

8:313 × 10−4

P = Peak ordinate

3 Part 2: Energy of Living Forms by “Logpar” Power Curves 59

1588; Spanish Armada Defeated British Empire’s expansion starts

1898; Philippines, Cuba, Puerto Rico seized from Spain

British Empire

USA empire

1947; After World Wars One and Two, India gets independent

1870 Napoleon III defeated Third Republic starts World Wars One and Two

s = Decline = senility time

1972; moon landings, 2001; 9/11 terrorist 1969–72: America leads the attacks: decline. Obama world 2009

1904; British Empire peak. Top British Science: Faraday, Maxwell, Darwin, Rutherford

1812; Napoleon I rules continental Europe and reaches Moscow

p = Peak time

They are shown in Fig. 11 as nine b-lognormal probability density functions (pdfs)

1524; Verrazano first in New York bay. Cartier in Canada, 1534

b = Birth time

French Empire

Table 1 (continued)

2035; Singularity? Will the USA yield to China?

1974; Britain joins the EEC and loses most of her colonies

1962; Algeria lost as most colonies. Fifth Republic starts in 1958

d = Death time

0:013

8:447 × 10–3

4:279 × 10−3

P = Peak ordinate

60 Evo-SETI Mathematics: Part 1: Entropy of Information. …

σ = 0:440

σ = 0:181

μ = 6:018

776 bc = –776 First Olympic Games, from which Greeks compute years

753 bc = –753 μ = 6:801 Rome founded Italy seized by Romans by 270 bc, Carthage and Greece by 146 bc, Egypt by 30 bc. Christ 0

Ancient Greece

Ancient Rome

σ = 0:277

σ = 0:366

μ = 5:583

1250 Frederick II dies Middle Ages end. Free Italian towns start Renaissance

1419 μ = 5:828 Madeira island Discovered African coastline explored by 1498

Renaissance Italy

Portuguese Empire

σ = 0:242

μ = 7:632

3100 bc = –3100 Lower and Upper Egypt unified. First Dynasty.

Ancient Egypt

σ computed by the history formulae

μ computed by the history formulae

b = Birth time

Table 2 Birth = b, μ and σ of the nine Historic Western Civilizations (3100 bc-2035 ad) shown in Fig. 11 and Table 1

H_Portugal = 9:004 bits/individual (continued)

H_Italy = 8:217 bits/individual

H_Rome = 9:390 bits/individual

H_Greece = 9:548 bits/individual

H_Egypt = 11:011 bits/individual

H = Shannon ENTROPY of the relevant b-lognormal pdf

3 Part 2: Energy of Living Forms by “Logpar” Power Curves 61

1402 Canary islands are conquered by 1496 In 1492 Columbus discovers America

1524 Verrazano first in New York bay Cartier in Canada, 1534

1588 Spanish Armada Defeated. British Empire’s expansion starts

1898 Philippines, Cuba and Puerto Rico seized by the USA from Spain

French Empire

British Empire

USA Empire

b = Birth time

Spanish Empire

Table 2 (continued) σ computed by the history formulae σ = 0:116

σ = 0:051

σ = 0:073

σ = 0:396

μ computed by the history formulae μ = 5:994

μ = 5:981

μ = 5:831

μ = 4:462

H_USA = 7:148 bits/individual

H_Britain = 6:677 bits/individual

H_France = 6:388 bits/individual

H_Spain = 7:586 bits/individual

H = Shannon ENTROPY of the relevant b-lognormal pdf

62 Evo-SETI Mathematics: Part 1: Entropy of Information. …

– – – – – –

Italy

Portugal

Spain

France

Britain

USA

–

–

–

–

–

–

–

2.397

2.868

3.157

1.958

0.541

1.328

–

–

–

–

–

–

0

2.242

2.713

3.002

1.804

0.386

1.173

0.154

–

–

–

–

–

1.069

1.540

1.829

0.630

– 1.856

– 2.327

– 2.616

– 1.418

0.7872

0.386

0.541

2.008

–

–

–

0

0.438

0.909

1.198

1.418

0.630

1.804

1.958

3.425

Portugal Spain

0.7872 0

0

1.173

1.328

2.795

Italy

0.759

0.289

0

1.198

2.616

1.829

3.002

3.157

4.624

0.289

0.471

0

–

0.909

2.327

1.540

2.713

2.868

4.335

0

–

–

0.471

0.759

0.438

1.856

1.069

2.242

2.397

3.864

France Britain USA

These numbers are the (Shannon) Entropy DIFFERENCES in bits/individual among the nine Historic Western Civilizations (3100 bc–2035 ad) shown in Fig. 11 and Table 1. A table like this one is called by mathematicians “a skew-symmetric matrix”. Its properties are found, for instance at the site https://en.wikipedia. org/wiki/Skew-symmetric_matrix. This mathematical remark might have much more profound meanings for Evo-SETI Theory, but we just don’t have the time to explore them now (January 3rd, 2019)

3.864

4.335

4.624

3.425

2.008

2.795

1.622

0.154

–

Rome

0

–

Greece 1.467

1.467

0

Egypt

1.622

Greece Rome

Information gaps among the 9 historic civilizations shown in Fig. 11 and Egypt Table 1 in bits/individual

Table 3 Information Gaps, i.e. Gaps in the levels of Evolution according to Evo-SETI theory

3 Part 2: Energy of Living Forms by “Logpar” Power Curves 63

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Evo-SETI Mathematics: Part 1: Entropy of Information. …

Fig. 12 Representation of the History of the Roman civilization as a LOGPAR finite curve. Rome was funded in 753 bc, i.e. in the year −753 in our notation, or b = −753. Then the Roman republic and empire (the latter since the first emperor, Augustus, roughly after 27 bc) kept growing in conquered territory until it reached its peak (maximum extension, up to Susa in current Iran) in the year 117 ad, i.e. p = 117, under emperor Trajan. Afterwards it started to decline and loose territory until the final collapse in 476 ad. (d = 476, Romulus Augustulus, last emperor). Thus, just three points in time are necessary to summarize the History of Rome: b = −753, p = 117, d = 476. No other intermediate point, like senility in between peak and death, is necessary at all since we now used a Logpar rather than a b-lognormal, as this author had done prior to 2017. The numbers along the vertical axis will be explained in Sects. 3.15 and 3.16

3.2 Finding the Parabola Equation of the Right Part of the Logpar We shall now cast into appropriate mathematics the above popular description of what a Logpar curve is. Consider the equation of a parabola in the time t having vertical axis along the t = p vertical line: y = α (t − p)2 + β (t − p) + γ

(77)

where α, β and γ are the three coefficients of the time that we must determine according to the assumptions shown in Fig. 1. To find them, we must resort to the three conditions that we know to hold by virtue of a glance to Fig. 1: 1. CONDITION: the height of the peak is P, just the same as the height of the peak of the b-lognormal on the left in Fig. 1. Thus, inserting the two equations of the peak, namely 

into (77), the latter yields immediately

t=p y=P

(78)

3 Part 2: Energy of Living Forms by “Logpar” Power Curves

P=γ

65

(79)

that, when inserted back into (77), changes it into y = α (t − p)2 + β (t − p) + P.

(80)

2. CONDITION: the tangent straight line at both the b-lognormal and the parabola at the peak abscissa p is horizontal. In other words, the first derivative of (80) at t = p must equal zero. Differentiating (80) with respect to t, equalling that to zero and then solving for β yields β = − 2 α (t − p).

(81)

Inserting (81) into (80), the latter is turned into y = − α (t − p)2 + P.

(82)

3. CONDITION: at the death time d, one must have y = 0, yielding from (82) the equation 0 = − α (d − p)2 + P.

(83)

P . (d − p)2

(84)

Solving (83) for α one gets α=

Finally, inserting (84) into (82) the desired equation of the parabola is found

(t − p)2 y(t) = P 1 − . (d − p)2

(85)

As confirmation, one may check that (85) immediately yields the two conditions 

y( p) = P y(d) = 0 .

(86)

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Evo-SETI Mathematics: Part 1: Entropy of Information. …

3.3 Finding the b-Lognormal Equation of the Left Part of the Logpar As for the b-lognormal between birth and peak, making up the left part of the Logpar curve, we already know all its mathematical details from the previous many papers published by this author on this topics, but we shall summarize here the main equations for the sake of completeness. The equation of the b-lognormal starting at b reads b_lognormal(t; μ, σ, b) = √

e−

(log(t−b)−μ)2 2σ 2

2π · σ · (t − b)

.

(87)

Tables listing the main equations that can be derived from (87) were given by this author in Refs. [6, 7] and we shall not re-derive them here again. We just confine ourselves to reminding that: 1. The abscissa p of the peak of (87) is given by p = b + eμ−σ . 2

(88)

Proof Take the derivative of (87) with respect to t and set it equal to zero. Then solve the resulting equation for t, that now becomes p, and (88) is found. 2. The ordinate P of the peak of (87) is given by σ2

e 2 −μ P=√ . 2π σ

(89)

Proof Rewrite p instead of t in (87) and then insert (88) instead of p. Then simplify to get (89). 3. The abscissa of the adolescence point (that should actually be better named “puberty point”) is the abscissa of the ascending inflexion point of (87). It is given by a = b + e−

σ

√

2 σ 2 +4 − 3σ2 2

+μ

(90)

Proof Take the second derivative of (87) with respect to t and set it equal to zero. Then solve the resulting equation for t, that now becomes a, and (90) is found. 4. The ordinate of the adolescence point is given by e−

σ

√

2 σ 2 +4 + σ4 4

√

2π σ

−μ−

1 2

(91)

3 Part 2: Energy of Living Forms by “Logpar” Power Curves

67

Proof Just rewrite a instead of t in (87) and then insert (90) and simplify the result. Let us now notice that, within the framework of the Logpar theory described in this chapter, we may NOT say that (87) fulfills the normalization condition ∞ b_lognormal(t; μ, σ, b) dt = 1

(92)

b

since (87) here is only allowed to range between b and p. Rather than adopting (92), we must thus replace (92) by the integral of (87) between b and p only. Fortunately, it is possible to evaluate this integral in terms of the error function defined by 2 er f (x) = √ π

x

e−t dt. 2

(93)

0

In fact, the integral of the b-lognormal (87) between b and p turns out to be given by p b_lognormal(t; μ, σ, b) dt b

p = b

−

(log(t−b)−μ)2 2σ 2

e dt = √ 2π σ (t − b)

1 + er f

 √ 2 ln( p−b)− 2 μ 2σ

√

2

(94)

Now, inserting (88) instead of p into the last erf argument, a remarkable simplification occurs: μ and b both disappear and only σ is left. In addition, the erf property er f (−x) = − er f (x) allows us to rewrite p

p

−

(log(t−b)−μ)2 2σ 2

1 + er f

 √ 2 ln( p−b)− 2 μ 2σ

√

e dt = √ 2 2π σ (t − b) b     1 − er f √σ2 1 + er f − √σ2 = . = 2 2

b_lognormal(t; μ, σ, b) dt = b

(95)

In conclusion, the area under the b-lognormal between birth and peak is given by p b_lognormal(t; μ, σ, b) dt = b

1 − er f 2



σ √ 2

 .

(96)

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Evo-SETI Mathematics: Part 1: Entropy of Information. …

3.4 Area Under the Parabola on the Right Part of the Logpar Between Peak and Death We already proved that the parabola on the right part of the Logpar curve has Eq. (85). Now we want to find the area under this parabola between peak and death, that is  d 

(t − p)2 P 1− dt (d − p)2 p

P = P(d − p) − (d − p)2 = P(d − p) −

d (t − p)2 dt p

P (d − p)3 3 (d − p)2

2 P(d − p)

3 4 1 P(d − p) = · 2 3 ⎡ ⎤ Ar ea o f Ar chimedes 1 ⎣ = · parabolic segment, ⎦. 2 pr oved < 212 BC =

(97)

The great Ancient Greek mathematician Archimedes (circa 287 bc—212 bc) of Syracuse (Sicily) already “knew” the last integral result even before the Calculus was discovered by Newton and Leibniz after 1660. More appropriately, (97) is a special case of Cavalieri’s quadrature formula (published in 1635, https://en.wikipedia.org/ wiki/Cavalieri%27s_quadrature_formula). Actually, Archimedes used the “method of exhaustion” to compute the area of a segment of the parabola, as very neatly described at the site https://en.wikipedia.org/ wiki/The_Quadrature_of_the_Parabola. In conclusion, the area under our parabola between peak and death is given by (97), that we now rewrite as  d 

2 P(d − p) (t − p)2 P 1− dt = . 2 3 (d − p) p

(98)

3 Part 2: Energy of Living Forms by “Logpar” Power Curves

69

3.5 Area Under the Full Logpar Curve Between Birth and Death We are now in a position to compute the full area A under the Logpar curve, that is given by the sum of Eqs. (96) and (98), that is 1 − er f



σ √ 2

 +

2

2 P(d − p) =A 3

(99)

This is possibly the most important equation in this chapter. In fact, if we want the Logpar be a true probability density function (pdf), we must assume in (99) A = 1.

(100)

But, surprisingly, we shall NOT do so! Let us rather ponder over what we are doing: 1. We are creating a “Mathematical History” model where the “unfolding History” of each Civilization in the time is represented by a Logpar curve. 2. The knowledge of only three points in time is requested in this model: b, p and d. 3. But the area under the whole curve depends on σ as well as on μ, as we see upon inserting (89) instead of P into (99), that is 1 − er f



σ √ 2



σ2

e 2 −μ 2 (d − p) +√ = A(μ, σ ). · 3 2π σ

2

(101)

4. Also p is to be replaced by its expression (88) in terms of σ and μ, yielding the new equation 1 − er f



σ √ 2

2



σ2 2

e +√

−μ

2π σ

·

  2 2 d − b − eμ−σ 3

= A(μ, σ ).

(102)

5. The meaning of (102) is that birth and death are fixed, but the position of the peak may move according to the different numeric values of σ and μ. 6. In addition to that, we “dislike” the presence of the error function er f in (102) since this is not an “ordinary” function, i.e. it is one of the functions that mathematicians call “higher transcendental functions”, having complicated formulae describing them. Thus, we would rather get rid of er f . How may we do so?

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Evo-SETI Mathematics: Part 1: Entropy of Information. …

3.6 The Area Under the Logpar Curve Depends on Sigma Only, and Here Is the Area Derivative W.R.T. σ The simple answer to the last question (6) is “by differentiating both sides of (102) with respect to σ ”. In fact, the derivative of the er f function (93) is just the “Gaussian” exponential 2 d er f (x) 2 = √ · e−x . dx π

(103)

and so the er f function itself will disappear by differentiating (102) with respect to σ . In fact, the derivative of the first term on the left hand side of (102) simply is, according to (103), ⎡ d ⎣ dσ

1 − er f



σ √ 2

2

⎤

−

σ2 2

⎦ = − e√ . 2π

(104)

As for the derivative with respect to σ of the second term on the left hand side of (102) we firstly notice that σ appears three times within that term. Thus, the relevant derivative is the sum of three terms, each of which includes the derivative of one of the three terms multiplied by the other two terms unchanged. In equations, one has: ⎤  μ−σ 2 2 d − b − e e d ⎣ ⎦ +√ · dσ 2 3 2π σ  σ2 √  μ−σ 2 3 σ2 σ2 − − 2 − e + d − b e 2 −μ 22 e 2 e 2 = √ −√ − √ 3 π 3 πσ2 2π   √ σ2 2 2 − eμ−σ + d − b e 2 −μ + . √ 3 π ⎡

1 − er f



σ √ 2



σ2 2

−μ

(105)

Several alternative forms of this Eq. (105) are possible, and that is rather confusing. However, using a symbolic manipulator (this author did do by virtue of Maxima), a few steps lead to the following form of (105): d A(σ ) d A(μ(σ ), σ ) ≡ dσ dσ √ √ σ2 σ2 3 σ2 σ2 2(d − p)e− 2 2(d − p)e− 2 2 2 e− 2 e− 2 =− √ + √ −√ + √ 3 π ( p − b)σ 2 3 π ( p − b) 3 π 2π − σ2  2 2 2 p σ − 2 d σ + b σ − 2p + 2 d e 2 = . (106) √ 3 2π ( p − b)σ 2

3 Part 2: Energy of Living Forms by “Logpar” Power Curves

71

This (106) is the requested derivative of the full-Logpar area with respect to σ .

3.7 Exact “History Equations” for Each Logpar Curve We now take a further, crucial step in our analysis of the Logpar curve: we impose that the derivative of the area with respect to sigma, i.e. (106), is zero d A(σ ) = 0. dσ

(107)

What does that mean? Well, sorry to use an emphatic expression, but “hold your breath, please”: (107) is the Evo-SETI equivalent of the Least Action Principle in physics! This mind-boggling conclusion does not show up at the moment, but it will at the end of this chapter. For the time being we content ourselves with the “crude mathematics” of rewriting the imposed condition (107) by virtue of the last expression in (106) that, getting rid of both the exponential and the denominator, immediately boils down to p σ 2 − 2 d σ 2 + b σ 2 − 2 p + 2 d = 0.

(108)

But this is just a quadratic equation in σ σ 2 ( p − 2d + b) = 2( p − d)

(109)

and so we finally get σ2 =

2(d − p) . 2 d − (b + p)

(110)

This is the most important new result discovered in the present chapter. It is the LOGPAR HISTORY EQUATION FOR σ √ √ 2 d−p σ =√ . 2 d − (b + p)

(111)

In other words, given the input triplet (b, p, d) then (110) immediately yields the exact σ 2 of the b-lognormal left part of the Logpar curve. It was discovered by this author on November 22, 2015, and led not only to this chapter, but to the introduction of the ENERGY spent in a Lifetime by a living creature, or by a whole civilization whose “power-vs-time” behaviour is given by the Logpar curve, as we will understand better in the coming sections of this chapter. At the moment, for reasons that will become obvious later, we confine ourselves to taking the limit of both sides of (111) for d → ∞, with the result

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Evo-SETI Mathematics: Part 1: Entropy of Information. …

√ √ √ √ 2 d−p 2 d lim σ = lim √ = lim √ = 1. d→∞ d→∞ d→∞ 2 d − (b + p) 2d

(112)

Since we already know that σ must be positive, (112) really shows that σ may range between zero and one only (but both 0 and 1 are “singular values” out of the “ordinary range” of σ in between them). We will discuss this “delicate point” in future papers, and now we just confines ourselves to the “ordinary” σ 0 < σ < 1.

(113)

Next to (111) one of course has a similar LOGPAR EQUATION FOR μ, that is immediately derived from (88) and (111). To this end, just take the log of (88) to get μ = ln( p − b) + σ 2

(114)

that, invoking (110), yields the desired Logpar equation for μ μ = ln( p − b) +

2(d − p) . 2 d − (b + p)

(115)

In conclusion, our key two Logpar history equations are 

√ √

2 d− p σ = √2 d−(b+ p) μ = ln( p − b) +

2(d− p) . 2 d−(b+ p)

(116)

3.8 Considerations on the Logpar History Equations Some considerations on the Logpar History Formulae (116) are now of order: 1. All these formulae are exact, i.e. no Taylor series expansion was used to derive them. 2. But they were obtained by equalling to zero the derivative with respect to σ of the total area under the Logpar curve given by (102). 3. Therefore the Logpar History Formulae (116) are the equations of a minimum (we shall later show that this is indeed a minimum and not a maximum) of the A(σ ) function expressing the total area (102) as a function of σ . 4. One further question might be: μ and σ are independent variables in the Gaussian (and so in the lognormal, that is just e(Gaussian) ). Since we differentiated (102) with respect to σ already, why don’t we try differentiating it with respect to μ also? The answer is: because differentiating (102) with respect to μ leads to the ABSURD result b = d i.e. one dies just when born! We leave the calculation to readers as an exercise, how funny!

3 Part 2: Energy of Living Forms by “Logpar” Power Curves

73

3.9 Logpar Peak Coordinates Expressed in Terms of (b, p, d) Only Of particular importance for all future Logpar applications is the expression of the peak coordinates ( p, P) expressed in terms of the input triplet (b, p, d) only. Since the peak abscissa p is assumed to be known, we only have to derive the formula for the peak ordinate P. That is readily obtained by inserting the Logpar History Formulae (116) into the peak height expression (89). After a few rearrangements, it is found to be given by √ 2 d − (b + p) · e− P= √ √ 2 π d − p( p − b)

(d− p) 2 d−(b+ p)

.

(117)

3.10 History of Ancient Rome as an Example of How to Use the Logpar History Formulae Let us go back to the History of Rome as summarized in the caption to Fig. 12. First of all, let us write down neatly the key three numeric input values in the History of Rome that were already mentioned in the caption to Fig. 12: ⎧ ⎨ b = −753 Rome_input_triplet = p = 117 ⎩ d = 476.

(118)

Then the Logpar History Eq. (116) immediately yield numerical values of the Logpar σ and μ for Rome, that we shall hereafter denote by σ R and μR, respectively  Rome_logpar_doublet

σ R = 0.672 μR = 7.221.

(119)

Next the study of the Logpar peak comes. We already know from (118) that the abscissa of the peak of the Roman civilization was in 117 ad under Trajan p Rome = 117.

(120)

But the Logpar peak ordinate for Rome must be found by virtue of (117). One thus gets PRome = 5.44 × 10−4 = 0.000544. This is precisely the peak value shown in Fig. 12.

(121)

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Evo-SETI Mathematics: Part 1: Entropy of Information. …

3.11 Area Under Rome’s Logpar and Its Meaning as “Overall Energy” of the Roman Civilization Let us go back to (102), i.e. the total area under the Logpar curve: 1 − er f



σ √ 2



2

  μ−σ 2 σ2 e 2 −μ 2 d − b − e +√ = A(μ, σ ). · 3 2π σ

(122)

If we insert the Logpar History Formulae inside this equation, we obviously get the expression of the total area under the Logpar as a function of just the input triplet (b, p, d) only. After some rearranging, this area formula turns out to be the rather complicated (but exact!) area equation:

A(b, p, d) =

1 − er f



 √ d− p √ 2 d−(b+ p) 2

+

√ √ d − p 2 d − (b + p) − ·e √ 3 π ( p − b)

d− p 2 d−(b+ p)

.

(123) What is the meaning of this area in Physics? If we consider the Logpar curve as the curve of the power (measured in watts) of the Roman civilization along the whole of its history course, then the area under this curve, i.e. the integral of the Logpar between birth and death, is the total energy (measured in joules) spent by that civilization in its whole Lifetime: ENERGY_spent_in_the_Civilization_LIFETIME d =

POWER_of_that_Civilization(t) dt b

d =

logpar_POWER_CURVE_of_that_Civilization(t) dt.

(124)

b

In other words still, if we know the power curve of any living being that lived in the past, like a cell, or an animal, or a human, or a Civilization of humans or of any other living forms (including ExtraTerrestrials), the integral of that power curve between birth and death is just the total energy spent by that living form during the whole of its Lifetime. More: if we assume that all Humans have potentially the same amount of energy to spend during their whole Lifetime, then the Logpar of great men who “died young” (like Mozart, for instance) must have the same area below their Logpar and so a much higher peak since they lived shorter than others. Agree? Let us now go back to Rome’s Civilization.

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75

Upon inserting the Rome’s input triplet (118) into the area Eq. (123) the number is found ARome = A(−753, 117, 476) = 0.381.

(125)

This is far from being equal to 1, i.e. the numeric value that would have made the Rome Logpar curve to be a true probability density. So, we have abandoned the use of probability densities (as all b-lognormals were, that this authors considered prior to 2017) but we have now our free hand to consider the ENERGY spent during a given Lifetime. And the energy is “something” profoundly different from the entropy! This author had previously considered ENTROPY, for instance with reference to his theorem that the (Shannon) Entropy of a Geometric Brownian Motion (GBM) is a LINEAR function of the time, just as the molecular clock is a linear function of the time also (that is Motoo Kimura’s neutral theory of molecular evolution (1968), https://en.wikipedia.org/wiki/Neutral_theory_of_molecular_evolution). What a great step ahead we made: by releasing the normalization condition typical of probability densities, we were able to introduce ENERGY into the Evo-SETI theory. But does that mean that we have abandoned ENTROPY? Not at all! Entropy and the Peak Locus Theorem supporting it, are still valid in that the Peak is the junction point belonging to both the b-lognormal and the parabola. And so the ENTROPY is computed by virtue of the left part of the curve only (i.e. the lognormal part in between birth and peak, that remains unchanged), while the ENERGY is computed by taking into account both the left and the right parts under the Logpar! Magic of mathematics.

3.12 The Energy Function of d Regarded as a Function of the Death Instant d, Hereafter Renamed D Now we make a further “terrible step”, since it’s about death! All of us would like to live as long as possible. The Conservation Instinct means just that. Then it makes sense to study the function (123) as a function of d only, meaning that we may increase the Logpar area as much as Life allows, i.e. as much as d keeps “going to the right” along the time axis. In fact, we are now NOT working with probability densities any more, and no normalization condition is blocking us any longer. Consider thus the new function of d that we call energy (or, more correctly LifetimeEnergy)

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Evo-SETI Mathematics: Part 1: Entropy of Information. …

Energy(d) = A(b, p, d)  √  d− p √ √ 1 − er f √2 d−(b+ d − p 2 d − (b + p) − p) + = ·e √ 2 3 π ( p − b)

d− p 2 d−(b+ p)

. (126)

What is the mathematical analysis of this as a function of d? First of all, (126) starts at the peak abscissa p because one’s death cannot come earlier than one’s peak p: more precisely, even in the worse cases of someone being “suddenly killed”, we may always say that the peak of his Life was the instant “just before his death”, and so the mathematical definition applies d ≥ p.

(127)

Then, inserting d = p into (126) and noticing from (93) that er f (0) = 0 we find Energy( p) =

1 − er f



 √ 0 √ 2 d−(b+ p) 2

+ 0 · e− 0 =

1 . 2

(128)

Why should the someone’s Energy peak be equal to just ½, and not to any other positive value (in joules)? Well, worry not, please: we will solve this matter of extending the Energy at peak from ½ to any other positive value in the next section of this chapter. But, for the time being, please just content yourself of using the conventional value 1/2. Thanks. The next question is: what is the first derivative of (126) with respect to d? The answer is that, since d appears five times in (126), its derivative is the sum of five terms such that each term contains the derivative with respect to one precise d out of the five terms, and this derivative is multiplied by the other four terms unchanged. We did this calculation by virtue of the Maxima symbolic manipulator (web site: http://maxima.sourceforge.net/) and the final result will be given below as Eq. (129). But now let us look the first derivative of (126) as provided by Maxima without any rearrangement: d− p d− p √ p − d%e p−2 d+b p − 2d + b%e p−2d+b + √ √ √ √ 3 π ( p − b) p − 2d + b 6 π ( p − b) p − d   d− p √ √ 2(d− p) 1 p−2d+b p − 2d + b p − d p−2d+b + ( p−2d+b) 2 %e − √ 3 π ( p − b)  √  p−d p−d 1 √ √ %e− p−2d+b − 3/2 ( p−2d+b) 2 p−2d+b p−d − √ π

√

(129)

By equalling to zero the above equation, we may check whether the LifetimeEnergy (126) has any maxima or minima. The result given by Maxima is that, equalling to zero the above first derivative (129), one gets

3 Part 2: Energy of Living Forms by “Logpar” Power Curves

77

p−d √ p − d( p 2 + (4b − 6d) p + 4d 2 − 2bd − b2 ) e− p−2d+b =0 √ √ π p − 2d + b(3 p 3 − 9dp 2 + (6d 2 + 6bd − 3b2 ) p − 6bd 2 + 3b2 d) (130)

In turn, (130) amounts to p 2 + (4b − 6d) p + 4d 2 − 2bd − b2 = 0.

(131)

This is a quadratic equation in d that, once solved for d, yields two roots. Discarding the one root having a minus sign in front, we finally are left with

dtime_of_Minimum_Energy =

√   √  5+3 p+ 1− 5 b 4

.

(132)

So, this (132) is the abscissa of the minimum of the Energy, i.e. the time at which the minimum of a Civilization’s Energy occurs after that Civilization had previously reached its Peak (We could prove (132) really is the time of the minimum, rather than of a maximum, by computing the second derivative of (126) with respect to d and then insert (132) there and show that the result is a purely positive number, but we shall not do so here for the sake of brevity).

3.13 Discovering an Oblique Asymptote of the Energy Function, Energy(D), While the Death Instant D Is Increasing Indefinitely Around April 2016, this author discovered that the Energy (126) has an oblique asymptote for D → ∞. Before we derive the equation of this oblique asymptote, a careful understanding of what d means is of order. We always said that the Logpar theory described in this chapter necessitates the three inputs (b, p, d). But in this section we are considering higher and higher values of the death instant d so that the area under the Logpar, i.e. the energy of the phenomenon of which the Logpar is the power, may assume any assigned value. Thus, in this section of this chapter, the death time d becomes a sort of new independent variable D rather than just one of the three fixed inputs (b, p, d). In other words still, from now on, we will be careful to make the distinction between 1. The fixed, i.e. known, death instant d and 2. The movable, i.e. independent variable D allowing us to extrapolate into the future the Logpar having the two fixed input values (b, p).

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Evo-SETI Mathematics: Part 1: Entropy of Information. …

Having so said, the ENERGY (126) must more correctly be rewritten as a function of D rather than d   √ D− p √ √ 1 − er f √2 D−(b+ D− p D − p 2 D − (b + p) − 2 D−(b+ p) p) . + ·e Energy(D) = √ 2 3 π ( p − b) (133) Let now consider the definition of oblique asymptote given in elementary Calculus textbooks: if the limit lim [Energy(D) − (m D + q)]

D→∞

(134)

exists, then the Energy curve Energy(D) approaches more and more the straight line yoblique_asymptote (D) = m D + q.

(135)

Differentiating (135) with respect to D we immediately see that the angular coefficient m of the oblique asymptote is given by the limit for D → ∞ of the first derivative of the energy (126), that we now know to be given by the lengthy expression (129). Maxima yielded: √ d Energy(D) 2 . = √ √ m = lim D→∞ dD 3 π e( p − b)

(136)

The same would, of course, have been found had we considered the limit lim

D→∞

Energy(D) m D+q = lim = m. D→∞ D D

(137)

As for the asymptote’s intercept with the vertical axis, q, (135) shows that it is given by the limit q = lim [Energy(D) − m D]. D→∞

(138)

Thus, (126), (136) and Maxima yielded the result

q=

1 − er f 2



√1 2

 b+ p − √ . 3 2π e( p − b)

(139)

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79

Technical Note: Maxima was unable to compute the limit (138) in a single shot: we had to do separately the two limits 1 − er f lim

D→∞



√

√

D− p 2D−(b+ p)



2

=

1 − er f



√1 2

 (140)

2

and the ∞ − ∞ limit (requiring L’Hospital’s rule) √ lim

D→∞

√ D − p 2 D − (b + p) − ·e √ 3 π ( p − b) =

D− p 2 D−(b+ p)

 √ 2D − √ √ 3 π e( p − b)

−(b + p) . √ √ √ 3 2 π e( p − b)

(141)

In conclusion, the oblique asymptote to the Energy (126) is given by yoblique_asymptote (D) = m D + q   ⎡ ⎤ √ √1 1 − er f b + p 2D 2 +⎣ − √ = √ √ .⎦ 2 3 π e( p − b) 3 2π e( p − b)   1 − er f √12 2 D − (b + p) = √ + . 2 3 2π e( p − b)

(142)

That is 2 D − (b + p) yoblique_asymptote (D) = √ + 3 2π e( p − b)

1 − er f 2



√1 2

 .

(143)

3.14 The Oblique Asymptote for the “History of Rome” Case A few comments about this oblique asymptote of the Energy (126) are now of order for the “History of Rome” case: 1. Let us seek the year when the asymptote crosses the horizontal line having value ½ as in Fig. 13. In other words, let us look for the year after the Middle Ages, i.e. in the Renaissance, when the Energy acquired again the same numerical value that it had had in the Roman Empire at the time of Trajan. This means replacing the left-hand side of (143) by ½ and then solving the resulting linear algebraic equation for D. One then gets

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Evo-SETI Mathematics: Part 1: Entropy of Information. …

Fig. 13 Total ENERGY i.e. total WORK produced each year by the Roman Empire starting just after its peak, that occurred in the year 117 under Emperor Trajan. This solid red curve is given by Eq. (133). We see that, after Trajan, the empire started to decline, producing less and less total energy and reaching its minimum in the year 385.844 AD ≈ 386 AD. These were the years (and actually decades, or even a few centuries) of the Barbaric Invasions inside the Western Roman Empire, after the Visigoths had inflicted the first severe defeat to the Romans at the battle of Hadrianople in 378 ad. Then, the “Dark Ages of the Western Civilizations”, or “Middle Ages”, lasted for about ten centuries, and it was not until about 1300 ad the Western Europe started flourishing again, reaching about the same Total Energy level that the former Roman Empire had had under Trajan. This level is shown in the above graph by the thin solid blue horizontal line. After roughly 1300 ad, the Italian Renaissance developed and then expanded into the whole of Western Europe in the following centuries. In addition, the dot-dot red line is the oblique asymptote to the Total Energy given by Eq. (143). Finally, while the horizontal time scale is in agreement with the historic facts, the vertical scale of this graph is completely ARBITRARY, and we shall RE-SCALE it to the correct Energy value (measured in Joules) in the coming sections of this chapter

D=

b+ p + 2

  √ 3 2π e( p − b)er f √12 4

.

(144)

Inserting the Rome values (118) into (144), we finally get the Renaissance year 1522.945 ∼ 1523 for the full recovery of the Energy back to the Trajan 117 ad value. 2. The numeric value ½ for the horizontal line in Fig. 2 is of course a “remnant” of the way we derived things in all previous sections, and does not apply to the true numeric values of Ancient Rome. To find these true values, please read first the Wikipedia site https://en.wikipedia.org/wiki/Roman_economy a part of which we now repeat just the same here for convenience. “Economic historians vary in their calculations of the gross domestic product (GDP) of the Roman economy during the Empire. In the sample years of 14, 100, and 150 ad, estimates of per-capita GDP range from 166 to 380 sesterces (denoted HS). The GDP per

3 Part 2: Energy of Living Forms by “Logpar” Power Curves

81

capita of Italy is estimated as 40–66 percent higher than in the rest of the Empire, due to tax transfers from the provinces and the concentration of elite income in the heartland. In the Scheidel–Friesen economic model, the total annual income generated by the Empire is placed at nearly 20 billion HS, with about 5% extracted by central and local government. Households in the top 1.5% of income distribution captured about 20% of income. Another 20% went to about 10 percent of the population who can be characterized as a non-elite middle. The remaining “vast majority” produced more than half of the total income, but lived near subsistence. All cited economic historians stress the point that any estimate can only be regarded as a rough approximation to the realities of the ancient economy, given the general paucity of surviving pertinent data.” So, the above ½ value must really be replaced by, say, 166 + 380 9 10 H s = 273 × 109 H s = 273 billion Sestertii. 2

(145)

3. The inclination of the oblique asymptote (143) is of course given by its angular coefficient (136) that we repeat here for convenience √ 0.1613138163461 2 = . m= √ √ ( p − b) 3 π e( p − b)

(146)

This is inversely proportional to ( p − b), that is the initial part of the Logpar when the Civilization grows up like as b-lognormal from birth to peak. If this ( p − b) is “long” (like Rome’s, where one had ( p − b)Rome = 117 − (−753) = 870

(147)

years) then the oblique asymptote is “only slightly inclined”, for Rome being m Rome = 1.8541817970815579 × 10−4 .

(148)

In the limit for an “infinitely long growth”, i.e. ( p − b) → ∞, the asymptote would be horizontal. On the contrary, for a “small” ( p − b) the asymptote would be “highly inclined”, and just vertical for ( p − b) → 0.

3.15 What if Hadn’t Rome Fallen? Discovering the Straight Line Parallel to the Asymptote but Starting at the Rome Power Peak What if hadn’t Rome fallen? This “silly question” isn’t that silly, as it will let us discover the straight line parallel to the energy asymptote (143) but starting at the

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Evo-SETI Mathematics: Part 1: Entropy of Information. …

power peak ( p, P). Let us first understand the new problem we are solving in this section. Carl Sagan in his seminal book “Cosmos” (Ref. [15], see in particular the diagram on page 335) correctly described the Middle Ages (or “Dark Ages”) “a poignant lost opportunity for the human Species that plagued the Western Civilization for about 1000 years” since the fall of Rome in 476 ad up to the Italian Renaissance starting around 1300 ad. However, these lost 1000 years become a “small time” if we consider the future development of the human civilization over periods of thousands of years if not even millions of years, which is what might already have happened to other ET Civilizations that our SETI astronomers are now looking for. For more details and two plots, please see Figs. 14 and 15. In conclusion, we might say that: 1. We only used three inputs (birth, peak, death) to define any LOGPAR power curve. The word “power” here is intended both in its physical sense (i.e. measured in watts) and in the loose sense of “how much power” a certain civilization (or a certain physical phenomenon, like a star) displays along its whole Lifetime from birth to death. 2. The AREA under the LOGPAR power curve is the ENERGY that was produced during that Lifetime. A striking result. 3. For all practical calculations, this ENERGY GROWS LINEARLY after the peak as given by (150) and its inverse (151) hereafter. Figure 14 shows the same Energy(D) curve given by (133) but for a much longer time scale extrapolated into the future: the start is again at the year 0 (Christ born) but it now extends to the year 5000 ad. Shown in Fig. 14 is the same solid red Energy curve as in Fig. 2 plus the very thick red straight line departing from the

Fig. 14 NoSetbackEnergy (thick red upper straight line) i.e. Energy of the Western = Romance Countries (successors to the Roman Empire) had Rome not fallen. We see that it starts at the time of Trajan (117 ad) and keeps growing continuously parallel to the oblique asymptote of the true “concave” curve of the actual fall-and-recovery Energy

3 Part 2: Energy of Living Forms by “Logpar” Power Curves

83

Fig. 15 Same as Fig. 14 but with Energy replaced by MONEY in billion Sestertii (Sestertium is denoted HS). The inclination is obviously different with respect to Fig. 14 because of the different scale on the vertical axis

same “Trajan” point of coordinates ( p, Energy_o f _Roman_Empir e_under _T ra jan) and increasing constantly being the parallel straight line to the oblique asymptote (143). The meaning of Fig. 14 is rather obvious: “in the long run” i.e. millennia after the “1000-years Dark Ages period”, the total energy produced by the Western Countries is now “nearly” just the same is it would have been hadn’t Rome fallen. We shall call the Energy of this new straight line the “NoSetbackEnergy” (NSE). Its equation is promptly derived: 1. Since the starting point has the abscissa p and the independent variable D starts at p (i.e. D ≥ p), we shall simply multiply (D − p) times the angular coefficient (136) of the parallel oblique asymptote. 2. The ordinate corresponding to the p abscissa is, by definition, given by E( p) = Roman_Empire_Energy_in_117_AD.

(149)

3. Then the equation of the N oSetback Energy(D) is given by √ 2(D − p) + E( p). N oSetback Energy(D) = E(D) = √ √ 3 π e( p − b)

(150)

4. Being linear in D, the last Eq. (150) has the advantage of being invertible, i.e. solvable for D √ √ 3 π e( p − b) D= p+ (151) [E(D) − E( p)]. √ 2

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Evo-SETI Mathematics: Part 1: Entropy of Information. …

This allows one to find the future date D at which the N oSetback Energy(D) = E(D) will reach any assigned value. In conclusion, over long time scales, like the ones considered in the Evo-SETI Theory for Civilizations, whether human or Alien, the N oSetback Energy(D) = E(D) is a simple and linear representation of how the Energy used (or produced) by that Civilization up to its peak will unroll in the times following the peak instant p, if E( p) is known.

3.16 Energy Output of the Sun as a G2 Star Over the About 10 Billion Years of Its Lifetime We now make a rather “dramatic intellectual jump”: forget about Ancient Rome and consider the Energy of the Sun as a G2 Star instead! First of all, let us neatly write down the key three numeric input values (measured in years) in the History of the Sun as they are described in the caption to Fig. 16 and on the ground of present-day astrophysics:

Fig. 16 History of the  Sun as a Logpar power curve, created only by assigning the three numeric input Logpar values b = −4.567 × 109 year, p = 0 year, d = 5 × 109 year . The Sun formed about 4.6 billion years ago from the collapse of part of a giant molecular cloud that consisted mostly of hydrogen and helium and that probably gave birth to many other stars. This age is estimated using computer models of stellar evolution and through nucleocosmochronology. The result is consistent with the radiometric date of the oldest Solar System material, at 4.567 billion years ago (see the Wikipedia site https://en.wikipedia.org/wiki/Sun#Formation). The Sun is now producing energy at about a constant rate, and it will keep doing so for at least a billion year in the future. The above graph reveals so since in the next billion year the energy production of the Sun will change by less that 10−11 (a femto). In the above graph the b-lognormal power curve (not a probability density any more, since not normalized to 1 any more) describing time between the birth of the Sun 4.567 billion years ago and nowadays is shown in blue (forget about its part between now and 5 billion years in the future). Superimposing that blue curve is the red parabola, holding good between now and the final death of the Sun in about 5 billion years. We do not have the time now to plot the future linear energy growth of the Sun

3 Part 2: Energy of Living Forms by “Logpar” Power Curves

Fig. 17 History of a 45-billion-years-lasting M star as a curve, created only by assigning the three numeric input  b = −4.567 × 109 year, p = 0 year, d = 5 × 109 year

85

Logpar Logpar

⎧ ⎨ b = − 4.567 × 109 Sun_input_triplet = p=0 ⎩ d = 5 × 109 .

power values

(152)

Then the Logpar History Eq. (116) immediately yield numerical values of the Logpar σ and μ for the Sun, that we shall hereafter denote by σ S and μS, respectively:  Sun_logpar_doublet

σ S = 0.829 μS = 22.929.

(153)

Next, the study of the Sun Logpar peak comes. Let us assume that the Sun, as a “device producing energy”, is having its own activity at the peak right now, that is p Sun = 0. (154) Then, the Logpar peak ordinate for the Sun must be found by virtue of (117). One thus gets PSun = 7.48 × 10−11 .

(154)

This is the Sun’s energy peak value shown in Fig. 16. Astrophysics textbooks would now provide today’s output energy of the Sun, and we should re-write the units of the vertical axis of Fig. 16 accordingly. But we don’t have the time to do so now. We leave the question to another forthcoming paper by this author (Fig. 17).

3.17 Energy Output of an M Star Over 45 Billion Years of Lifetime We live around the Sun and had enough time (4.5 billion years) to develop Life forms starting with the first RNA molecules up to Humankind nowadays. M stars are

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Evo-SETI Mathematics: Part 1: Entropy of Information. …

by far the most common stars. About 76% of the main-sequence stars in the solar neighborhood are M stars. However, the M main-sequence stars (red dwarfs) have such low luminosities that none are bright enough to be seen with the unaided eye, unless under exceptional conditions. But from the point of view of SETI, M stars are extremely interesting. In fact, they can live for much longer times than the just 10 billion years like our Sun: in fact, M stars may live up to tens of billions of years or more. Of course, planets revolving around M stars are usually much closer to their star than the Earth is from the Sun, but that does not change the high probabilities that Extra Terrestrial Civilizations may live around M stars more frequently than around G stars. Just as an exercise, let us plot the Logpar power curve of an M star living 45 billion years altogether. Let us also suppose that the peak energy production of that M stars is nowadays, and that it took exactly the same time for that M star to be created as it was the case for the Sun, i.e. 4.567 billion years. Thus, the three numeric input values in the History of that M star lasting altogether 45 billion years are: ⎧ ⎨ b = − 4.567 × 109 45_billion_years_M_star_input_triplet = p=0 ⎩ d = 40 × 109 .

(155)

Then the Logpar History Eq. (116) immediately yield numerical values of the Logpar σ M and μM for that 45 billion year M star, respectively  45_billion_year_M_star_logpar_doublet

σ M = 0.973 μM = 23.188.

(156)

Next the study of this M star Logpar peak comes. Let us assume that the peak in energy production is right now: p45_billion_year s_M_star = 0.

(157)

Then, the Logpar peak ordinate for the Sun must be found by virtue of (117). One thus gets P45_billion_year s_M_star = 5.596 × 10−11 .

(158)

This is precisely the peak value shown in Fig. 5. Again, astrophysics textbooks would now provide today’s output energy of this M star, and we should re-write the units of the vertical axis of Fig. 4 accordingly. But we don’t have the time to do so now.

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87

3.18 Mean Power in a Lifetime In this section we are going to consider the notion of mean power value given by a Logpar power curve. Having abandoned the normalization condition for our Logpar curves, clearly we may not use the same mean value definition of a random variable typical of probability theory. However it’s easy to use the Mean Value Theorem for Integrals given in elementary Calculus textbooks. This is a variation of the mean value theorem which guarantees that a continuous function has at least one point where the function equals the average value of the function. Figure 18 graphically explains that: the rectangle has the same area as the shaded region under the curve. To translate the Mean Value Theorem for Integrals into a mathematical equation holding for Logpar curves, we clearly have to start from the Area Eq. (123) with d replaced by D, and divide that area by the length of the (D − b) segment in order to get the point along the vertical axis such that the area of the rectangle equals the Area (123). This is the required Mean Power Value over a Lifetime and is given by Mean_Power_over_a_lifetime =  1−er f

=

 √ D− p √ 2 D−(b+ p) 2

+

A(b, p, D) D−b √

√ D− p 2 D−(b+ p) √ 3 π( p−b)

· e−

D− p 2 D−(b+ p)

D−b

.

(159)

It is interesting to consider the limit of the Mean Energy over a Lifetime (159) for D → ∞. The calculation implies the use of L’Hospital’s rule, and the result is Asymptotic_Mean_Power_over_a_lifetime = lim (Mean_Power_over_a_lifetime) D→∞ √ 2 . = √ √ 3 π e( p − b)

(160)

Fig. 18 Mean Value Theorem for integrals: a continuous function has at least one point where the function equals the average value of the function. In other words: the rectangle has the same area as the shaded region under the curve

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Evo-SETI Mathematics: Part 1: Entropy of Information. …

By this we have completed the study of the mean along the vertical axis, i.e. the power axis. However, one might still wish to find, in some sense, “the mean value of what lies on the horizontal axis”, i.e. the Lifetime mean value. That is done in the next section.

3.19 Lifetime Mean Value It is natural to seek for some mathematical expression yielding the mean value of a Lifetime, meaning the mean value along the time axis of the (D − b) time segment representing the Lifetime of a living organism, or a civilization or even an ET civilization. We propose the following definition of such a Lifetime mean value: Lifetime_Mean_Value p

D t · b_lognormal(t; μ, σ, b) dt +

=

t · parabola(t) dt

(161)

p

b

inserting the b-lognormal (87) and the parabola (85) into (161), the latter is turned into p = b

(log(t−b)−μ)2

e− 2σ 2 t· √ dt + 2π σ (t − b)

D p



(t − p)2 dt. t · P 1− (D − p)2

(162)

The first integral may be computed in terms of the error function er f (x) given by (93), and the result is p b

(log(t−b)−μ)2

e− 2σ 2 t· √ dt 2π σ (t − b)     2   σ2 p−b)+μ log( p−b)−μ √ √ b 1 − er f e 2 +μ 1 − er f σ −log( 2σ 2σ + . = 2 2

(163)

This (163) may be further simplified by resorting to (88), with the result p b

−

(log(t−b)−μ)2 2σ 2

e t· √ dt = 2π σ (t − b)

e

σ2 2

+μ



1 − er f 2

√

2σ

 +

   b 1 − er f √σ2 2

. (164)

Re-expressing now (164) in terms of the History Formulae (116), it finally takes the form

3 Part 2: Energy of Living Forms by “Logpar” Power Curves

p

89

(log(t−b)−μ)2

e− 2σ 2 t· √ dt 2π σ (t − b)   √   √   3(D− p) 2 D− p D− p √ b 1 − er f ( p − b) e 2D−(b+ p) 1 − er f √2D−(b+ p) 2D−(b+ p) + . = 2 2 (165)

b

As for the second integral in (162), i.e. the parabola integral, it is promptly computed as follows D p



P(D − p)(3D + 5 p) (t − p)2 dt = . t · P 1− 12 (D − p)2

(166)

Inserting for P its expression (117), after some rearranging we conclude that the parabola integral is given by D p

√ √ 2 D − (b + p) D − p(3D + 5 p) − (t − p)2 dt = ·e t · P 1− √ 24 π ( p − b) (D − p)2

(D− p) 2 D−(b+ p)

.

(167) In conclusion, the mean Lifetime is found by summing (165) and (167) and reads Lifetime_Mean_Value =

  √  3(D− p) 2 D− p ( p − b) e 2D−(b+ p) 1 − er f √2D−(b+ p)

+

  √ 2  D− p b 1 − er f √2D−(b+ p) 2

√ +

√ 2 D − (b + p) D − p(3D + 5 p) − ·e √ 24 π ( p − b)

(D− p) 2 D−(b+ p)

(168) Just to give a numerical example, let us find the mean Lifetime of the Civilization of Rome. The first integral (165), by virtue of the Rome input triplet (118), yields the numeric value of the mean b-lognormal, i.e. Mean_Value_of_Rome_b - lognormal = −35.599

(169)

This means four years before the battle of Actium, fought on 2 September 31 bc: a crucial event that saw Cleopatra’s Egypt being absorbed as just one more Roman province. On the other hand, the parabola integral (167) and the Rome input triplet (118) yield for the parabola mean value the year

.

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Evo-SETI Mathematics: Part 1: Entropy of Information. …

Rome_parabola_Mean_Value = 32.758.

(170)

This 33 ad was a year falling during the empire of Tiberius (14–37 ad), and, most important, is just the year when Jesus Christ was crucified in Jerusalem. So, by summing up the two Eqs. (169) and (170),we reach the important conclusion that the mean value of the overall Rome Logpar power curve is just −2.840, i.e. Mean_Value_of_Rome_LOGPAR_History = −2.840 ≈ 3 B.C.

(171)

We regard this result as the “final proof” that our Evo-SETI Theory, in the Logpar form described in this paper, is basically CORRECT. In fact, (171) predicts that the “most important year in the History of Rome was… just the birth of Jesus Christ!

3.20 Logpar Power Curves Versus b-Lognormal Probability Densities Twenty months (November 2015 to July 2017) were necessary to this author to discover the Logpar power curves and their properties as described in the present chapter. In fact, prior to 2017, this author had built his Evo-SETI Theory on the notion of b-lognormal probability densities only. But now, the advent of Logpar power curves enables a profound study of energy within the Evo-SETI Theory that previously was not allowed. Energy means here both the energy displayed by a Civilization in time (like the Roman one, as exemplified in this chapter) as well as the energy produced by a star (like the Sun in the example briefly described earlier). All this was impossible to do in the b-lognormal approach to Evo-SETI Theory studied by this author in the years prior to 2017, i.e. in roughly in the years 2011–2015. Then, what about the Shannon Entropy as the measure of evolution intended as ever increasing amount of information (in bits) and complexity? Well, the Peak-Locus Theorem proved around 2013–2014 by this author even for a generic mean value curve other than just the simple exponential mean curve (typical of Geometric Brownian Motion (GBM)) is STILL VALID in the Logpar Evo-SETI Theory. In fact, the b-lognormal part of the Logpar (between birth and peak) REMAINS even in the Logpar approach. And so the proof remains of the key result asserting that the existence of the Molecular Clock (the fundamental discovery of molecular genetics made over 50 years ago) is derived in Evo-SETI Theory as a mathematical consequence of the Theory itself, just as Kepler’s laws are derived in Newtonian mechanics as a mathematical consequence of Newton’s Law of Gravitation. This is possible because the peak point is the junction point belonging to both the preceding b-lognormal (from which the Peak Locus Theorem and the Molecular Clock are

3 Part 2: Energy of Living Forms by “Logpar” Power Curves

91

derived) and to the ensuing parabola (from which the energy-related theorems are derived). A summary of the advantages of the Logpar power curves approach over the b-lognormal probability density approach is given in Table 4.

3.21 Conclusions About Logpars More and more exoplanets are now being discovered by astronomers either by observations from the ground or by virtue of space missions, like “CoRot”, “Kepler”, “Gaia”, and other future space missions. As a consequence, a recent estimate sets at 40 billion the number of Earth-sized planets orbiting in the habitable zones of sunlike stars and red dwarf stars within the Milky Way galaxy. With such huge numbers of “possible Earths” in sight, Astrobiology and SETI are becoming research fields more and more attractive to a number of young scientists. Mathematically innovative papers like the Evo-SETI ones, revealing unsuspected relationships like the one between the Molecular Clock and the Entropy of b-lognormals in Evo-SETI Theory, should thus be welcome. But in this chapter we did more than just in all previous Evo-SETI papers. While just preserving all the advantages of the b-lognormal probability density functions, we kept these b-lognormals good only for the first part of the curve: the one between birth and peak. The second part, between peak and death, was replaced by just a simple descending half-parabola, thus avoiding any inflexion point like the “senility” point typical of b-lognormals that was so difficult to estimate numerically in most cases. Thus LOGPAR curves have greatly simplified the description of any finite phenomenon in time like the Lifetime of a cell, or a human, or a civilization (like the Rome one used in this paper as an example) or even like an ET civilization. In addition to all this, we abandoned the normalization condition of b-lognormals retaining just their shape, and not their numbers. This transformed the Logpars into power curves, both in the popular sense where “power” means “political & military power” and in the strictly physical sense, where “power” means a curve measured in watts. And the area under such a Logpar is indeed the ENERGY associated to the Logpar phenomenon between birth and death. So, for the first time in the creation of our Evo-SETI Theory, we were able to add ENERGY to the ENTROPY previously considered already. And energy and entropy are the two pillars of classical Thermodynamics thus making Evo-SETI even more neatly applicable to stars, as the two examples of the Sun as a Logpar and of a 45-billion year M star suggest. Finally, there is one more crucial step ahead that we made by introducing Logpars. Without mentioning it so far, we actually “stumbled” into the PRINCIPLE OF LEAST ENERGY. This is reminiscent of the Principle of Least Action, i.e. the #1 mathematical tool of all theoretical physicists: just think of both particle physics and all the unified theories of gravitation, where a certain action function is postulated, then the Least Action Principle (or Hamilton’s Principle) and then the relevant EulerLagrange differential equations are derived, and finally (hopefully) solved, yielding

birth b, senility (=descending inflexion) s, death d (tangent at s intercept with time axis). DISADVANTAGE: estimating s is hard

b

2π σ (t−b)

Normalization Condition in terms Given the three inputs b, s, d one has of the three input parameters 2 *∞ e− [ln(t−b)−μ] 2 σ2 √ dt = 1

Normalization Condition in terms Total Area under b-lognormal equals ONE of the only truly independent 2 *∞ e− [ln(t−b)−μ] 2 σ2 variable σ (μ is a FALSE √ dt = 1 2π σ (t−b) independent variable) b

3 numeric inputs

b-lognormal probability density in the time (used by this author 2012–2017)



σ √ 2



σ2

−μ + · e√2 2π σ

· 3

  2 2 d−b−eμ−σ

Energy(d) = A(b, p, d)   √ d− p 1 − er f √2 d−(b+ p) = 2 √ √ d − p 2 d − (b + p) − + ·e √ 3 π ( p − b)

. (continued)

d− p 2 d−(b+ p)

=A

Given the three inputs b, p, d one has

2

1−er f

Total Area under the LOGPAR curve is A

birth b, peak p, death d (parabola intercept with the time axis). ADVANTAGE: b, p, d are EASILY found

LOGPAR power curve in the time (used by this author since 2017)

Table 4 b-lognormal probability density functions versus Logpar power curves: logpars are the superior Evo-SETI tool, since they enable the computation of the LIFETIME ENERGY for every living form, and that just by assigning THREE numerical inputs only: birth time b, peak time p and death time d

92 Evo-SETI Mathematics: Part 1: Entropy of Information. …

Asymptotic energy for high values of Death D

0

e

−

M L (t) = ln(m L (t)) −

–

2 σ L2 (t−ts)

[ln(n)−M L (t)]2

σ L2 (t − ts). 2

dn = e M L (t) e n· √ √ 2π σ L t − ts · n

with its inverse

m L (t) ≡

∞

starting at the initial time ts σ L2 2

(t−ts)

Mean value

for the lognormal stochastic Process L(t)

Mean Value Theorem

History Formulae (d−b) (s−b)

b-lognormal probability density in the time (used by this author 2012–2017) ⎧ ⎨ σ = √ d−s√ d−b s−b ⎩ μ = ln(s − b) + (d−s)(b+d−2s) .

Table 4 (continued)

yoblique_ asymptote (D) =

2√D−(b+ p) 3 2π e( p−b)

+

 2

1−er f

√1 2



.

LOGPAR lifetime_mean_value  √  3(D− p)  2 D− p ( p − b) e 2D−(b+ p) 1 − er f √2D−(b+ p) = 2   √  D− p b 1 − er f √2D−(b+ p) + 2 √ √ (D− p) 2 D − (b + p) D − p(3D + 5 p) − 2 D−(b+ p) . + ·e √ 24 π ( p − b)

2 d−(b+ p)

LOGPAR power curve in the time (used by this author since 2017) ⎧ √ √ ⎨ σ = √ 2 d− p 2 d−(b+ p) ⎩ μ = ln( p − b) + 2(d− p) .

3 Part 2: Energy of Living Forms by “Logpar” Power Curves 93

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Evo-SETI Mathematics: Part 1: Entropy of Information. …

the trajectory of particles. Well, the ACTION has the dimension of an ENERGY MULTIPLED BY THE TIME, and this is precisely what we did when finding the area under the Logpar and considering the Logpar integral in between birth and death. So we claim that… the Logpar is the optimal trajectory of our Evo-SETI Theory, also in regard to the Least Action Principle. But an adequate description of this result would require one or more papers giving more profound justifications, of course.

4 Part 3: Before and After the Singularity According to Evo-SETI Theory 4.1 Every Exponential in Time Has just a Single Knee: The Instant at Which Its Curvature Is Highest Consider the easiest possible exponential curve as a function of the time t having the equation y(t) = et .

(172)

Now ask the question: what is the expression of the curvature κ for such a simple exponential in time? The answer may be found in textbooks about the Calculus, as well as at the Wikipedia site https://en.wikipedia.org/wiki/Curvature and reads

κ(t) =

+ 2 + +d y+ + dt 2 + 1+



dy dt

2 23

.

(173)

If we insert (172) into (173), we get + t+ +e +

et κsimple_exponential (t) =  =  3 . 3 1 + e2 t 2 1 + (et )2 2

(174)

Now, the radius of curvature R(t) is defined as the reciprocal of the curvature (173), that is

 2 23 1 + dy dt + 2 + . R(t) = +d y + + dt 2 +

(175)

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95

Then, in the particular case of the simple exponential (172), its radius of curvature provided by (175) reads 3  1 + e2 t 2 . Rsimple_exponential (t) = et

(176)

where the absolute value at the denominator of (175) disappeared since the exponential is a positive curve. Let us now ask the fundamental question: for the simple exponential (172), what is the minimum radius of curvature? To answer this question we must compute the derivative of (176) with respect to t, set it equal to zero, and then solve the resulting equation with respect to t, thus finding the time t of the minimum curvature. Let us do so: after a few steps, one gets d Rsimple_exponential (t) = 0= dt

3 2



1 + e2 t

 21

3  2e2 t − 1 + e2 t 2 et e2 t

(177)

leading to

 1 1 + e2 t = 0 that is 2 e2t − 1 = 0 1 + e2 t 2 · 3 et − et

(178)

and finally tsimple_exponential_knee = −

  1 ln(2) = − 0.346. = ln √ 2 2

(179)

We call (179) the abscissa of the knee of the simple exponential (172) because for t < tsimple_exponential_knee the growth of the exponential (172) is very slow, while for t > tsimple_exponential_knee the growth of the exponential (172) is very fast. Thus, the simple exponential’s knee time is the “dividing time” in between very slow and very fast growth. Better still, the knee time is the tangent point between the simple exponential (172) and its osculating circle https://en.wikipedia.org/wiki/Osc ulating_circle having the minimum radius of curvature, or, if you so prefer, the maximum curvature. And the calculations we have made so far show that every exponential only has a single knee, and not many. Actually we may now complete the calculation of the only exponential knee point by inserting (179) into (172) and so obtaining  ln(2) 1 ysimple_exponential tsimple_exponential_ knee = etsimple_exponential_knee = e− 2 = √ = 0.707. 2 (180)

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Evo-SETI Mathematics: Part 1: Entropy of Information. …

Fig. 19 The simple exponential y = et (left to right: the raising blue curve) and the osculating red circle having the minimum radius of curvature, i.e. the maximum curvature, just for the value tsimple_exponential_knee = − ln(2) 2 = − 0.346.

As for the numeric value of the radius of curvature of the simple exponential at its knee, upon inserting (179) into (176), after a few reductions one finds

  3 2 2 ln √12 3   3 1 + e 2t 2 1+e [3] 2   = 2.60. = = Rsimple_exponential (t) = et 2 ln √12 e

(181)

This value of 2.60 for the radius of curvature is evidently correct if one just has a look to Fig. 19. We will skip here the calculations of the coordinates of the centre of the osculating circle.

4.2 GBM Exponential as Mean Value of the Increasing Number of Species Since the Origin of Life on Earth As we pointed out already, the key idea of Evo-SETI theory is that the increasing number of Species living on Earth since the time of the Origin of Life (about 3.5 billion years ago) is a lognormal stochastic process L(t) with an arbitrary (i.e. to be defined experimentally) mean value denoted m L (t) (Please see Refs. [7, 8] for many more mathematical details). In the present chapter, however, we shall only assume the much easier model having the exponential (rather than the arbitrary one) mean value given by m GBM (t) = N s · e B (t−ts)

⎧ ⎨ t ≥ ts with Ns > 0 ⎩ B > 0.

(182)

4 Part 3: Before and After the Singularity According to Evo-SETI Theory

97

This is the so-called Geometric Brownian Motion (GBM) sub-case of the general lognormal process L(t), i.e. of the general theory developed in Refs. [7, 8]. We assume the time t to start at ts (“time of start”) and we assume the mean value at this starting time ts to be equal to the known positive number N s. In fact (182) yields m GBM (ts) = N s.

(183)

If, as in all papers previously published by this author, ts means the time of the Origin of Life on Earth, then (183) has N s = 1, meaning that there was only ONE Species on Earth when Life started, and this only Species is today assumed by Evolution specialists to be RNA. The first question posed by the GBM exponential mean value (182) is: what is the practical meaning of the positive constant B? The answer is easily found by considering (182) nowadays, namely by inserting t = 0 into (182). One then gets m GBM (0) = Average_Number_of_Species_NOW = N s · e− B ts .

(184)

Solving (184) for B yields the meaning of B:  B=

ln

Average_Number_of_Species_NOW Ns

− ts

 .

(185)

In all his papers published prior the 2017, this author assumed the following two CONVENTIONAL values for ts and B  ts = − 3.5 × 109 years Our_CONVENTIONAL_values == Number_of_Species_NOW = 50 million. (186) In terms of B, inserting (186) into (185) one gets  Our_CONVENTIONAL_values =

ts = − 3.5 × 109 years B=

5.0650095895406915×10 years

−9

.

(187)

In the sequel of this chapter we shall discuss the validity of the two numbers appearing in (186) and (187).

4.3 Deriving the Knee Time for GBMs The next question we face is: at what time does the KNEE occur for the GBM exponential (182)? To answer this question we must insert (182) instead of y(t) into the expression (175) for the radius of curvature. The result is

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Evo-SETI Mathematics: Part 1: Entropy of Information. …

 2  23  1 + dtd N s · e B (t−ts) + + . RGBM (t) = + d2 + + dt 2 N s · e B (t−ts) +

(188)

Noticing that the derivative of any increasing exponential certainly is a positive quantity, we may get rid of the absolute value at the denominator of (188), getting

RGBM (t) =

 2  23  1 + dtd N s · e B (t−ts) d2 dt 2

N s · e B (t−ts)

.

(189)

Since the knee is the value of t for which (189) is minimum, we must then differentiate (189) with respect to t and set the relevant derivative equal to zero. Upon differentiating the fraction in (189) just with respect to, N s · e B (t−ts) initially, and then multiplying this times the derivative of N s · e B (t−ts) with respect to t, one gets  d N s · e B (t−ts) d RG B M (t) d RG B M (t) d RG B M (t) · N s B e B (t−ts) =  =  0= B (t−ts) dt dt d Ns · e d N s · e B (t−ts) ⎧   2  21 d 2 2 ⎪ ⎨ 23 1 + dtd N s · e B (t−ts) 2 dt N s · e B (t−ts) dtd 2 N s · e B (t−ts) · dtd 2 N s · e B (t−ts) =  2 2 ⎪ d ⎩ B(t−ts) N s · e dt 2 3    d B (t−ts) 2 2 d 3 B (t−ts) ⎫ ⎪ ⎬ 1 + dt e · dt 3 e − . (190)  2 2 ⎪ d ⎭ B (t−ts) e 2 dt

N s·B e B (t−ts)

Since (190) must equal zero, only the quantity within braces must equal zero, yielding  2  21  3 d d B (t−ts) Ns · e 0= 2 N s · e B (t−ts) 1+ 2 dt dt 3    2 2 d 3 1 + dtd N s · e B (t−ts) · dt 3 N s · e B (t−ts) − .  2 2 d B (t−ts) Ns · e dt 2

(191)

Upon performing all derivatives shown in (191) one gets 3   1 1 + N s B 2 e2B (t−ts) 2 · N s 3 B 3 e B (t−ts) 2 2 2B (t−ts) 2 B (t−ts) 0 = 3 1 + Ns B e Ns B e − . N s 4 B 4 e2B (t−ts) (192)

4 Part 3: Before and After the Singularity According to Evo-SETI Theory

99

1  Again, since (192) must equal zero, the whole factor 1 + N s 2 B 2 e2B (t−ts) 2 · N s · e B (t−ts) may be taken out, leaving just 0 = 3N s B −

1 + N s 2 B 2 e2B (t−ts) . N s B e2B (t−ts)

(193)

The last Eq. (193) must be solved for t, thus yielding the knee abscissa, t_knee, of the exponential (182). Rearranging (193) one gets 2N s 2 B 2 e2B (t−ts) = 1

(194)

1 . 2N s 2 B 2

(195)

yielding e2B (t−ts) =

Now (195) is a quadratic equation in e B(t−ts) yielding the two roots   1 B (t − ts) = ln ± √ 2N s B

(196)

that is, rewriting now tGBM_ knee instead of t tGBM_knee = ts +

  ln ± √2N1 s B B

.

(197)

Clearly, the minus sign inside the log in (197) must be discarded (since any log argument must always be positive) and so (197) becomes

tGBM_knee = ts −

ln

√  2N s B B

(198)

This is the fundamental GBM knee equation, at the root of all further discussions in the sequel of this chapter. Please notice that four parameters exist in this Eq. (198): ts, N s, B, tGBM_knee . The sequel of this chapter is a discussion of the “four-parameter equation” (198) in relationship to what really has happened to the number of living Species on Earth in the past, and perhaps in the future…

4.4 Knee-Centered Form of the GBM Exponential It is interesting to re-cast the GBM exponential (182) in a new form where the time t is centered around the GBM knee time (198), rather than around the time of the

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Evo-SETI Mathematics: Part 1: Entropy of Information. …

Origin of Life on Earth, i.e. the time-of-start ts. To do so, just solve (198) for ts

ts = tGBM_knee +

ln

√  2N s B (199)

B

and insert (199) into (182). The result is B (t−ts)

m GBM (t) = N s e = Ns e e B (t−tGBM_knee ) = . √ 2B

Bt

·e

−B·ts

= Ns e

Bt

·e

  √ ln( 2 N s B ) −B tGBM_knee + B

(200)

That is, the knee-centered GBM exponential has the simple form m GBM (t) =

e B (t−tGBM_knee ) . √ 2B

(201)

This form (201) of the GBM exponential shows very neatly that: 1. For all times before the knee time, tGBM_knee , the exponential growth is very slow, i.e. like the negative exponential e−t for t → − ∞. 2. For all times after the knee time, tGBM_knee , the exponential growth is very fast, i.e. like the positive exponential et for t → ∞. 3. The constant √12B is just a scale factor that does not change the essence of both remarks (1) and (2) at all.

4.5 Finding WHEN the GBM Knee Will Occur According to the Author’s Conventional Values for ts and B As pointed out in (186), prior to 2017 this author always assumed that the time of the origin of Life on Earth was -3.5 billion years and that the current number of Species living on Earth was 50 millions. Was this actually the case? WHEN will the GBM knee occur? In the past? Or nowadays? Or in the future? To find out, let us insert the assumed numeric values (186) into the GBM knee Eq. (198) with the particular B given by the lower Eq. (187). The surprising result is tGBM_knee_for_this_author’s_assumed_inputs = 2.0272472028487825 × 108 years ≈ 200 million years in the future!

(202)

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101

Clearly, the author’s assumed values (186) MUST BE WRONG if the Singularity is near to nowadays, as Kurzweil suggested. So, how may we get out of this paradox?

4.6 Ray Kurzweil’s 2006 Book “the Singularity Is Near” In 2006 the seminal book “The Singularity Is Near” was published by the American inventor and futurist Ray Kurzweil about artificial intelligence and the future of humanity. Summarizing (too much!) this book’s content, the Singularity is the time when computers will take over humans by virtue of their own superior intellectual capabilities (artificial intelligence = AI) and so, loosely said, poor humans “made of meat” will stop existing, automatic machines will dominate the Earth, and finally they will expand into space. This author’s admiration for Kurzweil’s book led him to create the mathematical model presented in this chapter within the framework of this author’s Evo-SETI (Evolution and SETI) mathematical theory (see Refs. [5] through [8]).

4.7 Kurzweil’s Singularity Is the Same as Our GBM’s Knee in Our Evo-SETI Theory Now the reader is ready to understand why we developed the mathematical theory of the GBM knee in Sects. 4.1–4.6 of the present chapter. We claim that Kurzweil’s Singularity is just the (only) GBM’s KNEE in our Evo-SETI Theory. This claim is rather obvious if you look at the exponential shown in Fig. 20. It depicts the GBM exponential assuming that, since about 10 billion years ago, the number of Civilizations in the Universe (i.e. not just in the Milky Way Galaxy, as assumed by the famous Drake equation of traditional SETI) “exploded”. These are just guesses of course, but the behaviour of the exponential before and after its own knee is obvious and immediately reveals that the knee is one only.

4.8 Measuring the Pace of Evolution B by the Average Number m0 of Species Living on Earth NOW In Evo-SETI Theory the present time of Biological Evolution is the zero instant t = 0. So, if we wish to “merge” the Evo-SETI Theory with Kurzweil’s claim that “The Singularity Is Near”, we simply have to let tGMB_knee = 0

(203)

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Evo-SETI Mathematics: Part 1: Entropy of Information. …

Fig. 20 i.e. Figure 5 of the author’s paper “SETI as a Part of Big History”, Acta Astronautica, 101 (2014), 67–80

into the GBM knee Eq. (198). Thus, the GBM knee Eq. (198) becomes

ts =

ln

√

 2N s B

.

B

(204)

Better still, since the number of living Species at the time of the origin of Life ts was just one (i.e. RNA), letting N s = 1 into (204), changes the latter into ts =

ln

√  2B B

.

(205)

This we like to call the Singularity-Evo-SETI (SES) equation. Contrary to the four-parameter Eq. (198), the SES Eq. (205) only has two parameters (ts and B) and we are now going to “play” with them to see what the conclusions are. In addition, the Singularity-centered GBM exponential is now even easier than the knee-centered exponential (201). In fact, by virtue of (203), (201) now becomes eB t m SINGULARITY - CENTERED_EXPONENTIAL (t) = √ . 2B

(206)

This Eq. (206) we will call the “Singularity-Centered-Exponential” (SCE) Equation. In the sequel of this chapter we will use it to find out the b-lognormals that it

4 Part 3: Before and After the Singularity According to Evo-SETI Theory

103

envelops, i.e. the so-called family of Running-b-Lognormals (RbLs, in the language of Evo-SETI theory). Finally, consider the very important numeric value of the Singularity-CenteredExponential (206) at the present time t = 0. This numeric value is very important because, happening now, i.e. occurring now, we may hope to be able to measure it experimentally as a part of current Biology. Upon defining the new experimental quantity m0 as the current value of the average number of Species living on Earth, i.e. upon defining m0 ≡ m SINGULARITY - CENTERED_EXPONENTIAL (0) = m L (0)

(207)

then the exponential equals 1 in (206) and we are left with m0 = √

1

.

(208)

1 B=√ . 2 m0

(209)

2B

Solving (208) for B, one gets

This is an important new result of our Singularity-Evo-SETI Theory inasmuch as it relates to each other two numerical quantities like m0 and B that were regarded as completely unrelated prior to this chapter. In other words still, we have discovered how to numerically measure the PACE of Evolution in the Singularity-Evo-SETI Theory if we just can measure, or at least estimate, the average number m0 = m L (0) of Species living on Earth right now. This is why we call (209) “the Pace equation”.

4.9 An Unexpected Discovery: The “Origin-to-Now” (“OTN”) Equation Relating the Time of the Origin of Life on Earth (t s) to m0 (the Average Number of Species Living on Earth Right Now) Let us stop a moment to ponder over what we did so far in this paper: 1. We assumed that Biological Evolution was just a realization of the stochastic process called Geometric Brownian Motion (GBM) in the number of Species living on Earth since the origin of Life. This assumption is the same as assuming that the mean value of the number of living Species is an exponential increasing in time from the initial value of 1 when RNA appeared on Earth to the (unknown but measurable in principle) current value m0.

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Evo-SETI Mathematics: Part 1: Entropy of Information. …

2. We then merged our Evo-SETI GBM Theory with Kurzweil’s claim that the Singularity is near, practically nowadays Eq. (203). 3. As a consequence, we discovered the simple pace Eq. (209) relating the pace B to the current average value m0 of Species living on Earth. 4. But… we did not yet relate the time of the origin of Life on Earth, ts, to m0. Well, we will do so right now, and solve the resulting numerical equation in m0 that proves to be the “final, resolving equation” of our Singularity-Evo-SETI Theory. 5. To do so, just re-write the SES Eq. (204) in terms of the pace Eq. (209). One then gets the new “Origin-To-Now”(OTN) equation

ts =

ln

√

 2B

B

=

√ 2 m0 ln



1 m0

 =−

√ 2 m0 ln(m0).

(210)

Since ts is a negative number, one might prefer to take the minus sign in front of it, thus rewriting (210) as follows − ts √ = m0 ln(m0) 2

(211)

thus making it plainly clear that, if we want to solve (211) for m0 in terms of the better-known ts, we can solve (211) numerically only, and not analytically.

4.10 Solving the “Origin-to-Now” Equation NUMERICALLY for the Two Cases of −3.5 and −3.8 Billion Years of Life on Earth We now proceed to solve (211) graphically for the two different cases of −3.5 and −3.8 billion years of Life development on Earth. Let us rewrite (211) with x instead of m0 and so in the form: √

1=

2 x ln(x). − ts

(212)

One might say that, solving (212) numerically is the same as finding the numerical value of the “independent variable” x for which the function of m0 on the right-handside of (212) equals just 1. This function of m0 is shown in Fig. 21 hereafter. One immediately notices that the function of x given by (212) is increasing and reaches the value of 1 just at x = 1.333 ∗ 108 . In other words, the SINGULARITY-EVO-SETI Theory predicts that, if Life originated 3.5 billion years ago, NOWADAYS there should be 133 million living Species on Earth, on the average.

4 Part 3: Before and After the Singularity According to Evo-SETI Theory

105

Fig. 21 Solving (212) numerically to find out that, if Life started 3.5 billion years ago, then the average number m0 of living Species today should be 133 million, or, better, 132.4 million

And a little more refined numerical solution to (212), that we omit for the sake of brevity, replaced this 133 million by 132.4 billion Species living on Earth on the average right now. In this case, the pace of evolution, B, derived from (209), would be Bfor_ts = - 3.5_billion_years =

5.341 × 10−9 . year

(213)

Let us now suppose that Life originated on Earth 3.8 billion years ago. Then Fig. 22 (a graph absolutely similar to Fig. 21 with just 3.8 billion years ago replacing the previous 3.5 billion years ago) shows that the average number m0 of Species living on Earth today should be around 143.1 million. Clearly, having Life had 0.3 billion years (300 million years) more time to evolve, Life would have increased the average number of Species living now from 132.4 million to 143.1 million. In the latter case, the pace of evolution, B, derived from (209), would be slightly less than (213), namely Bfor_ts = - 3.8_billion_years =

4.941 × 10−9 year

(214)

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Evo-SETI Mathematics: Part 1: Entropy of Information. …

Fig. 22 Solving (213) numerically to find out that, if Life started 3.8 billion years ago, then the average number m0 of living Species today should be 142.9 million, or, better, 143.1 million

4.11 But… Biologists Are UNABLE to Measure m0 Experimentally! But… unfortunately the biologists are in wild disagreement among themselves when it comes to estimate m0 numerically: it could range between 8.7 million ± 1.3 million according to the United Nations Environmental Program, to even 1 trillion or more according to dissenting Biologists. The whole discussion seems to be summarized by this sentence, that we reproduce here from the last site we mentioned: “Calculating how many Species exist on Earth is a tough challenge. Researchers aren’t even sure how many land animals are out there, much less the numbers for plants, fungi or the most uncountable group of all: microbes.”

4.12 Lognormal pdf of the GBM Up to this point, we only dealt with the mean value of the number of living Species and imposed that it must be an exponential, i.e. the relevant stochastic process must be a GBM. But we did not consider both the upper and lower standard deviation curves above and below this exponential, respectively, that are crucial in order to estimate how much each different realization of the GBM “spreads” around its own mean value. Now we do so.

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The relevant mathematics is a little more complicated than those of the previous Sections, and so the reader might find it helpful to take advantage of a symbolic manipulator to calculate especially the relevant integrals: Maxima was used by this author, but Mathematica and Maple would be helpful too. We must first of all recall from, say, Refs. [7–10], that the probability density function (pdf) of the GBM is the lognormal in the number n ≥ 0 of living Species, is given by the equation 

GBM_lognormal_pdf(n; t, ts, N s, B, σ ) =

e

−

2 2 ln(n)−ln(N s)−B(t−ts)+ σ (t−ts) 2 2 σ 2 (t−ts)

√

√ 2 π n σ t − ts

.

(215)

This pdf has one more new positive parameter σ > 0 whose numerical value we will finally determine at the end of this Section in terms of the previously determined numerical value of ts, N s = 1, B and m0. Then, the mean value of the pdf (215) is given by the integral 

∞ n·

e

−

0

2 2 ln(n)−ln(N s)−B(t−ts)+ σ (t−ts) 2 2 σ 2 (t−ts)

√

√ 2 π n σ t − ts

eB t dn = N s e B (t−ts) = √ . 2B

(216)

This is just the same as (182) and (200). It is actually possible to compute all (i.e. k = 1, 2, . . .) the moments of the lognormal pdf (215), getting 

∞ nk ·

e

−

0

ln(n)−ln(N s)−B(t−ts)+

√

σ 2 (t−ts) 2

2 σ 2 (t−ts)

√

2 π n σ t − ts

2

dn = N s k e−

2 B k(t−ts)+k(k−1) σ 2 (t−ts) 2

.

(217)

For k = 0, (217) yields the normalization condition of the pdf (215). For k = 1, the mean value (216) is found again. From (217) one might derive much of the descriptive statistics of the lognormal pdf (215), that we omit here for the sake of brevity. We are just interested in the k = 2 case, yielding the mean value of the square, that is 

∞ n2 · 0

e

−

2 2 ln(n)−ln(N s)−B(t−ts)+ σ (t−ts) 2 2 σ 2 (t−ts)

√

√ 2 π n σ t − ts

dn = N s 2 e2B (t−ts)+σ

2

(t−ts)

, = GBM2 .

(218)

Now the GBM variance comes, that, by virtue of (216) and (218) turns out to be , 2 = GBM2 − GBM 2 σGBM

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Evo-SETI Mathematics: Part 1: Entropy of Information. …

= N s 2 e2B (t−ts)+σ (t−ts) − N s 2 e2B (t−ts)  2  = N s 2 e2B (t−ts) eσ (t−ts) − 1 . 2

(219)

The square root of (219) is the GBM standard deviation . . σGBM = N s e B (t−ts) eσ 2 (t−ts) − 1 = GBM · eσ 2 (t−ts) − 1.

(220)

This is one more function of the time, that, upon replacing the mean value (206), explicitly reads e B t . σ 2 (t−ts) e − 1. σGBM (t) = √ 2B

(221)

The square root factor in (220) .

eσ 2 (t−ts) − 1

(222)

is known to statisticians as the “variation coefficient” (since it gives the pace of how the ratio σGBM / GBM changes in time). Then, (222) shows that the variation coefficient vanishes at the initial GBM instant ts, which is the same as saying that all three curves of the mean value, upper standard deviation and lower standard deviation have the same initial point at ts, or, in other words still, the GBM starts at ts with probability 1. And all the above simply means that the two standard deviation curves have the two equations ⎧ ⎨ upper _st_dev(t) = ⎩ lower _st_dev(t) =



eB t √ 1 2B  Bt e √ 1 2B

 eσ 2 (t−ts) − 1  . − eσ 2 (t−ts) − 1 . +

.

(223)

4.13 Finding the GBM Parameter σ The standard deviation (221) still contains the unknown parameter σ that we are now going to find. To this end, consider the current (i.e. t = 0) value of the standard deviation curve (221). Letting t = 0 into (221) one gets . σGBM (0) =

e−ts· σ 2 − 1 √ 2B

(224)

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109

This equation may then be solved for the unknown parameter σ , yielding, after a few steps, /   ln 1 + 2B 2 (σGBM (0))2 σ = −ts

(225)

Let us now stop a moment and ponder over what we are doing: 1. σGBM (0) is the standard deviation nowadays, that is the (unknown) value above and below the value (supposed to be known to biologists, but not really so in the practice) m0 of the number of Species living on Earth nowadays. So, why don’t we re-write δm0 instead of σGBM (0) in (225)? 2. And why don’t we replace B by m0 in (225) by virtue of (209)? 3. Well, if we take both suggestions into account, (225) is turned into its final form #   $ $ ln 1 +  δm0 2 % m0

σ =

−ts

(226)

Yet, we don’t know the numerical value of δm0 in (226): how can we express it in terms of the numerical values of other known quantities? The answer lies in the OTN (Origin-To-Now) Eq. (210) that we rewrite here for convenience ts = −

√ 2 m0 ln(m0).

(227)

What we need is the error analysis of this Eq. (227). Differentiating it, one gets δts = −

√

2 [1 + ln(m0)] δm0.

(228)

Then, dividing (228) by (227) one gets the relationship among the relative errors on ts and m0 δts [1 + ln(m0)] δm0 = · ts ln(m0) m0

(229)

Solving (229) for the relative error on m0 yields δm0 ln(m0) δts = · . m0 1 + ln(m0) ts

(230)

Multiplying this by m0 yields the δm0, i.e. the standard deviation of the number of Species living on Earth nowadays δm0 =

m0 ln(m0) δts · . 1 + ln(m0) ts

(231)

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Evo-SETI Mathematics: Part 1: Entropy of Information. …

As for σ , (230) is the expression to be inserted into (226) in order to get the numeric value of σ , since all other quantities are now numerically known. By doing so, we get for σ the final expression

σ =

#

$  $ ln(m0) $ ln 1 + 1+ln(m0) · % −ts

δts ts

2 .

(232)

4.14 Numerical Standard Deviation Nowadays for the Two Cases of Life Starting 3.5 and 3.8 Billion Years Ago The time is now ripe to get down to the numbers: 1. The standard deviation on ts, δts, in between the two Cases of Life starting − 3.5 and −3.8 billion years ago, respectively, is clearly the average between them δts =

−(3.5 + 3.8) × 109 = −3.65 × 109 . 2

(233)

2. As for Case 1 (i.e. ts = −3.5 × 109 years ago), with m0 given by the results of Fig. 21, namely m0 = 132.4 million = 1.324 × 108

(234)

then (231) yields a standard deviation of δm0 = 1.3106591479540047 × 108 .

(235)

But… this standard deviation is nearly as much big as the mean value (234) itself. Does this make sense? We think it does. In fact, it simply stresses the huge uncertainly affecting the number of living Species nowadays… just to give a hand to the “poor” Biologists, who don’t even know if m0 is in the order of the hundreds of millions (as we claim) or (crazy!) even up to trillions! 3. As for Case 2 (ts = −3.8 × 109 years ago), with m0 given by the results of Fig. 22, namely m0 = 143.1 million = 1.341 × 108 then (231) yields a standard deviation of

(236)

4 Part 3: Before and After the Singularity According to Evo-SETI Theory

δm0 = 1.4168786255929899 × 108 .

111

(237)

Same comments as for Case 1.

4.15 Numerical σ for the Two Cases of Life Starting 3.5 and 3.8 Billion Years Ago Finally σ . We know from the theory that σ is positive, and we know from the practice (see Ref. [7], in particular pages 231–233) that it is “usually smaller than 1”. So, what do we expect from (232) in the two respective Cases 1 and 2? As for Case 1, (232) yields σ = 1.3970082115699428 × 10−5

(238)

while, for Case 2, (232) yields σ = 1.3972208698944956 × 10−5 .

(239)

These two numbers are very small, and they differ from each other only on the fifth significant figure: “surprises” of the Singularity-EvoSETI Theory.

4.16 Conclusions At this point we have to face the crucial question: “What is the meaning of the b-lognormals before and after the Singularity?”. The answer to this question comes along these lines: 1. As for the time before the Singularity, the vertical axis is the overall number of Species living on Earth at a certain time of Biological Evolution. This interpretation seems to be correct for the whole of the 3.5 (or 3.8) billion years of Biological evolution before present. And that implied a numerical value for B in the order of 10−9 years (see Ref. [7]). 2. However, when this author described the b-lognormals of the most important Historic Western Civilizations (see Ref. [7]), the value of B was about 10−5 , i.e. four orders of magnitude higher. In other words, the two Bs of Biological Evolution and of the History of human civilizations differ by about four orders of magnitude, the latter being about 10,000 times faster than the former. 3. Not to mention the value of B after the Singularity, that is of course “experimentally” unknown to us. In addition, after the Singularity, the living Species must be replaced by generations of computers, meaning that each b-lognormal represents a more and more advanced family of computers ruling “Life” on Earth after the Singularity.

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4. And… Aliens (i.e. Alien machines) visiting the Earth would be astonished to find out that, before the Singularity, i.e. until 2045 or so (as Kurzweil claimed), the inhabitants of Earth were still made out of meat, as in the Science Fiction story by Terry Bisson https://en.wikipedia.org/wiki/They%27re_Made_Out_of_Meat 5. After this point, one more paper should be written to describe… how the blognormal’s “width” would correctly describe the “average duration in time” of each Species (before the Singularity) and of each computer Species (after the Singularity)… 6. …but this is “too much to be done now”, and so we have leave it to a new, forthcoming paper, hoping that computers… will not take over too soon.

References 1. L. Grinin, A. Markov, A. Korotayev, On similarities between biological and social evolutionary mechanisms: mathematical modeling. Cliodynamics 4, 185–228 (2013) 2. A.V. Korotayev, A.V. Markov, L.E. Grinin, Mathematical modeling of biological and social phases of big history, in Teaching and Researching Big History: Exploring a New Scholarly Field (Uchitel: Volgograd, Russia, 2014), pp. 188–219 3. A.V. Korotayev, A.V. Markov, Mathematical modeling of biological and social phases of big history, in Teaching and Researching Big History—Exploring a New Scholarly Field ed. by L. Grinin, D. Baker, E. Quaedackers, A. Korotayev (Uchitel Publishing House: Volgograd, Russia, 2014), pp. 188–219 4. C. Maccone, The statistical drake equation. Acta Astronaut. 67(2010), 1366–1383 (2010) 5. C. Maccone, A mathematical model for evolution and SETI”. Orig Life Evol. Biospheres (OLEB) 41(2011), 609–619 (2011) 6. C. Maccone, Mathematical SETI (Praxis-Springer, 2012). ISBN-10: 3642274366 | ISBN13: 978-3642274367 | Edition: 2012. 7. C. Maccone, SETI, evolution and human history merged into a mathematical model. Int. J. Astrobiol. 12(3), 218–245 (2013) 8. C. Maccone, Evolution and mass extinctions as lognormal stochastic processes. Int. J. Astrobiol. 13(4), 290–309 (2014) 9. C. Maccone, SETI as a part of big history. Acta Astronaut. 101(2014), 67–80 (2014) 10. C. Maccone, Evo-SETI entropy identifies with molecular clock. Acta Astronaut. 115, 286–290 (2015) 11. A.V. Markov, A.V. Korotayev, Phanerozoic marine biodiversity follows a hyperbolic trend. Palaeoworld 16, 311–318 (2007) 12. A.V. Markov, A.V. Korotayev, The dynamics of phanerozoic marine biodiversity follows a hyperbolic trend. Zhurnal Obschei Biologii 68, 3–18 (2007) 13. A.V. Markov, A.V. Korotayev, Hyperbolic growth of marine and continental biodiversity through the Phanerozoic and community evolution. Zh. Obshch. Biol. 69, 175–194 (2008) 14. A.V. Markov, V.A. Anisimov, A.V. Korotayev, Relationship between genome size and organismal complexity in the lineage leading from prokaryotes to mammals. J. Paleontol. 44, 363–373 (2010) 15. C. Sagan, Cosmos (Random House, New York, 1980)

SETI, Evolution and Human History Merged into a Mathematical Model

Abstract In this Chapter “SETI, Evolution and Human History Merged into a Mathematical Model” we propose a new mathematical model capable of merging Darwinian Evolution, Human History and SETI into a single mathematical scheme: (1) Darwinian Evolution over the last 3.5 billion years is defined as one particular realization of a certain stochastic process called Geometric Brownian Motion (GBM). This GBM yields the fluctuations in time of the number of species living on Earth. Its mean value curve is an increasing exponential curve, i.e. the exponential growth of Evolution. (2) In 2008 this author provided the statistical generalization of the Drake equation yielding the number N of communicating ET civilizations in the Galaxy. N was shown to follow the lognormal probability distribution. (3) We call “b-lognormals” those lognormals starting at any positive time b (“birth”) larger than zero. Then the exponential growth curve becomes the geometric locus of the peaks of a one-parameter family of b-lognormals: this is our way to re-define Cladistics. (4) b-lognormals may be also be interpreted as the lifespan of any living being (a cell, or an animal, a plant, a human, or even the historic lifetime of any civilization). Applying this new mathematical apparatus to Human History, leads to the discovery of the exponential progress between Ancient Greece and the current USA as the envelope of all b-lognormals of Western Civilizations over a period of 2500 years. (5) We then invoke Shannon’s Information Theory. The b-lognormals’ entropy turns out to be the index of “development level” reached by each historic civilization. We thus get a numerical estimate of the entropy difference between any two civilizations, like the Aztec-Spaniard difference in 1519. (6) In conclusion, we have derived a mathematical scheme capable of estimating how much more advanced than Humans an Alien Civilization will be when the SETI scientists will detect the first hints about ETs. Keywords Darwinian evolution · Statistical drake equation · Lognormal probability densities · Geometric Brownian Motion · Hilbert space · Entropy

© Springer Nature Switzerland AG 2020 C. Maccone, Evo-SETI, https://doi.org/10.1007/978-3-030-51931-5_3

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1 SETI and Darwinian Evolution Merged Mathematically 1.1 Introduction: The Drake Equation (1961) as the Foundation of SETI In 1961, the American astronomer Frank D. Drake tried to estimate the number N of communicating civilizations in the Milky Way galaxy by virtue of a simple equation now called the Drake equation. N was written as the product of seven factors, each a kind of filter, every one of which must be sizable for there to be a large number of civilizations: Ns, the number of stars in the Milky Way Galaxy; fp, the fraction of stars that have planetary systems; ne, the number of planets in a given system that are ecologically suitable for life; fl, the fraction of otherwise suitable planets on which life actually arises; fi, the fraction of inhabited planets on which an intelligent form of life evolves (as in Human History); fc, the fraction of planets inhabited by intelligent beings on which a communicative technical civilization develops (as we have it today); and fL, the fraction of planetary lifetime graced by a technical civilization (a totally unknown factor). Written out, the equation reads N = N s · f p · ne · f l · f i · f c · f L .

(1)

All the f ’s are fractions, having values between 0 and 1; they will pare down the large value of Ns. To derive N, we must estimate each of these quantities. We know a fair amount about the early factors in the equation, the number of stars and planetary systems. We know very little about the later factors, concerning the evolution of life, the evolution of intelligence or the lifetime of technical societies. In these cases, our estimates will be little better than guesses. It has to be said that the original formulation of (1) by Frank Drake in 1961 was slightly different, namely N = R ∗ · f p · ne · f l · f i · f c · L .

(2)

In (2), R* is the average rate of star formation per year in the Galaxy and L is the length of time for which civilizations in the Galaxy release detectable signals into space. However, the number of stars in the Galaxy, Ns, is related to the star formation rate R* by T Galaxy

Ns =

R ∗ (t)dt,

(3)

0

where T Galaxy is the age of the Galaxy. Assuming for simplicity that R* is constant in time, then (3) yields

1 SETI and Darwinian Evolution Merged Mathematically

N s = R ∗ · TGalaxy i.e.R ∗ =

115

Ns

,

(4)

L . TGalaxy

(5)

TGalaxy

that, inserted into (2), changes it into N = N s · f p · ne · f l · f i · f c · Then (5) becomes just (1) if one identifies fL =

L TGalaxy

(6)

as the fraction of planetary lifetime (as a part of the whole Galaxy existence T Galaxy ) graced by a technical civilization. In the 60 years that have elapsed since Drake proposed his equation, a number of scientists and writers have tried either to improve it or criticize it in many ways. For instance, Walters et al. [22] suggested inserting a new parameter in the equation taking interstellar colonization into account. Wallenhorst [21] tried to prove that there should be an upper limit of about 100 to the number N. Ksanfomality [7] again asked for more new factors to be inserted into the Drake equation, this time in order to make it compatible with the peculiarities of planets of Sun-like stars. Also, the ´ temporal aspect of the Drake equation was stressed by Cirkovi´ c [3]. However, while these authors were concerned with improving the Drake equation, others simply did not consider it useful and preferred to forget about it, like Burchell [2]. Also, it has been correctly pointed out that the habitable part of the Galaxy is probably much smaller than the entire volume of the Galaxy itself (the important relevant references are Gonzalez et al. [5], Lineweaver et al. [8] and Gonzalez [6]). For instance, it might be a sort of torus centred around the so-called ‘corotation circle’, i.e. a circle around the Galactic Bulge such that stars orbiting around the Bulge and within such a torus never fall inside the dangerous spiral arms of the Galaxy, where supernova explosions would probably fry any living organism before it could develop to the human level or beyond. Fortunately for Humans, the orbit of the Sun around the Bulge is just a circle staying within this torus for 5 billion years or more [1, 16]. In all cases, the final result about N has always been a sheer number, i.e. a positive integer number ranging from 1 to thousands or millions. This ‘integer or real number’ aspect of all variables making up the Drake equation is what this author regarded as ‘too simplistic’. He extended the Drake equation so as to embrace Statistics in his 2008 paper [9]. This paper was later published in Acta Astronautica [10], and more mathematical consequences were derived in Maccone [11] and [13].

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1.2 Statistical Drake Equation (2008) Consider Ns, the number of stars in the Milky Way Galaxy, i.e. the first independent variable in the Drake equation (1). Astronomers tell us that approximately there should be about 350 billion stars in the Galaxy. Of course, nobody has counted all the stars in the Galaxy! There are too many practical difficulties preventing us from doing so: just to name one, the dust clouds that do not allow us to see even the Galactic Bulge (central region of the Galaxy) in visible light, although we may ‘see it’ at radio frequencies like the famous neutral hydrogen line at 1420 MHz. Hence, it does not really make much sense to say that Ns = 350 × 109 , or similar fanciful exact integer numbers. Scientifically we say that the number of stars in the Galaxy is 350 billion plus or minus, say, 50 billions (or whatever values the astronomers may regard as more appropriate). It thus makes sense to REPLACE each of the seven independent variables in the Drake equation (1) by a mean value (350 billions, in the above example) plus or minus a certain standard deviation (50 billions, in the above example). By doing so, we moved a step ahead: we have abandoned the too-simplistic equation (1) and replaced it by something more sophisticated and scientifically serious: the statistical Drake equation. In other words, we have transformed the simplistic classical Drake equation (1) into a statistical tool capable of investigating a host of facts hardly known to us in detail. In other words still: (1) we replace each independent variable in (1) by a random variable, labelled Di (from Drake); (2) we assume the mean value of each Di to be the same numerical value previously attributed to the corresponding input variable in (1); (3) but now we also add a standard deviation σ Di on each side of this mean value, as provided by the knowledge obtained by scientists in the discipline covered by each Di . Having done so, we wonder: how can we find out the probability distribution for each Di ? For instance, shall that be a Gaussian, or what? This is a difficult question, for nobody knows, for instance, the probability distribution of the number of stars in the Galaxy, not to mention the probability distribution of the other six variables in the Drake equation (1). In 2008, however, this author found a way to get around this difficulty, as explained in the next section.

1.3 The Statistical Distribution of N Is Lognormal The solution to the problem of finding the analytical expression for the probability density function (pdf) of the positive random variable N is as follows: (1) Take the natural logs of both sides of the statistical Drake equation (1). This changes the product into a sum.

1 SETI and Darwinian Evolution Merged Mathematically

117

(2) The mean values and standard deviations of the logs of the random variables Di may all be expressed analytically in terms of the mean values and standard deviations of Di [9]. (3) The central limit theorem (CLT) of statistics, states that (loosely speaking) if you have a sum of independent random variables, each of which is arbitrarily distributed (hence, also including uniformly distributed), then, when the number of terms in the sum increases indefinitely (i.e. for a sum of random variables infinitely long) … the sum random variable approaches a Gaussian. (4) Thus, the ln(N) approaches a Gaussian. (5) Namely, N approaches the lognormal distribution (as discovered back in the 1870s by Sir Francis Galton). Table 1 shows the most important statistical properties of a lognormal. (6) The mean value and standard deviations of this lognormal distribution of N may be expressed analytically in terms of the mean values and standard deviations of the logs of Di already found previously, as shown in Table 1. Table 1 Summary of the properties of the lognormal distribution that applies to the random variable N = number of ET communicating civilizations in the Galaxy Random variable

N = number of communicating ET civilizations in Galaxy

Probability distribution

lognormal

pdf

Mean value

f N (n) =

− 1 √1 n 2πσ e

N  = eμ e

(ln(n)−μ)2 2σ2

(n ≥ 0)

σ2 2

Variance

 2  2 σ2N = e2μ eσ eσ − 1

Standard deviation All the moments, i.e. kth moment

 σ2 2 σ N = e μ e 2 eσ − 1  k 2 σ2 N = ekμ ek · 2

Mode (=abscissa of the lognormal peak)

n mode ≡ n peak = eμ e−σ

Value of the mode peak

f N (n mode ) =

Median (=fifty–fifty probability value for N)

Median = m = eμ  2  2 K3 σ +2 eσ − 1 3 = e

Skewness

2

√ 1 e−μ e 2πσ

σ2 2

(K 2 ) 2

Kurtosis

K4 (K 2 )2

Expression of μ in terms of the lower (ai ) and upper (bi ) limits of the Drake uniform input random variable Di

μ= 7

Expression of σ2 in terms of the lower (ai ) and upper (bi ) limits of the Drake uniform input random variable Di

2 σ 7=

2

2

2

= e4σ + 2e3σ + 3e2σ − 6

i=1 Yi 

2 i=1 σYi

= =

7 i=1

7 i=1

bi [ln(bi )−1]−ai [ln(ai )−1] bi −ai



1−

ai bi [ln(bi )−ln(ai )]2 (bi −ai )2



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For all the relevant mathematical proofs, more mathematical details and a few numerical examples of how the Statistical Drake Equation works, please see Maccone [10].

1.4 Darwinian Evolution as Exponential Increase of the Number of Living Species Consider now Darwinian Evolution. To assume that the number of species increased exponentially over 3.5 billion years of evolutionary time span is certainly a gross oversimplification of the real situation, as proven, for instance, by Rohde and Muller [18]. However, we will assume this exponential increase of the number of living species in time just for a moment in order to cast the theory into a mathematically simple and fruitful form. The introduction of Geometric Brownian Motion (GBM) in the next section of this chapter will solve this difficulty. In other words, we assume that 3.5 billion years ago there was on Earth only one living species, whereas now there may be (say) 50 million living species or more. Note that the actual number of species currently living on earth does not really matter as a number for us: we just want to stress the exponential character of the growth of species. Thus, we shall assume that the number of living species on Earth increases in time as E(t) (standing for ‘exponential in time’): E(t) = Ae Bt ,

(7)

where A and B are two positive constants that we will soon determine numerically. This assumption of ours is obviously in agreement with the classical Malthusian theory of population growth. However, it also is in line with the more recent ‘Big History’ viewpoint about the whole evolution of the Universe, from the Big Bang up to now, requesting that progress in evolution occurs faster and faster, so that only an exponential growth is compatible with the requirements that (7) approaches infinity for t → ∞ and all its time derivatives are exponentials too, apart from constant multiplicative factors. Let us now adopt the convention that the current epoch corresponds to the origin of the time axis, i.e. to the instant t = 0. This means that all the past epochs of Darwinian Evolution correspond to negative times, whereas the future ahead of us (including finding ETs) corresponds to positive times. Setting t = 0 in (7), we immediately find E(0) = A

(8)

proving that the constant A equals the number of living species on earth right now. We shall assume A = 50 million species = 50 × 107 species.

(9)

1 SETI and Darwinian Evolution Merged Mathematically

119

To also determine the constant B numerically, consider two values of the exponential (7) at two different instants t 1 and t 2 , with t 1 < t 2 , that is

E(t1 ) = Ae Bt1 , E(t2 ) = Ae Bt2 .

(10)

Dividing the lower equation by the upper one, A disappears and we are left with an equation in B only: E(t2 ) = e B(t2 −t1 ) . E(t1 )

(11)

ln(E(t2 )) − ln(E(t1 )) . t2 − t1

(12)

Solving this for B yields B=

We may now impose the initial condition stating that 3.5 billion years ago there was just one species on Earth, the first one (whether this was RNA is unimportant in the present simple mathematical formulation):

t1 = −3.5 × 109 years, E(t1 ) = 1 where ln(E(t1 )) = ln(1) = 0

(13)

The final condition is of course that today (t 2 = 0) the number of species equals A given by (9). Upon replacing both (9) and (13) into (12), the latter becomes: 

ln 5 × 107 1.605 × 10−16 ln(E(t2 )) = . =− B=− t1 −3.5 × 109 year sec

(14)

Having thus determined the numerical values of both A and B, the exponential in (7) is thus fully specified. This curve is plotted in Fig. 1 just over the last billion years, rather than over the full range between −3.5 billion years and now.

1.5 Introducing the ‘Darwin’ (D) Unit, Measuring the Amount of Evolution that a Given Species Reached In all sciences ‘to measure is to understand’. In physics and chemistry this is done by virtue of units such as the metre, second, kilogram, coulomb, etc. Hence, it appears useful to introduce a new unit measuring the degree of evolution that a certain species has reached at a certain time t of Darwinian Evolution, and the obvious name for such a new unit is the ‘Darwin’, denoted by a lower case ‘d’. For instance, if we adopt the exponential evolution curve described

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Fig. 1 Exponential curve representing the growing number of species on Earth up to now, without taking the well-known mass extinctions into any consideration at all

in the previous section, we might say that the dominant species on Earth right now (Humans) have reached an evolution level of 50 million Darwins. How many Darwins may have an alien civilization already reached? Certainly more than 50 millions, i.e. more than 50 Md, but we will not check out until SETI scientists will possibly detect the first extraterrestrial civilization. We are not going to discuss further this notion of measuring the ‘amount of evolution’, since we are aware that endless discussions might come out of it. However, it is clear to us that such a new measuring unit (and ways to measure it for different species) will sooner or later have to be introduced to make Evolution a fully quantitative science.

1.6 Darwinian Evolution Is just a Particular Realization of Geometric Brownian Motion in the Number of Living Species Consider again the exponential curve described in the previous section. The most frequent question that non-mathematically minded persons ask this author is: ‘then you do not take the mass extinctions into account’. The answer to this objection is that our exponential curve is just the mean value of a certain stochastic process that may run above and below that exponential in a totally unpredictable way. Such a stochastic process is called Geometric Brownian Motion (abbreviated GBM) and is described, for instance, at the web site: http://en.wikipedia.org/wiki/Geometric_Bro wnian_motion, from which Fig. 2 is taken. In other words, mass extinctions that occurred in the past are indeed taken into account as unpredictable fluctuations in the number of living species that occurred

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121

Fig. 2 GBM. Two particular realizations of the stochastic process called Geometric Brownian Motion (GBM) taken from the Wikipedia site http://en.wikipedia.org/wiki/Geometric_Brownian_ motion. Their mean values are the exponential (7) with different values of A and B for each shown stochastic process

in the particular realization of the GBM between −3.5 billion years and now. Hence, extinctions are ‘unpredictable vertical downfalls’ in that GBM plot that may indeed happen from time to time. Also notice that: (1) The particular realization of GBM occurred over the last 3.5 billion years is very much unknown to us in its numerical details, but … (2) We would not care either, inasmuch as the theory of stochastic processes only cares about such statistical quantities like the mean value and the standard deviation curves, that are deterministic curves in time with known equations.

2 GBM as the Key to Stochastic Evolution of All Kinds 2.1 The N(t) GBM as Stochastic Evolution On 8 January, 2012, this author came to realize that his Statistical Drake Equation, previously described is the special static case (i.e. ‘the picture’, so as to say) of a more general time-dependent statistical Drake equation (i.e. ‘the movie’, so as to say) that we study in this section. In other words, this result is a powerful generalization in time of all results described in sections: ‘SETI and Darwinian Evolution merged mathematically’ and ‘GBM as the key to stochastic evolution of all kinds’. This section is thus an introduction to a new, exciting mathematical model that one may call ‘Exponential Evolution in Time of the Statistical Drake Equation’.

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To be precise, the number N in the statistical Drake equation (1), yielding the number of extraterrestrial civilizations now existing and communicating in the Galaxy, is replaced in this section by a stochastic process N(t), jumping up and down in time like the number e raised to a Brownian motion, but actually in such a way that its mean value keeps increasing exponentially in time as N (t) = N0 eμt .

(15)

In (15), N 0 and μ are two constants with respect to the time variable t. Their meaning is, respectively: (1) N 0 is the number of ET communicating civilizations at time t = 0, namely ‘now’, if one decides to regard the positive times (t > 0) as the future history of the Galaxy ahead of us, and the negative times (t < 0) as the past history of the Galaxy. (2) μ is a positive (if the number of ET civilizations increases in time) or negative (if the number of ET civilizations decreases in time) parameter that we may call ‘the drift’. To fix the ideas, and to be optimistic, we shall suppose μ > 0. This evolution in time of N(t) is just what we expect to happen in the Galaxy, where the overall number N(t) of ET civilizations does probably increase (rather than decrease) in time because of the obvious technological evolution of each civilization. However, this N(t) scenario is a stochastic one, rather than a deterministic one, and certainly does not exclude temporary setbacks, like the end of civilizations due to causes as diverse as: (a) asteroid and comet impacts, (b) rogue planets or stars, arriving from somewhere and disrupting the gravitational stability of the planetary system, (c) supernova explosions that would ‘fry’ the entire nearby ET civilizations (think of AGN, the Active Nucleus Galaxies and ask: how many ET civilizations are dying in those galaxies right now?), (d) ET nuclear wars, and (e) possibly more causes of civilization end that we do not know about yet. Mathematically, we came to define the pdf of this exponentially increasing stochastic process N(t) as the lognormal N (t)_pdf(n, N0 , μ, σ, t)

=√

1

−

√ e 2πσ tn

2 2 ln(n)− ln N0 +μ t− σ2 t 2 σ2 t

for 0 ≤ n ≤ ∞.

(16)

2 GBM as the Key to Stochastic Evolution of All Kinds

123

It is easy to prove that this lognormal pdf obviously fulfills the normalization condition ∞ N (t)_pdf(n, N0 , μ, σ, t) dn 0

∞ =

√ 0

1

−

√ e 2π σ t n

2 2 ln(n)− ln N0 +μ t− σ2 t 2 σ2 t

dn = 1.

(17)

Also, the mean value of (16) indeed yields the exponential curve (15) ∞ n · N (t)_pdf(n, N0 , μ, σ, t) dn 0

∞ =

n√ 0

1

−

√ e 2π σ t n

2 2 ln(n)− ln N0 +μ t− σ2 t 2 σ2 t

dn = N0 eμt .

(18)

The proof of (17) and (18) is given in Appendix 11.A as the Maxima file ‘GBM_as_N_of_t_v33’ of Maccone [15]. Table 2 summarizes the main properties of GBM, namely of this N(t) stochastic process.

2.2 Our Statistical Drake Equation Is the Static Special Case of N(t) In this section, we prove the crucial fact that the lognormal pdf of our Statistical Drake Equation given in Table 1 is just ‘the picture’ case of the more general exponentially growing stochastic process N(t) (‘the movie’) having the lognormal pdf (16) as given in Table 2. To make things neat, let us denote by the subscript ‘GBM’ for both the μ and σ appearing in (16). The latter thus takes the form: N (t)_pdf(n, N0 , μGBM , σGBM , t) = √ 

−

×e

 σ2 t ln(n)− ln N0 +μGBM t− GBM 2 2 2 σGBM t

2

1

√ 2π σGBM t n

for 0 ≤ n ≤ ∞.

(19)

Similarly, let us denote by the subscript ‘Drake’ for both μ and σ appearing in the lognormal pdf given in the third line of Table 1 (this is also equation (1.B.56) of Maccone [15]), namely the pdf of our statistical Drake equation:

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Table 2 Summary of the properties of lognormal distribution that applies to the stochastic process N(t) = exponentially increasing number of ET communicating civilizations in the Galaxy, as well as the number of living species on earth over the last 3.5 billion years Stochastic process

N (t) =  1)N umber o f E T Civili zations (in S E T I ). 2)N umber o f Living Species (in Evolution).

Probability distribution

Lognormal distribution of the GBM

pdf

N (t)_pdf(n, N0 , μ, σ, t) = 2

√

1√ e 2πσ tn

−

ln N0 +μt σ2 t 2σ2 t

for n ≥ 0

All the moments, i.e. kth moment

N (t) = N0 eμt

 σ2N (t) = N02 e2μt eσ2t − 1  2 σ N (t) = N0 eμt eσ t − 1

2  σ2 t  k  N (t) = N0k ekμt e k −k 2

Mode (=abscissa of the lognormal peak)

n mode ≡ n peak = N0 eμt e−

Value of the mode peak

f N (t) (n mode ) =

Mean value Variance Standard deviation

Median (=fifty–fifty probability value for N(t)) Skewness

√ 1 √ e−μt 2πσ t

meadian =m =N0 eμt e−

3 (K 2 ) 2

K4 (K 2 )2

eσ

2t

σ2 t 2

 2  2 = eσ t + 2 eσ t − 1

K3

Kurtosis

N0

3σ2 t 2

2

2

2

= e4σ t + 2e3σ t + 3e2σ t − 6

Clearly, these two different GBM stochastic processes have different numerical values of N0 , μ and σ, but the equations are the same for both processes

lognormal_ pdf_ of_ Statistical_ Drake_ Eq(n, μDrake , σDrake ) (ln(n)−μ Drake )2 for 0≤n≤∞. 1 2 σ2 =√ e− 2π σDrake n

(20)

Now, a glance at (19) and (20) reveals that they can be made to coincide if and only if the two simultaneous equations hold 

2 2 σGBM t = σDrake , ln N0 + μGBM t −

2 σGBM t 2

= μDrake .

(21)

On the other hand, when we pass (so as to say) ‘from the movie to the picture’, the two σ must be the same thing, and so must be the two μ, that is, one must have:

2 GBM as the Key to Stochastic Evolution of All Kinds



σGBM = σDrake = σ, μGBM = μDrake = μ.

125

(22)

Checking thus the upper equation (22) against the upper equation (21), we are only left with t = 1

(23)

Hence, t = 1 is the correct numeric value of the time leading ‘from the movie to the picture’. Replacing this into the lower equation (21), and keeping in mind the upper equation (22), the lower equation (21) becomes ln N0 + μGBM −

σ2 = μDrake . 2

(24)

Since the two μ also must be the same because of the lower equation (22), then (24) further reduces to ln N0 −

σ2 = 0, 2

(25)

that is σ2

N0 = e 2

(26)

and the problem of ‘passing from the movie to the picture’ is completely solved. In conclusion, we have proven the following ‘movie to picture’ theorem: The stochastic process N(t) reduces to the random variable N if, and only if, one inserts ⎧ ⎪ t = 1, ⎪ ⎪ ⎨σ GBM = σDrake = σ, (27) μGBM = μDrake = μ, ⎪ ⎪ ⎪ 2 σ ⎩ N0 = e 2 into the lognormal probability density (16) of the stochastic process N(t).

2.3 GBM as the Key to Mathematics of Finance But what is this N(t) stochastic process reducing to the lognormal random variable N in the static case? Well, N(t) is no less than the famous GBM, of paramount importance in the mathematics of finance. In fact, in the so-called Black–Scholes models, N(t) is related to the log return of the stock price. Huge amounts of money

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all over the world are handled at Stock Exchanges according to the mathematics of the stochastic process N(t), that is differently denoted S t there (‘S’ from Stock). But we would not touch these topics here, since this chapter is about Evolution and SETI, rather than about stocks. We just content ourselves to have proven that the GBM used in the mathematics of finance is the same thing as the exponentially increasing process N(t) yielding the number of communicating ET civilizations in the Galaxy!

3 Darwinian Evolution Re-defined as a GBM in the Number of Living Species 3.1 A Concise Introduction to Cladistics and Cladograms Cladistics is the science describing when new forms of life developed in the course of Evolution. Cladistics is thus the science of lineages, i.e. phylogenetic trees, like the one shown for instance in Fig. 3, and it is today strongly based on computer codes, in turn based on high-level mathematics. Our innovative contribution to cladistics and cladograms like the one in Fig. 3 is to put the horizontal axis of time below them, and then realize that the cladograms branches are exponential functions of the time. In other words, these exponential arches are either increasing in time, or decreasing, or just staying constants (i.e. they are just horizontal lines, like the ones in Fig. 3), but the length of these exponential arches is as long as the species they represent survived during the course of evolution. This mathematical representation of the whole of evolution is: (1) Easy, inasmuch as exponential functions like (7) are the easiest possible functions in mathematics. (2) Clear, inasmuch as we know pretty well when a new species appeared in the course of evolution. (3) GBM-based, inasmuch as the exponential arches indeed are the mean values in time of the corresponding ‘unpredictable’ GBMs yielding the number of members of that species living at a certain time in evolution. At last, the study of the mathematical properties of GBMs is now open to scientists, rather than only to bankers and businessmen, as it happened in the last 48 years (1973–2021). Note that, in 1997, the Nobel Prize in Economics was assigned to Robert C.

Fig. 3 A horizontal cladogram (taken from http://en.wikipedia.org/wiki/Cladogram) with the ancestor (not named) to the left

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127

Merton and Myron Scholes (Fischer Black had already died in 1995) for their mathematical discoveries (Black–Scholes–Merton models) based on GBMs. Perhaps, new Nobel Prizes will be assigned for applying GBMs to evolution and astrobiology.

3.2 Cladistics: Namely the GBM Mean Exponential as the Locus of the Peaks of b-Lognormals Representing Each a Different Species Started by Evolution at Time t = b How is it possible to ‘match’ the GBM mean exponential curve with the lognormals appearing in the Statistical Drake Equation? Our answer to this question is ‘by letting the GBM mean exponential become the envelope of the b-lognormals representing the cladistic branches, i.e. the new species that were produced by evolution at different times as long as evolution unfolded’. Let us now have a look at Fig. 4. The envelope shown in Fig. 4 is not really an envelope in the strictly mathematical sense explained in calculus textbooks. However, it is ‘nearly the same thing in practice’ because it actually is the geometric locus of the peaks of all b-lognormals. We shall now explain this in detail. First of all, let us write down the equation of the b-lognormal, i.e. of the lognormal starting at any instant t = b (while ordinary lognormals all start only at zero); in other words, (t − b) replaces n in the first equation in Table 1: 

b_ lognormal(t, μ, σ, b) =

√

1 2πσ·(t−b)

holding for t > b and up to t = ∞.

e−

(ln(t−b)−μ)2 2 σ2

(28)

Then, notice that its peak falls at the abscissa p and ordinate P given by, respectively (as given by the 8th and 9th line in Table 1):

Fig. 4 Darwinian exponential as the envelope of b-lognormals. Each b-lognormal is a lognormal starting at a time (t = b = birth time) larger than zero and represents a different species ‘born’ at time b of Darwinian evolution

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p = b + eμ−σ = b_ lognormal_ peak_ abscissa, 2

P=

σ2 e 2 −μ

√

2πσ

= b_ lognormal_ peak_ ordinate.

(29)

Can we match the second equation (29) with the Darwinian exponential (7)? Yes, if we set at time t = p: 

A=

√1 , 2πσ σ2 2 −μ

e Bp = e

 ,

that is

A= Bp

√1 , 2πσ 2 = σ2 −

μ.

(30)

The last system of two equations may now be inverted, i.e. exactly solved with respect to μ and σ: 

1 σ = √2π , A μ = −Bp +

(31)

1 , 4πA2

showing that each b-lognormal in Fig. 4 (i.e. its μ and σ) is perfectly determined by the Darwinian exponential (namely by A and B) plus a precise value of the blognormal’s peak time p. In other words, this is a one-parameter (the parameter is p) family of curves that are all constrained between the time axis and the Darwinian exponential. Clearly, as long as one moves to higher values of p, the peaks of these curves become narrower and narrower and higher and higher, since the area under each b-lognormal always equals 1 (normalization condition).

3.3 Cladogram Branches Are Increasing, Decreasing or Stable (Horizontal) Exponential Arches as Functions of Time It is now possible to understand how cladograms shape up in our mathematical theory of evolution: they depart from the time axis at birth time (b) of the new species and then either: (1) Increase if the b-lognormal of the ith new species has (keeping in mind the convention pi < 0 for past events, i.e. events prior to now): ⎧ ⎨ Ai = ⎩B = i

√1 2πσi σi2 2

−μi pi

, > 0 that is μi >

σi2 . 2

(32)

(2) Decrease if the same b-lognormal has (keeping in mind the convention pi < 0 for past events):

3 Darwinian Evolution Re-defined as a GBM in the Number of Living Species

⎧ ⎨ Ai = ⎩B = i

√1 2πσi σi2 2

−μi pi

129

, < 0 that is μi
b. On the other hand, b-lognormals like (28) are infinite in time, i.e. spanning from t = b to t = + ∞, so one might immediately wonder how (28) might possibly represent a finite lifespan. Well, the answer to such a question will be given later in the next section, when we will introduce the notion of ‘death instant’ t = d as the intersection point between the tangent to (28) in its descending inflexion point and the time axis. At the moment, we content ourselves with studying some mathematical properties of the infinite b-lognormal pdf (28). This was done in a highly innovative editorial way in the author’s book entitled ‘Mathematical SETI’, Maccone [15]. In fact, the mathematical proof of each of the theorems proven there was hardly demonstrated line-by-line in the text. On the contrary, the hardest calculations were performed by aid of Maxima, the powerful computer algebra code (also called Macsyma) created by NASA and MIT in the 1960s and now freely downloadable from the web site http://maxima.sourceforge. net/.

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Hence, the reader may find them in the Maxima file ‘b_lognormals_inflexion_points_and_DEATH_time.wmx’ that is reprinted in Appendix 6.A. to the author’s 2012 book. From now on, we shall simply state the equation numbers in that Maxima file proving a certain result about b-lognormals, and the interested reader will then find the relevant proof by reading the corresponding Maxima command lines (‘i’ = input lines) and output lines (‘o’ = output lines). This way of proving ‘electronically’ the mathematical results simplifies things greatly, if compared with the ‘ordinary’ lengthy proofs of traditional books, and students and researchers will be able to download for free the corresponding Maxima symbolic manipulator from the site: http://maxima.sourceforge.net/.

4.2 Infinite b-Lognormals Again, a b-lognormal simply is a lognormal pdf starting at any positive value b > 0 (called ‘birth’) rather than at the origin. As such, a b-lognormal has the following equation in the independent variable t (time) and with the three independent parameters μ, σ and b, of which μ is a real number, while both σ and b are positive numbers: 

b_ lognormal(t, μ, σ, b) =

√ 1 e 2πσ(t−b)

2

− (ln(t−b)−μ) 2 σ2

holding for t > b and up to t = ∞.

(36)

This we call the infinite b-lognormal, meaning that it extends to the right up to infinity. Its main mathematical properties are basically the same as those of the ordinary lognormals starting at zero and given in Table 1, with only one exception: all formulae representing an abscissa have the same expression as for ordinary lognormals with a + b term added because of the right-shift of magnitude b. In other words, all infinite b-lognormals have the formulae given in Table 4 (a formal, analytical proof of all results in Table 4 can be found in Appendix 6.A of the author’s book ‘Mathematical SETI’, Maccone [15]).

4.3 From Infinite to Finite b-Lognormals: Defining the Death Time, d, as the Time Axis Intercept of the b-Lognormal Tangent Line at Senility s The b-lognormal extends up to t = + ∞ and this is in sharp contrast with the fact that every living being sooner or later dies at the finite time d (‘death’) such that 0 < b < d < ∞. We thus must somehow define this finite death time d in order to let the b-lognormals become a realistic mathematical model for the life-and-death of every living being.

4 Lifespans of Living Beings as b-Lognormals

133

Table 4 Properties of the b-lognormal distribution, namely the infinite b-lognormal distribution given by (36) Probability distribution pdf

Abscissa of the ascending inflexion point Ordinate of the ascending inflexion point Abscissa of the descending inflexion point Ordinate of the descending inflexion point Mean value Variance

b-lognormal, namely the infinite b-lognormal 2 1 1 − (ln(t−b)−μ) 2σ2 f b-lognormal (t; μ, σ, b) = √ e · (t ≥ b) 2πσ (t − b)

Adolescence ≡ a = b + e−

σ

√

σ2 +4 3σ2 − 2 2

f b-lognormal (adolescence) ≡ A = Senility ≡ s = b + e

σ

√

σ2 +4 3σ2 − 2 2

f b-lognormal (senility) ≡ S =

e

σ

e−

σ

+μ

√

σ2 +4 2

−μ+ √ e 2πσ

σ2 − 1 4 2

+μ

√

σ2 +4 −μ+ σ2 − 1 2 4 2 e

√

2πσ

σ2 2

b_lognormal = b + eμ e  2  2 2 σb-lognormal = e2μ eσ eσ − 1 σ2 2

 2 eσ − 1

Standard deviation

σb-lognormal = eμ e

Peak abscissa = mode

b-lognormal peak ≡ b-lognormalmode ≡ p = b + eμ e−σ = 2

b + eμ−σ

2

Peak ordinate = value of the mode peak

 f b-lognormal b-lognormalmode =

Median (=fifty–fifty probability abscissa)

Median = m = b + eμ

√1 2πσ

· e−μ · e

σ2 2

=

√1 2πσ

·e

σ2 2

−μ

These are both statistical and geometric properties of the pdf (36), whose importance will become evident later

We solved this problem by defining the death time t = d as the intercept point between the time axis and the straight line tangent to the b-lognormal at its descending inflexion point t = s, i.e. the tangent line to the lognormal curve at senility. And, from now on, we shall call finite b-lognormal any such truncated b-lognormal, ending just at t = d. This section is devoted to the calculation of the equation yielding the d point in terms of the b-lognormal’s μ and σ, and the whole procedure is described at the lines %i45 thru %o56 of the file ‘b-lognormals_inflexion_points_and_DEATH_time. wxm’ in Appendix 6.B of Maccone [15]. Let us start by recalling the simple formula yielding the equation of the straight line having an angular coefficient m and tangent to the curve y(t) at the point having the coordinates (t 0 , y0 ): y − y0 = m(t − t0 ).

(37)

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Then, the value of y0 clearly is the value of the b-lognormal at its senility time, given by the sixth line in Table 4, that is, rearranging: y0 =

e

σ

√

σ2 +4 2

σ2

e−μ+ 4 − 2 . √ 2πσ 1

(38)

On the other hand, the abscissa of the senility time t = s is given by the fifth line in Table 4, that is t0 = b + e

σ

√

2 σ2 +4 − 3σ2 2

+μ

.

(39)

Finally, we must find the expression of the angular coefficient m at the senility time, and this involves finding the b-lognormal’s derivative at senility. Then Maxima has no problem to find this, and the lines %i48 and %o48 show that one gets, after some rearranging √ m=−

2e

7 σ2 4

√

 σ√σ2 +4+8μ+2 4 σ 2 + 4 − σ e− . √ 2 4 πσ

(40)

Inserting (38), (39) and (40) into (37) one obtains the equation of the desired straight line tangent to the b-lognormal at senility:

y−

e

σ

√

σ2 +4 2

2

−μ+ σ4 − 21

e √ 2πσ

√

2e

7 σ2 4

√

=−

 √ 2 σ σ2 +4 − 3σ2 +μ 2 × t −e −b .

 σ√σ2 +4+8μ+2 4 σ2 + 4 − σ e− √ 2 4 πσ (41)

In order to find the abscissa of the death point t = d, we just need to insert y = 0 into Eq. (41) and solve for the resulting t. Then Maxima yields at first a rather complicated result (%o52). However, keeping in mind that the term in b must obviously appear ‘alone’ in the final equation since the b-lognormal is only an ordinary lognormal shifted to make it start at b, the way to further simplify (41) becomes obvious, and the final result simply is √ d =b+

2 σ√σ2 +4 3σ2 σ2 + 4 + σ e 2 − 2 +μ 4

.

This is the ‘death time’ of all living beings born at any time b .

(42)

4 Lifespans of Living Beings as b-Lognormals

135

4.4 Terminology About Various Time Instants Related to a Lifetime The reader is now asked to look carefully at Fig. 5 to familiarize with mathematical notations and their meaning describing the lifetime of all living beings: Obvious are the definitions of the instants of: (1) (2) (3) (4) (5)

birth (b = starting point on the time axis) adolescence (a = ascending inflexion abscissa, with ordinate A) peak (p = maximum point abscissa, with ordinate P) senility (s = descending inflexion abscissa, with ordinate S) and death (d = death abscissa = intercept between the time axis and the straight line tangent to the b-lognormal at the descending inflexion point).

Fig. 5 Lifetime of all living beings, i.e. finite b-lognormal: definitions of the basic instants of birth (b = starting point on the time axis), adolescence (a = ascending inflexion abscissa, with ordinate A), peak (p = maximum point abscissa, with ordinate P), senility (s = descending inflexion abscissa, with ordinate S) and death (d = death abscissa = intercept between the time axis and the straight line tangent to the b-lognormal at the descending inflexion point). Also defined are the obvious single-time-step-spanning segments called childhood (C = a − b), youth (Y = p − a), maturity (M = s − p), decline (D = d − s). In addition, also defined are the multiple-time-step-spanning segments of the all-covering lifetime (L = d − b), vitality (V = s − b) (i.e. lifetime minus decline) and fertility (F = s − a) (i.e. adolescence to senility)

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4.5 Terminology About Various Time Spans Related to a Lifetime Also defined in Fig. 5 are the obvious time segments called: (1) (2) (3) (4) (5) (6) (7)

Childhood (C = a − b), Youth (Y = p − a), Maturity (M = s − p), Decline (D = d − s), Fertility (F = s − a), Vitality (V = s − b), Lifetime (L = d − b).

Then, from all these definitions and from the mathematical properties of the b-lognormals listed in Table 4, one obtains immediately the following equations: −σ

√

Childhood ≡ C = a − b = e

Youth ≡ Y = p − a = eμ−σ − e− 2

Maturity ≡ M = s − p = e √ Decline ≡ D = d − s = −e

σ

√

2 σ2 +4 − 3 2σ 2

+μ

=e

σ

√

σ

2 σ2 +4 − 3 2σ 2

σ

√

2 σ2 +4 − 3σ2 2

σ2 + 4 + σ

2 σ2 +4 − 3 2σ 2

√

+μ

σ

+μ

2 σ2 +4 − 3σ2 2

+μ

σ

√

σ2 +4

3 σ2

Vitality ≡ V = s − b = e

σ

,

2 e

σ

√

− e−

σ

(45)

2 σ2 +4 − 3 2σ 2

√4  σ2 + 4 + σ

+μ

(44)

2

√

+μ

.

2 σ2 +4 − 3 2σ 2

(46) +μ

(47) √

2 σ2 +4 − 3 2σ 2

+μ

Lifetime = L = d − b √ 2 σ√σ2 +4 3 σ2 σ2 + 4 + σ e 2 − 2 +μ . = 4 Obviously one also has

+μ

(43)

− eμ−σ .

2

Fertility ≡ F = s − a = e 2 − 2  √  2 σ σ2 + 4 − 3 2σ +μ =2e sin h , 2

,

,

(48)

(49)

4 Lifespans of Living Beings as b-Lognormals

137

Lifetime = Vitality + Decline = (s − b) + (d − s) √ 2 σ√σ2 +4 3 σ2 σ2 + 4 + σ e 2 − 2 +μ . =d −b = 4

(50)

as one may check analytically by adding (48) and (46), and checking the result against (49). In addition, dividing (46) by (48), all exponentials disappear and one obtains the important new equation σ Decline = Vitality

√  σ2 + 4 + σ 2

.

(51)

This we shall use later in the section ‘mathematical history of civilizations’ in connection with the ‘golden ratio’ and ‘golden b-lognormals’.

4.6 Normalizing to One All the Finite b-Lognormals Finite b-lognormals are positive functions of time, as requested for any pdf, but they are not normalized to one yet, as it is also demanded for any pdf. This is because: (1) If one computes the integral of the b-lognormal (36) between birth b and senility s one obtains

s b_lognormal(t, μ, σ, b)dt b eσ

√

2 σ2 +4 − 3 2σ 2



=

√ b

=

+μ

1 + 2

1 2πσ(t − b)

e

−(ln(t−b)−μ)2 2 σ2

 √ √  erf 42 σ2 + 4 − 3 σ 2

,

dt

(52)

where erf (x) is the well-known error function of probability and statistics, defined by the integral 2 erf(x) = √ π

x 0

e−z dz. 2

(53)

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Notice that, during the integration in (52), the independent variable μ disappeared, leaving a result depending on σ only. We shall not prove (52) here: the proof can be found in Appendix 6.B of Maccone [15], lines %i78 through %o79. (2) If we add to (52) the integral of the descending straight line tangent to the blognormal at s, taken between s (given by the fifth line in Table 4) and d (given by (42)), we obtain

√

d

σ2 +4+σ

b+

2

σ

√

e



σ2 +4 3 σ2 − 2 +μ 2

4

y_ from_ eq._ (36)dt = s

√ y_ from_ eq._ (36)dt =

b+e

σ

√

σ2 +4 3 σ2 − 2 +μ 2

 3 σ√σ2 +4 5σ2 1 σ2 + 4 + σ e 4 − 4 − 2 . 5√ 22 π

(54)

Once again μ disappeared, leaving a result depending on σ only. Again, we shall not prove (54) here: the proof can be found in Appendix 6.A of Maccone [15], lines (%i85) through (%087). (3) In conclusion, adding (52) and (54), one gets the area under the finite b-lognormal (from b to d)

Area_ under_ FINITE_ b-lognormal d =

FINITE_ b-lognormal dt = K (σ)

(55)

b

with √

 3 σ√σ2 +4 5σ2 1 − 4 −2 2+4+σ e 4 σ 1 K (σ) = + 5√ 2 22 π  √ √  erf 42 σ2 + 4 − 3 σ . + 2

(56)

In the practice, it will be sufficient to compute the numeric value of K(σ) for a given σ and divide the corresponding finite b-lognormal by this value to have it normalized to one.

4 Lifespans of Living Beings as b-Lognormals

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4.7 Finding the b-Lognormals Given b and Two Out of the Four a, p, s, d The question is now: having introduced the five points b, a, p, s, d, do some equations exist enabling one to determine the b-lognormal’s μ and σ in terms of the birth time b (supposed to be always known) and any two more points out of the remaining four (a, p, s and d)? This author was able to discover several such pairs of equations, yielding μ and σ exactly (and not as numeric approximations) and they are all listed in Table 5. The mathematical proofs are given in Appendix 6.B of Maccone [15], and will not be repeated here. The most important out of all these equations are our brandnew history formulae, given by the two equations: 

d−s √ , σ = 4 √d−b s−b

μ = ln(s − b) +

2s 2 −(3d+b)s+d 2 +bd . (d−b)s−bd+b2

(57)

Essentially, these two equations allow us to find a b-lognormal when its birth, senility and death times are given. This is precisely what happens in the study of human history, since we certainly know when a past civilization was born (for instance when a new town was founded and later became the capital of a new empire), and when it died (because of war, usually). Less precisely we may know the time when its decline began (after reaching its peak), which is the s appearing in the fifth line of Table 4. However, if one manages to find that out in history books, then the b-lognormal (36) is fully determined by our history formulae (57).

5 Golden Ratios and Golden b-Lognormals 5.1 Is σ Always Smaller Than 1? So far, we have derived a number of properties of the b-lognormals given by (36) and representing the life of a living being. However, one question remains: is there any specific reason why σ should be smaller or larger than one? More precisely, while we know σ to be necessarily positive, no ‘plausible’ reason seems to exist for it to be smaller than one, as it appears to be numerically in majority of life forms. To explore this topic a little more, consider a trivial rectangular triangle having catheti equal to 2 and σ, respectively. √ Owing to the well-known Pythagorean theorem, the hypotenuse obviously equals σ 2 + 4. Since the hypotenuse always is longer than any of the catheti, we conclude that  σ2 + 4 > σ.

(58)

No exact formula exists, only numeric approximations

No exact formula exists, only numeric approximations

  ⎧ ⎪ ln s−b ⎪ p−b ⎪ ⎪σ=  ⎨   1−ln s−b p−b ⎪ ⎪  ⎪ ⎪ 2 2 ⎩ μ = ln(s − b) − σ σ2 +4 + 3σ2

–

p

  ⎧ p−b ⎪ ln a−b ⎪ ⎪ ⎪σ=  ⎨   p−b 1+ln a−b ⎪ ⎪  ⎪ ⎪ 2 2 ⎩ μ = ln(a − b) + σ σ2 +4 + 3σ2

⎧ d−s ⎪ ⎨ σ = √d−b√s−b History formulae 2 2 ⎪ μ = ln(s − b) + 2s +(−3d−b)s+d +bd ⎩ (d−b)s−bd+b2

–

  ⎧ ⎪ ln s−b ⎪ p−b ⎪ ⎪  ⎨σ=   1−ln s−b p−b ⎪ ⎪  ⎪ ⎪ 2 2 ⎩ μ = ln(s − b) − σ σ2 +4 + 3σ2

  √ 2 √   σ= 2 ln √s−b +1−1 s−b ⎪ ⎪ ⎪ 2 ⎩ μ = ln[(a−b)(s−b)] + 3σ 2 2

s

⎧ ⎪ ⎪ ⎪ ⎨

(i.e. finding both its μ and σ) given the birth time, b, and any two out of the four instants a = adolescence, p = peak, s = senility, d = death

d

s

  ⎧ p−b ⎪ ln a−b ⎪ ⎪ ⎪  ⎨σ=   p−b 1+ln a−b ⎪ ⎪  ⎪ ⎪ 2 2 ⎩ μ = ln(a − b) + σ σ2 +4 + 3σ2 ⎧   √ 2 ⎪  ⎪ ⎪ ⎨ σ = √2  ln √s−b +1−1 s−b ⎪ ⎪ ⎪ 2 ⎩ μ = ln[(a−b)(s−b)] + 3σ 2 2

a

p

a

–

Given

Table 5 Finding the b-lognormal d

–

⎧ d−s ⎪ ⎨ σ = √d−b√s−b 2 2 ⎪ ⎩ μ = ln(s − b) + 2s +(−3d−b)s+d +bd (d−b)s−bd+b2 History formulae

No exact formula exists, only numeric approximations

No exact formula exists, only numeric approximations

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5 Golden Ratios and Golden b-Lognormals

141

Now insert (58) into (51). The result is Decline = Vitality

σ

√

σ2 + 4 + σ 2

 >

σ (σ + σ ) σ (2σ ) 2σ 2 = = = σ 2. 2 2 2

(59)

Since all variables in this inequality are positive, we may rewrite it as Decline > σ2 · Vitality.

(60)

Now, in the majority of known life forms, it appears that the vitality time (i.e. the time between birth b and senility s), i.e. (s − b) is longer, or much longer than the decline time (i.e. the time between senility s and death d, i.e. (d − s)). Thus, the only way to let (60) apply to biological reality is to conclude that it must be σ2 < 1 or σ2  1

(61)

from which one finally infers (for all life forms known to humans) 0 < σ < 1.

(62)

Actually, in Section ‘Extrapolating history into the past: Aztecs’ of this chapter and in Chap. 7 of Maccone [15] the numerical value of σ is estimated for b-lognormals of the most important historic Western Civilizations (Ancient Greece, Ancient Rome, Italian Renaissance, Portuguese Empire, Spanish Empire, French Empire, British Empire and finally American (USA) Empire), and in all cases the numerical value of σ turned out to be smaller than 1. This will be re-proven here in Section ‘Extrapolating history into the past: Aztecs’ as we just said. Hence, one would tend to think that this 0 < σ < 1 result must be a ‘law of nature’ of some kind, though we cannot offer any better proof. There might, however, be ‘pathological cases’ of forms of life for which σ > 1 and so their decline would be larger or much larger than their vitality: just think of some science fiction movies like Star Wars, where some living being declares to be 900 years old or more … Anyway, the dividing line between ‘good’ and ‘bad’ values of σ seems to be the σ = 1 case. Is this case significant? Yes, very much, as we discover in the next section.

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5.2 Golden Ratios and Golden b-Lognormals If one lets σ = 1 into (51) one obtains σ

√  σ2 + 4 + σ

√  1 5+1

Decline = = Vitality 2 2 = golden ratio = 1.6180339887 . . . = φ.

√ 1+ 5 = 2 (63)

This is the famous ‘golden ratio’, hailed by artists, architects and mathematicians as aesthetically pleasing for over 2000 years. In the Renaissance (1509), the Italian, Luca Pacioli (1445–1517) wrote a book about it by the Latin title of ‘De Divina Proportione’ (The Divine Proportion), with illustrations by Leonardo Da Vinci. Hence, let us go back to (63). We now wish to prove that the following ‘divine proportion’ holds among Lifetime, Vitality and Decline (but only for those life forms having σ = 1, of course): √ Lifetime Decline 1+ 5 = = Golden_ Ratio ≡ φ ≡ Decline Vitality 2 = 1.618 . . . .

(64)

For the proof, admit for a moment that (64) holds good. Then, because of (50), the supposed (64) may be rewritten as (Table 6) Vitality + Decline Vitality Lifetime = = +1 Decline Decline Decline 1 1 = Decline + 1 = + 1. φ Vitality

φ=

(65)

Thus, if this is correct, we reach the conclusion that φ must fulfill the equation φ=

1 + 1 that is φ2 − φ − 1 = 0. φ

(66)

Solving this quadratic equation in φ yields φ=

−(−1) ±

√ 2

1 − 4(−1)

√ 1± 5 = . 2

(67)

Discarding the negative root in (67) (since the ratio of positive quantities may only yield a new positive quantity) leaves the positive root only, and this is just the golden ratio appearing in (63). Thus, we met with no contradiction in assuming the ‘divine proportion’ (64) to be true, and so it is true indeed.

5 Golden Ratios and Golden b-Lognormals

143

Table 6 Golden b-lognormal distribution Probability distribution

Golden b-lognormal

pdf

f Golden_ b-lognormal (t; μ, b) =

Abscissa of the ascending inflexion point Ordinate of the ascending inflexion point Abscissa of the descending inflexion point Ordinate of the descending inflexion point

Adolescence ≡ a = b + e

(ln(t−b)−μ) 1 2 √1 e− 2π (t−b)

√

√

d =b+

√ 1+ 5

4 e−μ− √ 2π

5−3 2

f Golden_ b-lognormal (Senility) ≡ S =

Abscissa of the death point

(t ≥ b)

√ μ− 3+2 5

f Golden_ b-lognormal (adolescence) ≡ A = Senility ≡ s = b + eμ+

2

√ 1− 5

4 e−μ− √ 2π

√ 2 5−3 5+1 eμ+ 2

4 1

Golden_ b-lognormal = b + eμ+ 2

Mean value Variance Standard deviation Peak abscissa = mode Peak ordinate = value of the mode peak Median (=fifty–fifty probability value) Skewness

2 2μ+1 (e − 1) σGolden_ b-lognormal = e 1√ σGolden_ b-lognormal = eμ+ 2 e − 1

Golden_ b-lognormalpeak ≡ Golden_ b-lognormalmode ≡ p = b + eμ−1

 1 f Golden_ b-lognormal Golden_ b-lognormalmode = √1 e 2 −μ 2π

Meadian = m = b + eμ K3

3 (K 2 ) 2

Kurtosis

K4 (K 2 )2

√ = (e + 2) e − 1 = 6.185 . . . = e4 + 2e3 + 3e2 − 6 = 110.936 . . .

I.e. the b-lognormal having σ = 1, and its statistical properties

As a consequence, it appears quite natural to call golden b-lognormal the particular case σ = 1 of (36), that is 

golden_ b_ lognormal(t, μ, b) =

(ln(t−b)−μ) √ 1 e− 2 2π(t−b)

holding for t > b and up to t = ∞.

2

(68)

This is a ‘new’ statistical distribution, whose main statistical properties are listed in Table 5, of course derived by setting σ = 1 into the corresponding entries of Table 4. Actually, rather than being only a single curve, (68) is a one-parameter family of curves in the (t, Golden_b-lognormal) plane, the parameter being μ. One is thus led to wonder what properties might this family of Golden b-lognormals possibly have. Then, this author discovered a simple theorem: all the golden b-lognormals (68) have

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Fig. 6 The geometric locus of the peaks of all golden b-lognormals (in the above diagram starting all at b = 2) as the parameter μ takes on all positive values (0 ≤ μ ≤ ∞), is the equilateral hyperbola given by (69)

their peaks lying on the equilateral hyperbola of equation Golden_ b-lognormal_ PEAK_ LOCUS(t, b) = √

1 . √ 2π e(t − b)

(69)

The proof is easy: just solve for μ the equation (line 11 in Table 5) yielding the peak abscissa of (68). The result is μ = ln( p − b) + 1.

(70)

Inserting (70) into the expression for the peak height P given in line 12 of Table 5, (69) is found, and the theorem is thus proven. Figure 6 shows immediately the equilateral hyperbola (69) for the case b = 2. But all these considerations about the golden ratio and the golden b-lognormals appear to be only an iceberg’s tip if one thinks of the many known results relating the golden ratio to the Fibonacci numbers, Lucas numbers, and so on. Hence, much more work is needed certainly in this new field we have uncovered.

6 Mathematical History of Civilizations 6.1 Civilizations Unfolding in Time as b-Lognormals Centuries of human history on Earth should have taught us something. Basically, civilizations are born, fight against each other and ‘die’, merging, however, with newer civilizations.

6 Mathematical History of Civilizations

145

To cast all this in terms of mathematical equations is hard. The reason nobody has done so is because the task is so daunting. Indeed, no course on ‘Mathematical History’ is taught at any university in the world. In this section, we will have a stab at this. Our idea is simple: any civilization is born, reaches a peak, then declines … just like a b-lognormal!

6.2 Eight Examples of Western Historic Civilizations as Finite b-Lognormals We now offer eight examples of such a view: the historic development of the civilizations of: (1) (2) (3) (4) (5) (6) (7) (8)

Ancient Greece Ancient Rome Renaissance Italy Portuguese Empire Spanish Empire French Empire British Empire American (USA) Empire.

Other historic empires (for instance the Dutch, German, Russian, Chinese, and Japanese ones, not to mention the Aztec and Incas Empires, or the Ancient ones, like the Egyptian, Persian, Parthian, or the medieval Mongol Empire) should certainly be added to such a picture, but we regret we do not have the time to carry on those studies in this chapter. Those historic-mathematical studies will be made at a later stage of development of this new research field that, in our view, is ‘Mathematical History’: the mathematical view of human history based on b-lognormal probability distributions. To summarize this section’s content, for each one of the eight civilizations listed above, we define: (a) Birth b, namely the year when that civilization was supposed to be ‘born’, even if only approximately in time. (b) Senility s, namely the year of an historic event that marked the beginning of the decline of that civilization. (c) Death d, namely the year when an historic event marked the ‘official passing away’ of that civilization from history. Then, consider the two equations (57). For each civilization, these two equations allow us to compute both μ and σ in terms of the three assigned numbers (b, s, d). As a consequence, the time of the given civilization peak is found immediately from the upper equation (29), that is Peak_ time = abscissa_ of_ the_ maximum = p

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= b + eμ−σ . 2

(71)

Also, we can then write down the equation of the corresponding b-lognormal immediately. The plot of this function of time gives a clear picture of the historic development of that civilization, though, to save space, we prefer not to reproduce here the above eight b-lognormals separately. Inserting the peak time (71) into (36), the peak ordinate of the civilization is found, namely ‘how civilized that civilization was at its peak,’ and this is explicitly given by the lower equation (29), namely: σ2

e 2 −μ . peak_ ordinate = P = √ 2πσ

(72)

Table 7 summarizes the three input data (b, s, d) drawn by the author from history textbooks, and then the two output data (p, P) of that Civilization’s peak, namely its best legacy to other subsequent Civilizations.

6.3 Plotting All b-Lognormals Together and Finding the Trends Having determined the b-lognormal for each civilization we wish to study, the time is ripe to plot all of them together and ‘see what the trends are’. This is done in Fig. 7. We immediately notice some trends: (1) The first two civilizations in time (Greece and Rome) are separated from the six modern ones by a large, 1000 years gap. This is of course the Middle Ages, i.e. the Dark Ages that hampered the development of Western Civilization for about 1000 years. Carl Sagan said, ‘the millennium gap in the middle of the diagram represents a poignant lost opportunity for the human species’, Sagan [19]. (2) While the first two civilizations of Greece and Rome lasted more than 600 years each, all modern civilizations lasted much less: 500 years at most, but really less, or much less indeed. (3) Since b-lognormals are pdfs, the area under each b-lognormal must be the same, i.e. just 1 (normalization condition). Thus, the shorter a civilization lives, the highest its peak must be! This is obvious from Fig. 7: Greece and Rome lasted so long, and their peak was so much smaller than the British or the American peak! (4) In other words, our theory accounts for the ‘higher level of the more recent historic civilizations’ in a natural fashion, with no need to introduce further free parameters. Not a small result, we think. (5) All these remarks lead to the Appendix 7.A file in Maccone [15] and the Figures therewith, starting with Fig. 7 hereafter.

1250 Frederick II dies. Middle Ages end. Free Italian towns

1419 Madeira island discovered

1492 Columbus discovers America

1524 Verrazano first in New York bay

1588 Spanish Armada Defeated

Renaissance Italy

Portugal

Spain

France

Britain

1962 Algeria lost, as most colonies

1898 Last colonies lost to the USA

1999 Last colony Macau lost

1660 1600 Bruno burned, 1642 Galileo dies. 1667 Cimento Academy Shut

476 AD Western Roman Empire ends. Dark Ages start

30 BC Cleopatra’s death: last Hellenistic queen

d = Death time

1914 World War One won at 1973 The UK joins a high cost European EEC

1815 Napoleon defeated at Waterloo

1805 Spanish fleet lost at Trafalgar

1822 Brazil independent, colonies retained

1564 Council of Trent. Tough Catholic and Spanish rule

753 BC Rome founded. 235 AD Military Anarchy Italy seized by Romans starts. Rome not capital any by 270 BC more

Ancient Rome

323 BC Alexander the Great’s death. Hellenism starts

600 BC Mediterranean Greek coastal expansion

s = Senility time

Ancient Greece

b = Birth time 2.488 × 10−3

p = Peak ordinate

4.279 × 10−3

5.938 × 10−3

3.431 × 10−3

5.749 × 10−3

(continued)

1868 Victorian Age. Science: 8.447 × 10−3 Faraday and Maxwell

1732 French Canada and India conquest tried

1741 California to be settled by Spain, 1759–76

1716 Black slave trade to Brazil at its peak

1497 Renaissance art and architecture. Science. Copernican revolution

59 AD Christianity preached 2.193 × 10−3 in Rome by Saints Peter and Paul against slavery

434 BC Pericles’ Age. Democracy peak. Arts and science peak

p = Peak time

Table 7 Finding the b-lognormals of eight among the most important civilizations of the Western world: Ancient Greece, Ancient Rome, Renaissance Italy, Portugal, Spain, France, Britain and the USA

6 Mathematical History of Civilizations 147

b = Birth time

s = Senility time

1898 Philippines, Cuba, 2001 9/11 terrorist attacks Puerto Rico seized

2050? Will the USA yield to China?

d = Death time 1973 Moon landings, 1969–72

p = Peak time 0.013

p = Peak ordinate

For each such civilization three input dates are assigned on the basis of historic facts: (1) the birth time, b; (2) the senility time, s, i.e. the time when the decline began, and (3) the death time, d, when the civilization reached a formal end. From these three inputs and the two equations (57) the b-lognormal of each civilization may be computed. As a result, that civilization’s peak is found, as shown in the last two columns. In general, this peak time turns out to be in agreement with the main historical facts

USA

Table 7 (continued)

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6 Mathematical History of Civilizations

149

Fig. 7 Showing the b-lognormals of eight civilizations in Western history, with two exponential envelopes for them

6.4 b-Lognormals of Alien Civilizations So much about the past. But what about the future? What are the b-lognormals of ET civilizations in this Galaxy? Nobody knows, of course. And nobody will know as long as the SETI scientists are unable to detect the first signs of an extraterrestrial civilization. Science fiction fans, however, might take pleasure in casting the Star Trek timeline into the mathematical language of b-lognormals. In this regard, interesting is ‘The Star Trek Chronology’, by Okuda and Okuda [17]. Also, the interested readers should get a copy of the great book by Finney and Jones [4]. This book is ‘revolutionary’, inasmuch as it re-reads the history of many human past civilizations with the glasses of the new science of SETI. We learned a lot from this book, but … no mathematics is there, just words. Our achievement was to “convert that book into equations”.

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7 Extrapolating History into the Past: Aztecs 7.1 Aztecs–Spaniards as an Example of Two Suddenly Clashing Civilizations with Large Technology Gap The only example we know for sure about two suddenly clashing civilizations with very different technological levels comes from human history. That was in 1519, when the Spaniard Hernán Cortés, with 600 men, 15 horsemen, 15 cannons and hundreds of indigenous carriers and warriors, was able to subdue the Aztec empire of Montezuma II, numbering some 20 million people. How was that possible? Well, we claim that basically there was a psychological breakdown in the Aztecs due to their obvious technological inferiority to the Spaniards, causing the Aztecs to regard the Spaniards as ‘Semi-Gods’, or ‘Gods’. We also claim that this is precisely what might happen to humans when they meet for the first time with a much more technologically advanced alien civilization in the Galaxy: humans might be shocked and paralysed by Alien superiority, thus simply surrendering to alien will. We also claim, however, that this human–alien sudden clash might be somehow softened were humans able to make a mathematical estimate of how much more advanced than us aliens will be. This mathematical theory of the technological civilization level is now developed in this section with a reference to the Aztecs–Spaniards example.

7.2 ‘Virtual Aztecs’ Method to Find the ‘True Aztecs’ b-Lognormal First of all, this author has developed a mathematical procedure to correctly locate the b-lognormal of past human civilizations in time. Consider the Aztec–Spaniard case: how much were the Spaniards more technologically developed than the Aztecs? Well, we claim that the answer to this question comes from the consideration of wheels. The use of wheels was unknown to the Aztecs. However, although they did not develop the wheel proper, the Olmec and certain other western hemisphere cultures seem to have approached it, as wheel-like worked stones have been found on objects identified as children’s toys. This is just the point: we assume that the Aztecs ‘were on the verge’ of discovering wheels when the Spaniards arrived in 1519. But then, when had wheels been discovered by the Asian–European civilizations? Evidence of wheeled vehicles appears from the mid-4th millennium BC, nearsimultaneously in Mesopotamia, the Northern Caucasus (Maykop culture) and Central Europe, so the question of which culture originally invented the wheeled vehicle remains unresolved and under debate.

7 Extrapolating History into the Past: Aztecs

151

The earliest well-dated depiction of a wheeled vehicle (a wagon–four wheels, two axles), is on the Bronocic pot, a ca. 3500–3350 B.C. clay pot excavated in a Funnelbeaker culture settlement in southern Poland. The wheeled vehicle spread from the area of its first occurrence (Mesopotamia, Caucasus, Balkans, Central Europe) across Eurasia, reaching the Indus Valley by the 3rd millennium BC. During the 2nd millennium BC, the spoke-wheeled chariot spread at an increased pace, reaching both China and Scandinavia by 1200 B.C. In China, the wheel was certainly present with the adoption of the chariot in ca. 1200 B.C. To fix the numbers, we shall thus assume that wheels had been discovered by the Asian–Europeans about 3500 B.C. Hence, summing 3500 plus 1519 (when the wheelless Aztecs clashed against the wheel-aware Spaniards), we obtain about 5000 years of technological difference of level among these two civilizations. And 5000 years means 50 centuries, and not just ‘a few centuries’ of Aztecs inferiority, as historians having no mathematical background have superficially claimed in the past: our blognormal theory is quantitatively much more precise than just ‘words’! However, let us now extend into the past, up to 3800 B.C., the older diagram shown in Fig. 7. Adding the Greece-to-Spain b-lognormal (shown in Fig. 8), the newer, resulting diagram extending to 3800 B.C. is shown in Fig. 9. (1) The Ancient-Greece-peak (434 BC) to Britain’s peak (1868) exponential, namely the dash-dot black curve. (2) The Ancient-Greece-peak (434 BC) to USA peak (1973) exponential, namely the solid black curve. (3) The Ancient-Greece-peak (434 BC) to Spain peak (1741) exponential, namely the dot–dot black curve. The Greece-to-Spain exponential was introduced since it is needed to understand the clash between the Aztecs and the Spaniards (1519–1521), as described by the ‘Virtual Aztec’ b-lognormal, going back 50 centuries before 1519 (see Fig. 9). In Fig. 9, the virtual Aztec b-lognormal is the b-lognormal peaking at the time in the past when the Western civilizations discovered the wheel, i.e. about 3500 B.C. in Mesopotamia, Southern Caucasus and Central Europe. This b-lognormal is the dash–dash black curve in Fig. 9. The Aztecs started their expansion in central Mexico in 1325, so when Cortez arrived in 1519 they were a civilization 1519 − 1325 = 194 years old. Reporting this 194 years lapse before the year 3500 B.C., we find that the virtual Aztecs had been ‘born’ 194 years earlier, namely in 3694 BC, which is thus the b-value of the virtual Aztec b-lognormal bV A = −3694 .

(73)

Then we have to find the b-lognornal itself, i.e. its μVA and σVA . In other words, we have to find μVA and σVA knowing only the two peak coordinates, pVA = −3500 and PVA (the numeric value of the peak height PVA is obviously known, since it equals the value of the Greece-to-Spain exponential, the dot–dot curve in Fig. 9):

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Fig. 8 Showing the b-lognormals of eight Western civilizations over the 5000 years (=50 centuries) time span from 3800 B.C. to 2200 A.D. In addition, three exponential ‘envelopes’ (or, more precisely, three ‘loci of the maxima’) are shown

⎧ 2 μVA −σVA ⎪ , Greece_ to_ Spain_ EXPONENTIAL ⎪ ⎨ − 3500 = pVA = bVA + e ⎪ ⎪ ⎩

2 σVA

= 7.305 × 10

−4

= pVA

(74)

e 2 −μVA =√ . 2πσVA

One may let μVA disappear from the above two equations by multiplying them side-by-side and then finding the following new equation in σVA only, that must thus be solved for σVA : 2 σVA

Numerically_ known = ( pVA − bVA )PVA

e− 2 =√ . 2πσVA

(75)

Unfortunately, it is not possible to solve this equation for σVA exactly. The best we can do is to expand its right-hand side into a MacLaurin power series for σVA (which is acceptable since we know that 0 < σ < 1 for all b-lognormals representing

7 Extrapolating History into the Past: Aztecs

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Fig. 9 The virtual Aztec b-lognormal is the b-lognormal peaking at a time in the past when the Western civilizations discovered the wheel, i.e. about 3500 B.C. in Mesopotamia, Southern Caucasus and Central Europe. This b-lognormal is the dash–dash black curve in the above diagram. The Aztecs started their expansion in 1325, so when Cortez arrived in 1519 they were a civilization 1519 − 1325 = 194 years old. Reporting this 194 years lapse before the year 3500, we find that the virtual Aztecs had been ‘born’ 194 years earlier, namely in 3694 B.C., which is thus the b-value of the virtual Aztec b-lognormal. Then we have to find the b-lognornal itself, i.e. its μ and σ. Its peak lies on the Greece-to-Spain exponential curve but, unfortunately, not exactly upon it since the system of two simultaneous equations (71) and (72) cannot be solved exactly for μ and σ. Thus, this approximated numerical solution, corresponding to the quadratic (77), is reflected in the diagram by positioning the virtual Aztec b-lognormal slightly above the Greece-to-Spain exponential. Finally, the true Aztec b-lognormal is just the same thing as the virtual Aztec b-lognormal except that its peak is shifted in time by an amount of (3500 + 1519) years = 5019 years into the future, so that its peak falls at 1519, when the Spaniards arrived

life-spans), thus getting (the series is truncated at power 2 in σVA ): σ2

Numerically_ known = ( pVA − bVA )PVA

1 − 2VA ≈√ . 2πσVA

(76)

Solving this for σVA leads to the quadratic in σVA √ 2 σVA + 2 2π( pVA − bVA )PVA σVA − 2 = 0,

(77)

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whose two roots are √  √ 2 σVA = − 2π( pVA − bVA )PVA ± 2 π( pVA − bVA )2 PVA + 1.

(78)

Discarding the negative root, this leads to the only positive root for σVA √  √ 2 σVA = − 2π( pVA − bVA )PVA + 2 π( pVA − bVA )2 PVA +1

(79)

and the problem is (approximately) solved, since μVA is then found from σVA by solving the upper equation (74): 2 + ln( pVA − bVA ). μVA = σVA

(80)

Replacing the numerical values given by (73) and (74) into (79) and (80), the latter yield, respectively:

σVA = 1.103, μVA = 6.484.

(81)

The numeric value for σVA slightly higher than 1 is a bit surprising, and shows once again that we are working in a numerically approximated solution of the transcendental equation (75): a more accurate numeric solution of (75) would be needed. The same fact is revealed graphically in Fig. 9, inasmuch as the virtual Aztec b-lognormal peak lies a little bit above the Greece-to-Spain exponential curve. In conclusion, the Virtual Aztec b-lognormal equation reads Virtual Aztec b-lognormal (t, μVA , σVA , bVA ) 1

2 ln t−b −μ − ( ( VA2) VA )

=√ e 2πσVA (t − bVA )

2σVA

.

(82)

As for the true Aztec b-lognormal, that is just the same as the virtual Aztec blognormal except that it is shifted in time so as to start in 1325, when the Aztec expansion in central Mexico started. By construction, the peak of such true Aztec b-lognormal falls exactly in the year 1519, when the Spaniards arrived: True_ Aztec_ b-lognormal(t, μVA , σVA , bTA ) ln t−b −μ − ( ( TA2) VA ) 1 2σVA =√ . e 2πσVA (t − bTA ) 2

(83)

This true Aztec b-lognormal is also shown in Fig. 11, just below all other blognormals, immediately revealing that the Aztecs were by far technologically inferior to all other European civilizations of the time. We shall now explore a little more in detail the last statement.

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Consider Figs. 10 and 11, which are enlarged portions of Fig. 9 but limited to the years between 1000 and 2200 (Fig. 10) and, even better, between the years 1300 and 1520 (Fig. 11). Then one immediately infers that:

Fig. 10 Enlarged portion of Fig. 9 limited to the years between 1000 and 2200

Fig. 11 Enlarged portion of Fig. 10 limited to the years between 1300 and 1520. If we assume the technological level of the Spaniards to equal 100%, then the technological level of the Aztecs is only about 18%, i.e. the Aztec b-lognormal is about one-fifth of the Spaniard b-lognormal height in 1520. No wonder the Spaniards crushed the Aztecs, then. Yet, in the next section, we claim that Shannon’s Information Theory provides an even better way to measure the cultural and technological gap between Aztec and Spaniards: this is what physicists have long been calling Entropy

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(1) Around 1497, the culturally leading country in the world was Renaissance Italy (green solid b-lognormal). (2) In 1519, however, continental Spain (black dot-dot Greece-to-Spain exponential curve) had virtually reached the same cultural and technological level as the (already starting to decline) Renaissance Italy. (3) In 1519, the Portuguese Empire (brown solid b-lognormal) was at its beginnings, since it had started in 1419. (4) In 1519, the Spanish Empire (orange solid b-lognormal) was at its beginnings, since it had started in 1492. (5) In 1519, the Aztec Empire (black solid b-lognormal) was at its top: ready to be crushed by the Spaniards, owing to the huge cultural and technological inferiority of the Aztecs (18.7%) to the Spaniards (assumed 100% in comparison).

8 b-Lognormal Entropy as ‘Civilization Amount’ 8.1 Introduction: Invoking Entropy and Information Theory We now take a more profound mathematical step ahead than just using b-lognormals: we resort to information theory, firstly put forward by Claude Shannon (1916–2001) in 1948. In particular, we now need Shannon’s notion of differential entropy H of an assigned probability density f X (x), defined by the integral ∞ H =−

f X (x). ln f X (x)dx.

(84)

−∞

Essentially, this is a measure of ‘how much peaked’ a pdf is (small entropy) in contrast to a ‘largely spread’ pdf (high entropy), and so a pdf entropy is also called its ‘uncertainty’ (no relationship to Heisenberg’s uncertainty principle). In particular, we need the expression of differential entropy of the b-lognormal. Let us thus start by finding the differential entropy of the ordinary lognormal (i.e. starting at zero), that is: ∞ Hlognormal = −

√ 0

∞ =−

√ 0

1 2πσx

1 2πσx

e−

e−

(ln x−μ)2 2σ2

(ln x−μ)2 2σ2

 (ln x−μ)2 1 ln √ e 2σ2 dx 2πσx

 √  (ln x − μ)2 − ln 2πσ − ln x − dx 2σ2

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∞ √  ∞ 1 x−μ)2 (ln x−μ)2 1 − (ln 2σ 2 = ln 2πσ dx + ln x √ e e− 2σ2 dx √ 2πσx 2πσx 0

∞ + 0

0

x−μ)2 1 (ln x − μ)2 − (ln 2σ 2 e √ 2σ2 2πσx

(85)

now the subtitution ln x = z changes all three integrals into well-known integrals of normal distribution, equal to 1 (normalization condition), μ (mean value definition) and σ2 (variance definition), respectively: ∞  ∞ 1 √ 2 (z−μ)2 1 − (z−μ) z√ = ln 2πσ e 2σ2 dz + e− 2σ2 dz √ 2πσ 2πσ −∞

−∞

∞

2 1 (z − μ) − (z−μ) 2σ2 dx · e √ 2σ2 2πσ −∞ √   √ 1 1 = ln 2πσ + μ + 2 σ2 = ln 2πσ + μ + . 2σ 2

+

As for the differential entropy of the b-lognormal, it is just the same as (85), since the b-lognormal simply is a lognormal shifted to a new origin b along the x-axis, and so all infinite support integrals in (85) remain unchanged, and the proof is just the same as (85). Thus, in conclusion, the differential entropy of both the ordinary lognormal and the b-lognormal is given by Hlognormal = Hb-lognormal = ln

√  1 2πσ + μ+ . 2

(86)

Notice also that (86) yields the b-lognormal differential entropy in nats, i.e. in natural logarithms. If one wants to express (86) in bits, then one must divide (86) by ln 2 = 0.693… In other words, one has    √ 1 1 ln 2πσ + μ + Hlognormal_ in_ bits = Hb-lognormal_ in_ bits = ln 2 2    √ 1 (87) ≈ 1.443 . . . ln 2πσ + μ+ . 2 The proof of (87) from (86) simply follows from the well-known change-of-base formula for the logarithms log2 N =

ln N ln N ≈ ≈ 1.443 ln N ln 2 0.693

(88)

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applied to definition (84) of differential entropy with ln (…) replaced by log2 (…).

8.2 Exponential Curve in Time Determined by Two Points Only Let us now rewrite the exponential curve in time: E(t) = Ae Bt .

(89)

Now suppose that the exponential curve (89) is passing through two (and only two) assigned points of coordinates (p1 , P1 ) and (p2 , P2 ), respectively. In other words, we assume that the two simultaneous equations hold

P1 = Ae B p1 P2 = Ae B p2 .

(90)

This system (90) of two equations may be solved with respect to the two constants A and B in a few simple steps that we omit here (but they are found in the Appendix 30. A of Maccone [15], see equations %02 through %07). The result is: ⎧ ⎪ ⎪ ⎨ A = ⎪ ⎪ ⎩B =

P1

p2 p2 − p1 p2 p −p

P2 2 1 P ln P2 1

p2 − p1

(91) .

Thus, the two equations (91) completely solve the problem of finding the exponential curve in time (89) passing through the two assigned points of coordinates (p1 , P1 ) and (p2 , P2 ).

8.3 Assuming that the Exponential Curve in Time Is the GBM Mean Value Curve We are now ready to take the usual step ahead in our stochastic representation of Darwinian theory (Evolution, as described above) and of historical progress (Mathematical History, as described in ‘Extrapolating history into the past: Aztecs’ section) assuming that the exponential curve in time (89) is the same thing as the exponential mean value of GBM in time, given by (15) (and (11.1) in Maccone [15]), that is N (t) = N0 eμt = N0 eμGBM t .

(92)

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Evolution and human progress now cease to be deterministic chains of events and rather become a stochastic (random) sequence of events, as indeed it is in reality, with all their ups and downs, although constrained to have a deterministic exponential overall average increase. This ‘mathematical representation of Darwinian Evolution and Historic Human Progress as Geometric Brownian Motion’ is thus the #1 message put forward by this chapter and by the Maccone [15] book, while the use of entropy is the #2 message. In Eq. (92) we have set μ = μGBM

(93)

to remind that the drift parameter μ is now the drift parameter of the GBM, denoted by μGBM . Having so said, let us now check (89) against (92). One then obtains

A = N0 , B = μGBM .

(94)

On the other hand, it will be remembered from (26) (and (11.12) of Maccone [15]) 2 that the relationship between N 0 and σGBM is N0 = e

2 σGBM 2

.

(95)

Thus, upon inserting (95) into the upper equation (94), the latter may be rewritten 

2 σGBM

A=e 2 , B = μGBM .

(96)

2 Solving then (96) for both μGBM and σGBM one obtains:



μGBM = B, 2 = 2 ln A. σGBM

(97)

These two equations may be rewritten in terms of the two points of coordinates (p1 , P1 ) and (p2 , P2 ) by inserting (91) into (97). With a little rearranging, the result is   ⎧ P ln P2 ⎪ 1 ⎨ μGBM = p2 −

p1p2  (98) P 2 ln 1p1 ⎪ ⎩ 2 P2 σGBM = p2 − p1 .

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In conclusion, we have: (1) Determined the exponential curve in time passing through the two assigned points of coordinates (p1 , P1 ) and (p2 , P2 ). These two points will later be identified as the two peaks of the initial b-lognormal (p1 , P1 ) and final b-lognormal (p2 , P2 ), respectively. (2) Assumed that the above exponential curve in time is the mean value of a certain GBM. This allows for the representation of Darwinian evolution and later human history as a GBM, with all its ups and downs, but constrained by definition to have an exponential increase in the ‘level of evolution’, as 3.5 billion years of evolution and progress on Earth plainly show. (3) Found the two equations (98) fully expressing this GBM in terms of the two initial and final peaks only. This is in full agreement with the statistical Drake equation static case, as described in ‘GBM as the key to stochastic evolution of all kinds’ and ‘Darwinian Evolution re-defined as a GBM in the number of living species’ section (and in Chaps. 1, 3, 6, 7, 8 and 11 of the author’s book, Maccone [15]). The way is thus paved to outline a full mathematical (statistical) theory of Darwinian evolution and human history, later to be extended to alien civilizations after the first ‘Contact’, even if these ETs will be ‘post-biological’ (namely based on artificial intelligence, rather than on ‘flesh’).

8.4 The ‘No-Evolution’ Stationary Stochastic Process This short section is devoted to the special case when the two peak ordinates, P1 and P2 , are equal to each other: P2 = P1 .

(99)

Then, (99) and (91), with a little rearrangement, yield

A = P1 = P2 , B = 0.

(100)

This means that the ‘former exponential’ is now a straight line parallel to the time axis and located at a certain positive ordinate A = P1 = P2 above it. In other words, the former GBM stochastic process has now become a stationary stochastic process, showing that ‘No-Evolution’ occurs between the two times p1 and p2 : for millions or billions of years, living beings are born, reproduce and die generation after generation, with no evolution at all.

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2 Also, the drift parameter μGBM vanishes, and the σGBM is a constant in time equal to twice the natural log of the constant peak ordinate (P1 , P2 ), as one immediately may see by inserting (99) into (98):



μGBM = 0, 2 = 2 ln(P1 ) = 2 ln(P2 ). σGBM

(101)

8.5 Entropy of the ‘Running b-Lognormal’ Peaked at the GBM Exponential Mean Much more interesting than the ‘No-Evolution’ case just considered is of course the truly exponential evolution case. It is summarized at-a-glance in Fig. 4 that intuitively shows the basic mechanism that we propose in this paper, in order to unify the notions of Darwinian evolution, human historical progress and SETI. However, all this is just qualitative. If we want to go quantitative, the only way is to resort to Shannon’s information theory and then resort to the notion of b-lognormal differential entropy described in Section ‘Introduction: invoking entropy and information theory’, as we now describe in detail. To start, let us call ‘running b-lognormal’ the generic b-lognormal peaked at the generic instant t = p. It will be remembered that the key equations describing the running b-lognormal are the two equations (31) that we reproduce here conveniently re-written in this section’s notation (the subscript ‘RbL’ means ‘running b-lognormal’): 

1 μ RbL = 4πA 2 − p B, 1 σ RbL = √2πA .

(102)

Therefore, the differential entropy of the running b-lognormal in bits is found by inserting (102) into (87), with the immediate result

Hrunning_b-lognormal_ in_ bits = =

− ln(A) +

1 4πA2

ln 2

ln

√

−p B+

 2πσ RbL + μ RbL +

1 2

ln 2 1 2

.

(103)

This equation expresses the running b-lognormal’s differential entropy in terms of the two constants A and B of the exponential curve (89) and of the abscissa in time (i.e. p) at which the running b-lognormal’s peak is positioned. Hence, in reality, the only ‘free parameter’ in (103) is indeed this ‘free’ b-lognormal’s peak abscissa

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p. Let us write this fact neatly as follows: Hrunning_b-lognormal_ in_ bits ( p) =

− ln(A) +

1 4πA2

−p B+

1 2

ln 2

.

(104)

We may now rewrite the last equation in terms of the coordinates of the two points (p1 , P1 ) and (p2 , P2 ) by inserting (91) into (104). The result of this straight substitution is: Hrunning_b-lognormal_ in_ bits ( p) ⎡ ⎛ ⎞ p2 ⎢ p2 − p1 ⎢ P 1 ⎢ = ⎢− ln⎝ 1 p1 ⎠ + ln 2 ⎢ p −p P2 2 1 ⎣

⎤  

 4π

1 P1 P2

p2 p2 − p1 p1 p2 − p1

⎥ 1⎥ ⎥ + ⎥. 2 − p p2 − p1 2⎥ ⎦ P2 P1

ln

(105)

This formula may only slightly be simplified as follows: Hrunning_b-lognormal_ in_ bits ( p)  p2    ⎡ ⎤ 2 p1 P1 P2 p2 − p1 ln ln p1 P1 1 ⎣ P2 1 P2 = + −p + ⎦. − 2 p2 ln 2 p2 − p1 p − p 2 2 1 p −p 4πP1 2 1

(106)

This is the differential entropy in bits of the running b-lognormal peaked at p. It may more conveniently be rewritten as   Hrunning_b-lognormal_ in_ bits ( p) = − p

ln

P2 P1

( p2 − p1 ) ln 2 + Part_ not_ depending_ on_ p.

(107)

We will use (107) in a moment!

8.6 Decreasing Entropy for an Exponentially Increasing Evolution: Progress! We now reach the conclusion of this chapter, as well as the Epilogue of the book on ‘Mathematical SETI’. Young students at university courses on thermodynamics are still made to learn that ‘entropy always increases’, meaning that the second law of thermodynamics rules so. However, thermodynamics was born in 1700–1800 days upon the discovery

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163

of the laws of gases, and quite a few scientists have later (1900–2000 days) come to realize that the sentence ‘entropy always increases’ may hardly apply to the evolution of intelligent living species as it occurred on Earth over the last 3.5 billion years. This author belongs to the latter category of scientists, and is proud to claim that his theory, outlined in this Chapter “SETI, Evolution and Human History Merged into a Mathematical Model” neatly shows that entropy decreases (rather than always increasing) when it comes to describe the evolution of life up to the current time, i.e. when it comes to describe progress! Our proof is as follows. Consider (87), i.e. the entropy of the running b-lognormal. If we consider the entropy change between the initial b-lognormal peaked at (p1 , P1 ) and the final one peaked at (p2 , P2 ), this entropy change is, by definition, given by  H _in_ bits = Hrunning_b-lognormal_ in_ bits ( p2 ) − Hrunning_b-lognormal_ in_ bits ( p1 ).

(108)

When rewritten in terms of (107), this entropy change is simpler than (107) since the two ‘Parts_not_depending_on_p’ cancel against each other, and the result is just:  H _in_ bits = Hrunning_b-lognormal_ in_ bits ( p2 ) − Hrunning_b-lognormal_ in_ bits ( p1 )     ln PP21 ln PP21 + p1 . = − p2 ( p2 − p1 ) ln 2 ( p2 − p1 ) ln 2

(109)

However, the last equation may be further simplified as follows:    H _in_ bits = − p2

ln

 

P2 P1

( p2 − p1 ) ln 2   ln PP21

ln

+ p1

P2 P1

( p2 − p1 ) ln 2     ln PP21 ln PP21 = . =− = −( p2 − p1 ) ln 2 ln 2 ( p2 − p1 ) ln 2

(110)

Thus, we have reached a result of adamantine beauty: the entropy change, when passing from a lower civilization to a higher civilization, is simply given by the log of the ratio between the lower civilization peak and the higher civilization peak (apart from the factor ln 2 at the denominator, necessary to measure the entropy in bits):    H _in_ bits =

ln

P1 P2

ln 2

.

(111)

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Since in the actual unfolding of evolution we always had P1 < P2 , the entropy change (111) in passing from the lower civilization to the higher civilization is the log of a number smaller than 1, i.e. it is a negative number! Thus, entropy decreases in passing from a lower civilization to a higher civilization, and that proves that progress amounts to a decrease in entropy. One more important result derived from (107) is found upon rewriting it explicitly as a function of p as follows: Hrunning_b-lognormal_ in_ bits ( p) =

− ln(A) +

1 4πA2

−p B+

1 2

ln 2 −B p + part_ not_ depending_ on_ p. = ln 2

(112)

Then, (108) and (112) yield immediately  H _in_ bits = −

B ( p2 − p1 ). ln 2

(113)

This also is a new result of adamantine beauty, since it yields the entropy change in bits by virtue of the difference in time between the two peak abscissas (p2 − p1 ) and the constant B expressing the exponential increase (i.e. the mean exponential drift in the GBM) of (89). Just to distinguish the two adamantine results (111) and (113) from each other, we might call (111) the ‘entropy change in evolution by virtue of the peak ordinates only’, and (113) the ‘entropy change in evolution by virtue of the peak abscissas and the average GBM exponential drift, B, only’.

8.7 Six Examples: Entropy Changes in Darwinian Evolution, Human History Between Ancient Greece and Now, and Aztecs and Incas Versus Spaniards Appendix 30.A of Maccone [15], not only provides full mathematical proofs of all the results we have derived so far in this chapter: but also offers six examples showing how entropy actually decreased in six different cases of past life on Earth. These six cases are, respectively: (1) Darwinian Evolution on Earth over the last 3.5 billion years. This is shown to correspond to an entropy decrease of 25.57 bits per each living being, if today’s number of living species is assumed to be 50 million. Were there more than 50 million species living on Earth right now, our Eqs. (111) and (113) would yield the corresponding entropy decrease accordingly. As for the proofs, please refer to equations (%i45) through (%o52) of Appendix 30.A of Maccone [15]. (2) Entropy changes in human history from ancient Greece to the end of the British Empire, i.e. to nearly nowadays. According to the b-lognormal theory that we

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described in Section ‘Extrapolating history into the past: Aztecs’, the year 434 B.C. (i.e. p1 = − 434 · years) corresponds to the peak of the age of Pericles in Athens, while the year 1868 (i.e. p2 = 1868 · years) corresponds to the peak of the Victorian age in Britain (Maxwell equations mastering electromagnetism published just 4 years earlier, in 1864). The entropy change between these two peaks is computed in equations (%i53) through (%o59) of Maccone [15], and amounts to an entropy decrease of 1.76 bits per individual. (3) Entropy changes in human history from ancient Greece to the end of the American Empire, assumed to yield to China about the year 2050. Again, according to the b-lognormal theory that we described in Section ‘Extrapolating history into the past: Aztecs’, the year 434 B.C. (i.e. p1 = −434 · years) corresponds to the peak of the age of Pericles in Athens, while the year 1973 (i.e. p2 = 1973 · years) corresponds to the peak of the American Empire (Americans had just landed on the Moon on July 20th, 1969, and stopped landing on 14 December 1972). Equations (%i60) through (%o66) of Maccone [15] show that the corresponding entropy decrease between the two peaks amounts to 2.38 bits per individual, thus a higher number than for the Greece-to-Britain GBM exponential mean, as intuitively obvious. (4) Computing the entropy difference between the Aztecs and the Spaniards when they came suddenly in touch (with no previous contact) in 1519, when Cortez landed in Vera Cruz and started invading the Aztec Empire. This example is particularly important for SETI inasmuch as it ‘might resemble’ what could happen in case humanity came physically in touch with an Alien Civilization all of a sudden (as shown in the movie ‘Independence Day’). Well, in section “Virtual-Aztecs’ method to find the ‘True-Aztecs’ b-lognormal’ we found a way to compute ‘how backward the Aztecs were’ by exploiting the fact that they were just one the verge of discovering the use of wheels. Wheels were used by Aztec children in their toys, but not by adults in warfare, and this is just what had already happened in Mesopotamia, Northern Caucasus and Central Europe about 3500 years before Christ. Thus, we set p1 = −3500 · years to describe the Aztec technological backwardness with respect to the Spaniards, and of course we set p2 = 1521 · years to mean that Cortez completed the conquest of the Aztec Empire in 1521. Again, we assumed the Greece-to-Britain exponential to be the right one (since it refers to the actually occurred facts, rather than future facts also, as in the Greece-to-USA case). In conclusion, (111) and (113) then yield a entropy difference of 3.84 bits per individual between the Aztecs and the Spaniards. This is higher than the 1.76 bits difference between the Victorian Britons and the Pericles Greeks, of course: nearly twice as much, owing to the huge 5000-years difference in evolution between the Aztecs and the Spaniards, versus the just about 2300 years in evolution between Pericles and the Victorian age. Let us also compute the entropy difference between the Incas and the Spaniards when they came suddenly in touch (with no previous contact) in 1532, when Pizarro landed in Tumbes and started invading the Incas Empire. Again, the wheel was unknown to the Incas, and so we assumed p1 = −3500 · years.

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Naturally, we now assumed p2 = 1532 · years, and the result given by (111) and (113) is an entropy difference of 3.85 bits per individual. Just very slightly higher than for the Aztec–Spaniard case, meaning that … (5) The Aztecs were very slightly more technologically advanced than the Incas when they both were subdued by the Spaniards. In fact, if you assume that the Aztec Empire had its start around the year 850 A.D. (roughly at the peak of the Maya Empire, so that, in some sense, the Aztec ‘inherited’ part of the Maya civilization), and then you assume that the Incas Empire was founded around 1250 (when the Incas reached Cuzco), then, assuming again the Greece-toBritain exponential as the true exponential of technological development, (111) and (113) yield an entropy difference of 0.3 bits per individual in favour of the Aztecs (more technologically advanced) over the Incas. A subtle quantitative statement that may be the current historical knowledge about both people is possibly unable to ponder over. Hence, with these final six entropy measurements, we hope to have been the first author to be able to give a quantitative description of both Darwinian evolution and human history, based on our new discoveries about the mathematical properties of the finite and infinite b-lognormals. This we did to be able to quantitatively estimate how much an alien civilization might be more advanced than us.

8.8 b-Lognormals of Alien Civilizations So much about the past, but what about the future? What are the b-lognormals of ET civilizations? Nobody knows, of course, and nobody will until the SETI scientists will detect the first signs of an ET civilization. A good book to read, however, is ‘Interstellar Migration and the Human Experience’, by Finney and Jones [4]. Also, science fiction fans might take pleasure in casting the Star Trek timeline into the mathematical language of our b-lognormals. Interesting is also ‘The Star Trek Chronology’ by Okuda and Okuda [17], but …no mathematics is there. We need the mathematics that we will develop in our next chapter by extrapolating exponentials and entropy of the human past into the future, with reference to the Fermi Paradox.

9 Conclusion: Summary of Technical Concepts Described As a conclusion to this chapter, we would like to summarize the new technical concepts we had to introduce here.

9 Conclusion: Summary of Technical Concepts Described

(1)

(2)

(3)

(4)

(5)

(6)

167

The Drake equation, describing 10 billion years of evolution in this Galaxy (stars to humans), was transformed from the simple product of seven factors into the product of any number of random variables. This statistical Drake equation is more ‘serious’ scientifically, and leads to the conclusion that, if the number of input random variables is increased more and more, then the pdf of the number of civilizations in the Galaxy must be lognormal. Darwinian evolution on Earth was re-defined mathematically as a stochastic process (i.e. a random function of time) yielding the number of living species on Earth over the last 3.5 billion years. This definition allows for sudden lows in the number of living species (mass extinctions) but, apart from those, the overall mean behaviour of the number of species of Earth in time must be increasing exponentially. Today, some 50 million species are supposed to live on Earth, while 3.5 billion years ago there was just one (RNA?), thus fixing the exponential mean curve perfectly. Geometric Brownian Motion (GBM) is exactly the right stochastic process fulfilling all the above requirements representing the evolution of life on Earth (and elsewhere in the universe, like life on extrasolar planets). However, GBMs were so far studied only in Financial Mathematics (Black–Scholes models, leading to the Nobel Prize in Economy assigned in 1997 to Scholes and Merton for exploiting GBMs), and so it is high time for evolutionary scientists and astrobiologists to realize the key merits of GBMs, primarily their mean value increasing in time exponentially. We took one more step ahead by introducing lognormal distributions (blognormals) starting at any time b rather than just at zero, as ordinary lognormals do. Then, these b-lognormals were ‘matched’ to GBMs by forcing all their peaks to stay just on the GBM exponential mean value curve. This leads to a one-parameter family of b-lognormals (the parameter is the peak abscissa p) representing a new living species that appeared on Earth exactly at time b. Cladistics, the science of Evolution Phylogenetic Trees, then is reduced to a simple game of b-lognormals departing from the main exponential curve of evolution (i.e. the GBM mean value) and then either increasing, or decreasing, or keeping constants in time in a stochastic fashion (this is our ‘NoEv’ new pdf, that is, not a lognormal any more). In other words, we have accounted for prospering species, or extinct species (decreasing exponential arches that, sooner or later, reach a numeric value above zero but less than one, meaning extinction), or even just ‘stationary’ species (like insects, for instance, that keep being the same as they were about 400 million years ago). The lifetime of any living being may also be represented by a made-finite blognormal. In fact, every living being is born at time b, reaches puberty at time a (adolescence, i.e. the abscissa of the b-lognormal increasing inflexion point), then goes to the peak of his living capabilities (abscissa p of the b-lognormal’s peak) and starts declining. He/she then reaches the non-return decline point (abscissa s of the ‘senility’ point, the decreasing inflection point abscissa) and dies at time d when the straight line tangent to the b-lognormal at senility intercepts the time axis.

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(7)

The ‘golden ratio’ was long hailed by artists, architects and men of literature as ‘symbol of visual perfection’. Well, surprisingly enough, this author discovered a class of b-lognormals strongly related to the golden ratio (golden b-lognormals). The future will show whether this discovery is just an iceberg’s tip, leading to many more discoveries related to the golden ratio’s already quite rich literature. (8) However, b-lognormals cannot be used to describe the lifetime of a living being only. They may be used to describe the lifetime of societies also. There is a profound theorem behind all this, stating that (in easy terms) ‘just as the sum of two independent Gaussian distributions is one more Gaussian, similarly the product of two independent lognormal distributions is one more lognormal’. We thus could use b-lognormals to study the historic course of human civilizations on Earth (i.e. the ‘f sub i’ factor in the Drake equation). (9) The history of Ancient Greece, Ancient Rome, Renaissance Italy, and then Portugal, Spain, France, Britain and the USA Empires were then cast into the language of b-lognormals. This was possible since the author discovered two equations (‘History Formulae’) that allow the computation of the b-lognormal’s μ and σ given the birth time b, the death time d and the intermediate value of ‘senility s’ (incipient decline), where the (infinite) b-lognormal hinges with the straight line going to death, thus making the infinite b-lognormal a finite one, as all lives are. Also, the b-lognormal of the Aztec civilization was computed as an essay in mathematical history. (10) However, the most important result achieved by this author is undoubtedly his study of Entropy as ‘Civilization Amount’. In fact, each b-lognormal has a precise entropy (or ‘uncertainty’) value in the sense of Shannon’s information theory and so, for instance, it is possible to assign entropy values to all historic civilizations previously represented by virtue of b-lognormals: Aztecs, Greece, Rome, Renaissance Italy, Portugal, Spain, France, Britain and the USA. This explains by entropy values, rather than by just words, ‘how much’ a civilization was more or less ‘organized’ (i.e. ‘advanced’) than another one. For instance, the entropy difference between Aztecs and Spaniards, when they clashed in 1519-20-21, turned out to equal about 3.85 bits per individual, while the entropy difference between the first living being of 3.5 billion years ago (RNA?) and today’s humans turned out to equal 27.57 bits for each living being, thus providing a direct numeric measure of different evolving species or civilizations. This author plans to investigate these new results more in depth in forthcoming papers. In conclusion, this author thinks he could make true progress by casting Evolution, Human History and SETI into his unified statistical framework made up of b-lognormals and GBMs. Quite simply, this was possible since he avoided sterile philosophical debates like ‘What is Life?’ and replaced them by the theme ‘WHEN did Life occur?’.

References

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References 1. B. Balazs, The galactic belt of intelligent life, in Biostronomy—The Next Steps, ed. by G. Marx (Kluwer Academic Publishers, 1988), p. 61–66 2. M.J. Burchell, W(h)ither the Drake equation? Int. J. Astrobiol. 5, 243–250 (2006) ´ 3. M.M. Cirkovi´ c, On the temporal aspect of the Drake equation and SETI. Astrobiology 4, 225–231 (2004) 4. B.R. Finney, E.M. Jones, Interstellar Migration and the Human Experience (University of California Press, Berkeley, CA, 1986) 5. G. Gonzalez, D. Brownlee, P. Ward, The galactic habitable zone: galactic chemical evolution. Icarus 152, 185–200 (2001) 6. G. Gonzalez, Habitable zones in the universe. Origin Life Evol. Biosph. 35, 555–606 (2005) 7. L.V. Ksanfomality, The Drake equation may need new factors based on peculiarities of planets of Sun-like starsm, in Planetary Systems in the Universe. Proceedings of IAU Symposium 202 (2004), p. 458 8. C.H. Lineweaver, Y. Fenner, B.K. Gibson, The galactic habitable zone and the age distribution of complex life in the Milky Way. Science 303, 59–62 (2004) 9. C. Maccone, The statistical Drake equation. Paper #IAC-08-A4.1.4 presented on 1 October, 2008, at the 59th International Astronautical Congress (IAC) held in Glasgow, Scotland, UK, 29 Sept–3 Oct 2008 10. C. Maccone, The statistical Drake equation. Acta Astron. 67, 1366–1383 (2010) 11. C. Maccone, The statistical Fermi paradox. J. Br. Interplanet. Soc. 63, 222–239 (2010) 12. C. Maccone, The KLT (Karhunen-Loève Transform) to extend SETI searches to broad-band and extremely feeble signals. Acta Astron. 67, 1427–1439 (2010) 13. C. Maccone, SETI and SEH (statistical equation for habitables). Acta Astron. 68, 63–75 (2011) 14. C. Maccone, A mathematical model for evolution and SETI. Origins Life Evol. Biosph. 41, 609–619 (2011) 15. C. Maccone, Mathematical SETI (Praxis-Springer, Berlin, 2012) 16. L.S. Marochnik, L.M. Mukhin, Belt of life in the galaxy, in Biostronomy—The Next Steps, ed. by G. Marx (Kluwer Academic Publishers, Dordrecht, 1988), pp. 49–59 17. M. Okuda, D. Okuda, The Star Trek chronology (1996). Available from Amazon.com 18. R.A. Rohde, R.A. Muller, Cycles in fossil diversity. Nature 434, 208–210 (2005) 19. C. Sagan, Cosmos (Random House, New York, 1980). See in particular page 335 and the caption to the diagram there 20. A. Szumski, Finding the interference—the Karhunen–Loève transform as an instrument to detect weak RF signals. InsideGNSS (“Working Papers” section), May–June 2011 issue (2011), p. 56–63 21. S.G. Wallenhorst, The Drake equation reexamined. QJRAS 22, 380 (1981) 22. C. Walters, R.A. Hoover, R.K. Kotra, Interstellar colonization: a new parameter for the Drake equation? Icarus 41, 193–197 (1980)

Evolution and Mass Extinctions as Lognormal Stochastic Processes

Abstract In a series of papers and in a book, this author put forward a mathematical model capable of embracing the search for extra-terrestrial intelligence (SETI), Darwinian Evolution and Human History into a single, unified statistical picture, concisely called Evo-SETI. The relevant mathematical tools are: 1. Geometric Brownian motion (GBM), the stochastic process representing evolution as the stochastic increase of the number of species living on Earth over the last 3.5 billion years. This GBM is well known in the mathematics of finances (Black–Sholes models). Its main features are that its probability density function (pdf) is a lognormal pdf, and its mean value is either an increasing or, more rarely, decreasing exponential function of the time. 2. The probability distributions known as b-lognormals, i.e. lognormals starting at a certain positive instant b > 0 rather than at the origin. These b-lognormals were then forced by us to have their peak value located on the exponential mean-value curve of the GBM (Peak-Locus theorem). In the framework of Darwinian Evolution, the resulting mathematical construction was shown to be what evolutionary biologists call Cladistics. 3. The (Shannon) entropy of such b-lognormals is then seen to represent the ‘degree of progress’ reached by each living organism or by each big set of living organisms, like historic human civilizations. Having understood this fact, human history may then be cast into the language of b-lognormals that are more and more organized in time (i.e. having smaller and smaller entropy, or smaller and smaller ‘chaos’), and have their peaks on the increasing GBM exponential. This exponential is thus the ‘trend of progress’ in human history. 4. All these results also match with SETI in that the statistical Drake equation (generalization of the ordinary Drake equation to encompass statistics) leads just to the lognormal distribution as the probability distribution for the number of extra-terrestrial civilizations existing in the Galaxy (as a consequence of the central limit theorem of statistics). 5. But the most striking new result is that the well-known ‘Molecular Clock of Evolution’, namely the ‘constant rate of Evolution at the molecular level’ as shown by Kimura’s Neutral Theory of Molecular Evolution, identifies with growth rate of the entropy of our Evo-SETI model, because they both grew linearly in time since the origin of life. © Springer Nature Switzerland AG 2020 C. Maccone, Evo-SETI, https://doi.org/10.1007/978-3-030-51931-5_4

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6. Furthermore, we apply our Evo-SETI model to lognormal stochastic processes other than GBMs. For instance, we provide two models for the mass extinctions that occurred in the past: (a) one based on GBMs and (b) the other based on a parabolic mean value capable of covering both the extinction and the subsequent recovery of life forms. 7. Finally, we show that the Markov and Korotayev [10, 11] model for Darwinian Evolution identifies with an Evo-SETI model for which the mean value of the underlying lognormal stochastic process is a cubic function of the time. In conclusion: we have provided a new mathematical model capable of embracing molecular evolution, SETI and entropy into a simple set of statistical equations based upon b-lognormals and lognormal stochastic processes with arbitrary mean, of which the GBMs are the particular case of exponential growth. Keywords Darwinian Evolution · Entropy · Geometric Brownian motion · Lognormal probability densities · Molecular clock · Statistical Drake equation

1 Introduction: Mathematics and Science Sir Isaac Newton published his law of universal gravitation in 1687. In the following 250 years (~1700–1950) many eminent mathematicians developed celestial mechanics (i.e. the theory of orbits) manually. The result of all those ‘difficult calculations’ was seen after the advent of computers in 1950: as of 2021 we have a host of spacecrafts of all types flying around and beyond the solar system: in other words, the space age is the outcome of both the law of universal gravitation and of lots of mathematics. Similarly, between 1861 and 1862 James Clerk Maxwell first published ‘the Maxwell equations’. These summarized all previous experimental work in electricity and magnetism and paved the mathematical way to all subsequent discoveries, from radio waves to cell phones. Again a bunch of equations was the turning point in the history of humankind, although very few people know about ‘all those mathematical details’ even in today’s Internet age. Having so said, this author believes that the time is ripe for a brand-new mathematical synthesis embracing the whole of Darwinian Evolution, human history and the search for extra terrestrial intelligence (SETI) into a bunch of simple equations. But these equations have to be statistical, rather than deterministic as it was in the case of Newton’s and Maxwell’s equations. In fact, the number of possible examples covered by these equations is huge and thus it may be handled only by virtue of statistics. Just imagine the number of Earth-type exoplanets existing in the Milky Way in 2021 is estimated to be around 40 billion. Then, if we are going to predict what stage in the evolution of life a certain newly discovered exoplanet may have reached, our predictions may only be statistical. The Evo-SETI theory, outlined in this book, is intended to be the correct mathematical way to let humans ‘classify’ any newly discovered exoplanet or even an alien

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civilization (as required by SETI) according to its own ‘degree of evolution’, given by the entropy of the associated b-lognormal probability density function (pdf), as we shall see in the section ‘Entropy as the evolution measure’ of this chapter. Also, the index of evolution (Evo-Index) defined in the section ‘Entropy as the evolution measure’ measures the positive evolution starting from zero at the time of the origin of life on that exoplanet, and so we presume that we have found a way to quantify progress in the evolution of life, from RNA to humans and on to extraterrestrials.

2 A Summary of the ‘Evo-SETI’ Model of Evolution and SETI This book describes recent developments in a new statistical theory casting evolution and SETI into mathematical equations, rather than just using words only: this we call the Evo-SETI model of evolution and SETI. Our final goal is to prove that the Evo-SETI model and the well-known molecular clock of evolution are in agreement with each other. In fact, the (Shannon) entropy of the b-lognormals in the Evo-SETI model decreases linearly with time, just as the molecular clock increases linearly with time. Apart from constants with respect to the time, b-lognormals entropy and molecular clock are the same. However, the calculations required to prove them are lengthy. To overcome this obstacle, the Appendix (available in the supplementary material) gives a summary of all the analytical calculations that this author performed by the Maxima symbolic manipulator especially to prove the Peak-Locus theorem described in section ‘PeakLocus theorem’. It is interesting to point out that the Macsyma symbolic manipulator or ‘computer algebra code’ (of which Maxima is a large subset) was created by NASA at the Artificial Intelligence Laboratory of MIT in the 1960s to check the equations of celestial mechanics that had been worked out manually by a host of mathematicians in the previous 250 years (1700–1950). Actually, those equations might have contained errors that could have jeopardized the Moon landings of the Apollo Program, and so NASA needed to check them by computers, and Macsyma (nowadays Maxima) did a wonderful job. Today, anyone can download Maxima for free from the website http://maxima.sourceforge.net/. The Appendix of this chapter (available in the supplementary material) is written in Maxima language and the conventions applied for denoting the input instructions by (%i[equation number]) and the output results by (%o[equation number]), as we shall see shortly. Going now back to the general lognormal stochastic process L(t) (standing for ‘life at t’, and also for ‘lognormal stochastic process at time t’), let us first point out that: 1. L(t) starts at a certain time t = ts with certainty, i.e. with probability one. For instance, if we wish to represent the evolution of life on Earth as a stochastic process in the number of species living on Earth at a certain time t, then the starting time t = ts will be the time of the origin of life on Earth. Although we do

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not know exactly when that occurred, we approximately set it at ts = −3.5 billion years, with the convention that past times are negative times, the present time is t = 0 and future times will be positive times. 2. We now make the basic mathematical assumption that the stochastic process L(t) is a lognormal process starting at t = ts, namely its pdf is given by the lognormal

L(t)_pdf(n; M L (t), σ L , t) = √  with

n>0 t  ts

 and

e

ln(n)−M L (t)]2 2σ L2 (t−ts)

−[

√ 2πσ L t − tsn

σ L  0, M L (t) = arbitrary function of t.

(1)

Profound ‘philosophical justifications’ exist behind the assumption summarized by Eq. (1): for instance, the fact that ‘lognormal distributions are necessarily brought into the picture by the central limit theorem of statistics’ [3, 4]. However, we will not be dragged into this mountain of philosophical debates since this author is a mathematical physicist wanting to make progress in understanding nature, rather than wasting time in endless debates. Therefore, we now go on to our third basic assumption. 3. The mean value of the process L(t) is an arbitrary (and continuous) function of the time denoted by mL (t) in the sequel. In equations, that is, one has, by definition m L (t) = L(t).

(2)

In other words, we analytically compute the following integral, yielding the mean value of the pdf (1), getting (for proof, see (%o5) and (%o6) in the Appendix available in the supplementary material) ∞ m L (t) =

n√ 0

ln(n)−M L (t)]2 2σ L2 (t−ts)

−[

e

dn = e M L (t) e √ 2πσ L t − ts n

σ L2 2

(t−ts)

.

(3)

There are two functions of the time in (3): mL (t) and M L (t). Since we assumed mL (t) to be an arbitrary continuous function of the time, it follows from (3) that M L (t) must also be so, and they may be freely interchanged, since (3) may be exactly solved with respect to either of them. Also, please do not worry about ‘the variance σL of L(t)’ in (3): we shall shortly see at point (7) that σL is determined by both the arbitrary mean value function mL (t) and the standard deviation of the process L(t) at the end time te by δNe = Δ(te). At this point, knowing the pdf (1) and the mean value (3), it is just a mathematical exercise to derive all the statistical properties of the stochastic process L(t). Doing so, however, would require several pages of lengthy calculations (even by Maxima) that cannot be compressed in this chapter. Thus, this author recommends readers to have a look

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175

at his recently published papers and his book [4, 7–9], where the calculations yielding the statistical properties of L(t) are described more in detail. Here, we have just summarized them in Table 1. Table 1 Summary of the properties of the lognormal distribution that applies to the stochastic process L(t) = lognormally changing number of ET communicating civilizations in the Galaxy, as well as the number of living species on Earth over the last 3.5 billion years Stochastic process

L(t) =  (1) Number of ET civilizations(in SETI) (2) Number of living species(in evolution)

Probability distribution

Lognormal distribution of all lognormal stochastic processes, i.e. the lognormal stochastic processes with arbitrary mean M L (t)

Probability density function

−

[ln(n)−M L (t)]2 2

2σ L (t−ts) e L(t)_pdf(n; M L (t), σ L , t) = √ √ 2πσ L t − ts n ⎧ ⎪ n>0 ⎪ ⎪ ⎪ ⎨ t  ts with ⎪ σL  0 ⎪ ⎪ ⎪ ⎩ M L (t) = arbitrary function of t

Mean value Variance

σ L2

L(t) ≡ m L(t) = e M L (t) e 2 (t−ts) 2

2 2 σ L(t) = e2M L (t) eσ L (t−ts) eσL (t−ts) − 1 σ L2 2

 2 eσL (t−ts) − 1

Standard deviation

σ L(t) = e M L (t) e

Upper standard deviation curve

m L(t) + σ L(t) =

 σ L2 2 e M L (t) e 2 (t−ts) 1 + eσL (t−ts) − 1

Lower standard deviation curve

m L(t) − σ L(t) =

 σ L2 2 e M L (t) e 2 (t−ts) 1 − eσL (t−ts) − 1

All the moments, i.e. kth moment



Mode (=abscissa of the lognormal peak)

n mode ≡ n peak = e M L (t) e−σ L (t−ts)

(t−ts)

 2  σ L2  L k (t) = ek M L (t) e k −k 2 (t−ts) 2

Value of the mode peak f L(t) (n mode ) =

σ L2

L (t) e 2 (t−ts) e−M √ √ 2πσ L t−ts

Median (=fifty-fifty probability value for L(t)) median = e M L (t)  2

Skewness 2 K3 eσ L (t−ts) − 1 eσ L (t−ts) + 2 3 = (K 2 ) 2

(continued)

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Table 1 (continued) Stochastic process

L(t) =  (1) Number of ET civilizations(in SETI) (2) Number of living species(in evolution)

Kurtosis

K4 K 22

= e4σ L (t−ts) + 2e3σ L (t−ts) + 3e2σ L (t−ts) − 6 2

2

2

Clearly, these two different L(t) lognormal stochastic processes may have two different time functions for M L (t) and two different numerical values for σ L , but the equations are the same for both processes, i.e. for the number of ET civilizations in the Galaxy and for the number of living species in the past of Earth. This is the general lognormal growth, not necessarily Malthusian

4. At this point, the vertical axis of the (t, L(t)) plot still is undermined up to an arbitrary multiplicative constant. At the initial instant ts, we may thus denote by Ns (‘number at start’) the numerically certain (i.e. with probability 1) starting value of the stochastic process L(t). A few easy steps from (3) then show that, introducing Ns, the mean value (3) must be replaced by the ‘re-normalized’ one m L (t) = N s e M L (t)−M L (ts) e

σ L2 2

(t−ts)

(4)

(for the proof of this result, see (%i8) through (%o13) in the Appendix available in the supplementary material). In fact, if you set t = ts into (4), both exponentials become 1, and one just gets the initial condition m L (ts) = N s.

(5)

One may say that the two assumed numeric values (ts, Ns) are the initial boundary conditions of the stochastic process L(t). 5. Similarly, one must specify the two numbers (te, Ne) (‘end time’ and ‘number at end time’) representing the final boundary conditions, with N e = N s e M L (te)−M L (ts) e

σ L2 2

(te−ts)

(6)

(see (%o23) in the Appendix available in the supplementary material). 6. Replacing the mean value (3) by virtue of the renormalized mean value (4) would lead to a new Table similar to Table 1 that, however, would contain the new term Ns. That new Table we will skip for the sake of brevity. 7. The initial (5) and final (6) conditions only affect the mean value curve (4). That is not enough, however, since every stochastic process is determined not only by its mean value, but also by its two upper and lower standard deviation curves inferred from the higher moments of the lognormal pdf (1) and from (3), i.e. (see (%i14) through (%o21) in the Appendix available in the supplementary material)

2 A Summary of the ‘Evo-SETI’ Model of Evolution and SETI

 L (t) = e M L (t) e

σ L2 2

(t−ts)

177

 eσL (t−ts) − 1 2

(7)

Calling δNe = L (te) the final standard deviation (namely the standard deviation at the end time te) this becomes one more input that must be assigned in addition to the four boundary conditions (ts, Ns) and (te, Ne) plus the arbitrary function M L (t). Then from (6) one may derive the promised σL (see (%i22) through (%o27) in the Appendix available in the supplementary material)    2  − 2M L (ts) ln e2M L (ts) + (δN e)2 NN es σL = √ te − ts

(8)

to be inserted into the lognormal pdf (1), which is thus completely determined by the M L (t) arbitrary function plus the five numbers (ts, Ns, te, Ne and δNe). This completes the description of the L(t) process.

3 Important Special Cases of mL (t) 1. The particular case of (3) where the mean value is given by the generic exponential m GBM (t) = N0 eμGBM t or, more easily, = A e Bt

(9)

is called geometric Brownian motion (GBM), and is widely used in financial mathematics, where it is the ‘underlying process’ of the stock values (Black– Scholes models (1973), or Black–Scholes–Merton models, with the Nobel prize in Economics awarded in 1997 to Sholes and Merton only since Black had unfortunately passed away in 1995). This author used the GBM in his previous mathematical models of evolution and SETI [4–9], since it was assumed that the growth of the number of ET civilizations in the Galaxy, or, alternatively, the number of living species on Earth over the last 3.5 billion years, grew exponentially (Malthusian growth). Notice also that, upon equating the two right-hand sides of (3) and (9), we find that e MGBM (t) e

2 σGBM 2

(t−ts)

= N0 eμGBM (t−ts) .

(10)

Solving this equation for M GBM (t) yields

2 σGBM MGBM (t) = ln N0 + μGBM − (t − ts). 2

(11)

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Evolution and Mass Extinctions as Lognormal Stochastic Processes

This is (with ts = 0) just the ‘mean value showing at the exponent’ of the wellknown ordinary (i.e. starting at t = 0) GBM pdf, i.e. 

GBM(t)_ pdf(n; N0 , μGBM , σGBM , t) =

e

−

 2   σ2 t ln(n)− ln N0 + μGBM − GBM 2

√

⎧ n > 0, ⎪ ⎪ ⎨ t  0, with ⎪ N > 0, ⎪ ⎩ 0 σGBM  0.

2 2σGBM t

√ 2πσGBM tn (12)

A summary of the statistical properties of the GBMs is given in Table 2. We conclude this short description of the GBM as the exponential sub-case of the general lognormal process (1) by warning that GBM is a misleading name, since GBM is a lognormal process and not Gaussian one, as the Brownian motion is indeed. 2. Another particularly interesting case of the mean value function mL (t) in (3) is when it equals a generic polynomial in t, namely 

degree

m polynomial (t) =

ck t k ,

(13)

k=0

ck being the coefficient of the kth power of the time t in the polynomial (13). We just confine ourselves to mention here that the case where (13) is a second-degree polynomial (i.e. a parabola in t) may be used to describe the mass extinctions that plagued life on Earth over the last 3.5 billion years, as we shall see in the section ‘Mass extinctions described by an adjusted parabola branch’ of this chapter. A summary of the statistical properties of the L(t) process when its mean value is the polynomial (13) is given in Table 3.

4 Introducing b-lognormals We must also introduce the notion of b-lognormal pdf, namely a lognormal pdf (in the time variable as independent variable), which rather than starting at t = 0, starts at any time t = b. Therefore, the b-lognormal pdf is given by [ln(t−b)−μ]2

e− 2σ2 . b-lognormal_ pdf(t; μ, σ, b) = √ 2πσ(t − b)

(14)

4 Introducing b-lognormals

179

Table 2 Summary of the properties of the lognormal distribution that applies to the GBM stochastic process N(t) as the exponentially increasing number of ET communicating civilizations in the Galaxy, as well as the number of living species on Earth over the last 3.5 billion years (Malthusian or exponential growth)  Stochastic process (1) Number of ET civilizations(in SETI) N (t) = (2) Number of living species(in evolution) Probability distribution

Lognormal distribution of the geometric Brownian motion (GBM),i.e. the lognormal stochastic process with exponential mean 

Probability density function GBM(t)_ pdf(n; σ L , ts, N s, t) =

e

−

⎧ ⎪ n > 0, ⎪ ⎪ ⎪ ⎨ t  0, with ⎪ ⎪ N0 > 0, ⎪ ⎪ ⎩ σGBM  0. Mean value Variance

  σ2 ln(n)− ln N s+ μGBM − GBM 2 2σ2L (t−ts)

GBM(t) ≡ m GBM(t) = N s eμ(t−ts) 2

2 σGBM(t) = N s 2 e2μ(t−ts) eσGBM (t−ts) − 1

Upper standard deviation curve Lower standard deviation curve



 2 m GBM(t) − σGBM(t) = N s eμ(t−ts) 1 − eσGBM (t−ts) − 1

All the moments, i.e. kth moment

  GBMk (t) = N s k e



  σ2 kμ− k−k 2 GBM (t−ts) 2

Mode (=abscissa of the lognormal peak)

n mode ≡ n peak = N s e

Value of the mode peak

f GBM(t) (n mode ) =

Median (=fifty-fifty probability value for GBM(t))

median = N s e

Skewness

K3

3 (K 2 ) 2

Kurtosis

K4 K 22

=

μ−3

2 σGBM 2

(t−ts)

e(√σGBM −μ)(t−ts) √ N s 2πσGBM t−ts

σ2 μ− GBM (t−ts) 2 2

 2

2 eσGBM (t−ts) − 1 eσGBM (t−ts) + 2

= e4σGBM (t−ts) + 2e3σGBM (t−ts) + 3e2σGBM (t−ts) − 6 2

2

(t−ts)

√ √ 2πσGBM t − ts n

 2 σGBM(t) = N s eμ(t−ts) eσGBM (t−ts) − 1

 2 m GBM(t) + σGBM(t) = N s eμ(t−ts) 1 + eσGBM (t−ts) − 1

Standard deviation



2

2

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Evolution and Mass Extinctions as Lognormal Stochastic Processes

Table 3 Summary of the properties of the polynomial lognormal distribution that applies to the stochastic process P(t) as the lognormally changing number of ET communicating civilizations in the Galaxy, as well as the number of living species on Earth over the last 3.5 billion years, if the mean value is a polynomial in the time  Stochastic process (1) Number of ET civilizations(in SETI) P(t) = (2) Number of living species(in evolution) Probability distribution Probability density function

Lognormal distribution of stochastic processes with polynomial mean ⎛



⎞

degree

P(t)_pdf⎝n;

ck (t − ts) , σpolynomial , t ⎠ k

k=0 ⎡ −⎣ln(n)−log

⎤

σ2 degree polynomial k (t−ts)⎦ 2 k=0 ck (t−ts) + 2 2σ2polynomial (t−ts)

e

=

⎧ ⎪ ⎪ ⎨n > 0 with t  ts ⎪ ⎪ ⎩σ

√ √ 2πσpolynomial t − ts n

polynomial

Mean value curve

degree

P(t) ≡ m P(t) =

ck (t − ts)k = e Mpolynomial (t) e

2 σpolynomial

2

0

(t−ts)

k=0 degree

σ2P(t) = Standard deviation

2



Variance

ck (t − ts)k



2 σpolynomial (t−ts)

e

−1

k=0



 σ P(t) =

degree

ck (t − ts)k

e

2 σpolynomial (t−ts)

−1

k=0

Upper standard deviation curve



degree

m P(t) + σ P(t) =

 ck (t − ts)k

 1+

e

2 (t−ts) σpolynomial

 −1

k=0

Lower standard deviation curve



degree

m P(t) − σ P(t) =

ck (t − ts)k

 

σ2 (t−ts) −1 1 − e polynomial

k=0

All the moments, i.e.   jth moment P j (t) =

j



degree

ck (t − ts)

k



e

j2− j

2  σpolynomial 2

(t−ts)

k=0

Mode (abscissa of the lognormal peak)

 n mode ≡ n peak =

degree

 ck (t − ts)

k

e−3

2 σpolynomial 2

(t−ts)

k=0

Value of the mode peak

f N (t) (n mode ) =



2 σpolynomial (t−ts)

e degree

ck (t−ts)k

 √

√ 2πσpolynomial t−ts

k=0

  2 Median (=fifty-fifty degree σGBM k e− 2 (t−ts) − ts) median = c (t probability value for k k=0 P(t))  2

Skewness σ2 σ K3 e polynomial (t−ts) − 1 e polynomial (t−ts) + 2 3 = (K 2 ) 2

(continued)

4 Introducing b-lognormals Table 3 (continued) Stochastic process

Kurtosis

181

 P(t) = K4 K 22

=e

(1) Number of ET civilizations(in SETI) (2) Number of living species(in evolution)

2 4σpolynomial (t−ts)

+ 2e

2 3σpolynomial (t−ts)

+ 3e

2 2σpolynomial (t−ts)

−6

It describes the lifetime of any living being, be it a cell, a plant, a human, a civilization of humans or even an ET civilization. Interested readers should please read Maccone [9], particularly pp. 227–245 where the notion of finite (in time) blognormal [as opposed to the infinite (in time) b-lognormal given by (14)] was also introduced. Professional statisticians sometime call ‘three-parameter lognormal’ the pdf (14). This is because (14) embodies the third parameter b in addition to the two classical ones, μ and σ of the ordinary lognormal, i.e. (14) with b = 0. Statisticians are obviously interested in the numerical estimation of μ and σ, and also of b, by virtue of the ‘maximum Likelihood’ techniques of statistics. Although that is certainly an important topic for the application of b-lognormals to real cases, we are not going to face these issues in this chapter: their study has to be delayed to a further research paper.

5 Peak-Locus Theorem The Peak-Locus theorem is a new mathematical discovery of ours which plays a central role in Evo-SETI theory. In its most general formulation, it holds good for any lognormal process L(t) and any arbitrary function M L (t) [or mean value mL (t)]. In words, and utilizing the simple example of the Peak-Locus theorem applied to GBMs, the Peak-Locus theorem states what is shown in Fig. 1: the family of all b-lognormals ‘trapped’ between the time axis and the growing exponential of the GBMs, where all the b-lognormal peaks lie, can be exactly (i.e. without any numerical approximation) described by three equations yielding the three parameters μ, σ and b as three functions of the peak abscissa, p, only. In equations, the Peak-Locus theorem states that the family of b-lognormals having each of its peak exactly located on the mean value curve (4), is given by the following three equations, specifying the parameters μ(p), σ(p) and b(p), appearing in (14) as three functions of the single ‘independent variable’ p, i.e. the abscissa (i.e. the time) of the b-lognormal’s peak: ⎧ ⎪ ⎪ ⎨ μ( p) =

2

eσ L ts e−2[ M L ( p)−M L (ts)] − 4π N s 2 σ2L ts −[ M L ( p)−M L (ts)] 2 e e√ , 2πN s μ( p)−[σ( p)]2

σ( p) = ⎪ ⎪ ⎩ b( p) = p − e

.

p

σ2L , 2

(15)

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Evolution and Mass Extinctions as Lognormal Stochastic Processes

Fig. 1 Darwinian exponential as the geometric locus of the peaks of b-lognormals. Each blognormal is a lognormal starting at a time (t = b = birth time) in general different from zero and represents a different species that originated at time b of the Darwinian Evolution. That is Cladistics in our Evo-SETI model. It is evident that, the more the generic ‘running b-lognormal’ moves to the right, its peak becomes higher and higher and narrower and narrower, since the area under the b-lognormal always equals 1. Then, the (Shannon) entropy of the running b-lognormal is the degree of evolution reached by the corresponding species (or living being, or civilization, or ET civilization) in the course of evolution

This general form of the Peak-Locus theorem is proven in the Appendix available in the supplementary material by equations (%i28) through (%o44). The remarkable point about all this seems to be the exact separability of all the equations involved in the derivation of (15), a fact that was unexpected to this author when he discovered it around December 2013. And the consequences of this new result are in the applications: 1. For instance in the ‘parabola model’ for mass extinctions that will be studied in the section ‘Mass extinctions described by an adjusted parabola branch’ of this chapter. 2. For instance the Markov–Korotayev cubic that will be studied in the section ‘Markov–Korotayev biodiversity regarded as a lognormal stochastic process having a cubic mean value’ of this chapter. 3. And finally in the many stochastic processes each having a cubic mean value that are just the natural extension into statistics of the deterministic cubics studied by this author in Chap. 10 of his book ‘Mathematical SETI’ [8]. But the study of the entropy of all these cubic lognormal processes has to be deferred to a future research paper. Notice now that, in the particular case of the GBMs having mean value (9) with μGBM = B, and starting at ts = 0 with N 0 = Ns = A, the Peak-Locus theorem (15) boils down to the simpler set of equations

5 Peak-Locus Theorem

183

⎧ 1 ⎪ ⎨ μ( p) = 4π A2 − Bp, 1 σ( p) = √2π A , ⎪ ⎩ b( p) = p − eμ( p)−σ 2 .

(16)

In this simpler form, the Peak-Locus theorem was already published by the author in Maccone [7–9], while its most general form (15) is new for this paper.

6 Entropy as the Evolution Measure The (Shannon) entropy of the running b-lognormal HL ( p) =

# $

√ 1 1 ln 2πσ( p) + μ( p) + ln(2) 2

(17)

is a function of the peak abscissa p and is measured in bits, as common in Shannon’s information theory. By virtue of the Peak-Locus theorem (15), it becomes [see the Appendix available in the supplementary material, (%o45) through (%o50)] 2

HL ( p) =

eσ L ts−2[ M L ( p)−M L (ts)] 4πN s 2

−

σ2L ( p−ts)+2[M L ( p)−M L (ts)] 2

+

ln(2)

1−2 ln(N s) 2

.

(18)

More precisely, (18) is the entropy of the family of ∞1 running b-lognormals (the family’s parameter is p) that are peaked upon the mean value curve (3). Although (3) is not the ‘envelope’ of the b-lognormals (14) in a strict mathematical sense, yet, in the practice, it is approximately the same thing, since it ‘almost envelopes’ all the b-lognormals. This is ‘the greatest result’ of our Evo-SETI model inasmuch as, for instance, in the case of the history of the Western civilizations since the Greeks up to 2200 AD (represented each by a b-lognormal), as shown in Fig. 2, then the ‘enveloping exponential’ is just the GBM mean value exponential (see Maccone [8, 9] for more historic and mathematical details). So it also happens for Darwinian Evolution (see Maccone [7]). The b-lognormal entropy (17) is thus the measure of evolution amount of that b-lognormal: it measures ‘the decreasing disorganization in time of what that blognormal represents’, let it be a cell, a plant, a human or a civilization of humans (since ‘the product of many b-lognormals is one more b-lognormal’), or even of ETs. Entropy is thus disorganization decreasing in time. But would it not be more intuitive to use a measure of ‘increasing organization’ in time? Of course yes. Our Evo_ Index L ( p) = −[HL ( p) − HL (ts)]

(19)

184

Evolution and Mass Extinctions as Lognormal Stochastic Processes

Fig. 2 Shown here are the eight leading civilizations of the Western world in the historic timespan between 800 BC and 2200 AD. Each one is represented by a different b-lognormal given by an Eq. (14), and the ‘envelope’ of all of them is approximately given by the GBM exponential (9) (Peak-Locus theorem). Then, the (Shannon) entropy (17) of each b-lognormal becomes the measure of the degree of evolution reached by each civilization. Please see Maccone [9, pp. 233–239] for more detailed descriptions and calculations

Evo-Index is a function of p that, however, has a minus sign in front, thus changing the decreasing trend of the (Shannon) entropy (17) into the increasing trend of our Evo-Index (19). Also, our Evo-Index starts at zero at the initial time ts: Evo_ Index L (ts) = 0

(20)

Please see the Maxima code in the Appendix to Chapter “OVERCOME Theorem, that is PEAK-LOCUS Theorem, Proven by Maxima”, in particular equation (%o101) there. In the GBM case, i.e. when the mean value (3) is the (Malthusian) exponential curve (9), the b-lognormal entropy (18) becomes just a linear function of time p, HGBM ( p) =

− ln(A) +

1 4πA2

− pB +

1 2

ln(2)

.

(21)

Then, the Evo-Index of GBM simply is a new linear function of time p also Evo_ IndexGBM ( p) =

B ( p − ts). ln(2)

(22)

6 Entropy as the Evolution Measure

185

That is, of course, a straight line starting at the time ts of the origin of life on Earth and increasing linearly thereafter, since it has the positive constant derivative B dEvo_ IndexGBM ( p) = = a positive constant. dp ln(2)

(23)

But this is precisely the linear growth in time of the molecular clock also! So, we have discovered that the entropy of our Evo-SETI model and the molecular clock are the same thing, apart for multiplicative constants (depending on the adopted units, such as bits, seconds, etc.). This is a great conclusion proving that our GBM model for Darwinian Evolution, described in Maccone [7–9] is correct. We are ‘grateful’ to Emil Zuckerkandl, Linus Pauling, Motoo Kimura and his pupils Tomoko Ohta and Takeo Maruyama (‘Neutral Theory of Molecular Evolution’) for bringing the molecular clock to light [12, 13].

7 Evo-SETI Every day astronomers are discovering new extra-solar planets, either by observations from the Earth or by space missions, like ‘CoRoT’ and ‘Kepler’. ‘Gaia’ is now on its way to the Lagrangian point L2 of the Sun–Earth system, and will measure the parallaxes (=distances) of a billion stars in our Galaxy. A recent estimate sets at 40 billion the number of Earth-sized planets orbiting in the habitable zones of Sun-like stars and red dwarf stars within the Milky Way. Thus the assumption that ‘we are alone in the Galaxy’ (let alone the Universe), is becoming simply more and more foolish. SETI is a branch of science trying to detect signatures of intelligent life: either by picking up a radio signal or by detecting an optical pulsating laser, or even by detecting a pulsating beam of neutrinos. SETI scientists are usually elected members of the SETI Permanent Committee of the International Academy of Astronautics (IAA) and, on 3 October 2012, this author was elected Chair of that Committee. So, the author now bears the responsibility to coordinate the world-wide SETI activities, and in this position, formulated the mathematical model of Evo-SETI summarized in this chapter. Why? The model was developed because we only have one example of a civilization (ourselves) that evolved to the point of developing technological capabilities like those required by SETI (radio and optical top instrumentation, supercomputers, etc.). The Drake equation (1961) was the first step in theoretical SETI, and in Maccone [3] the author gave its mathematical extension into probability and statistics. But this is not enough. Suppose that one day the SETI scientists detect an alien message or a proof that Aliens exist, perhaps not too far from us in the Galaxy. What would we do then?

186

Evolution and Mass Extinctions as Lognormal Stochastic Processes

As Chair of the IAA SETI Permanent Committee, this author would firstly ask: how much more advanced than us are those ‘guys’? Especially in technology, if they were able to send signals, or other artefacts capable of being discovered by us. Thus, we need some scientific criterion capable of letting us know the technology gap between us and them, even if only approximately. This author thinks that the answer to this question is the entropy of the running b-lognormal, as described earlier. For instance, in Maccone [9], the author estimated that the technology gap between Aztecs and Spaniards, when they suddenly met in 1519, was about fifty centuries, corresponding to an entropy gap of 3.84 bits per individual. It was this gap that made 20 million Aztecs to have a psychological breakdown and collapse in front of a few thousand ‘much superior’ Spaniards. We must think of that if we want to prepare for the first contact with an alien civilization.

8 Mass Extinctions of Darwinian Evolution Described by a Decreasing GBM 8.1 GBMs to Understand Mass Extinctions of the Past In this section, we describe the use of GBMs to model the mass extinctions that occurred on Earth several times in its geological past. The most notable example probably is the mass extinction of dinosaurs 64 million years ago, now widely recognized by scientists as caused by the impact of a ~10 km sized asteroid where the Chicxulub crater in Yucatan, Mexico, is now found [1, 2]. Incidentally, in 2007 this author was part of a NASA team in charge of studying a space mission capable of deflecting an asteroid off its collision course against the Earth, should this event unfortunately occur again in the future: so he got a background in planetary defense. Let us now go straight to the GBMs and consider the mean value given in the fourth line of Table 2 again, that is the mean value of a GBM increasing in time to simulate the rise of more and more species in the course of evolution, so μ > 0 for it. But in modelling mass extinctions, we clearly must have a decreasing GBM, i.e. μ < 0, over a much shorter time lapse, just years or some centuries instead of billions of years, as in Darwinian Evolution. So, the starting time now is the impact time, ts = t Impact , and our GMB mean value becomes mean_ value(t) = C eμ(t−tImpact ) ,

(24)

where C is a constant that we now determine. Just think that, at the impact time, (24) yields   mean_ value tImpact = C.

(25)

8 Mass Extinctions of Darwinian Evolution Described by a …

187

On the other hand, at the same impact time, one has   mean_ value tImpact = NImpact ,

(26)

where N Impact is the number of living species on Earth just seconds before the asteroid impact time. Thus, (25) and (26) immediately yield C = NImpact .

(27)

This, inserted into (24), yields the final mean value curve as a function of the time mean_ value(t) = NImpact eμ(t−tImpact ) .

(28)

Let us now consider what happens after the impact, namely the death of many living species over a period of time called ‘nuclear winter’ and caused by the debris thrown into the Earth’s atmosphere by the asteroid ejecta. Nobody seems to know exactly how long the nuclear winter lasted after the impact that actually killed all dinosaurs and other species, but not the mammals, who, being much smaller and so much more easy-fed, could survive the nuclear winter. Mathematically, let us call t = t End the time when the nuclear winter ended, so that the overall time span of the mass extinction is given by tEnd − tImpact .

(29)

At time t End , a certain number of living species, say N End , survived. Replacing this into (28) yields NEnd = mean_ value(tEnd ) = NImpact eμ(tEnd −tImpact ) .

(30)

Solving (30) for μ yields the first basic formula for our GBM model of mass extinctions:

N ln NImpact End μ=− . (31) t End − tImpact Notice that in (31) are four input variables   tImpact , NImpact tEnd , NEnd ,

(32)

which we must assign numerically in order determine μ for that particular mass extinction. Let us also remark that it is convenient to introduce two new variables, time_lapse and t Extinction , respectively, defined as the overall amount of time during which the extinction occurs, and the middle instance in this overall time lapse, namely

188

Evolution and Mass Extinctions as Lognormal Stochastic Processes



Time_ Lapse = tEnd − tImpact , t +t tExtinction = Impact2 End .

(33)

Clearly (31), by virtue of (33), becomes μ=−

ln

NImpact NEnd



Time_ Lapse

.

(34)

This version of (31) is easier to differentiate as it only has three independent variables instead of four. Thus, the total differential of (34) is found (but we will not write all the steps here), and, once divided by (34), yields the relative error on μ expressed in terms of the relative errors on N Impact , N End and Time_Lapse: δμ δTime_ Lapse =− μ Time_ Lapse δNImpact δNEnd 1 1



+ − . NImpact N Impact N NEnd Impact ln NEnd ln NEnd

(35)

Let us now find σ. To this end, we must introduce a fifth input [besides the four input variables given by (32)], denoted δN End and representing the standard deviation affecting the number of living species on Earth at the end of the nuclear winter, i.e. when life starts growing again. This means that we must now consider the GBM as a standard deviation function of the time, Δ(t), given by the sixth line in Table 2, which in this case, takes the form  2 (t) = NImpact eμ(t−tImpact ) eσ (t−tImpact ) − 1.

(36)

At the end time, t End (36) becomes  2 (tEnd ) = NImpact eμ(tEnd −tImpact ) eσ (tEnd −tImpact ) − 1.

(37)

But this equals δN End by the very definition of δN End , and so we get the new equation δNEnd = NImpact e

μ(tEnd −tImpact )

 2 eσ (tEnd −tImpact ) − 1.

(38)

This is basically the equation in σ we were seeking. We only have to replace μ into (38) by virtue of (34), and then solve the resulting equation for σ. By doing so (we omit the relevant steps for the sake of brevity), we finally get the sought expression of σ:

8 Mass Extinctions of Darwinian Evolution Described by a …

% # &

2 $ & End & ln 1 + δN NEnd ' σ= . Time_ Lapse

189

(39)

This is the GBM σ for the mass extinctions. Notice that the special δN END = 0 case of (39), immediately yields σ = 0. This is the special case where the GBM ‘converges’ (so as to say) into a single point at t = t END , namely, with probability one there will be exactly N End species that survived the nuclear winter after the impact. This is just like the initial condition of ordinary Brownian motion, B(0) = 0, which is always fulfilled with probability one. But in this case it is a final condition, rather than an initial condition. As such, this particular case of (39) is hardly realistic in the true world of an after-impact. Nevertheless, we wanted to point it out just to show how subtle the mathematics of stochastic processes can be. Another remark following from (39) is about the expression of the relative error on σ, namely δσ/σ, expressed in terms of the four inputs (32) plus δN End . The relevant expression is long and complicated, and we will not rewrite it here. Finally, it must be mentioned that the upper standard deviation curve given by (28) plus (36), i.e. #  $ 2 upper_ st_ dev(t) = NImpact eμ(t−tImpact ) 1 + eσ (t−tImpact ) − 1

(40)

has its maximum at the just-after-impact time

⎡ ( ⎤ 2 2 2 4 1 ⎣ 2μ μ − 2μσ − σ + σ + 3μ ⎦ tImpact + 2 ln . 2  σ σ2 +2μ

(41)

Again, we will not rewrite here all the steps leading to (41), and just confine 2 ourselves to mentioning that one gets a quadratic in eσ t that, solved for t, yields (41). Having given the mathematical theory of mass extinctions provided by GBMs, we now proceed to show a numerical example. Naturally, the chosen example is about the Cretaceous–Paleogene (K–Pg) impact and the ensuing nuclear winter, which we assume to have lasted a thousand years after the impact itself, though other shorter time lapses could be considered as well.

190

Evolution and Mass Extinctions as Lognormal Stochastic Processes

8.2 Example: The K–Pg Mass Extinction Extending Ten Centuries After Impact Readers should be warned that the numeric example and graph we now present is just an exercise, and we do not claim that it really shows what happened 64 million years ago during the K–Pg impact and the consequent mass extinction. Yet it provides useful hints about how the GBMs work in the simulations of true mass extinctions, and not just those of the past, but also those of the future, should an asteroid hit the Earth again and cause millions or billions of human casualties: planetary defence is a ‘must’ for us! So, let us assume that: 1. The K–Pg impact occurred exactly 64 million years ago (just to simplify the calculations): tImpact = −64 × 106 year.

(42)

2. At impact, there were 100 living species on Earth, NImpact = 100.

(43)

Again, this is likely to be very roughly underestimated, but we use 100 so as to immediately draw the percentage of surviving species as described at the next point (3). 3. At the end of the impact effects, there were only 30 living species, and only the 30% survived, NEnd = 30.

(44)

4. We also assume that the error on the value of (44) is about 33.3%. In other words, we assume δNEnd = 10.

(45)

5. Finally, we assume that the impact effects lasted for a 1000 years, i.e. tEnd − tImpact = 1000 year,

(46)

from which, by virtue of (42), we infer tEnd = −63.999 × 106 year.

(47)

These are our five input data. The two outputs then are μ = −3.815 × 10−11 s−1 = −1.204 × 10−3 year−1

(48)

8 Mass Extinctions of Darwinian Evolution Described by a …

191

Fig. 3 The K–Pg mass extinction as a decreasing GBM in the number of living species over 1000 years after impact. The maximum of the upper standard deviation curve has the numeric value −6.3999983 × 107 years given by (41)

and σ2 = 3.339 × 10−12 s−1 = 1.054 × 10−4 year−1 .

(49)

Figure 3 shows the mean value curve (solid blue curve), and the two upper and lower standard deviation curves (solid thin blue curves) for the corresponding GBM in the decreasing number of living species on Earth as the consequence of the impact. In conclusion, Table 4 summarizes all results about the decreasing GBM representing the decreasing number of species on Earth during a mass extinction. In the future, these ideas should be extended not just to the analysis of all mass extinctions that occurred in the geological past of Earth, but also to crucial events in human history such as wars, famines, epidemics and so on, when mass extinctions of humans occurred. An excellent topic to describe mathematically large sections of history that so far were mostly described by means of words only. By doing so, we would significantly contribute to the studies on mathematical history.

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Evolution and Mass Extinctions as Lognormal Stochastic Processes

Table 4 Summary of the properties of the lognormal distribution that applies to the stochastic process N DEC (t) as the exponentially decreasing number of living species on Earth during a mass extinction Stochastic process

NDECREASING (t) ≡ NDEC (t) = Number of living species (in a mass extinction)

Probability distribution

Lognormal distribution of the adjusted and decreasing GBM starting at tImpact   NDEC (t)_pdf n; μ, σ, tImpact , t = ⎧   2 2 σ (t−tImpact ) ⎪n > 0 ln(n)− ln( NImpact )+μ(t−tImpact )− ⎪ 2 ⎪ ⎪ − ⎨t  t 2σ2 (t−tImpact ) Impact e √ √ with 2πσ t−tImpact n ⎪ NImpact > 0 ⎪ ⎪ ⎪ ⎩ σ0

 2  MDEC (t) = ln NImpact + μ − σ2 t − tImpact

Probability density function

Particular MGBM (t) function Mean value curve Variance Standard deviation Upper standard deviation curve Lower standard deviation curve

NDEC (t) ≡ m DEC (t) = NImpact eμ(t−tImpact ) 2

σ2 = N2 e2μ(t−tImpact ) eσ (t−tImpact ) − 1 DEC(t)

Impact

( 2 σ NDEC (t) = NImpact eμ(t−tImpact ) eσ (t−tImpact ) − 1   ( 2 m NDEC (t) + σ NDEC (t) = NImpact eμ(t−tImpact ) 1 + eσ (t−tImpact ) − 1   ( 2 m NDEC (t) − σ NDEC (t) = NImpact eμ(t−tImpact ) 1 − eσ (t−tImpact ) − 1  2  σ2 (t−tImpact )  k k 2 NDEC ekμ(t−tImpact ) e k −k (t) = NImpact

All the moments, i.e. kth moment



Mode (=abscissa of the lognormal peak)

n mode ≡ n peak = NImpact eμ(t−tImpact ) e−

Value of the mode peak

f NDEC(t) (n mode ) =

Median (=fifty-fifty probability value for N(t)) Skewness

(

3σ2 t−tImpact 2

)

2 √ 1√ e−μ(t−tImpact ) eσ (t−tImpact ) NImpact 2πσ t−tImpact

σ2 (t−tImpact ) 2 median = m = NImpact eμ(t−tImpact ) e−  2

σ2 σ K3 e polynomial (t−ts) − 1 e polynomial (t−ts) + 2 3 =

(K 2 ) 2

Kurtosis

K4 (K 2 )2

2 2 2 = e4σ (t−tImpact ) + 2e3σ (t−tImpact ) + 3e2σ (t−tImpact ) − 6

9 Mass Extinctions Described by an Adjusted Parabola Branch

193

9 Mass Extinctions Described by an Adjusted Parabola Branch 9.1 Adjusting the Parabola to the Mass Extinctions of the Past The mass extinction model described in the previous section and based on an adjusted and decreasing GBM has a flaw: the tangent straight line to its mean value curve at the end time is not horizontal. Thus, it is not a realistic model inasmuch as its end time cannot correctly represent the starting point after which the number of living species on Earth started growing once again. On the contrary, the mass extinction model that we build in this section does not have any such flaw: its end time is both the end time of the decreasing number of living species on Earth and also its starting time for increasing living species numbers. Its tangent straight line is indeed horizontal, as required. Getting now over to the mathematics, consider the adjusted mean value curve of L(t) given by the parabola (i.e. second-order polynomial in the adjusted time (t − t Impact ))  2   m parabola (t) = c2 t − tImpact + c1 t − tImpact + c0 .

(50)

In order to find its three unknown coefficients c0 , c1 and c2 , we must resort to the initial and final conditions (i.e. the two boundary conditions of the problem): 

  m parabola tImpact = NImpact , m parabola (tEnd ) = NEnd.

(51)

Inserting (50) into (51), the latter takes the form 

  NImpact = m parabola tImpact = c0 ,  2   NEnd = c2 tEnd − tImpact + c1 tEnd − tImpact + c0 .

(52)

The last two equations reduce to the single one  2   NEnd − NImpact = c2 tEnd − tImpact + c1 tEnd − tImpact .

(53)

On the other hand, the time derivative of the mean value (50) is   dm parabola (t) = 2c2 t − tImpact + c1 . dt

(54)

Equating this to zero, and replacing the time by the end time, we impose that the tangent straight line at the end time must be horizontal. Thus, from (54) one gets:

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Evolution and Mass Extinctions as Lognormal Stochastic Processes

  2c2 tEnd − tImpact + c1 = 0,

(55)

which, solved for c1 and matched to (53), yields 

 2 NEnd − NImpact  = −c2 tEnd  − tImpact , c1 = −2c2 tEnd − tImpact .

(56)

These two linear equations in c1 and c2 may immediately be solved for them, with the result ⎧ ⎨ c2 = NImpact −NEnd2 , (tEnd −tImpact ) (57) ⎩ c1 = −2( NImpact −NEnd ) . tEnd −tImpact Finally, inserting both (57) and the upper Eq. (52) into the mean value parabola (50), the latter takes its final form 

m parabola (t) = NImpact − NEnd



   2 t − tImpact t − tImpact + NImpact .  2 − 2 tEnd − tImpact tEnd − tImpact (58)

One may immediately check that the two boundary conditions (51) are indeed fulfilled by (58). Also, the minimum of the parabola (58) (i.e. the zero of its first time derivative) falls at the end time tend , obviously by construction, i.e. because of (54). So, the parabola (58) is indeed the right curve with a horizontal tangent line at the end, which we were seeking. As for the standard deviation, it is given by the seventh row in Table 3, of course ‘adjusted’ by replacing the time t appearing in Table 3 by the new time difference (t − tImpact ) appearing in the mean value curve (58) already. Thus, the standard deviation for the parabolic mass extinction model is given by  2 σparabola (t) = m parabola (t) eσ (t−tImpact ) − 1.

(59)

Consequently, the upper standard deviation curve is $ #  σ2 (t−tImpact ) m parabola (t) + σparabola (t) = m parabola (t) × 1 + e −1 .

(60)

and the lower standard deviation curve is $ #  2 m parabola (t) − σparabola (t) = m parabola (t) × 1 − eσ (t−tImpact ) − 1 .

(61)

Table 5 shows the statistical properties of our parabolic mass extinction model.

9 Mass Extinctions Described by an Adjusted Parabola Branch

195

Table 5 Summary of the properties of the lognormal distribution that applies to the stochastic process Pparabola (t) = decreasing number of living species on Earth during a mass extinction whose mean value decreases like the left-branch of a parabola between t Impact and t End (the parabola minimum, thus having a horizontal line tangent at t = t End ) Stochastic process

Pparabola (t) = Number of living species (in a parabolic mass extinction)

Probability distribution

Lognormal distribution of the adjusted and parabolic process starting at t Impact

Probability density function Particular Mparabola (t) function Mean value curve (i.e. the parabola) Variance Standard deviation curve Upper standard deviation curve Lower standard deviation curve All the moments, i.e. kth moment





Pparabola (t)_ pdf n; Mparabola (t), σ, t =



− √ 1√ e 2πσ tn

2 ln(n)−Mparabola (t) 2 2σ t

for n > 0

M parabola (t) = $

#  (t−tImpact )2   2 t−tImpact + NImpact − σ2 t − tImpact ln NImpact − NEnd 2 − 2 tEnd −tImpact (tEnd −tImpact ) # $   (t−tImpact )2 t−tImpact m parabola (t) = NImpact − NEnd + NImpact 2 − 2 tEnd −tImpact (tEnd −tImpact ) 2

2 σparabola(t) = m 2parabola (t) eσ (t−tImpact ) − 1 ( 2 σparabola (t) = m parabola (t) eσ (t−tImpact ) − 1   ( 2 m parabola (t) + σparabola (t) = m parabola (t) 1 + eσ (t−tImpact ) − 1

  ( 2 m parabola (t) − σparabola (t) = m parabola (t) 1 − eσ (t−tImpact ) − 1

)

* k Pparabola (t) = $

k  #  σ2   (t−tImpact )2 2 t−tImpact NImpact − NEnd + NImpact e k −k 2 (t−tImpact ) 2 − 2 tEnd −tImpact (tEnd −tImpact )

Mode (=abscissa of the lognormal peak)

n mode ≡ n peak = # $

3σ2 t−t ( Impact )  (t−tImpact )2  t−tImpact 2 + NImpact e− NImpact − NEnd 2 − 2 tEnd −tImpact (tEnd −tImpact )

Value of the mode peak

f NDEC (t) (n mode ) = √

2πσ

√

t−tImpact



e (

)   (t−tImpact )2 −2 t−tImpact +N Impact 2 tEnd −tImpact (tEnd −tImpact )

σ2 t−tImpact

( NImpact −NEnd )

(continued)

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Evolution and Mass Extinctions as Lognormal Stochastic Processes

Table 5 (continued) Stochastic process

Pparabola (t) = Number of living species (in a parabolic mass extinction)

Median (=fifty-fifty probability value for NFIX (t))

median =m= $

σ2 t−t # ( Impact )  (t−tImpact )2  t−tImpact 2 + NImpact e− NImpact − NEnd 2 − 2 tEnd −tImpact (tEnd −tImpact )

Skewness

K3 (K 2 )3/2

Kurtosis

K4 (K 2 )2

(  2 2 = eσ (t−tImpact ) + 2 eσ (t−tImpact ) − 1

2 2 2 = e4σ (t−tImpact ) + 2e3σ (t−tImpact ) + 3e2σ (t−tImpact ) − 6

9.2 Example: The Parabola of the K–Pg Mass Extinction Extending Ten Centuries After Impact At this point it is natural to check our parabolic mass extinction model against the corresponding exponential (i.e. GBM-based) mass extinction model. In order to allow for the perfect match between the two relevant plots, we shall assume that the five numeric input values given in the subsection ‘Important special cases of mL (t) (2)’ for the GBM model are numerically kept just the same for the parabolic model also. Thus, Fig. 4 is obtained for the parabolic K–Pg mass extinction. Actually, we may now superimpose the two plots given by Figs. 3 and 4, respectively, thus obtaining Fig. 5.

10 Cubic as the Mean Value of a Lognormal Stochastic Process 10.1 Finding the Cubic When Its Maximum and Minimum Times Are Given, in Addition to the Five Conditions to Find the Parabola Having completely solved the problem of deriving the equations of the lognormal process L(t) for which the mean value is an assigned parabola, the next step is to derive the cubic (i.e. the third-degree polynomial in t) now assumed to be the mean value of the lognormal process L(t). The relevant calculations are longer than for the parabola case, but still manageable. Unfortunately, similar calculations turn out to be too long and complicated for even higher polynomials like a quartic or a quintic: namely, analytic solutions appear to be prohibitive for polynomials higher than the

10 Cubic as the Mean Value of a Lognormal Stochastic Process

197

Fig. 4 The K–Pg mass extinction as a decreasing parabola in the number of living species over 1000 years after impact. The five numeric input values for this plot are just the same as those used for the construction of Fig. 3 in order to allow a perfect comparison between the two models, exponential (i.e. GBM-based) and parabolic

Fig. 5 Figures 3 and 4 superimposed in order to allow for the perfect comparison between the two models (exponential (i.e. GBM-based) and parabolic) of the K–Pg mass extinction as a decreasing lognormal stochastic process in the number of living species over 1000 years after impact

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Evolution and Mass Extinctions as Lognormal Stochastic Processes

Fig. 6 If we double the horizontal axis time window of Fig. 5, then the result is the current Fig. 6. It clearly shows that the parabolic model (in red) allows for the recovery of life on Earth after the nuclear winter, while the GBM does not so. Thus, the parabolic lognormal process is a better model than the decreasing exponential (GBM) process

cubic considered in this section, and we shall not describe here our failed attempts in this regard. Let us start by writing down the cubic beginning with the starting time ts: Cubic(t) = a(t − ts)3 + b(t − ts)2 + c(t − ts) + d.

(62)

We must determine the cubic’s four coefficients (a, b, c, d) in terms of the following seven inputs: 1. the initial time (starting time) ts; 2. the initial numeric value Ns of the stochastic process L(t) at ts, namely L cubic (ts) = Ns. To be precise, we assume that it is certain (i.e. with probability 1) that the process L(t) will take up the value Ns at the initial time t = ts, and so will its mean value, with a zero standard deviation there; 3. the final time (ending time) te of our lognormal L(t) stochastic process; 4. the final numeric value Ne of the mean value of the stochastic process L(t) at te, namely we define L cubic (te) = N e;

(63)

10 Cubic as the Mean Value of a Lognormal Stochastic Process

199

5. in addition to the assumption (63), we also must assume that L(t) will have a certain standard deviation δNe above and below the mean value (63) at the endtime t = te. These first five inputs are just the same as the five inputs described in the subsection ‘Important special cases of mL (t) (1)’ for the GBMs, and in the subsection ‘Example: the parabola of the K–Pg mass extinction extending ten centuries after impact’ for the parabola model, that in both cases we then used to describe the mass extinctions as stochastic lognormal processes. For the cubic case we introduce two more inputs: 6. the time of the cubic’s maximum, t_Max; and 7. the time of the cubic’s minimum, t_min. It is intuitively clear that, in order to handle the four-coefficient cubic (62), more conditions are necessary than just the previous five conditions, necessary to handle both the two-parameter GBM (9) and the three-coefficient parabola (50). However, it was not initially obvious to this author how many more conditions would have been necessary and especially which ones. The answers to these two questions came out only by doing the actual calculations, as we now describe for the particular case when the two conditions (6) and (7) reveal themselves sufficient to determine the cubic (62) completely. This particular way of determining the cubic is important in the study of Darwinian Evolution as described by the contemporary Russian scientist Andrey Korotayev and his colleague Alexander V. Markov, which we shall study in the next section. Going over to the actual calculations, we notice that, because of the two initial conditions (1) and (2), the cubic (62) yields, respectively 

Cubic(ts) = d, Cubic(ts) = N s.

(64)

These, inserted into the cubic (62), change it to Cubic(t) = a(t − ts)3 + b(t − ts)2 + c(t − ts) + N s.

(65)

We then invoke the two final conditions (3) and (4) that translate into the single equation Cubic(te) = N e.

(66)

In other words, (66) changes the cubic (62) to N e − N s = a(te − ts)3 + b(te − ts)2 + c(te − ts).

(67)

The only three unknowns in (67) are the three still unknown cubic coefficients (a, b, c). But actually (67) is a relationship among these three coefficients (a, b, c). Thus, in reality, we only need two more conditions yielding, for instance, both b and

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Evolution and Mass Extinctions as Lognormal Stochastic Processes

c as functions of a, respectively, and we would then insert them both into (67) that would then become an equation with the only unknown a. Solving that equation for a would then solve the problem completely, yielding then both b and c as functions of all known quantities. So, let us now look for these two still missing conditions on (a, b, c). To this end, the key idea is that every cubic has both a maximum and a minimum. To find them, the cubic’s (62) first derivative with respect to t must be set equal to zero: dCubic(t) = 3a(t − ts)2 + 2b(t − ts) + c = 0. dt

(68)

Solving this quadratic for t yields the two roots: 

t1 = ts + t2 = ts +

√ −b− b2 −3ac √3a −b+ b2 −3ac 3a

= ts + X 1 , = ts + X 2 ,

(69)

having set 

X1 = X2 =

√ −b− b2 −3ac , √3a −b+ b2 −3ac . 3a

(70)

Note that the two Eqs. (69) yield the abscissas (i.e. the instants) of the two stationary points of the quadratic (68), but we do not know which ones, i.e. we do not know which one is the maximum and which one is the minimum. If we suppose that the abscissas (i.e. the instants) of the maximum and the minimum of the cubic (62) are assigned, i.e. they are known, then X1 and X2 are also known, since they are the same as the maximum and the minimum except for the additional time ts, the starting time of the cubic (62). By doing so, we have indeed taken the two conditions (6) and (7) into account. Adding the equations in (70) and then solving for b yields b=−

3a(X 1 + X 2 ) , 2

(71)

that is the expression of b as a function of a that we were seeking. Similarly, multiplying the equations in (70) and then solving for c yields the required expression of c as a function of a: c = 3a X 1 X 2 .

(72)

So, by substituting the two Eqs. (71) and (72) into (67), we get an equation in the only unknown a that is

10 Cubic as the Mean Value of a Lognormal Stochastic Process

201

$ # 3a(X 1 + X 2 ) N e − N s = (te − ts) × a(te − ts)2 − (te − ts) + 3a X 1 X 2 . 2 (73) Solving (73) for a yields a=

2(N e − N s) ,. + 2 (te − ts) 2(te − ts) − 3(X 1 + X 2 )(te − ts) + 6X 1 X 2

(74)

Next we find b by substituting (74) into (71) b=

−3(X 1 + X 2 )(N e − N s) ,, + (te − ts) 2(te − ts)2 − 3(X 1 + X 2 )(te − ts) + 6X 1 X 2

(75)

and we also find c by substituting (74) into (72) c=

6X 1 X 2 (N e − N s) ,. + (te − ts) 2(te − ts)2 − 3(X 1 + X 2 )(te − ts) + 6X 1 X 2

(76)

Thus, the cubic (65) is now obtained by substituting (74)–(76) into (65), with the result Cubic(t) = (N e − N s) ×

, + (t − ts) 2(t − ts)2 − 3(X 1 + X 2 )(t − ts) + 6X 1 X 2 , + N s. + (te − ts) 2(te − ts)2 − 3(X 1 + X 2 )(te − ts) + 6X 1 X 2

(77)

A glance to (77) immediately reveals that both the boundary conditions given by the lower Eqs. (64) and (66), respectively, and indeed fulfilled. But the important point is to notice that the cubic (77) is symmetric in X 1 and X 2 , namely that the cubic (77) does not change at all if X 1 and X 2 are interchanged. Again, this is another way to say that ‘we do not know which one, out of X 1 and X 2 , corresponds to the abscissa of the maximum and the abscissa of the minimum’. The answer to this apparent ‘surprise’ is that it all depends on the factor (Ne–Ns) in front of the fraction in (77): 1. if Ne > Ns then the cubic’s coefficient of t 3 in (77) is positive. Then, the cubic ‘starts’ at −∞, grows up to its maximum, then goes down to its minimum, and finally ‘climbs up again on the right’. In other words, the maximum is reached before the minimum. And this will be the case of the Markov–Korotayev’s cubic of evolution that we shall study in the next section. 2. if Ne < Ns it is the other way round. That is, the cubic ‘starts’ at +∞, gets down to its minimum first, then it climbs up to its maximum, and finally gets down to −∞ on the right. In other words, its minimum comes before its maximum. But there is still a better form of (77) that we wish to point out. This comes from the replacement of X 1 and X 2 in (77) by virtue of the explicit abscissa of the maximum, t_Max, and of the minimum, t_min, related to X 1 and X 2 via (69), that is (assuming for simplicity that Ne > Ns as in the Markov–Korotayev case):

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Evolution and Mass Extinctions as Lognormal Stochastic Processes



√

tMax = ts + −b−√3ab −3ac = ts + X 1 , 2 tmin = ts + −b+ 3ab −3ac = ts + X 2 , 2

(78)

from which one gets 

X 1 = tMax − ts, X 2 = tmin − ts.

(79)

Thus, inserting (79) into (77), we reach our final version of the cubic mean value of the L(t) lognormal stochastic process      (t − ts) 2(t − ts)2 − 3 tMax + tmin − 2ts (t − ts) + 6(tMax − ts) tmin − ts  Cubic(t) = (N e − N s)     + N s. (te − ts) 2(te − ts)2 − 3 tMax + tmin − 2ts (te − ts) + 6(tMax − ts) tmin − ts

(80) Our reader might have noticed that the condition (5) was not used to derive the cubic (80). This is because the condition (5) does not affect the cubic (80): it only affects the standard deviation σCubic (t) and the two corresponding upper and lower standard deviation curves above and below the mean value cubic (80). This fact is evident from Eq. (3) clearly showing that the time function M L (t) and the positive parameter σL have nothing to do with each other, i.e. they are independent of each other, just as the mean value and the variance of the Gaussian (normal) distribution are totally independent of each other. Thus, going back to Eq. (39), we conclude that it does not hold good for GBMs only, but rather it applies to all lognormal stochastic processes, whatever their mean value mL (t) might possibly be. In conclusion, the positive parameter σ is determined by (39) just rewritten in this section’s notation:

σ=

%   & & ln 1+ δN e 2 ' Ne te − ts

.

(81)

We are now ready to write down the two equations of the upper and lower standard deviation curves. They are actually the same as the two equations at the seventh and eighth lines in Table 1, which we re-write here in the current ‘cubic’ notation: ⎧ curve(t) = m Cubic(t) + σCubic(t) ⎪ ⎪ upper_ standard_  deviation_  ( ⎪ ⎪ ⎨ = Cubic(t) 1 + eσ2 (t−ts) − 1 , ⎪ lower_ standard_ curve(t) = m Cubic(t) − σCubic(t) ⎪  deviation_  ( ⎪ ⎪ ⎩ = Cubic(t) 1 − eσ2 (t−ts) − 1 .

(82)

11 Markov–Korotayev Biodiversity Regarded as a Lognormal …

203

11 Markov–Korotayev Biodiversity Regarded as a Lognormal Stochastic Process Having a Cubic Mean Value 11.1 Markov–Korotayev’s Work on Evolution Let us now refer to the interesting Wikipedia site http://en.wikipedia.org/wiki/And rey_Korotayev, whom we quote verbatim. According to this, in 2007–2008 the Russian scientist Andrey Korotayev, in collaboration with Alexander V. Markov, showed that a ‘hyperbolic’ mathematical model can be developed to describe the macrotrends of biological evolution. These authors demonstrated that changes in biodiversity through the Phanerozoic correlate much better with the hyperbolic model (widely used in demography and macrosociology) than with the exponential and logistic models (traditionally used in population biology and extensively applied to fossil biodiversity as well). The latter models imply that changes in diversity are guided by a first-order positive feedback (more ancestors, more descendants) and/or a negative feedback arising from resource limitation. Hyperbolic model implies a second-order positive feedback. The hyperbolic pattern of the world population growth has been demonstrated by Korotayev to arise from a second-order positive feedback between the population size and the rate of technological growth. According to Korotayev and Markov, the hyperbolic character of biodiversity growth can be similarly accounted for by a feedback between the diversity and community structure complexity. They suggest that the similarity between the curves of biodiversity and human population probably comes from the fact that both are derived from the interference of the hyperbolic trend with cyclical and stochastic dynamics [10, 11]. This author was astounded by Fig. 7 (taken from Wikipedia) showing the increase, but not monotonic increase, of the number of genera (in thousands) during the 542 million years making up for the Phanerozoic. Thus, this author came to wonder whether the red curve in Fig. 7 could be regarded as the cubic mean value curve of a lognormal stochastic process, just as the exponential mean value curve is typical of GBMs. This author’s answer to the above question is ‘yes’: we may indeed use our cubic (80) to represent the red line in Fig. 5, thus reconciling the Markov–Korotayev theory with our theory requiring that the profile curve of evolution must be the cubic mean value curve of a certain lognormal stochastic process (and certainly not a GBM in this case). Let us thus consider the following numerical inputs to the cubic (80) that we derive ‘by a glance to Fig. 7’ (the precision of these numerical inputs is really unimportant at this early stage of ‘matching’ the two theories, ours and the Markov– Korotayev’s, since we are just looking for the ‘proof of concept’, and better numeric approximations might follow in the future):

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Evolution and Mass Extinctions as Lognormal Stochastic Processes

Fig. 7 During the Phanerozoic the biodiversity shows a steady but not monotonic increase from near zero to several thousands of genera

⎧ ts = −530, ⎪ ⎪ ⎨ N s = 1, ⎪ te = 0, ⎪ ⎩ N e = 4000.

(83)

In other words, the first two equations in (83) mean that the first of the genera appeared on Earth about 530 million years ago, i.e. before that time the number of genera on Earth was zero. Also, the last two equations in (83) mean that at the present time t = 0, the number of genera on Earth is 4000 on average. Now, ‘on average’ means that, nowadays, a standard deviation of about 1000 (plus or minus) affects the average value of 4000. This is shown in Fig. 7 by the grey stochastic process called ‘all genera’. And this is re-phrased mathematically by invoking the condition (5) of subsection ‘Finding the cubic when its maximum and minimum times are given, in addition to the five conditions to find the parabola’, and assigning the fifth numeric input δN e = 1000.

(84)

Then, as a consequence of the four numeric boundary inputs (83) plus the standard deviation on the current value of genera (84), Eq. (81) yields the numeric value of the positive parameter σ,

σ=

%   & & ln 1 +  δN e 2 ' Ne te − ts

= 0.011.

(85)

11 Markov–Korotayev Biodiversity Regarded as a Lognormal …

205

Having thus assigned numeric values to the first five conditions of the subsection ‘Finding the cubic when its maximum and minimum times are given, in addition to the five conditions to find the parabola’, only conditions (6) and (7) remain to be assigned. These are the two abscissae of the maximum and minimum, respectively, which at a glance at Fig. 7 makes us establish as (of course in millions of years ago) 

tMax = −400, tmin = −220.

(86)

Inserting these seven numeric inputs into the cubic (80) and into both the Eq. (82) of the upper and lower standard deviation curves, the final plot shown in Fig. 8 is produced.

Fig. 8 Our cubic mean value curve (thick red solid curve) plus and minus the two standard deviation curves (thin solid blue curves) give more mathematical information than just the previous Fig. 7. In fact, we now have the two standard deviation curves of the lognormal stochastic process L(t) that are completely missing in the Markov–Korotayev theory and in their plot shown in Fig. 7. We thus claim that our cubic mathematical theory of the lognormal stochastic process L(t) is a ‘more profound mathematization’ than the Markov–Korotayev theory of evolution since it is stochastic, rather than just deterministic. This completes our ‘stochastic extension’ of the Markov–Korotayev evolution model

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Evolution and Mass Extinctions as Lognormal Stochastic Processes

12 Conclusions Let us finally reach the conclusions of this chapter: 1. In section ‘A summary of the ‘Evo-SETI’ model of evolution and SETI’ we showed how to ‘construct’ a lognormal stochastic process L(t) whose mean is an assigned and ‘arbitrary’ function of the time mL (t). This is of paramount importance for all future applications. 2. In the practice, this ‘arbitrary’ mean time mL (t) may be either an exponential N 0 eμt , and then the corresponding lognormal process L(t) is the well-known polynomial_degr ee ck t k , GBM, or it may be a polynomial function of the time, k=0 and then we have shown how to compute all the statistical properties of the corresponding lognormal process L(t). 3. In particular, we have given an important application of this duality (either exponential or polynomial assumed as mean value) in the case of the mass extinctions that plagued the development of life on Earth several times over the last 3.5 billion years. Our result is that the parabolic model is preferable to the GBM model for mass extinctions, inasmuch as the possibility of the recovery of life (as indeed it always happened on Earth) is in the parabolic model, but not in the GBM one. 4. Finally, we compared our last ‘stochastic cubic’ result with the Markov–Korotayev model of evolution of life on Earth based on a cubic-shaped mean value function for L(t). We conclude that their model and ours agree quite well, but ours is ‘mathematically more profound’ since it also provides both upper and lower standard deviation curves that are not present in the Markov–Korotayev model since it is deterministic, rather than stochastic, like ours. In conclusion, we have uncovered an important generalization of GBMs into a lognormal stochastic process L(t) having an arbitrary mean, rather than just an exponential one. These results should pave the way for a future understanding of the evolution of life on exoplanets on the basis of the way the evolution of life unfolded on Earth over the last 3.5 billion years. That will be the goal of our research papers.

Supplementary Material Supplementary materials of this paper is available at http://dx.doi.org/10.1017/S14 7355041400010X.

References

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References 1. L.W. Alvarez, W. Alvarez, F. Asaro, H.V. Michel, Extraterrestrial cause for the CretaceousTertiary extinction. Science 208(4448), 1095–1108 (1980) 2. W. Alvarez, In the Mountains of Saint Francis: Discovering the Geologic Events that Shaped Our Earth, a popular book available in Kindle edition (2008) 3. C. Maccone, The statistical Drake equation. Paper #IAC-08-A4.1.4 presented on 1st Oct 2008, at the 59th International Astronautical Congress (IAC) held in Glasgow, Scotland, UK, 29 Sept–3 Oct 2008 4. C. Maccone, The statistical Drake equation. Acta Astronaut. 67, 1366–1383 (2010) 5. C. Maccone, The statistical Fermi paradox. J. Br. Interplanet. Soc. 63, 222–239 (2010) 6. C. Maccone, SETI and SEH (statistical equation for habitables). Acta Astronaut. 68, 63–75 (2011) 7. C. Maccone, A mathematical model for evolution and SETI. Orig. Life Evol. Biosph. 41, 609–619, available online Dec 3rd (2011) 8. C. Maccone, Mathematical SETI, 2012 edn. (Praxis-Springer, Chichester, Heidelberg, 2012), p. 724. ISBN-10:3642274366 | ISBN-13: 9783642274367 9. C. Maccone, SETI, evolution and human history merged into a mathematical model. Int. J. Astrobiol. 12(3), 218–245 (2013). Available online since 23 Apr 2013 10. A. Markov, A. Korotayev, Phanerozoic marine biodiversity follows a hyperbolic trend. Palaeoworld 16(4), 311–318 (2007) 11. A. Markov, A. Korotayev, Hyperbolic growth of marine and continental biodiversity through the Phanerozoic and community evolution. J. Gen. Biol. 69(3), 175–194 (2008) 12. M. Nei, Mutation-Driven Evolution (Oxford University Press, Oxford, 2013) 13. M. Nei, K. Sudhir, Molecular Evolution and Phylogenetics (Oxford University Press, Oxford and New York, 2000)

New Evo-SETI Results About Civilizations and Molecular Clock

Abstract In two papers (Maccone in Int J Astrobiol 12(3):218–245, 2013 [2]; Maccone in Int J Astrobiol 13(4):290–309, 2014 [3]) as well as in the book (Maccone in Mathematical SETI. Praxis, Chichester; Springer, Berlin, in the Fall of 2012, 724 p, 2012 [1]), this author described the Evolution of life on Earth over the last 3.5 billion years as a lognormal stochastic process in the increasing number of living Species. In (Maccone in Mathematical SETI. Praxis, Chichester; Springer, Berlin, in the Fall of 2012, 724 p, 2012 [1]; Maccone in Int J Astrobiol 12(3):218–245, 2013 [2]), the process used was ‘Geometric Brownian Motion’ (GBM), largely used in Financial Mathematics (Black-Sholes models). The GBM mean value, also called ‘the trend’, always is an exponential in time and this fact corresponds to the so-called ‘Malthusian growth’ typical of population genetics. In (Maccone in Int J Astrobiol 13(4):290–309, 2014 [3]), the author made an important generalization of his theory by extending it to lognormal stochastic processes having an arbitrary trend mL (t), rather than just a simple exponential trend as the GBM have. The author named ‘Evo-SETI’ (Evolution and SETI) his theory inasmuch as it may be used not only to describe the full evolution of life on Earth from RNA to modern human societies, but also the possible evolution of life on exoplanets, thus leading to SETI, the current Search for ExtraTerrestrial Intelligence. In the Evo-SETI Theory, the life of a living being (let it be a cell or an animal or a human or a Civilization of humans or even an ET Civilization) is represented by a b-lognormal, i.e. a lognormal probability density function starting at a precise instant b (‘birth’) then increasing up to a peaktime p, then decreasing to a senility-time s (the descending inflexion point) and then continuing as a straight line down to the death-time d (‘finite b-lognormal’). 1. Having so said, the present chapter describes the further mathematical advances made by this author in 2014–2015, and is divided in two halves: Part One, devoted to new mathematical results about the History of Civilizations as b-lognormals, and 2. Part Two, about the applications of the Evo-SETI Theory to the Molecular Clock, well known to evolutionary geneticists since 50 years: the idea is that our EvoEntropy grows linearly in time just as the molecular clock.

© Springer Nature Switzerland AG 2020 C. Maccone, Evo-SETI, https://doi.org/10.1007/978-3-030-51931-5_5

209

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New Evo-SETI Results About Civilizations and Molecular Clock

(a) Summarizing the new results contained in this paper: In Part One, we start from the History Formulae already given in [1, 2] and improve them by showing that it is possible to determine the b-lognormal not only by assigning its birth, senility and death, but rather by assigning birth, peak and death (BPD Theorem: no assigned senility). This is precisely what usually happens in History, when the life of a VIP is summarized by giving birth time, death time, and the date of the peak of activity in between them, from which the senility may then be calculated (approximately only, not exactly). One might even conceive a b-scalene (triangle) probability density just centred on these three points (b, p, d) and we derive the relevant equations. As for the uniform distribution between birth and death only, that is clearly the minimal description of someone’s life, we compare it with both the blognormal and the b-scalene by comparing the Shannon Entropy of each, which is the measure of how much information each of them conveys. Finally we prove that the Central Limit Theorem (CLT) of Statistics becomes a new ‘E-Pluribus-Unum’ Theorem of the Evo-SETI Theory, giving formulae by which it is possible to find the b-lognormal of the History of a Civilization C if the lives of its Citizens C i are known, even if only in the form of birth and death for the vast majority of the Citizens. (b) In Part Two, we firstly prove the crucial Peak-Locus Theorem for any given trend mL (t) and not just for the GBM exponential. Then we show that the resulting Evo-Entropy grows exactly linearly in time if the trend is the exponential GMB trend. (c) In addition, three Appendixes (online) with all the relevant mathematical proofs are attached to this paper. They are written in the Maxima language, and Maxima is a symbolic manipulator that may be downloaded for free from the web. In conclusion, this chapter further increases the huge mathematical spectrum of applications of the Evo-SETI Theory to prepare Humans for the first Contact with an Extra-Terrestrial Civilization. Keywords Darwinian evolution · Entropy · Geometric Brownian motion · Lognormal probability densities · Molecular clock · SETI

1 Part I: New Results About Civilizations in Evo-SETI Theory 1.1 Introduction Two mathematical papers were published by this author in 2013 and 2014, respectively:

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1. ‘SETI, Evolution and Human History Merged into a Mathematical Model’, International Journal of Astrobiology, vol. 12, issue (3), pp. 218–245 (2013) (this will be called [2] in the sequel of the current chapter) and 2. ‘Evolution and Mass Extinctions as Lognormal Stochastic Processes’, International Journal of Astrobiology, vol. 13, issue (4), pp. 290–309 (2014) (this will be called [3] in the sequel of the current chapter). They provide the mathematical formulation of the ‘Evo-SETI Theory’, standing for ‘a unified mathematical Theory of Evolution and SETI’. Hoverer, the calculations required to prove all Evo-SETI results are lengthy, and this circumstance may unfortunately ‘scare’ potential readers that would love to understand Evo-SETI, but do not want to face all the calculations. To get around this obstacle, the three Appendixes at the end of this chapter are a printout of all the analytical calculations that this author conducted by the Maxima symbolic manipulator, especially to prove the Peak-Locus Theorem described in Section ‘Peak-Locus Theorem’. It is interesting to point out that the Macsyma symbolic manipulator or ‘computer algebra code’ (of which Maxima is a large subset) was created by NASA at the Artificial Intelligence Laboratory of MIT in the 1960s to check the equations of Celestial Mechanics that had been worked out by hand by a host of mathematicians in the previous 250 years (1700–1950). Actually, those equations might have contained errors that could have jeopardized the Moon landings of the Apollo Program, and so NASA needed to check them by computers, and Macsyma (nowadays Maxima) did a wonderful job. Today, everyone may download Maxima for free from the website http://maxima.sourceforge.net/. The Appendixes of this paper are written in Maxima language and the conventions apply of denoting the input instructions by (%i[equation number]) and the output results by (%o[equation number]), as we shall see in a moment. In conclusion, in order to allow non-mathematically trained readers to appreciate this unified vision of how life developed on Earth over the last 3.5 billion years, a ‘not-too mathematical’ summary of the content of these two papers is now provided, also enabling readers to grasp the wide spectrum of Evo-SETI applications.

1.2 A Simple Proof of the b-Lognormal’s pdf This chapter is based on the notion of a b-lognormal, just as are [2, 3]. To let this paper be self-contained in this regard, we now provide an easy proof of the b-lognormal equation as a probability density function (pdf). Just start from the well-known Gaussian or normal pdf e−(x−μ) /(2σ ) . √ 2πσ 2

Gaussian_ or_ normal(x; μ, σ ) =

2

(1)

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New Evo-SETI Results About Civilizations and Molecular Clock

This pdf has two parameters: 1. μ turns out to be the mean value of the Gaussian and the abscissa of its peak. Since the independent variable x may take up any value between −∞ and +∞, i.e. it is a real variable, so μ must be real too. 2. σ turns out to be the standard deviation of the Gaussian and so it must be a positive variable. 3. Since the Gaussian is a pdf, it must fulfil the normalization condition ∞ −∞

e−(x−μ) /(2σ ) dx = 1 √ 2πσ 2

2

(2)

and this is the equation we need in order to ‘discover’ the b-lognormal. Just perform in the integral (2) the substitution x = lnt (where ln is the natural log). Then (2) is turned into the new integral 

∞ 0

e−(ln t−μ) /(2σ ) dt = 1. √ 2πσt 2

2

(3)

But this (3) may be regarded as the normalization condition of another random variable, ranging ‘just’ between zero and +∞, and this new random variable we call ‘lognormal’ since it ‘looks like’ a normal one except that x is now replaced by ln t and t now also appears at the denominator of the fraction. In other words, the lognormal pdf is 

−(ln(t)−μ)2 /(2σ2 )

lognormal(t; μ, σ) = e √2πσ·t , holding for 0 ≤ t < ∞, −∞ < μ < ∞, σ ≥ 0.

(4)

Just one more step is required to jump from the ‘ordinary lognormal’ (4) (i.e. the lognormal starting at t = 0) to the b-lognormal, that is the lognormal starting at any positive instant b > 0 (‘b’ stands for ‘birth’). Since this simply is a shifting along the time axis from 0 to the new time origin b > 0, in mathematical terms it means that we have to replace t by (t − b) everywhere in the pdf (4). Thus, the b-lognormal pdf must have the equation 

b_lognormal(t; μ, σ, b) =

2

/(2σ e−(ln(t−b)−μ) √ 2πσ·(t−b)

holding for t ≥ b and up to t = ∞.

2)

(5)

The b-lognormal (5) is called ‘three-parameter lognormal’ by statisticians, but we prefer to call it b-lognormal to stress its biological meaning described in the next section.

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1.3 Defining ‘Life’ in the Evo-SETI Theory The first novelty brought by our Evo-SETI Theory is our definition of life as a ‘finite b-lognormal in time’, extending from the time of birth (b) to the time of death (d) of the living creature, let it be a cell, an animal, a human, a civilization of humans or even an Extra-Terrestrial (ET) civilization. Figure 1 shows what we call a ‘finite b-lognormal’. On the horizontal axis is the time t ranging between b and d. But the curve on the vertical axis is actually made up by two curves: 1. Between b and the ‘senility’ time s (i.e. the descending inflexion point of the curve) on the vertical axis are the positive numerical values taken up by the pdf (5), that we prefer to call ‘infinite b-lognormal’ to distinguish it from the ‘finite b-lognormal’ shown in Fig. 1. 2. Between s and d the curve is just a straight line having the same tangent at s as the b-lognormal (5). We are not going to derive its equation since that would take too long, but its meaning is obvious: since nobody lives for an infinite amount of time, it was necessary to ‘cut’ the infinite b-lognormal (5) at the junction point s and continue it with a simple straight line finally intercepting the time axis at the death instant d. As easy as that.

Fig. 1 ‘Life’ in the Evo-SETI Theory is a ‘finite b-lognormal’ made up by a lognormal pdf between birth b and senility (descending inflexion point) s, plus the straight tangent line at s leading to death d

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New Evo-SETI Results About Civilizations and Molecular Clock

1.4 History Formulae Having so defined ‘life’ as a finite b-lognormal, this author was able to show that, given one’s birth b, death d and (somewhere in between) one’s senility s, then the two parameters μ (a real number) and σ (a positive number) of the b-lognormal (5) are given by the two equations 

d−s √ σ = √d−b , s−b μ = ln(s − b) +

(d−s)(b+d−2s) . (d−b)(s−b)

(6)

These were called ‘History Formulae’ by this author for their use in Mathematical History, as shown in the next section. The mathematical proof of (6) is found in [2, pp. 227–231] and follows directly from the definition of s (as descending inflexion point) and d (as interception between the descending tangent straight line at s and the time axis). In previous versions of his Evo-SETI Theory, the author gave an apparently different version of the History Formulae (6) reading 

σ=

√ d−s √ , d−b s−b

μ = ln(s − b) +

2s 2 −(3d+b)s+d 2 +b d . (d−b)s−b d+b2

(7)

This simply was because he had not yet factorized the fraction of the second equation (with apologies).

1.5 Death Formula One more interesting result discovered by this author, and firstly published by him in 2012 [1, Chap. 6, Eq. (6.30), p. 163] is the following ‘Death Formula’ (its proof is obtained by inserting the History Formulae (6) into the equation for the peak abscissa, 2 p = b + eμ−σ ): d=

s + b · ln(( p − b)/(s − b)) . ln(( p − b)/(s − b)) + 1

(8)

This formula allows one to compute the death time d if the birth time b, the peak time p and the senility time s are known. The difficulty is that, while b and p are usually well known, s is not so, thus jeopardizing the practical usefulness of the Death Formula (8).

1 Part I: New Results About Civilizations in Evo-SETI Theory

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1.6 Birth–Peak–Death (BPD) Theorem This difficulty of estimating s for any b-lognormal led the author to discover the following BPD theorem that he only obtained on April 4, 2015. Ask the question: can a given b-lognormal be entirely characterized by the knowledge of its birth, peak and death only? Yes is the answer, but no exact formula exists yielding s in terms of (b, p, d) only. Proof Start from the exact Death Formula (8) and expand it into a Taylor series with respect to s around p and, say, to order 2. The result given by Maxima is d = p + 2(s − p) −

3(s − p)2 + ··· 2(b − p)

(9)

Equation (9) is quadratic equation in s that, once solved for s, yields the secondorder approximation for s in terms of (b, p, d) √  2 − p 2 + (3d − b) p − 3bd + 2b2 + p + 2b s= . 3

(10)

In the practice, Eq. (10) is a ‘reasonable’ numeric approximation yielding s as a function of (b, p, d), and is certainly much better that the corresponding first-order approximation given by the linear equation d = p + 2(s − p) + · · ·

(11)

whose solution simply is s=

p+d , 2

(12)

i.e., s (to first approximation) simply is the middle point between p and d, as geometrically obvious. However, if one really wants a better approximation than the quadratic one (10), it is possible to expand the Death Formula (8) into a Taylor series with respect to s around p to third order, finding d = p + 2(s − p) +

5(s − p)3 3(s − p)2 + + ··· 2( p − b) 6( p − b)2

(13)

Equation (13) is a cubic (i.e. third-degree polynomial) in s that may be solved for s by virtue of the well-known Cardan (Girolamo Cardano 1501–1576) formulae that we will not repeat here since they are exact but too lengthy to be reproduced in this chapter.

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New Evo-SETI Results About Civilizations and Molecular Clock

As a matter of fact, it might even be possible to expand the Death Formula (8) to fourth order in s around p that would lead to the fourth-degree algebraic equation (a quartic) in s d = p + 2(s − p) +

5(s − p)3 (s − p)4 3(s − p)2 + + + ··· 2( p − b) 6( p − b)2 2( p − b)3

(14)

and then solve Eq. (14) for s by virtue of the exact four formulae of Lodovico Ferrari (1522–1565) (he was Cardan’s pupil) that are huge and occupy a whole page each one. However, this game may not go on forever: the fifth-degree algebraic equation is not solvable by virtue of radicals and so we must stop with degree 4. Then there is the problem of finding which one, out of the three (Cardan) or four (Ferrari) roots numerically is ‘the right one’. This author thus wrote a Maxima code given here as #1 Appendix to this paper where he solved several cases of finding s from (b, p, d) related to the important Fig. 2 of this paper. In other words, the inputs to Table 1 of this paper were (b, p, d) and not (b, s, d), as the author had always done previously, for instance in deriving the whole of Chap. 7 of [1] back in 2012. This improvement is remarkable since it allowed a fine-tuning of Table 1 with respect to all similar previous material. In other words still, ‘it is easier to assign birth, peak and death rather than birth, senility and death’. That’s why the Theorem described in this section was called BPD Theorem. The reader is invited to ponder over Appendix 1 as the key to all further, future developments in Mathematical History.

Fig. 2 The b-lognormals of nine Historic Western Civilizations computed thanks to the History Formulae (6) with the three numeric inputs for b, p and d of each Civilization given by the corresponding line in Table 1. The corresponding s is derived from b, p and d by virtue of the second-order approximation (10) provided by the solution of the quadratic equation in the BPD theorem

3100 bc Lower and Upper Egypt unified First Dynasty

776 bc First Olympic Games, from which Greeks compute years

753 bc Rome founded Italy seized by Romans by 270 bc, Carthage and Greece by 146 bc, Egypt by 30 bc. Christ born around the year 0

1250 Frederick II dies Middle Ages end Free Italian towns start Renaissance.

1419 Madeira island discovered African coastline explored by 1498

Ancient Egypt

Ancient Greece

Ancient Rome

Renaissance Italy

Portuguese Empire

b = birth time

s = decline = senility time

1562 Council of Trent ends in 1563 Catholic and Spanish rule.

273 ad Aurelian builds new walls around Rome after Military Anarchy, 235–270 ad

293 bc Alexander the Great dies in 323 Hellenism starts in Near East

1716 1822 Black slave trade to Brazil Brazil independent, other at its peak colonies retained Millions of blacks enslaved or killed

1497 Renaissance art and architecture Birth of Science Copernican revolution (1543)

117 ad Rome at peak: Trajan in Mesopotamia Christianity preached in Rome by Saints Peter and Paul against slavery by 69 ad

438 bc Pericles’ Age Democracy peak Arts and Science peak. Aristotle

1154 bc 689 bc Luxor and Karnak temples Assyrians invade Egypt in edified by Ramses II by 671 bc, leave 669 bc 1260 bc

p = peak time

2.193 × 10−3

2.488 × 10−3

8.313 × 10−4

P = peak ordinate

(continued)

1999 3.431 × 10−3 Last colony, Macau, lost to Republic of China

1660 5.749 × 10−3 Cimento Academy ended Bruno burned at stake in 1600 Galileo dies in 1642

476 ad Western Roman Empire ends Dark Ages start in West Not in East

30 bc Cleopatra dies: last Hellenistic queen

30 bc Cleopatra dies: last Hellenistic queen

d = death time

Table 1 Birth, peak, decline and death times of nine Historic Western Civilizations (3100 bc–2035 ad), plus the relevant peak heights

1 Part I: New Results About Civilizations in Evo-SETI Theory 217

1588 Spanish Armada Defeated British Empire’s expansion starts

1898 Philippines, Cuba, Puerto Rico seized from Spain

British Empire

USA Empire

They are shown in Fig. 2 as nine b-lognormal pdfs

1524 Verrazano first in New York bay Cartier in Canada, 1534

French Empire

b = birth time

1402 Canary islands are conquered by 1496 Columbus discovers America in 1492

Spanish Empire

Table 1 (continued)

1962 Algeria lost as most colonies Fifth Republic starts in 1958

1898 Last colonies lost to the USA: Philippines, Cuba and Puerto Rico

d = death time

2035 Singularity: computers ruling? Will the USA yield to China?

1947 1974 After World Wars One and Britain joins the EEC Two, India gets independent and loses most of her colonies

1870 Napoleon III defeated Third Republic starts World Wars One and Two

1844 Spanish fleet lost at Trafalgar in 1805

s = decline = senility time

1972 2001 Moon Landings, 1969–72: 9/11 Terrorist attacks: America leads the world decline. Obama 2009

1904 British Empire peak Top British Science: Faraday, Maxwell, Darwin, Rutherford

1812 Napoleon I dominates continental Europe and reaches Moscow

1798 Largest extent of Spanish colonies in America: California settled since 1769

p = peak time

0.013

8.447 × 10−3

4.279 × 10−3

5.938 × 10−3

P = peak ordinate

218 New Evo-SETI Results About Civilizations and Molecular Clock

1 Part I: New Results About Civilizations in Evo-SETI Theory

219

1.7 Mathematical History of Nine Key Civilizations Since 3100 BC The author called (6) the ‘History Formulae’ since in [2, pp. 231–235], Eq. (6), with the numerical values provided there, allowed him to draw the b-lognormals of eight leading civilizations in Western History: Greece, Rome, Renaissance Italy, and the Portuguese, Spanish, French, British and American (USA) Empires. Please notice that: 1. The data in Table 1 and the resulting b-lognormals in Fig. 2 are experimental results, meaning that we just took what described in History textbooks (with a lot of words) and translated that into the simple b-lognormals shown in Fig. 2. In other words, a new branch of knowledge was forged: we love to call it ‘Mathematical History’. More about this in future papers. 2. The envelope of all the above b-lognormals ‘looks like’ a simple exponential curve. In Fig. 2, two such exponential envelopes were drawn: the one going from the peak of Ancient Greece (the Pericles age in Athens, cradle of Democracy) to the peak of the British Empire (Victorian age, the age of Darwin and Maxwell) and to the peak of the USA Empire (Moon landings in 1969–1972). This notion of b-lognormal envelope will later be precisely quantified in our ‘Peak-Locus Theorem’. 3. It is now high time to introduce a ‘measure of evolution’ namely a function of the three parameters μ, σ and b accounting for the fact that ‘the experimental Fig. 2 clearly shows that, the more the time elapses, the more highly peaked, and narrower and narrower, the b-lognormals are’. In [2, pp. 238–243], this author showed that the requested measure of evolution is the (Shannon) entropy, namely the entropy of each infinite b-lognormal that fortunately has the simple equation √ 1 Hinfinite_b-lognormal (μ, σ) = ln( 2πσ) + μ + . 2

(15)

The proof of this result was given in [2, pp. 238–239]. If measured in bits, as customary in Shannon’s Information Theory, Eq. (15) becomes √ ln( 2πσ) + μ + 1/2 Hinfinite_ b−lognormal_ in_ bits (μ, σ) = . ln 2

(16)

This is the b-lognormal entropy definition that was used in [2, 3] and we are going to use in this paper also. In reality, Shannon’s entropy is a measure of the disorganization of an assigned pdf f X (x), rather than a measure of its organization. To change it into a measure of organization, we should just drop the minus sign appearing in front of the Shannon definition of entropy for any assigned pdf f X (x):

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New Evo-SETI Results About Civilizations and Molecular Clock

∞ H =−

f X (x) · ln f X (x)dx

(17)

−∞

We will do so to measure Evolution of life on Earth over the last 3.5 billion years. The final goal of all these mathematical studies is of course to ‘prepare’ the future of Humankind in SETI, when we will have to face other Alien Civilizations whose past may be the future for us.

1.8 b-Scalene (Triangular) Probability Density Having recognized that BPD (and not birth–senility–death) are the three fundamental instants in the lifetime of any living creature, we are tempted to introduce a new pdf called b-scalene, or, more completely, b-scalene triangular pdf. The idea is easy: 1. 2. 3. 4. 5. 6.

The horizontal axis is the time axis, denoted by t. The vertical axis is denoted by y. The b-scalene pdf starts at the instant b ‘birth’. The b-scalene pdf ends at the instant d ‘death’. Somewhere in between is located the pdf peak, having the coordinates (p, P). The pdf between (b, 0) and (p, P) is a straight line, hereafter called ‘first b-scalene’ (line). 7. The pdf between (p, P) and (d, 0) is a straight line, hereafter called ‘second b-scalene’ (line). Let us now work out the equations of the b-scalene. First of all its normalization condition implies that the sum of the areas of the two triangles equals 1: (d − p)P ( p − b)P + = 1. 2 2

(18)

Solving Eq. (18) for P we get P=

2 . d −b

(19)

Then, the equations of the two straight lines making up the b-scalene pdf are found to be, respectively: y= and

2(t − b) , for b ≤ t ≤ p (d − b)( p − b)

(20)

1 Part I: New Results About Civilizations in Evo-SETI Theory

y=

2(t − d) , for p ≤ t ≤ d. (d − b)( p − d)

221

(21)

Proof The proofs of Eqs. (20) and (21), as well as of all subsequent formulae about the b-scalene, are given in #2 Appendix to the present paper. We will simply refer to them with the numbers of the resulting equations in the Maxima code. Thus, Eq. (20) corresponds to (%o15) and Eq. (21) to (%o20). Also, it is possible to compute all moments (i.e. the kth moment) of the b-scalene immediately. In fact, Maxima yields [(%o27) and (%o28)] b_scalenek  =

2[(d − b) p k+2 + (bk+2 − d k+2 ) p + bd k+2 − bk+2 d] . (d − b)(k + 1)(k + 2)( p − b)( p − d)

(22)

Setting k = 0 into Eq. (22) yields of course the normalization condition (%o29) b_scalene0  = 1.

(23)

Setting k = 1 into Eq. (22) yields the mean value (%o30) b+ p+d . 3

b_scalene =

(24)

Setting k = 2 into Eq. (22) yields the mean value of the square (%o32) b_scalene2  =

b2 + p 2 + d 2 + b p + d p + b d . 6

(25)

Then, subtracting the square of Eq. (24) into (25), one gets the b-scalene variance (%o34) 2 σb_scalene =

(d − b)2 . 24

(26)

The square root of Eq. (26) is the b-scalene standard deviation (%o36) σb_scalene =

d −b . 3√ 22 3

(27)

We could go on to find more descriptive statistical properties of the b-scalene, but we prefer to stop at this point. Much more important, in fact, is to compute the Shannon Entropy of the b-scalene. Equations (%i37) through (%o41) show that the Shannon Entropy of the b-scalene is given by b_scalene_ Shannon_ ENTROPY_ in_ bits

222

New Evo-SETI Results About Civilizations and Molecular Clock

=

1 + ln((d − b)2 /4) . 2 ln 2

(28)

This is a simple and important result. Since p does not appear in Eq. (28), the Shannon Entropy of the b-scalene is actually independent of where its peak is! Also, one is tempted to make a comparison between the Entropy of the b-scalene and the Entropy of the UNIFORM distribution over the same interval (d − b) This will be done in the next section.

1.9 Uniform Distribution Between Birth and Death In the Evo-SETI Theory, the meaning of a uniform distribution over the time interval (d − b) simply is ‘we know nothing about that living being except when he/she/it was born (at instant b) and when he/she/it died (at instant d)’. No idea even about when the ‘peak’ p of his/her/its activity occurred. Thus, the uniform distribution is the minimal amount of information about the lifetime of someone that one might possibly have. The pdf of the uniform distribution over the time interval (d − b) is obviously given by the constant in time (%o43) f uniform_b_d (t) =

1 , for b ≤ t ≤ d. d −b

(29)

It is immediately possible to compute all moments of the uniform distribution (%o44) d b

d k+1 − bk+1 tk dt = . d −b (d − b)(k + 1)

(30)

The normalization condition of Eq. (30) is obviously found upon letting k = 0. The mean value is found by letting k = 1 into Eq. (30), (%o53), and is just the middle point between birth and death uniform_b_d =

b+d . 2

(31)

The mean value of the square is found by letting k = 2 into Eq. (30) and reads (%o54) uniform_b_d 2  =

b2 + b d + d 2 . 2

(32)

1 Part I: New Results About Civilizations in Evo-SETI Theory

223

Subtracting the square of Eq. (31) into (32), we get the uniform distribution variance (%o58) 2 σuniform_b_d =

(d − b)2 . 12

(33)

Finally, the uniform distribution standard deviation is the square root of (33) (%o59) σuniform_ in_b_d =

d −b √ . 2 3

(34)

We stop the derivation of the descriptive statistics of the uniform distribution at this point, since it is easy to find all other formulae in textbooks. Rather, we prefer to concentrate on the Shannon Entropy of the uniform distribution, that upon inserting the pdf (29) into the entropy definition (17), yields (%o62) Uniform_ distribution_ ENTROPY_ in_ bits =

ln(d − b) . ln 2

(35)

1.10 Entropy Difference Between Uniform and b-Scalene Distributions We are now in a position to find out the ‘Entropy Difference’ between the uniform and the b-scalene distributions. Subtracting Eq. (28) into (35) one gets (%o71) ln(d − b) 1 + ln((d − b)2 /4) − ln 2 2 ln 2 1 2[ln(d − b) − ln 2] ln(d − b) − − = ln 2 2 ln 2 2 ln 2 1 = 0.27865247955552. =1− 2 ln 2

(36)

One may say that in passing from just knowing birth and death to knowing birth, peak and death, one has reduced the uncertainty by 0.27865247955552 bits, or, if you prefer, the Shannon Entropy has been reduced by an amount of 0.27865247955552 bits. Again in a colourful language, if you just know that Napoleon was born in 1769 and died in 1821, and then add that the peak occurred in 1812 (or at any other date), than you have added 0.27865247955552 bits of information about his life.

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New Evo-SETI Results About Civilizations and Molecular Clock

Readers might now wish to ponder over statements like the last one about Napoleon in order build up a Mathematical Theory of History, simply called Mathematical History. We stop here now, but some young talent might wish to develop these ideas much more in depth, disregarding all criticism and just being bold, bold, bold, …

1.11 ‘Equivalence’ Between Uniform and b-Lognormal Distributions One more ‘crazy idea’ suggested by the Evo-SETI Theory is the ‘equivalence’ between uniform and lognormal distributions, as described in #2 Appendix. The starting point is to equate the two mean values and the two standard deviations of these two distributions and then… see what comes out! So, just equate the two mean values first, i.e. just equate Eq. (31) and the wellknown mean value formula for the lognormal distribution (see Table 2, fourth line) (%o2) Table 2 Summary of the properties of the b-lognormal distribution that applies to the random variable C = History in time of a certain Civilization Random variable

C = History in time of a certain civilization

Probability distribution

b-lognormal with parameters μ, σ and b

Probability density function

f C (t) =

Mean value

C = b + eμ+(σ

Variance Standard deviation

σC2 = e2μ eσ (eσ − 1)  2 2 σC = eμ eσ /2 eσ − 1

Mode (=abscissa of the b-lognormal peak)

tmode = tpeak = p = b + eμ−σ

Value of the Mode Peak

f C (tmode ) = f C (tpeak ) = P =

Median (=fifty-fifty probability value for C)

median_ of_ C = b + eμ  2 2 K3 eσ − 1(eσ + 2) 3/2 =

2

/(2σ e−[ln(t−b)−μ] √ 2πσ(t−b)

2

Skewness

2)

with t ≥ b = min bi

2 )/2

2

2 2

e(σ√ /2)−μ 2πσ

K2

Kurtosis

K4 K 22

Expression of μ in terms of the birth instants bi and death instants d i of the uniform input random variables C i

2

2

μ=

N  i=1

Yi  =

N   di ln(di )−bi ln(bi ) i=1

N N  Expression of in terms of the birth instants bi   1− σ2 = σY2 i = and death instants d i of the uniform input i=1 i=1 random variables C i

σ2

2

= e4σ + 2e3σ + 3e2σ − 6 di −bi

 −1

bi di [ln(di )− ln(bi )]2 (di −bi )2

This set of results we like to call ‘E-Pluribus-Unum Theorem’ of the Evo-SETI Theory



1 Part I: New Results About Civilizations in Evo-SETI Theory

b+d 2 = e(σ /2)+μ . 2

225

(37)

Similarly, we equate the uniform standard deviation (34) and the lognormal standard deviation (see Table 2, sixth line) and get (%o3)  d −b 2 √ = e(σ /2)+μ eσ2 −1 . 2 3

(38)

A glance to Eqs. (37) and (38) shows that we may eliminate μ upon dividing Eq. (38) by (37), and that yields the resolving equation in σ (%o4)  d −b = eσ2 −1 . √ 3(b + d)

(39)

After a few steps, Eq. (39) may be solved for the exponential, yielding (%o5) eσ = 2

4(b2 + bd + d 2 ) 3(b + d)2

(40)

4(b2 + b d + d 2 ) . 3(b + d)2

(41)

and finally, taking logs  σ2 = ln

Taking the square root, Eq. (41) becomes

 4(b2 + bd + d 2 ) . σ = ln 3(b + d)2

(42)

Then, inserting Eq. (41) into (38) and solving the resulting equation for μ, one finds for μ (%o10)  μ = ln

√

3(b + d)2



. √ 4 b2 + bd + d 2

(43)

In conclusion, we have proven that, if we are given just the birth and death times of the life of anyone, this uniform distribution between birth and death may be converted into the ‘equivalent’ lognormal distribution starting at the same birth instant and having the two parameters μ and σ given by, respectively ⎧   √ 3(b+d)2 ⎪ ⎨ μ = ln 4√b2 +bd+d 2 ,    2 +bd+d 2 ) ⎪ ⎩ σ = ln 4(b3(b+d) . 2

(44)

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New Evo-SETI Results About Civilizations and Molecular Clock

One may also invert the system √ of two simultaneous Eq. (44). In fact, multiplying Eq. (37) by 2 and Eq. (38) by 2 3 and then summing, b disappears and one is left with the d expression (%o13) d = e(σ

2

/2)+μ

  √  1 + 3 eσ2 −1 .

(45)

√ Similarly, multiplying Eq. (37) by 2 and Eq. (38) by 2 3 and then subtracting, d disappears and one is left with the b expression (%o14) b = e(σ

2

/2)+μ



1−

√  2  3 eσ −1 .

(46)

In conclusion, the inverse formulae of Eq. (44) are ⎧   √ √ ⎨ b = e(σ2 /2)+μ 1 − 3 eσ2 −1   √ √ ⎩ d = e(σ2 /2)+μ 1 + 3 eσ2 −1 .

(47)

Let us now find how much the Shannon Entropy changes when we replace the lognormal distribution to the uniform distribution between birth and death. We already know that the uniform distribution entropy is the largest possible entropy, and is given by Eq. (35). Then, we only need to know that the lognormal entropy is given by the expression (%o16) (for the proof, see, for instance, [1, Chap. 30, pp. 685–687]) √ ln( 2πσ) + μ + 1/2 lognormal_ entropy_ in_ bits = ln 2

(48)

Inserting Eq. (44) into (48) a complicated expression would be found (%o17) that we will not re-write here. Also the uniform entropy (35) may be rewritten in terms of μ and σ by inserting Eq. (44) into it, and the result is (%o20). At this point we may subtract the lognormal entropy to the uniform entropy and so find out how much information we ‘arbitrarily inject into the system’ if we replace the uniform pdf by the lognormal pdf. The result is given by (%o22) and reads uniform_ pdf_ ENTROPY − lognormal_ pdf_ ENTROPY   2 √ ln 12(eσ − 1) + 2μ + σ2 ln( 2πσ) + μ + 1/2 − = 2 ln 2  ln 2  ln 6(eσ − 1)/ψ2 + σ2 + 1 2

=

2 ln 2

.

(49)

Notice that, rather unexpectedly, Eq. (49) is independent of μ. Numerically, we may get an idea about Eq. (49) in the limit case when σ → 0, then finding

1 Part I: New Results About Civilizations in Evo-SETI Theory

  2 ln 6(eσ − 1)/ψ2 + σ2 + 1 lim

σ →0

2 ln 2 = −0.254 bits

=

227

ln(6/π) + 1 2 ln 2 (50)

Not too a big numeric error, apparently.

1.12 b-Lognormal of a Civilization’s History as CLT of the Lives of Its Citizens This and the following sections of Part 1 are most important since they face mathematically the finding of the b-lognormal of a certain Civilization in time, like any of the Civilizations shown in Fig. 2. We claim that the b-lognormal of a Civilization History is obtained by applying the CLT of Statistics to the lifetimes of the millions of Citizens that make and made up for that Civilization in time. Though this statement may appear rather obvious, the mathematics is not so, and we are going to explain it from scratch right now. Then: 1. Denote by C the random variable (in time) yielding the History of that Civilization in time. In the end, the pdf of C will prove to be a b-lognormal and we will derive this fact as a consequence of the CLT of Statistics. 2. Denote by C i the random variable (in time) denoting the lifetime of the ith Citizen belonging to that Civilization. We do not care about the actual pdf of the random variable C i : it could be just uniform between birth and death (in this case, C i is the lifetime of a totally anonymous guy, as the vast majority of Humans are, and certainly cells are too, and so forth for other applications). Or, on the contrary, it could be a b-scalene, as in the example about Napoleon, born 1769, died 1821, with peak in 1812, or this pdf could be anything else: no problem since the CLT allows for arbitrary input pdfs. 3. Denote by N the total number of individuals that made up and are making up and will make up for the History of that Civilization over its total existence in time, let this time be years or centuries or millions or even billions of years (for ET Civilizations, we suppose!). In general, this positive integer number N is going to be very large: thousands or millions or even billions, like that fact that Humans nowadays number about 7.8 billion people. In the practice, we may well suppose that N approaches infinity, i.e. N → ∞, which is precisely the mathematical condition requested to apply the CLT of Statistics, as we shall see in a moment. 4. Then consider the statistical equation N

C =  Ci . i=1

(51)

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New Evo-SETI Results About Civilizations and Molecular Clock

This we shall call ‘the Statistical Equation of each Civilization’ (abbreviated SEC). What is the meaning of this equation? Well, if we suppose that all the random variables C i are ‘statistically independent of each other’, then Eq. (51) is the ‘Law of Compound Probability’, well known even to beginners in statistical courses. And the lifespans of Citizens C i almost certainly are independent of each other in time: dead guys may hardly influence the life of alive guys! 5. Now take the logs of Eq. (51). The product is converted into a sum and the new form of our SEC is ln C =

N 

ln Ci .

(52)

i=1

6. To this Eq. (52) we now apply the CLT. In loose terms, the CLT states that ‘if you have a sum of a number of independent random variables, and let the number of terms in the sum approach infinity, then, regardless of the actual probability distribution of each term in the sum, ‘the overall sum approaches the normal (i.e. Gaussian) distribution’. 7. And the mean value of this Gaussian equals the sum of the mean values of the ln C i , while the variance equals the sum of the variances of the ln C i . In equations, one has ln C = normally_distributed_random_variable

(53)

with mean value given by μ = ln C =

N 

ln Ci 

(54)

i=1

and variance given by σln2 C =

N 

2 σln Ci .

(55)

i=1

(8) Let us now ‘invert’ Eq. (53), namely solve it for C. To do so, we must recall an important theorem that is proved in probability courses, but, unfortunately, does not seem to have a specific name. It is the transformation law (so we shall call it, see for instance [8, pp. 130–131]) allowing us to compute the pdf of a certain new random variable Y that is a known function Y = g(X) of another random variable X having a known pdf. In other words, if the pdf f X (x) of a certain random variable X is known, then the pdf f Y (y) of the new random variable Y, related to X by the functional relationship

1 Part I: New Results About Civilizations in Evo-SETI Theory

Y = g(X )

229

(56)

can be calculated according to the following rules: (a) First, invert the corresponding non-probabilistic equation y = g(x) and denote by x i (y) the various real roots resulting from this inversion. (b) Second, take notice whether these real roots may be either finitely- or infinitely-many, according to the nature of the function y = g(x). (c) Third, the pdf of Y is then given by the (finite or infinite) sum f Y (y) =

 f X (xi (y)) , |g (xi (y))| i

(57)

where the summation extends to all roots x i (y) and |g (x i (y))| is the absolute value of the first derivative of g(x) where the ith root x i (y) has been replaced instead of x. Going now back to (53), in order to invert it, i.e. in order to find the pdf of C, we must apply the general transformation law (57) to the particular transformation y = g(x) = ex .

(58)

That, upon inversion, yields the single root x1 (y) = x(y) = ln y.

(59)

On the other hand, differentiating Eq. (58) one gets g (x) = ex

(60)

g (x1 (y)) = eln y = y,

(61)

and

where Eq. (60) was already used in the last step. So, the general transformation law (57) finally yields just the lognormal pdf in y for the random variable C, the time History of that Civilization: f C (y) =

 f x (xi (y)) 1 = f Y (ln(y)) (x (y))| |g |y| i i

1 2 2 =√ e−(ln(y)−μ) /(2σ ) , for y > 0. 2πσy

(62)

230

New Evo-SETI Results About Civilizations and Molecular Clock

with μ given by Eq. (54) and σ given by Eq. (55). This is a very important result to understand the History of Civilizations mathematically: we now see why, for instance, all Civilizations shown in Fig. 2 are b-lognormals in their Historic development! The pdf (62) actually is a b-lognormal, rather than just an ordinary lognormal starting at zero. In fact, the instant b at with it starts may not be smaller than the birth instant of the first (Historically!) individual of the population. Thus, the true b-lognormal pdf of the C Civilization is e−[ln(t−b)−μ ]/(2σ √ 2πσ(t − b) 2

f C (t) =

2

)

with t ≥ b = min bi .

(63)

1.13 The Very Important Special Case of Ci Uniform Random Variables: E-Pluribus-Unum Theorem This author has discovered new, important and rather simple equations for the particular case where the input variables C i are uniformly distributed between birth and death, namely, the pdf of each C i is f Ci (t) =

1 , for bi ≤ t ≤ di . di − bi

(64)

In Eq. (64) bi is the instant when he/she/it was born, and d i is the instant when he/she/it died. We may not know them at all: just think of the millions of Unknown Soldiers died in World War One and in all wars (billions?). But that will not prevent us from doing the mathematics of Eq. (64). Our primary goal now is to find the pdf of the random variable Y i = ln C i as requested by Eq. (52). To this end, we must apply again the transformation law (57), this time applied to the transformation y = g(x) = ln(x).

(65)

Upon inversion, Eq. (65) yields the single root x1 (y) = x(y) = e y .

(66)

On the other hand, differentiating Eq. (65) yields g (x) = and

1 x

(67)

1 Part I: New Results About Civilizations in Evo-SETI Theory

231

1 1 = y, x1 (y) e

(68)

g (x1 (y)) =

where Eq. (66) was already used in the last step. Then, by virtue of the uniform pdf (64), the general transformation law (57) finally yields f Yi (y) =

 f x (xi (y)) 1 1 · = (x (y))| y| |g d − b |1/e i i i i ey . di − bi

=

(69)

In other words, the requested pdf of Y i = ln C i is f Yi (y) =

ey , for ln bi ≤ y ≤ ln di . di − bi

(70)

These are the probability density functions of the natural logs of all the uniformly distributed C i random variables. Namely, in the colourful language of the applications of the Evo-SETI Theory, Eq. (70) is the pdf of all UNKNOWN FORMS OF LIFE, about which we only known when each of them was born and when it died. Let us now check that the pdf (70) fulfils indeed its normalization condition ln di

ln di f Yi (y)dy =

ln bi

ln bi

=

ey dy di − bi

elndi − elnbi di − bi = = 1. di − bi di − bi

(71)

Next we want to find the mean value and standard deviation of each Y i , since they play a crucial role for future developments. The mean value of the pdf (70) is given by either of the following alternative forms: ln(d  i)

Yi  =

ln(d  i)

y · f Yi (y)dy = ln(bi )

ln(bi )

y · ey dy di − bi

di [ln(di ) − 1] − bi [ln(bi ) − 1] = di − bi di ln(di ) − bi ln(bi ) = −1 di − bi   ln (di )di /(bi )bi −1 = di − bi

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New Evo-SETI Results About Civilizations and Molecular Clock



(di )di /(di −bi ) = ln (bi )bi /(di −bi )

− 1.

(72)

This is thus the mean value of the natural log of all the uniformly distributed random variables C i (just to use a few of the above equivalent forms). Thus, the whole Civilization is a b-lognormal with the following parameters: μ=

N 

 N   di ln(di ) − bi ln(bi ) −1 di − bi i=1    N  (di )di /(di −bi ) ln −1 = (bi )bi /(di −bi ) i=1  N  (di )di /(di −bi ) ln = −N (bi )bi /(di −bi ) i=1 N  (di )di /(di −bi ) = ln −N (bi )bi /(di −bi ) i=1

Yi  =

i=1

(73)

The last form of μ shows that the exponential of μ is μ

−N

e =e

N  (di )di /(di −b) . (bi )bi /(di −bi ) i=1

(74)

In order to find the variance also, we must first compute the mean value of the square of Y i , that is 

Yi2



ln(d  i)

ln(d  i)

y2 · ey dy di − bi ln(bi ) ln(bi )     di ln2 (di ) − 2 ln(di ) + 2 − bi ln2 (bi ) − 2 ln(bi ) + 2 . = di − bi

=

y · f Yi (y) dy = 2

(75)

The variance of Y i = ln(C i ) is now given by Eq. (75) minus the square of Eq. (73), that, using the first form of Eq. (73) and after a few reductions, yields: bi di [ln(di /bi )]2 (di − bi )2 √ ⎫⎤2 ⎡ ⎧ bi di ⎨ d  di −b ⎬ i i ⎦ . = 1 − ⎣ln ⎩ bi ⎭

2 =1− σY2 i = σln(C i)

(76)

1 Part I: New Results About Civilizations in Evo-SETI Theory

233

Whence, using the first form of Eq. (76) and taking the square root, yields the standard deviation of Y i

σYi = σln(Di ) =

1−

bi di [ln(di ) − ln(bi )]2 . (di − bi )2

(77)

Like the μ given by Eqs. (73), (76) also may be rewritten in a few alternative forms. For instance σY2 =

N 

σY2 i =

i=1

N  

N 

2 σln(C i)

i=1

 bi di [ln(di /bi )]2 1− = (di − bi )2 i=1 √ ⎫⎤2 ⎡ ⎧ bi di N ⎨ d  di −b ⎬  i i ⎣ln ⎦ . =N− ⎩ bi ⎭

(78)

i=1

We stop at this point, for we feel we have really proven a new theorem, yielding the b-lognormal in time of the History of any Civilization. This new theorem deserves a new name. We propose to call it by the Latin name of ‘E-Pluribus-Unum’ Theorem. Indeed, ‘E-Pluribus-Unum’ stands for ‘Out of Many, just One’, and this was the official motto of the USA from 1782 to 1956, when replaced by ‘In God we trust’ (probably in opposition to atheists views then supported by the Soviet Union). In this author’s view, ‘E-Pluribus-Unum’ adapts well to what we have described mathematically in the first part of this paper about Civilizations in Evo-SETI Theory.

2 Part 2: New Results About Molecular Clock in Evo-SETI Theory 2.1 Darwinian Evolution as a Geometric Brownian Motion (GBM) In [2, pp. 220–227], this author ‘dared’ to re-define Darwinian Evolution as ‘just one particular realization of the stochastic process called GBM in the increasing number of Species living on Earth over the last 3.5 billion years’. Now, the GBM mean value is the simple exponential function of the time m GBM (t) = A e Bt

(79)

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New Evo-SETI Results About Civilizations and Molecular Clock

with A and B being the positive constants. Thus, A equals mGBM (0), the number of Species living on Earth right now, and m GBM (ts) = 1

(80)

represents the first ‘living Species’ (call it RNA?) that started life on Earth at the ‘initial instant’ ts (‘time of start’). In [2, 3] we assumed that life started on Earth 3.5 billion years ago, that is ts = −3.5 × 109 years

(81)

and that the number of Species living on Earth nowadays is 50 million A = 50 × 106 .

(82)

Consequently, the two constants A and B in Eq. (79) may be exactly determined as follows: % A = 50 × 106 (83) B = − ln(A) = 1.605 × 10−16 s−1 . ts Please note that these two numbers are to be regarded as experimental constants (valid for Earth only), just like the acceleration of gravity g = 9.8 m s−2 , the solar constant, and other Earthly constants. Also, some paleontologists claim that life on Earth started earlier, say 3.8 billion years ago. In this case, Eq. (83) is to be replaced by the slightly different %

A = 50 × 106 B = − ln(A) = 1.478 × 10−16 s−1 . ts

(84)

but B did not change much, and so we will keep Eq. (83) as the right values as it was done in [2, 3]. Figure 3 shows two realizations of GBM revealing ‘at a glance’ the exponential increasing mean value of this lognormal stochastic process (see [2, pp. 222–223] for more details, and [3, pp. 291–294]) for a full mathematical treatment). Assuming GBM as the ‘curve’ (a fluctuating one!) representing the increasing number of Species over the last 3.5 billion years has several advantages: 1. It puts on a firm mathematical ground the intuitive notion of a ‘Malthusian’ exponential growth. 2. It allows for Mass Extinctions to have occurred in the past history of life on Earth, as indeed it was the case. Mass Extinctions in the Evo-SETI Theory are just times when the number of living Species ‘decreased very much’ from its exponential mean value, for instance going down by 70% just 250 million years

2 Part 2: New Results About Molecular Clock in Evo-SETI Theory

235

Fig. 3 Two realizations (i.e. actual instances) of Geometric Brownian Motion taken from the GBM Wikipedia site http://en.wikipedia.org/wiki/ Geometric_Brownian_motion. Please keep in mind that the name ‘Brownian Motion’ is incorrect and misleading: in fact, physicists and mathematicians mean by ‘Brownian Motion’ a stochastic process whose pdf is Gaussian, i.e. normal. But this is not the case with GBM, whose pdf is lognormal instead. This incorrect denomination seems to go back to the Wall Street financial users of the GBM

ago, but not going down to zero, otherwise we would not be living now. In [3] this author did more modelling about Mass Extinctions. 3. After what we just said, the two curves called ‘upper’ and ‘lower standard deviation curve’ are clearly playing a major role in Evo-SETI Theory. They represent the average departure of the actual number of living Species from their exponential mean value, as shown in Fig. 4. In [3, pp. 292–293], the author proved that the upper (plus sign) and lower (minus sign) standard deviation curves of GBM (above and below the mean value exponential (79), respectively), are given by the equations upper_ &_ lower_ std_ curves_ of_ GBM(t)  & 2 = m GBM (t) · 1 ± eσGBM (t−ts) − 1 .

(85)

The new constant σGBM appearing in Eq. (85) [not to be confused with the simple σ of the b-lognormal (5)] is provided by the final conditions affecting the GBM at the final instant of its motion, namely zero (=now) in our conventions. Denoting by A the current number of Species on Earth, as we did in Eqs. (79) and (82), and by δA the standard deviation around A nowadays (for instance, we assumed A to be equal to 50 million but we might add an uncertainty of, say, ±10 million Species around that value), then the σGBM in Eq. (85) is given by

236

New Evo-SETI Results About Civilizations and Molecular Clock

Fig. 4 Darwinian evolution as the increasing number of living Species on Earth between 3.5 billion years ago and now. The red solid curve is the mean value of the GBM stochastic process, given by Eq. (79), while the blue dot–dot curves above and below the mean value are the two standard deviation upper and lower curves, given by Eq. (105). The ‘Cambrian Explosion’ of life that started around 542 million years ago, approximately marks in the above plot ‘the epoch of departure from the time axis for all the three curves, after which they start climbing up more and more’. Notice also that the starting value of living Species 3.5 billion years ago is ONE by definition, but it ‘looks like’ zero in this plot since the vertical scale (which is the true scale here, not a log scale) does not show it. Notice finally that nowadays (i.e. at time t = 0) the two standard deviation curves have exactly the same distance from the middle mean value curve, i.e. 30 million living Species above or below the mean value of 50 million Species. These are assumed values that we used just to exemplify the GBM mathematics: biologists might assume other numeric values

σGBM

&   ln 1 + (δA/A)2 = . √ −ts

(86)

2.2 A Leap Forward: For Any Assigned Mean Value mL (t) We Construct Its Lognormal Stochastic Process A profound message was contained in [3] for all future applications of lognormal stochastic processes (both GBM and other than GBM): for any assigned at will

2 Part 2: New Results About Molecular Clock in Evo-SETI Theory

237

mean value function of the time mL (t), namely for any trend, we are able to find the equations of the lognormal process that has exactly that mean value, i.e. that trend! This author was so amazed by this discovery (that he made between September 2013 and January 2014) that he could not give a complete account of it when he published [3] available in Open Access since October 2014. Thus, the present new chapter is a completion of [3], but also is a leap forward in another unexpected direction: the proof that the Molecular Clock, well-known to geneticists for more than 50 years, may be derived mathematically as a consequence of the Evo-SETI Theory.

2.3 Completing [3]: Letting ML (t) There Be Replaced Everywhere by mL (t), the Assigned Trend In [3] this author started by considering the general lognormal process L(t) whose pdf is the lognormal L(t)_ pdf(n; M L (t), σ L , t) = % with

n ≥ 0, t ≥ ts,

% and

  −[ln(n)−M L (t)]2 / 2σ2L (t−ts)

e

√

√ 2πσ L t − ts n

σ L ≥ 0, M L (t) = arbitrary function of t.

(87)

Equation (87) also is the starting point of all subsequent calculations in the #3 Appendix, where it has the number (%o6). Notice that the positive parameter σL in the pdf (87) is denoted sL in the #3 Appendix, simply because Maxima did not allow us to denote it σL for Maxima-language reasons too long to explain! Also, mL (t) is more simply denoted m(t) in the #3 Appendix, and M L (t) is more simply denoted M(t). The mean value, i.e. the trend, of the process L(t) is an arbitrary (and continuous) function of the time denoted by mL (t) in the sequel. In equations, that is, one has, by definition m L (t) = L(t).

(88)

In other words, we analytically compute the following integral, yielding the mean value of the pdf (87), getting (for the proof, see (%o5) and (%o6) in the #3 Appendix) ∞ 0

e−[ln(n)−M L (t)] /(2σL (t−ts)) dn √ √ 2πσ L t − ts n 2

n·

m L (t) ≡

= e M L (t) e(σL /2)(t−ts) . 2

2

(89)

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New Evo-SETI Results About Civilizations and Molecular Clock

This is (%o8) in the #3 Appendix, and from now on, we will drop the usual sentence ‘in the #3 Appendix’ and just report the #3 Appendix equation numbers corresponding to the equation numbers in this paper. We have thus discovered the following crucial mean value formula, holding good for the general lognormal process L(t) inasmuch as the function M L (t) is arbitrary, and so is the trend mL (t) (%o9) 2 m L (t) = e M L (t) e(σL /2)(t−ts) .

(90)

This was done by the author in [3] already, p. 292, Eq. [2]. But at that time this author failed to invert (90), i.e. to solve it for M L (t), with the result (%o10): M L (t) = ln[m L (t)] −

σ2L (t − ts). 2

(91)

Equation (91) shows that it is always possible to get rid of M L (t) by substituting Eq. (91) into any equation containing M L (t) and appearing in [3]. In other words, one may re-express all results of [3] in terms of the trend function mL (t) only, justifying the idea ‘you give me the trend mL (t) and I’ll give you all the equations of the lognormal process L(t) for which m(t) is the trend’. An immediate consequence of (91) is found by letting t = ts [(%o11) and (%o12)]: M L (ts) = ln[m L (ts)], i.e. m L (ts) = e M L (ts) .

(92)

For instance, Eq. (8) on p. 292 of [3] yields the σL in terms of both the initial input data (ts, Ns) and final input data (te, Ne, δNe):  σL =

ln[e2M L (ts) + (δN e)2 (N s/N e)2 ] − 2M L (ts) √ te − ts

(93)

Well, this equation simplifies dramatically once Eq. (92) and the initial condition (Eq. (5) on p. 292 of Maccone [3]) (%o13) m L (ts) = N s

(94)

are taken into account. In fact, a few steps starting from Eq. (93) show that, by virtue of Eqs. (92) and (94), it reduces to (%o31) &   ln 1 + (δN e/N e)2 σL = √ te − ts

(95)

Of course, the corresponding GBM special case of Eq. (95) is (86), obtained by letting (te = 0, Ne = A) into Eq. (95). Also, Eq. (95) may be formally rewritten as follows:

2 Part 2: New Results About Molecular Clock in Evo-SETI Theory

239

⎫ ⎧   2 1/(te−ts) ⎬  ⎨ 2 ln 1 + (δN e/N e) δN e σ2L = . = ln 1 + ⎭ ⎩ (te − ts) Ne

(96)

Taking the exponential of Eq. (96), one thus gets a yet unpublished equation that we shall use in a moment  σ2L (t−ts)

e



= 1+

δN e Ne

2 (t−ts)(te−ts) .

(97)

Going back to the general lognormal process L(t), in [3], Tables 1, 2 and 3, we also proved that the moment of order k (with k = 0, 1, 2, …) of the L(t) process is given by  2 [L(t)]k = [m L (t)]k ek(k−1)σL (t−ts)/2



(98)

The mathematical proof of this key result by virtue Maxima is given in the #3 Appendix, equations (%i16) through (%o21). A new discovery, presented in this paper for the first time, is that, by virtue of Eqs. (97), (98) may be directly rewritten in terms of the boundary conditions (ts, te, Ne, δNe): 





[L(t)] = [m L (t)] k

k

 1+

δN e Ne

2 k(k−1)(t−ts)/(2(te−ts)) .

(99)

For k = 0, both Eqs. (98) and (99) yield the normalization condition of L(t):   [L(t)]0 = 1.

(100)

For k = 1 both Eqs. (98) and (99) yield the mean value again  [L(t)]1 = L(t) = m L (t).



(101)

But for k = 2 [the mean value of the square of L(t)] the novelties start. In fact, Eq. (99) yields 





[L(t)] = [m L (t)] 2

2

1+



δN e Ne

2 (t−ts)/(te−ts) .

(102)

Since the variance of L(t) is given by the mean value of its square minus the square of its mean value, subtracting the square of Eq. (101) into (102) yields

In terms of mL (t) and σL (t) '

' 2 ( σ L /2 (t−ts)

that is (%o31)

)

⎧ ⎫

 ⎨ ⎬  2 (t−ts)/(te−ts) δN e 1 + Ne m L (t) 1 − −1 ⎩ ⎭

2

 & 2 m L (t) 1 − eσ L (t−ts) − 1 that is (%o42)

All moments, i.e. k-th moment

Standard Deviation

Lower Standard Deviation Curve

−1

⎧ ⎫

 ⎨ ⎬ 2 (t−ts)/(te−ts)  e m L (t) 1 + − 1 1 + δN Ne ⎩ ⎭

2 (t−ts)/(te−ts)

 & 2 m L (t) 1 + eσ L (t−ts) − 1 that is (%o34)

δN e Ne

Upper Standard Deviation Curve



  2 k(k−1)(t−ts)/(2(te−ts)) e L(t)k = [m L (t)]k 1 + δN Ne

1+



L(t)k = [m L (t)]k ek(k−1)σ L (t−ts)/2 that is (%o20)

[m L (t)]2

mL (t) (arbitrarily assigned, i.e. known)

σL =

&   ln 1+(δN e/N e)2 √ te−ts

  2 (t−ts)/(te−ts) e m L (t) 1 + δN −1 Ne

 2  [m L (t)]2 eσ L (t−ts) − 1 that is (%o25)

m L (t) = e M L (t) e

(

In terms of mL (t) and of (1) The two initial inputs (ts, Ns) (2) Plus the three final inputs (te, Ne, δNe)

& 2 m L (t) eσ L (t−ts) − 1 that is (%o27)

Variance

Mean value L(t) also called Trend

2 2 b-lognormal pdf starting e−[ln(n)−M L (t)] / 2σ L (t−ts) L(t)_ pdf(n; M L (t), σ L , t) = √ √ at ts (%o6) 2πσ L t − ts n   σ L ≥ 0, n ≥ 0, and with M L (t) = arbitrary function of t. t ≥ ts,

Lognormal stochastic process L(t)

Table 3 Summary of the most important mathematical properties of the lognormal stochastic process L(t)

(continued)

240 New Evo-SETI Results About Civilizations and Molecular Clock

K3 (K 2 )3/2

K4 (K 2 )2

Skewness

Kurtosis

2

that is (%o57)

= e4σ

2 (t−ts)

+ 2e3σ

2 (t−ts)

+ 3e2σ

2 (t−ts)

−6

that is (%o65)

that is (%o61)

 2  2 = eσ (t−ts) + 2 eσ (t−ts) − 1

eσ L (t−ts)

& m L (t)

Median (=fifty-fifty probability)

f L(t) (n mode ) =

Ordinate of the peak = Mode ordinate

2

m L (t) e(3/2)σ L (t−ts)

1 √ 2 √ m L (t) 2πσ L t−tseσ L (t−ts)

n mod e = n peak =

In terms of mL (t) and σL (t)

Mode (=peak abscissa)

Lognormal stochastic process L(t)

Table 3 (continued)

δN e Ne

2 (t−ts)/(te−ts) +2

)  1+



δN e Ne

2 (t−ts)/(te−ts)

 2 (t−ts)/(te−ts)  2 e Same with eσ L (t−ts) = 1 + δN Ne

1+



m L (t) 3(t−ts)/(2(te−ts)) 1+(δN e/N e)2

1 √ √ (t−ts)/(te−ts) m L (t) 2πσ L t−ts [1+(δN e/N e)2 ]



m L (t) (t−ts)/(te−ts)

1+(δN e/N e)2



&

f L(t) (n mode ) =

n mod e = n peak =

In terms of mL (t) and of (1) The two initial inputs (ts, Ns) (2) Plus the three final inputs (te, Ne, δNe)

−1

2 Part 2: New Results About Molecular Clock in Evo-SETI Theory 241

242

New Evo-SETI Results About Civilizations and Molecular Clock

⎫ ⎧  2 (t−ts)/(te−ts) ⎬ ⎨ δN e Variance of L(t) = [m L (t)]2 1 + −1 . ⎭ ⎩ Ne

(103)

The square root of Eq. (103) is of course the standard deviation of L(t): Standard_ Deviation_ of_ L(t) * +  (t−ts)(te−ts)  + δN e 2 , = m L (t) · − 1. 1+ Ne

(104)

This is a quite important formula for all future applications of our general lognormal process L(t) to the Evo-SETI Theory. Even more important for all future graphical representations of the general lognormal process L(t) is the formula yielding the upper (plus sign) and lower (minus sign) standard deviation curves as two functions of t. It follows immediately from the mean value mL (t) plus or minus the standard deviation (104): Two_ Standard_ Deviation_ CURVES_ of_ L(t) ⎫ ⎧ * + ⎪ ⎪ 2 (t−ts)(te−ts)  ⎬ ⎨ + δN e , = m L (t) · 1 ± −1 . 1+ ⎪ ⎪ Ne ⎭ ⎩

(105)

Just to check that our results are correct, from (105) one may immediately verify that: 1. Letting t = ts in (105) yields mL(ts) as the value of both curves. But this is the same value as the mean value at t = ts also. Thus, at t = ts the process L(t) starts with probability one, since all three curves are at the just the same point. 2. Letting t = te in (105) yields Two_ Standard_ Deviation_ CURVES_ at_ te % δN e = N e ± δN e = m L (te) · 1 ± Ne

(106)

where mL (te) = Ne was used in the last step. This result is correct inasmuch as the two curves intercept the vertical line at t = te exactly at those two ordinates. 3. Letting δNe = 0 in (105) makes the two curves coincide with the mean value mL (t), and that is correct. 4. As a matter of terminology, we add that the factor &

* +  (t−ts)(te−ts)  + δN e 2 2 , σ (t−ts) eL −1= −1 1+ Ne

(107)

2 Part 2: New Results About Molecular Clock in Evo-SETI Theory

243

is called ‘coefficient of variation’ by statisticians since it is the ratio between the standard deviation and the mean value for all time values of the L(t) process, and in particular at the end time t = te, when it equals δNe/Ne. 5. Finally, we have summarized the content of this important set of mathematical results in Table 3.

2.4 Peak-Locus Theorem The Peak-Locus Theorem is a new mathematical discovery of ours playing a central role in the Evo-SETI theory. In its most general formulation, it holds good for any lognormal process L(t) and any arbitrary mean value mL (t), as we show in this section. In words, and utilizing the simple example of the Peak-Locus Theorem applied to GBMs, the Peak-Locus Theorem states what shown in Fig. 5: the family of all b-lognormals ‘trapped’ between the time axis and the growing exponential of the GBMs (where all the b-lognormal peaks lie) can be exactly (i.e. without any numerical approximation) described by three equations yielding the three parameters μ(p), σ(p) and b(p) as three functions of the peak abscissa, p, only. In equations, the Peak-Locus Theorem states that the family of b-lognormals having each its peak exactly located on the mean value curve (88), is given by the following three equations, specifying the parameters μ(p), σ(p) and b(p), appearing in the b-lognormal (5) as three functions of the single ‘independent variable’ p, i.e. the abscissa (i.e. the time) of the b-lognormal’s peak:

Fig. 5 GBM exponential as the geometric LOCUS OF THE PEAKS of b-lognormals. Each blognormal is a lognormal starting at a time (b = birth time) larger than zero and represents a different SPECIES that originated at time b of Evolution. That is CLADISTICS in our Evo-SETI Model. It is evident that, the more the generic ‘Running b-lognormal’ moves to the right, its peak becomes higher and higher and narrower and narrower, since the area under the b-lognormal always equals 1 (normalization condition). Then, the (Shannon) ENTROPY of the running b-lognormal is the DEGREE OF EVOLUTION reached by the corresponding SPECIES (or living being, or civilization, or ET civilization) in the course of Evolution

244

New Evo-SETI Results About Civilizations and Molecular Clock

⎧ 2 σ2L eσ L · p ⎪ ⎪ ⎨ μ( p) = 4π[m L ( p)]2 − p 2 (σ2L /2)· p σ( p) = √e2πm ⎪ L ( p) ⎪ 2 ⎩ b( p) = p − eμ( p)−[σ( p)] .

(108)

This general form of the Peak-Locus Theorem is proven in the Appendix by equations (%i66) through (%o82). The remarkable point about all this seems to be the exact separability of all the equations involved in the derivation of Eq. (108), a fact that was unexpected to this author when he discovered it around December 2013. And the consequences of this new result are in the applications: 1. For instance in the ‘parabola model’ for Mass Extinctions that was studied in Sect. 10 of [3]. 2. For instance to the Markov–Korotayev Cubic that was studied in Sect. 12 of [3–5]. 3. And finally in the many stochastic processes having each a Cubic mean value that are just the natural extension into statistics of the deterministic Cubics studied by this author in Chap. 10 of his book ‘Mathematical SETI’ [1]. But the study of the Entropy of all these Cubic Lognormal Processes has to be differed to a future research paper. Notice now that, in the particular case of the GBMs having mean value eμGBM (t−ts) with μGBM = B, and starting at ts = 0 with N 0 = Ns = Ne = A, the Peak-Locus Theorem (108) boils down to the simpler set of equations ⎧ 1 ⎪ ⎨ μ( p) = 4πA2 − Bp 1 σ = √2πA ⎪ ⎩ b( p) = p − eμ( p)−σ2 .

(109)

In this simpler form, the Peak-Locus Theorem was already published by the author in Maccone [1], while its most general form (108) is now proven in detail. Proof Let us firstly call ‘Running b-lognormal’ (abbreviated ‘RbL’) the generic blognormal of the family, starting at b, having peak at p and having the variable parameters μ(p) and σ(p). Then the starting equation of the Peak-Locus Theorem (108) is the #3 Appendix equation (%o73) expressing the fact that the peak of the RbL equals the mean value (90), where, however, the old independent variable t must be replaced by the new independent variable p of the RbL, that is eσ /2−μ 2 = e M L ( p) e(σL /2)( p−ts) . √ 2πσ 2

(110)

This equation may ‘surprisingly’ be separated into the following two simultaneous equations (%o74)

2 Part 2: New Results About Molecular Clock in Evo-SETI Theory



e(σ

2

/2)−μ

√1 2πσ

= e(σL /2) p 2 = e M L ( p) e−(σL /2)ts .

245

2

(111)

There are two advantages brought in by this separation of variables: in the upper Eq. (111) the exponentials ‘disappear’ yielding (%o79) σ2 σ2 −μ= L p 2 2

(112)

while the lower Eq. (111) is in σ only, and thus it may be solved for σ immediately (%o76) e(σL /2)ts e−M L ( p) . √ 2π 2

σ=

(113)

We may now get rid of M(p) in Eq. (113) by replacing it by virtue of (91), getting, after a few steps and rewriting p instead of t, (%o78) e(σL /2) p 2

σ= √

2πm L ( p)

.

(114)

which is just the middle Eq. (108). Finally, Eq. (112) may be solved for μ (%o79). μ=

σ2 σ2 − Lp 2 2

(115)

so that, inserting Eq. (114) into (115), the final expression of μ is found also (%o80) eσL p σ2L p. − 2 4π[m L ( p)]2 2

μ=

(116)

Our general Peak-Locus Theorem (108) has thus been proven completely.

2.5 tsGBM and GBM Sub-cases of the Peak-Locus Theorem The general Peak-Locus Theorem proved in the previous section includes, as subcases, many particular forms of the arbitrary mean-value function mL (t). In particular, we now want to consider two of them: 1. The tsGBM, i.e. the GBM starting at any given time ts, like the origin of life on Earth, that started at ts = −3.5 billion years ago.

246

New Evo-SETI Results About Civilizations and Molecular Clock

2. The ‘ordinary’ GBM, used in the Mathematics of Finances, starting at ts = 0. Clearly, the ordinary GBM is, in its turn, a sub-subcase of the tsGBM. Then, the tsGBM is characterized by the equation m tsGBM ( p) = m tsGBM (ts)e B( p−ts)

(117)

having set in agreement with Eqs. (79) and (90), (%o84) B=

2 σtsGBM . 2

(118)

One may determine the numeric constant B in terms of both the initial and final conditions of the tsGBM by replacing into Eq. (117) p by te (the end-time, i.e. the time of the final condition) and then solving Eq. (117) for B (%o88) B=

ln(m tsGBM (te)/m tsGBM (ts)) . te − ts

(119)

In the Evo-SETI Theory we assume ts to be the time of the ‘beginning of life’, when there was only one living Species (the first one, probably RNA, at 3.5 billion year ago on Earth, but we do not know at what time on exoplanets) and so we have m tsGBM (ts) = 1

(120)

m tsGBM ( p) = e B( p−ts)

(121)

ln(m tsGBM (te)) . te − ts

(122)

Then Eq. (117) reduces to

and Eq. (119) reduces to B=

As we already did in the section ‘Death Formula’, we assume the number of living Species on Earth nowadays (i.e. at te = 0) to be equal to 50 million, namely m(te) = 50 million. Then Eq. (122) reduces to Eq. (83), as it must be. Finally, the ordinary GBM subcase of tsGBM and sub-subcase of L(t) is characterized by Eq. (118) and by m GBM (t) = A e Bt .

(123)

Then, inserting both Eqs. (118) and (123) into the general Peak-Locus Theorem (108), the latter yields Eq. (109) as shown by (%o95). This is the ‘old’ Peak-Locus Theorem, firstly discovered by this author in late 2011 and already published by him in Chap. 8 of his 2012 book ‘Mathematical SETI’ of 2012, pp. 218–219.

2 Part 2: New Results About Molecular Clock in Evo-SETI Theory

247

For more applications of the Peak-Locus Theorem to polynomial mean values, see Maccone [3, pp. 294–308].

2.6 Shannon Entropy of the Running b-Lognormal The Shannon Entropy of the Running b-lognormal is the key to measure the ‘disorganization’ of what that running b-lognormal represents, let it be a Species (in Evolution) or a Civilization (in Human History) or even an Alien Civilization (in SETI). As it is well known, the Shannon Entropy (17) (measured in bits) of the Running b-lognormal having its peak at time p and the three parameters μ, σ, b is given by (%o96) (for the proof of this key mathematical result, please see Chap. 30 of the author’s book ‘Mathematical SETI’, pp. 685–687, the idea behind the proof is to expand the log of the Shannon Entropy of the b-lognormal, so that the calculation is split into three integrals, each of which may actually be computed exactly): ln

H=

√

 2πσ + μ + 1/2 ln 2

.

(124)

Having so said, the next obvious step is to insert the μ and σ given by the PeakLocus Theorem (108) into (124). After a few steps, we thus obtain the Shannon Entropy of the Running b-lognormal [see (%o97) and (%o98)]: ln(m L ( p)) eσ L · p − 2 ln 2 4π ln 2 · [m L ( p)] 1 . + 2 ln 2 2

H ( p, m L ( p)) =

(125)

This is the fundamental Shannon Entropy H of the Running b-lognormal for any given mean value mL (t). Notice that H is a function of the peak abscissa p in two ways: 1. Directly, as in the term eσL · p , and 2. Through the assigned mean value mL (p). 2

2.7 Introducing Our… Evo-Entropy(p) Measuring How Much a Life Form Has Evolved The Shannon Entropy was introduced by Claude Shannon (1916–2001) in 1948 in his seminal work about Information Theory, dealing of course with telecommunications, channel capacities and computers. But… we need something else to measure ‘how

248

New Evo-SETI Results About Civilizations and Molecular Clock

evolved’ a life form is: we need a positive function of the time starting at zero at the time ts of the origin of life on a certain planet, and then increasing (rather than decreasing). This new function is easily found: it is just the Shannon Entropy (17) WITHOUT THE MINUS SIGN IN FRONT OF IT (so as to make it an increasing function, rather than a decreasing function) and WITH THE NUMERIC VALUE −H(ts) SUBTRACTED, so as it starts at zero at the initial instant ts. This new function of p we call EVO-ENTROPY (Evolution Entropy) and its mathematical definition is thus simply [see (%o101) and (%o102)]: EvoEntropy( p, m L ( p)) = −H ( p, m L ( p)) + H (ts, m L (ts)).

(126)

In some previous papers by this author about Evo-SETI Theory, the Evo-Entropy (126) was called ‘Evo-Index’ (Index of Evolution) or with other similar names, but we now prefer to call it Evo-Entropy to make it clear that it is just the Shannon Entropy with the sign reversed and with value zero at the origin of life. Next we compute the actual expression of Evo-Entropy as a function of the only variable p, the Running b-lognormal peak. To this end, we must first get the expression of (125) at the initial time ts. It is (%o99) ln(m L (ts)) e p·ts − 2 ln 2 4π ln 2 · [m L (ts)] 1 + . 2 ln 2

H (ts, m L (ts)) =

(127)

Subtracting Eq. (127) into (125) with the minus sign reversed, we get the for the final (126) form of our Evo-Entropy (%o100) )  2 2 eσL ·ts eσL · p 1 EvoEntropy( p, m L ( p)) = − 4π ln 2 [m L (ts)]2 [m L ( p)]2 +

ln(m L ( p)/m L (ts)) . ln 2

(128)

The Evo-Entropy (128) is thus made up by two terms: 1. The term )  2 2 1 eσL ·ts eσ L · p − 4π ln 2 [m L (ts)]2 [m L ( p)]2 we shall call the NON-LINEAR PART of the Evo-Entropy (128), while 2. The term

(129)

2 Part 2: New Results About Molecular Clock in Evo-SETI Theory

ln(m L ( p)/m L (ts)) ln 2

249

(130)

we shall call the LINEAR PART of the Evo-Entropy (128), as we explain in the next section.

2.8 The Evo-Entropy(p) of tsGBM Increases Exactly Linearly in Time Consider again the tsGBM defined by Eq. (117) with (118). If we insert Eq. (117) into the EvoEntropy (128), then two dramatic simplifications occur: 1. The non-linear term (129) vanishes, inasmuch as it reduces to  ) 2 1 eσL · p σ2L ·ts − σ2 · p −σ2 ·ts = 0. e 4π ln 2[m L (ts)]2 e L ·e L

(131)

2. The linear term (130) simplifies, yielding (%o104) ' ( ln m L (ts)e B( p−ts) /m L (ts) ln(m L ( p)/m L (ts)) = ln 2 ln 2 B = ( p − ts). ln 2

(132)

In other words, the Evo-Entropy of tsGBM simply is the LINEAR function of the Running b-lognormal peak p EvoEntropytsGBM ( p) =

B ( p − ts). ln 2

(133)

This is a great result! And it was already envisioned back in 2012 in Chap. 30 of the author’s book ‘Mathematical SETI’ when he found that the Evo-Entropy difference between two Civilizations ‘with quite different levels of technological development’ (like the Aztecs and the Spaniards in 1519) is given by the equation EvoEntropytsGBM ( p2 − p1 ) =

B ( p2 − p1 ). ln 2

(134)

(see Eq. (30.29) on p. 693 of that book, where the old minus sign in front of the Shannon Entropy still ruled because this author had not yet ‘dared’ to get rid of it, as he did now in the new definition (126) of EvoEntropy). But what is the graph of this famous linear increase of Evo-Entropy? It is given by Fig. 6. So, we have discovered that the tsGBM Entropy in our Evo-SETI model and the Molecular Clock (see Nei [6] and Nei and Kumar [7]) are the same linear time

250

New Evo-SETI Results About Civilizations and Molecular Clock

Fig. 6 EvoEntropy (in bits per individual) of the latest species appeared on Earth during the last 3.5 billion years. This shows that a Man (i.e. the leading Species nowadays) is 25.575 bits more evolved than the first form of life (call it RNA?) 3.5 billion years ago

function, apart for multiplicative constants (depending on the adopted units, such as bits, seconds, etc.). This conclusion appears to be of key importance to understand ‘where a newly discovered exoplanet stands on its way to develop LIFE’.

3 Conclusions More and more exoplanets are now being discovered by astronomers either by observations from the ground or by virtue of space missions, like ‘CoRot’, ‘Kepler’, ‘Gaia’, Tess, Cheops, Plato and other future space missions. As a consequence, a recent estimate sets at 40 billion the number of Earth-sized planets orbiting in the habitable zones of Sun-like stars and red dwarf stars within the Milky Way Galaxy. With such huge numbers of ‘possible Earths’ in sight, Astrobiology and SETI are becoming research fields more and more attractive to a number of scientists.

3 Conclusions

251

Mathematically innovative papers like this one, revealing an unsuspected relationship between the Molecular Clock and the Entropy of b-lognormals in Evo-SETI Theory, should thus be welcome. Acknowledgements The author is grateful to the Reviewers for accepting this paper just as he had submitted it, without asking for any change that would have required more time. Equally, the author is grateful to Dr. Rocco Mancinelli, Editor in Chief of the International Journal of Astrobiology (IJA), for allowing him to add to each of his IJA published papers one or more pdf files with all the relevant calculations done by Maxima, a rather unusual feature in the scientific literature. Finally, the full cooperation of Ms. Amanda Johns, Ms. Corinna Connolly McCorristine and the Typesetters is gratefully acknowledged.

Supplementary Material To view supplementary material for this article, please visit http://dx.doi.org/10. 1017/S1473550415000506.

References 1. C. Maccone, Mathematical SETI (Praxis, Chichester; Springer, Berlin, in the Fall of 2012, 2012), 724 p. ISBN-10: 3642274366 | ISBN-13: 978–3642274367 2. C. Maccone, SETI, evolution and human history merged into a mathematical model. Int. J. Astrobiol. 12(3), 218–245 (2013) 3. C. Maccone, Evolution and mass extinctions as lognormal stochastic processes. Int. J. Astrobiol. 13(4), 290–309 (2014) 4. A. Markov, A. Korotayev, Phanerozoic marine biodiversity follows a hyperbolic trend. Palaeoworld 16(4), 311–318 (2007) 5. A. Markov, A. Korotayev, Hyperbolic growth of marine and continental biodiversity through the Phanerozoic and community evolution. J. Gen. Biol. 69(3), 175–194 (2008) 6. M. Nei, Mutation-Driven Evolution (Oxford University Press, Oxford, 2013) 7. M. Nei, S. Kumar, Molecular Evolution and Phylogenetics (Oxford University Press, New York, 2000) 8. A. Papoulis, S.U. Pillai, Probability, Random Variables and Stochastic Processes, 4rth edn. (Tata McGraw-Hill, New Delhi, 2002). ISBN 0-07-048658-1

Life Expectancy and Life Energy According to Evo-SETI Theory

Abstract This Chapter “Life Expectancy and Life Energy According to Evo-SETI Theory” is profoundly innovative for the Evo-SETI (Evolution and SETI) mathematical theory. While this author’s previous papers were all based on the notion of a b-lognormal, that is a probability density function in the time describing one’s life between birth and “senility” (the descending inflexion point), in this paper the b-lognormals range between birth and peak only, while a descending parabola covers the lifespan after the peak and down to death The resulting finite curve in time is called a LOGPAR, a nickname for “b-LOGnormal and PARabola”. The advantage of such a formulation is that three variables only (birth, peak and death) ore sufficient to describe the whole Evo-SETI theory, and the senility is discarded forever and so is the normalization condition of b-lognormals: only the shape of the b-lognormals is kept between birth and peak, but not its normalization condition. In addition, further advantages exist: 1. The notion of ENERGY at last becomes part of Evo-SETI theory. This is in addition to the notion of ENTROPY already contained in the theory as the Shannon Information Entropy of b-lognormals, as it was explored in this author’s previous chapters. Actually, the LOGPAR may now be regarded as a POWER CURVE, i.e. a curve expressing the power of the living being to which it refers. And this power is to be understood both in the strict sense of physics (i.e. a curve measured in watts) and in the loose sense of “political power” if the logpar refers to a Civilization. 2. Then the integral in the time of this power curve is of course the ENERGY either absorbed or produced by the physical phenomenon that the LOGPAR is describing in the time. For instance, if the logpar shows the time evolution of the Sun over about 10 billion years, the integral of such a curve is the energy produced by the Sun over the whole of its lifetime. Or, if the logpar describes the life of a man, the integral is the energy that this man must use in order to live. 3. The PRINCIPLE OF LEAST ENERGY, i.e. the key stone to all Physics, also enters now into the Evo-SETI Theory by virtue of the so-called LOGPAR HISTORY FORMULAE, expressing the b-lognormal’s mu and sigma directly in terms of the three only inputs b, p, d. The optimization of the lifetime of a living creature, or of a Civilization, or of a star, is obtained by setting to zero the first derivative of the area under the logpar power curve with respect to sigma. © Springer Nature Switzerland AG 2020 C. Maccone, Evo-SETI, https://doi.org/10.1007/978-3-030-51931-5_6

253

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Life Expectancy and Life Energy According to Evo-SETI Theory

That yields the best value of both mu and sigma fulfilling the Principle of Least Energy for Evo-SETI Theory. 4. We also derive for the first time a few more mathematical equations related to the “adolescence” (or “puberty”) time, i.e. the time when the living organism acquires the capability of producing offsprings. This time is defined as the abscissa of ascending inflection point of the b-lognormal between birth and peak. In addition, we prove that the straight line parallel to the time axis and departing from the puberty time comes to mean the “Fertility Span” in between puberty and the EOF (End-Of-Fertility time), which is where the above straight line intersects the descending parabola. All these new results apply well to the description of Man as the living creature to which our Evo-SETI mathematical theory perfectly applies. In conclusion, this chapter really breaks new mathematical ground in Evo-SETI Theory, thus paving the way to further applications of the theory to Astrobiology and SETI. Keywords Lognormal probability distributions · Power · Energy · Entropy · SETI

1 Part 1: Logpar Curves and Their History Equations 1.1 Introduction to Logpar “Finite Lifetime” Curves If our reader would like to read just the first three pages (pp. 41–42–43) of our OPEN ACCESS published paper https://www.cambridge.org/core/journals/international-journal-of-astrobiology/ article/div-classtitlenew-evo-seti-results-about-civilizations-and-molecular-clo ckdiv/494A1FECFF9A5A08E91662A8D57573DE then he/she would immediately realize why we were forced to introduce the notion of FINITE LIFETIME as the new logpar curve. The idea is easy: we seek to represent any lifetime by virtue of just three points in time: birth, peak, death (BPD). No other point in between. That is, no other “senility point” s such as those appearing in all b-lognormals that this author had published in his Evo-SETI Theory prior to 2017 (Refs. [3–8]). In fact, it is easier and more natural to describe someone’s lifetime just in terms of birth, peak and death, than in terms of birth, senility and death, because when the senility arrives is rather uncertain and so hard to define in the practice for any individual when his/hers senility occurs. Now look at Fig. 1, please. The first part, the one shown in blue on the left, i.e. prior to the peak time p, has just the same shape as a b-lognormal: it starts at birth time b, climbs up to the adolescence time a (ascending inflexion point of the blognormal) (in reality the adolescence time should more properly be called “puberty time” since it marks the beginning of the reproduction capacity for that individual) and finally reaches the peak time at p (the maximum, i.e. the point of zero first

1 Part 1: Logpar Curves and Their History Equations

−5

8×10

History of the SUN as a LOGPAR power curve

−5

7×10 6×10 5×10 4×10

−5 −5 −5 −5

3×10 2×10

255

−5 −5

1×10

0 3 3 3 3 3 − 5×10 − 4×10 − 3×10 − 2×10 − 1×10

0

1×10

3

2×10

3

3×10

3

4×10

3

5×10

3

6×10

3

7×10

3

Time in MILLION years: blue=negative=PAST, 0 is NOW, red=positive=FUTURE Fig. 1 History of the Sun as a logpar power curve, created only by assigning  the three numeric input logpar values b = −4.567 × 109 year, p = 0 year, d = 5 × 109 year . The Sun formed about 4.6 billion years ago from the collapse of part of a giant molecular cloud that consisted mostly of hydrogen and helium and that probably gave birth to many other stars. This age is estimated using computer models of stellar evolution and through nucleocosmochronology. The result is consistent with the radiometric date of the oldest Solar System material, at 4.567 billion years ago (see the Wikipedia site https://en.wikipedia.org/wiki/Sun#Formation). The Sun is now producing energy at about a constant rate, and it will keep doing so for at least a billion year in the future. The above graph reveals so since in the next billion year the energy production of the Sun will change by less that 10−11 (a femto). In the above graph the b-lognormal power curve (not a probability density any more, since not normalized to 1 any more) is covering the time between the birth of the Sun 4.567 billion years ago and nowadays and is shown in blue. On the contrary, the logpar part between now and 5 billion years in the future is shown in red and is just a parabola with its vertex at the peak of the b-lognormal

derivative of the b-lognormal). All this is just ordinary b-lognormal stuff, as we have been “preaching” since about 2012, except that our present b-lognormal does not fulfill the normalization constant and so it is not a probability density any more. But now the real novelty comes, i.e. the second part, the one on the right, shown in red. That is just a parabola having its vertex exactly at the peak time p. Notice that this definition automatically implies that the tangent line at the peak is horizontal, i.e. the same for both the b-lognormal and the parabola. Notice also that, after the peak, the parabola plunges down until it reaches the time axis at the death time d. Therefore this new definition of death time d is different from the old definition of d applying to b-lognormals plus a descending straight line with junction at the senility point s (descending inflexion point of the b-lognormal), as we did prior to 2017. And this is the LOGPAR (b-LOGnormal plus PARabola) new CURVE FINITE IN TIME (namely ranging in time just between birth and death). In the present paper and we study the logpar curve with surprising results bringing in the extremely important physical notion of ENERGY.

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1.2 Finding the Parabola Equation of the Right Part of the Logpar We shall now cast into appropriate mathematics the above popular description of what a logpar curve is. Consider the equation of a parabola in the time t having its vertical axis along the t = p vertical line: y = α(t − p)2 + β(t − p) + γ

(1)

where α, β and γ are the three coefficients of the time that we must determine according to the assumptions shown in Fig. 1. To find them, we must resort to the three conditions that we know to hold by virtue of a glance to Fig. 1: 1. #1 CONDITION: the height of the peak is P, just the same as the height of the peak of the b-lognormal on the left in Fig. 1. Thus, inserting the two equations of the peak, namely 

t=p y=P

(2)

into (1), the latter yields immediately P=γ

(3)

that, when inserted back into (1), changes it into y = α(t − p)2 + β(t − p) + P.

(4)

2. #2 CONDITION: the tangent straight line at both the b-lognormal and the parabola at the peak abscissa p is horizontal. In other words, the first derivative of (4) at t = p must equal zero. Differentiating (4) with respect to t, equalling that to zero and then solving for β yields β = −2α(t − p).

(5)

Inserting (5) into (4), the latter is turned into y = −α(t − p)2 + P.

(6)

3. #3 CONDITION: at the death time d, one must have y = 0, yielding from (6) the equation 0 = −α(d − p)2 + P.

(7)

1 Part 1: Logpar Curves and Their History Equations

257

Solving (7) for α one gets α=

P . (d − p)2

(8)

Finally, inserting (8) into (6) the desired equation of the parabola is found   (t − p)2 . y(t) = P 1 − (d − p)2

(9)

As confirmation, one may check that (9) immediately yields the two conditions 

y( p) = P y(d) = 0.

(10)

1.3 Finding the b-Lognormal Equation of the Left Part of the Logpar As for the b-lognormal between birth and peak, making up the left part of the logpar curve, we already know all its mathematical details from the previous many papers published by this author on this topics, but we shall summarize here the main equations for the sake of completeness. The equation of the b-lognormal starting at b reads (log(t−b)−μ)2

e− 2σ 2 . b_ lognormal(t; μ, σ, b) = √ 2π σ (t − b)

(11)

Tables listing the main equations that can be derived from (11) were given by this author in Refs. [5, 6] and we shall not re-derive them here again. We just confine ourselves to reminding that: 1. The abscissa p of the peak of (11) is given by p = b + eμ−σ . 2

(12)

Proof Take the derivative of (11) with respect to t and set it equal to zero. Then solve the resulting equation for t, that now becomes p, and (12) is found. 2. The ordinate P of the peak of (11) is given by σ2

e 2 −μ P=√ . 2π σ

(13)

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Life Expectancy and Life Energy According to Evo-SETI Theory

Proof Rewrite p instead of t in (11) and then insert (12) instead of p. Then simplify to get (13). 3. The abscissa of the adolescence point (that should actually be better named “puberty point”) is the abscissa of the ascending inflexion point of (11). It is given by a =b+e

−σ

√

2 σ 2 +4 − 3σ2 2

+μ

(14)

Proof Take the second derivative of (11) with respect to t and set it equal to zero. Then solve the resulting equation for t, that now becomes a, and (14) is found. 4. The ordinate of the adolescence point is given by e−

σ

√

2 σ 2 +4 + σ4 4

√

−μ−

1 2

(15)

2π σ

Proof Just rewrite a instead of t in (11) and then insert (14) and simplify the result. Let us now notice that, within the framework of the logpar theory described in this paper, we may NOT say that (11) fulfills the normalization condition ∞ b_ lognormal(t; μ, σ, b)dt = 1

(16)

b

since (11) here is only allowed to range between b and p. Rather than adopting (16), we must thus replace (16) by the integral of (11) between b and p only. Fortunately, it is possible to evaluate this integral in terms of the error function defined by 2 er f (x) = √ π

x

e−t dt. 2

(17)

0

In fact, the integral of the b-lognormal (11) between b and p turns out to be given by p

p b_ lognormal(t; μ, σ, b)dt =

b

b

2

(log(t−b)−μ) 1 + er f e− 2σ 2 dt = √ 2π σ (t − b)

√

√ 2 ln( p−b)− 2μ 2σ

2

Now, inserting (12) instead of p into the last erf argument, a remarkable simplification occurs: μ and b both disappear and only σ is left. In addition, the erf property er f (−x) = −er f (x) allows us to rewrite

1 Part 1: Logpar Curves and Their History Equations

=

 1 + er f − √σ2 2

=

259

1 − er f



σ √ 2



2

.

(18)

In conclusion, the area under the b-lognormal between birth and peak is given by p b_ lognormal(t; μ, σ, b)dt =

1 − er f



σ √ 2

2

.

(19)

b

This result will prove to be of key importance for the further developments described in the present paper.

1.4 Area Under the Parabola on the Right Part of the Logpar Between Peak and Death We already proved that the parabola on the right part of the logpar curve has Eq. (9). Now we want to find the area under this parabola between peak and death, that is 

d   d P (t − p)2 P 1− dt = P(d − p) − (t − p)2 dt (d − p)2 (d − p)2 p

p

P (d − p)3 3 (d − p)2   2P(d − p) 1 4 = = · P(d − p) 3 2 3 ⎡ ⎤ Ar ea o f Ar chimedes 1 ⎣ = · parabolic segment, ⎦. 2 pr oved < 212 B.C.

= P(d − p) −

(20)

The great Ancient Greek mathematician Archimedes (circa 287 BC–212 BC) of Syracuse (Sicily) already “knew” the last integral result even before the Calculus was discovered by Newton and Leibniz after 1660. More appropriately, (20) is a special case of Cavalieri’s quadrature formula (1635, https://en.wikipedia.org/wiki/Cavali eri%27s_quadrature_formula). Actually, Archimedes used the “method of exhaustion” to compute the area of a segment of the parabola, as very neatly described at the site https://en.wikipedia.org/wiki/The_Quadrature_of_the_Parabola.

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Life Expectancy and Life Energy According to Evo-SETI Theory

In conclusion, the area under our parabola between peak and death is given by (20), that we now rewrite as 

d   2P(d − p) (t − p)2 P 1− dt = . 2 3 (d − p)

(21)

p

1.5 Area Under the Full Logpar Curve Between Birth and Death We are now in a position to compute the full area A under the logpar curve, that is given by the sum of Eqs. (19) and (21), that is 1 − er f



σ √ 2



2

+

2P(d − p) =A 3

(22)

This is a really important equation for this chapter. In fact, if we want the logpar be a truly probability density function (pdf), we must assume in (22) A=1

(23)

But, surprisingly, we shall NOT do so! Let us rather ponder over what we are doing: 1. We are creating a “Mathematical History” model where the “unfolding History” of each Civilization in the time is represented by a logpar curve. 2. The knowledge of only three points in time is requested in this model: b, p and d. 3. But the area under the whole curve depends on σ as well as on μ, as we see upon inserting (13) instead of P into (22), that is 1 − er f



2

σ √ 2



σ2

e 2 −μ 2(d − p) +√ = A(μ, σ ). · 3 2π σ

(24)

4. Also p is to be replaced by its expression (12) in terms of σ and μ, yielding the new equation 1 − er f 2



σ √ 2



σ2 2

e +√

−μ

2π σ

·

 2 2 d − b − eμ−σ 3

= A(μ, σ ).

(25)

1 Part 1: Logpar Curves and Their History Equations

261

5. The meaning of (25) is that birth and death are fixed, but the position of the peak may move according to the different numeric values of σ and μ. 6. In addition to that, we “dislike” the presence of the error function erf in (25) since this is not an “ordinary” function, i.e. it is one of the functions that mathematicians call “higher transcendental functions”, having complicated formulae describing them. Thus, we would rather get rid of erf . How may we do so?

1.6 The Area Under the Logpar Curve Depends on Sigma Only, and Here Is the Area Derivative w.r.t. Sigma The simple answer to the last question (6) is “by differentiating both sides of (25) with respect to σ ”. In fact, the derivative of the erf function (17) is just the “Gaussian” exponential 2 d er f (x) 2 = √ · e−x . dx π

(26)

and so the erf function itself will disappear by differentiating (25) with respect to σ . In fact, the derivative of the first term on the left hand side of (25) simply is, according to (26), ⎡ d ⎣ dσ

1 − er f



σ √ 2

⎤

2

σ2

e− 2 ⎦ = −√ . 2π

(27)

As for the derivative with respect to σ of the second term on the left hand side of (25) we firstly notice that σ appears three times within that term. Thus, the relevant derivative is the sum of three terms, each of which includes the derivative of one of the three terms multiplied by the other two terms unchanged. In equations, one has: ⎤  μ−σ 2 2 d − b − e e d ⎣ ⎦ +√ · dσ 2 3 2π σ σ2 √  μ−σ 2 2 2 3 − σ2 − σ2 2 −e + d − b e 2 −μ 2 2 e e = √ −√ − √ 2 3 π 3 πσ 2π σ2 √  μ−σ 2 2 −e + d − b e 2 −μ . + √ 3 π ⎡

1 − er f



σ √ 2



σ2 2

−μ

(28)

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Life Expectancy and Life Energy According to Evo-SETI Theory

Several alternative forms of this Eq. (28) are possible, and that is rather confusing. However, using a symbolic manipulator (this author did so by virtue of Maxima), a few steps lead to the following form of (28): d A(σ ) d A(μ(σ ), σ ) ≡ dσ dσ √ √ σ2 σ2 3 σ2 σ2 2(d − p)e− 2 2(d − p)e− 2 2 2 e− 2 e− 2 =− √ + √ + √ −√ 3 π( p − b)σ 2 3 π ( p − b) 3 π 2π  − σ2  2 2 2 pσ − 2dσ + bσ − 2 p + 2d e 2 = . (29) √ 3 2π ( p − b)σ 2 This (29) is the derivative of the area with respect to sigma.

1.7 Exact “History Equations” for Each Logpar Curve We now take a further, crucial step in our analysis of the logpar curve: we IMPOSE that the derivative of the area with respect to sigma, i.e. (29), is zero d A(σ ) = 0. dσ

(30)

What does that mean? Well, hold your breath: (30) is the Evo-SETI equivalent of the LEAST ACTION PRINCIPLE in physics! This shocking conclusion does not show up at the moment, but it will at the end of this chapter. For the time being with content ourselves with the “crude mathematics” of rewriting the imposed condition (30) by virtue of the last expression in (29) that, getting rid of both the exponential and the denominator, immediately boils down to pσ 2 − 2dσ 2 + bσ 2 − 2 p + 2d = 0.

(31)

This is just the quadratic equation in σ σ 2 ( p − 2d + b) = 2( p − d)

(32)

and so we finally get σ2 =

2(d − p) . 2d − (b + p)

(33)

1 Part 1: Logpar Curves and Their History Equations

263

This is the most important new result discovered in the present chapter. It is the LOGPAR HISTORY EQUATION FOR σ √ √ 2 d−p σ =√ . 2d − (b + p)

(34)

In other words, given the input triplet (b, p, d) then (33) immediately yields the exact σ 2 of the b-lognormal left part of the logpar curve. It was discovered by this author on November 22, 2015, and led not only to this chapter, but to the introduction of the ENERGY spent in a lifetime by a living creature, or by a whole civilization whose “power-vs-time” behaviour is given by the logpar curve, as we will understand better in the coming sections of this chapter. At the moment, for reasons that will become obvious later, we confine ourselves to taking the limit of both sides of (34) for d → ∞, with the result √ √ √ √ 2 d−p 2 d = lim √ = 1. lim σ = lim √ d→∞ d→∞ 2d − (b + p) d→∞ 2d

(35)

Since we already know that σ must be positive, (35) really shows that σ may range between zero and one only 0 ≤ σ ≤ 1.

(36)

Next to (34) one of course has a similar LOGPAR EQUATION FOR μ, that is immediately derived from (12) and (34). To this end, just take the log of (12) to get μ = ln( p − b) + σ 2

(37)

that, invoking (33), yields the desired logpar equation for μ μ = ln( p − b) +

2(d − p) . 2d − (b + p)

(38)

Having proved both (34) and (38), we still have to make an important remark about them. We will put the suffix “LH” (standing for “Logpar History”) to each of them to remind their users that they were derived under the assumption (30) that the derivative of the area under the logpar curve is ZERO, amounting to the LEAST ENERGY PRICIPLE for Evo-SETI Theory. In conclusion, our key two LOGPAR LEAST-ENERGY HISTORY FORMULAE are √ √  2 d− p σ L H = √2d−(b+ p) (39) 2(d− p) μ L H = ln( p − b) + 2d−(b+ . p)

264

Life Expectancy and Life Energy According to Evo-SETI Theory

1.8 Considerations on the Logpar Least-Energy History Formulae Some considerations on the logpar least-energy History Formulae (39) are now of order: 1. All these formulae are exact, i.e. no Taylor series expansion was ever used to derive them. 2. But they were obtained by equalling to zero the derivative (30) with respect to σ of the total area under the logpar curve given by (25). 3. Therefore the logpar History Formulae (39) are the equations of a minimum of the A(σ ) function expressing the total area (25) as a function of σ . 4. One further question might be: μ and σ are independent variables in the Gaussian (and so in the lognormal, that is just eˆ(Gaussian)). Since we differentiated (25) with respect to σ already, why don’t we try differentiating it with respect to μ also? The answer is: because differentiating (25) with respect to μ leads to the ABSURD result b = d i.e. one dies just when born! We leave the calculation to readers as an exercise.

1.9 Logpar Peak Coordinates Expressed in Terms of (b, p, d) Only Of particular importance for all future logpar applications is the expression of the peak coordinates ( p, P) expressed in terms of then input triplet (b, p, d) only. Since the peak abscissa p is assumed to be known, we only have to derive the formula for the peak ordinate P. That is readily obtained by inserting the logpar History Formulae (39) into the peak height expression (13). After a few rearrangements, it is found to be given by √ (d− p) 2d − (b + p) · e− 2d−(b+ p) . PL H = √ √ 2 π d − p( p − b)

(40)

Again, the suffix “LH” was added to this expression for P to remind the readers that it is a mathematical consequence of using the Logpar History Formulae (39) in the course of its derivation.

2 Part 2: Energy as the Area Under All Logpar Power Curves

265

2 Part 2: Energy as the Area Under All Logpar Power Curves 2.1 The Area Under a Logpar and Its Meaning as “Lifetime Energy” Let us go back to (25), i.e. the total area under the logpar curve: 1 − er f



σ √ 2



 μ−σ 2 σ2 e 2 −μ 2 d − b − e · +√ = A(μ, σ ) 3 2π σ

2

(41)

If we insert the logpar History Formulae inside this equation, we obviously get the expression of the total area under the logpar as a function of just the input triplet (b, p, d) only. After some rearranging, this area formula turns out to be the rather complicated (but exact!) area equation:

A L H (b, p, d) =

1 − er f



√ d− p √ 2d−(b+ p) 2

+

√ √ d− p d − p 2d − (b + p) − 2d−(b+ p) . ·e √ 3 π( p − b) (42)

What is the physical meaning of this area? If we consider the logpar curve as the curve of the power (measured in watts) of the Sun life along the whole of its history course, then the area under this curve, i.e. the integral of the logpar between birth and death, is the total energy (measured in joules) spent by the Sun in its whole lifetime: ENERGY_ spent_ in_ the_ LIFETIME d =

POWER_ of_ that_ LIFETIME(t)dt b

d =

logpar_ of_ that_ LIFETIME(t)dt.

(43)

b

In other words still, and in a much more general sense, if we know the power curve of any living being that lived in the past, like a cell, or an animal, or a human, or a Civilization of humans or of any other living forms (including ExtraTerrestrials), the integral of that power curve, i.e. logpar curve, between birth and death is the TOTAL ENERGY spent by that living form during the whole of its lifetime. Just an example regarding the last statement: if we assume that all Humans have potentially the same amount of energy to spend during their whole lifetime, then the logpar of great men

266

Life Expectancy and Life Energy According to Evo-SETI Theory

who “died young” (like Mozart, for instance) must have the same area below their logpar and so a much higher peak since they lived shorter than others. We will not insist more on these ideas right now, but in coming papers there will be a lot to say. Let us go back to the Sun lifetime. Upon inserting the Sun input triplet ⎧ ⎨ b = −4.567 × 109 year Sun_ input_ triplet = p = 0 year ⎩ d = 5 × 109 year

(44)

into the area Eq. (42) the number is found   A L H _Sun = A L H −4.567 × 109 , 0.5 × 109 = 0.45301181787684

(45)

This is far from being equal to 1, the numeric value that would have made the Sun logpar curve to be a true probability density. So, we have abandoned the use of probability densities (as all b-lognormals that this authors considered prior to 2017 were) but we have now our free hand to consider the ENERGY spent during a given lifetime. And the ENERGY is “something” profoundly different from the ENTROPY, that this author had previously considered [for instance with reference to his theorem that the (Shannon) Entropy of a Geometric Brownian Motion is a LINEAR function of the time, just as the MOLECULAR CLOCK is a LINEAR function of the time also (and that is Kimura’s theory of NEUTRAL evolution at the molecular level)] (Refs. [1, 2, 9–11]). So we made a really remarkable step ahead: by releasing the normalization condition typical of probability densities, we were able to introduce ENERGY into the Evo-SETI theory. But does that mean that we have abandoned ENTROPY? Not at all! Entropy and the Peak Locus Theorem supporting it, ARE STILL VALID in that the peak is the JUNCTION POINT BELONGING TO BOTH THE b-LOGNORMAL AND THE parabola. Wow! We have thus defined the new function of b, p, d, only that we call the ENERGY of the logpar Energy L H (b, p, d) = Area under the LOGPAR(b, p, d)  √ d− p 1 − er f √2d−(b+ p) = 2 √ √ d− p d − p 2d − (b + p) − 2d−(b+ p) . + ·e √ 3 π ( p − b)

(46)

3 Part 3: Mean Energy in a Lifetime and Lifetime Mean Value

267

3 Part 3: Mean Energy in a Lifetime and Lifetime Mean Value 3.1 Mean Energy in a Lifetime In this section we are going to consider the notion of mean value of a logpar power curve. Having abandoned the normalization condition for our logpar curves, clearly we may not use the same mean value definition of a random variable typical of probability theory. However it’s easy to use the mean value Mean Value Theorem for Integrals. This is a variation of the mean value theorem which guarantees that a continuous function has at least one point where the function equals the average value of the function. Figure 2 graphically explains that. To translate the Mean Value Theorem for Integrals into a mathematical equation holding for our logpar curves, we clearly have to start from the Area Eq. (42) with d replaced by D, and divide that area by the length of the (D − b) segment in order to get the point along the vertical axis such that the area of the rectangle equals the Area (42). This is the required Mean Energy Value over a lifetime and is given by Mean_ Energy L H _ over_ a_ lifetime =  1−er f

=

√

√ D− p 2D−(b+ p)

2

+

√

√ D− p 2D−(b+ p) √ 3 π( p−b)

D−b

A(b, p, D) D−b D− p

· e− 2D−(b+ p)

.

(47)

It is interesting to consider the limit of the Mean Energy over a lifetime (47) for D → ∞. The calculation implies the use of L’Hospital’s rule, and the result is

Fig. 2 Mean value theorem for integrals: a continuous function has at least one point where the function equals the average value of the function. Taken from the Google Mathwords site https://www.google.it/search?rlz=1C2OPRA_enIT590IT590&site=&source=hp&q=mean+ value+theorem+for+integrals&oq=mean+value+&gs_l=psy-ab.1.1.0l4.2302.4300.0.7092.13.12.0. 0.0.0.127.925.10j2.12.0….0…1.1.64.psy-ab..1.11.854.0..46j0i46k1.14EBMyVmgwk

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Life Expectancy and Life Energy According to Evo-SETI Theory

Asymptotic_ Mean_ Energy L H _ over_ a_ lifetime   = lim Mean_ Energy L H _ over_ a_ lifetime D→∞ √ 2 = √ √ . 3 π e( p − b)

(48)

By this we have completed the study of the mean along the vertical axis, i.e. the power axis. However, one might still wish to find, in some sense, “the mean value of what lies on the horizontal axis”, i.e. the lifetime mean value. That is done in the next section.

3.2 Lifetime Mean Value It is natural to seek for some mathematical expression yielding the mean value of a lifetime, meaning the mean value along the time axis of the (D − b) time segment representing the lifetime of a living organism, or a civilization or even an ET civilization. We propose the following definition of such a lifetime mean value: lifetime L H _ mean_ value p =

D t · b_lognormal(t; μ, σ, b)dt +

t · parabola(t)dt

(49)

p

b

inserting the b-lognormal (11) and the parabola (9) into (49), the latter is turned into p = b

(log(t−b)−μ)2

e− 2σ 2 t·√ dt + 2π σ (t − b)

D p

 (t − p)2 dt. t · P 1− (D − p)2 

(50)

The first integral may be computed in terms of the error function er f (x) given by (17), and the result is p b

(log(t−b)−μ)2

e e− 2σ 2 t·√ dt = 2π σ (t − b)

σ2 2

+μ

+

  2  p−b)+μ √ 1 − er f σ −log( 2σ

2    p−b)−μ b 1 − er f log( √ 2σ 2

that may be further simplified by invoking (12), with the result

(51)

3 Part 3: Mean Energy in a Lifetime and Lifetime Mean Value

p b

2

(log(t−b)−μ) e e− 2σ 2 t·√ dt = 2πσ (t − b)

σ2 2

+μ



1 − er f 2

√

269

 2σ

+

   b 1 − er f √σ2 2

.

(52)

Re-expressing now (52) in terms of the History Formulae (39), it finally takes the form   √  3(D− p) 2 D− p (log(t−b)−μ)2 p ( p − b)e 2D−(b+ p) 1 − er f √2D−(b+ 2σ 2 e p) t·√ dt = 2 2π σ (t − b) b   √  D− p b 1 − er f √2D−(b+ p) . (53) + 2 As for the second integral in (50), i.e. the parabola integral, it is promptly computed as follows D p

 P(D − p)(3D + 5 p) (t − p)2 dt = . t · P 1− 2 12 (D − p) 

(54)

Inserting for P its expression (40), after some rearranging we conclude that the parabola integral is given by D p

√ √  (D− p) 2D − (b + p) D − p(3D + 5 p) − 2D−(b+ (t − p)2 p) . dt = t · P 1− ·e √ 2 24 π( p − b) (D − p) 

(55) In conclusion, the mean lifetime is found by summing (53) and (55) and reads lifetime L H _ mean_ value   √  3(D− p) 2 D− p ( p − b)e 2D−(b+ p) 1 − er f √2D−(b+ p) =    √ 2 D− p b 1 − er f √2D−(b+ p) + 2 √ √ (D− p) 2D − (b + p) D − p(3D + 5 p) − 2D−(b+ p) . + ·e √ 24 π ( p − b)

(56)

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Life Expectancy and Life Energy According to Evo-SETI Theory

4 Part 4: Adolescence Formulae (Or Puberty Formulae) 4.1 Logpar’s Increasing Inflexion Time as Adolescence Time (Or Puberty Time for Living Beings) The point of inflexion of the b-lognormal making up for the left part of the logpar has a special meaning for us. Since it lies somehow in between the birth and the peak of a living being (or of a Civilization), we like to think of it as the “adolescence time” and denote it by a, as we already did in (14). More correctly, in case we are referring to a true living being rather than to a Civilization or to the lifetime of a star, we should refer to it as “puberty time”, since it appears to be the time when the reproductive capacities of that living being start. As early as 2012 (see Ref. [5], p. 160, (6.21)) had this author discovered that, if b, a, and p are assigned, then the b-lognormal’s μ and σ are exactly determined in terms of b, a, and p by the following ADOLESCENCE FORMULAE (that might be called Puberty Formulae as well, if referred to living beings other than populations of living beings or even stars). Notice that we will apply the suffix “LA” (standing for logpar adolescence) to the following formulae in order to distinguish them neatly from the corresponding optimized (least energy) History Formulae (39): ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ σL A =

 p−b ln a−b   p−b +1 ln a−b

⎪ ⎪ ⎪ ⎪ ⎩ μ L A = ln( p − b) +

  2 p−b ln a−b  p−b +1 ln a−b

(57)

Proof We start from the definition (14) of a. With a little rearranging, it may be rewritten as √ σ σ 2 + 4 3σ 2 − + μ. (58) ln(a − b) = − 2 2 On the other hand, the peak time (12) yields ln( p − b) = μ − σ 2 .

(59)

Subtracting (59)–(58) makes μ disappear and yields the resolving equation in σ only  ln

p−b a−b

 =

√ σ σ2 + 4 σ2 + . 2 2

(60)

4 Part 4: Adolescence Formulae (Or Puberty Formulae)

271

Isolating the radical yields   p−b − 2 ln a−b

σ2 2

σ

 =

 σ2 + 4

(61)

the square of which is turns out to be a quadratic in σ   p−b − 4 ln a−b

σ2 2

σ2

2 = σ 2 + 4.

(62)

Solving (62) for σ 2 , one gets   2 p−b ln a−b σ2 =  p−b +1 ln a−b

(63)

and finally the Adolescence Formula for σ  p−b ln a−b σ =  . p−b ln a−b + 1

(64)

The corresponding Adolescence Formula for μ is obtained by solving (12) for μ and then inserting (64) into the resulting equation. One thus gets   2 p−b ln a−b μ = ln( p − b) +  p−b +1 ln a−b

(65)

and the Adolescence Formulae (57) are thus proven. Just notice again that all equations derived so far in this paper are exact, i.e. no Taylor expansion was ever used. That helps building our LOGPAR and Evo-SETI Theory on solid mathematical ground.

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Life Expectancy and Life Energy According to Evo-SETI Theory

5 Part 5: Life Expectancy and Fertility in Logpars 5.1 Reconsidering the Death Time d as a Living Being’s Life Expectancy Consider Fig. 3, showing the LOGPAR of a Man’s life. In the previous section we only studied the mathematical properties of the b-lognormal’s part on the left of the logpar curve. But now we want to extend our mathematical considerations to the right part of the logpar also, that is the parabola. The first, obvious notion to introduce is that of life expectancy, that is how long a living being may hope to live. Optimistically, for Humans this might be assumed to be around 80 years, though the numbers change considerably between men and women and according to the different parts of the world. Clearly, the duration of the life expectancy for Humans also depend upon the different economic conditions in which those Humans live. To

Fig. 3 The life of a Man as a LOGPAR power curve. The horizontal axis shows the Man’s age in years since his birth. Then, three curves are shown in this graph: (1) The blue b-lognormal (fun1) departing from birth at time b = 0, intercepting the horizontal green line at the b-lognormal’s ascending inflextion point (at 11 years of Man’s age), then climbing up to the peak at 33 years of age. Imagine that you don’t see the rest of the descending b-lognormal beyond the peak. (2) The red parabola. Imagine that you don’t see the first ascending part of the parabola between birth and peak. Then, after the peak, the red parabola plunges down on the right up to the age 80, i.e. the Man’s death. (3) The green horizontal line intercepts the ascending b-lognormal at its ascending inflexion point. That is the puberty time for Man, i.e. the beginning of its reproductive capabilities at age 11. Then the horizontal green line continues to the right until it intercepts the red parabola at the End-Of-Fertility time (EOF time). (4) The green line segment in between puberty and EOF time is the FERTILITY Span

5 Part 5: Life Expectancy and Fertility in Logpars

273

fix the ideas, we shall assume in this chapter (d − b)Man = 80 year s.

(66)

When considering the lifetime of a generic Man, as in Fig. 3, it is easier and customary to compute the Man’s age starting from his birth, so that in (66) one may set b = 0, changing (66) into dMan = 80 year s.

(67)

5.2 Introducing the Living Being’s End-Of-Fertility (EOF) Time Have a look at Fig. 3 again. Since the point of increasing inflexion of the b-lognormal on the left is the puberty time, one may consider the straight line parallel to the time axis, departing from that inflexion point and finally reaching its intercept with the parabola. This intercept we call End-Of-Fertility (EOF) time. Correspondingly, the segment length between the puberty time and the EOF time is the FERTILITY period of that individual. And now we cast all that into equations. First of all, we must determine the abscissa of the EOF time. In other words, we have to equal the ordinate A of the puberty time (i.e. the ascending inflexion point of the b-lognormal on the left) to the parabola intercept on the right, and then solve the resulting equation for the intercept time, that is precisely the EOF time. Equating thus (15) to the parabola (9), one gets e−

σ

√

2 σ 2 +4 + σ4 4

√

2π σ

−μ− 21

  (t − p)2 . = P 1− (d − p)2

(68)

Solving (68) for t one gets the desired EOF time:

tEOF

 √  σ σ 2 +4 σ2 1  e− 4 + 4 −μ− 2  = p + (d − p) 1 − . √ 2π σ P

(69)

Next, we must let P disappear from (69) by inserting (13) into (69). The result is that μ cancels out in the exponent and, after a little rearranging, one gets  tEOF = p + (d − p) 1 − e−

σ

√

σ 2 +4+σ 2 +2 4

.

(70)

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Life Expectancy and Life Energy According to Evo-SETI Theory

To re-express (70) in terms of b, a and p only, the Adolescence Formula for σ [that is the upper (57)] must be invoked, and the conclusion is      tEOF = p + (d − p) 1 − e

ln −

   ln 

2

p−b  a−b ) +4   ( a−b ) (p−b p−b 2 ) −2 ( a−b )+1 − ln( a−b √ lnp−b p−b ln( a−b )+1 ln( a−b )+1 p−b

4

.

(71)

This is the abscissa of the intercept between the parabola and the horizontal line drawn from the puberty time. In other words, this is the time when Fertility ends.

5.3 Life Expectancy of Living Beings The Life Expectancy of any living being is how much time that being may expect to live. In the LOGPAR framework, it is clearly given by the difference between the death time d and the birth time b; Life_ Expectancy = d − b.

(72)

5.4 Fertility Span of Living Beings The Fertility Span (FS) of any living being is the amount of time during which that being has the capability of producing offsprings. In the LOGPAR framework, this FS is clearly given by the difference between the EOF time and the puberty time. In equations, and in terms of μ and σ also, we must subtract (14)–(70), i.e. Fertility_ Span = FS = EOF_ time − a  = p + (d − p) 1 − e− − b + e−

σ

√

2 σ 2 +4 − 3σ2 2

+μ

σ

.

√

σ 2 +4+σ 2 +2 4

(73)

The next step is of course the insertion of the Adolescence Formulae (57) into (73) so as to get rid of both μ and σ . The result is the rather awesome but exact equation Fertility_ Span = FS = tEOF − a

5 Part 5: Life Expectancy and Fertility in Logpars

275

    

 

ln −

= p + (d − p) 1 − e ln

− ( p − b) · e

−

2

 ln p−b p−b  a−b ) +4   ( a−b ) ln (p−b p−b 2 ) −2 ( a−b )+1 − ln( a−b √ p−b p−b ln +1 ( a−b ) ln( a−b )+1 4

    ln p−b 2  a−b    +4 p−b 2 p−b ln a−b ln a−b +1  −  p−b p−b 2 ln a−b +1 ln a−b +1

p−b ( a−b ) ( ) √ ( ) 2 ( )

( (

) )

− b.

(74)

Unfortunately, there appear to exist no way to simplify (74) any further.

5.5 A Numerical Example About the Most Important Case: Man We conclude our discussion about the LOGPAR Life Expectancy and Fertility Span by providing a numerical example about the most important case of all: Man. Suppose that: 1. The Man’s age is measured since its own birth. This means to assume that for Man bMan = 0.

(75)

2. We assume that Puberty occurs at age 11 (though it might be age 10 for girls and 12 for boys) aMan = 11.

(76)

3. We assume that the peak of Man’s activity occurs at age 33 (according to Dante’s Divine Comedy!) pMan = 33.

(77)

These are the three input data (“triplet”) sufficient to make the b-lognormal precise. 4. Now the parabola comes. Clearly, we must assign the death time d, that amounts to what we already called “life expectancy” in (72) (since now b Man = 0). Optimistically, we will assume that, for a modern and “wealthy” Man, the life expectancy is about 80 years dMan = 80.

(78)

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Life Expectancy and Life Energy According to Evo-SETI Theory

5. These four assumed inputs (75), (76), (77) and (78) completely determine the μ and σ given by the Adolescence Formulae (57). The relevant numerical values turn out to be  μ L A = 4.071625208175094, (79) σ L A = 0.75836511438002. 6. The Peak ordinate (13) computed by virtue of the logpar Adolescence Formulae (57) for Man turns out to be PL A = 0.011957284676721.

(80)

7. The Adolescence ordinate A (or Puberty ordinate) is determined via (15) and turns out to equal A L A = 0.0041872095966534.

(81)

8. The EOF (End-Of-Fertility) Time, given by (71), amounts to the numerical value of about 70 years after birth EOF L A _ time = 70.88734573704019

(82)

9. And, finally, the Fertility SPAN is given by (73) and reads Fertility_ Span = 59.88734573704019.

(83)

With all the above numerical values [let us repeat it: computed by virtue of the Adolescence Formulae (57) and not by virtue of the History Formulae (39)] the resulting logpar plot for Man is given in Fig. 3.

5.6 Checking Numerically the (Small) Difference Between History Formulae and Adolescence Formulae for Man For all future applications to Astrobiology of the mathematical results described in this chapter, it is important to realize that there is no big numerical difference between the numbers given by the History Formulae (39) and the Adolescence Formulae (57). For instance, in the case of the Man logpar shown in Fig. 3, the Adolescence Formulae (57) yield the numerical values (79) that we repeat here for convenience 

μ L A = 4.071625208175094, σ L A = 0.75836511438002.

(84)

5 Part 5: Life Expectancy and Fertility in Logpars

277

On the other hand, the History Formulae (39), with the numerical values given by (75), (77) and (78), yield 

μ L H = 4.236665041781441, σ L H = 0.86032405540875.

(85)

Thus, it plainly appears that the difference among them is conceptual, rather than numeric: the History Formulae are OPTIMIZED by virtue of the condition (30) that sigma must be minimum, whereas the Adolescence Formulae are not so.

6 Conclusions More and more exoplanets are now being discovered by astronomers either by observations from the ground or by virtue of space missions, like “CoRot”, “Kepler”, “Gaia”, and other future space missions. As a consequence, a recent estimate sets at 40 billion the number of Earth-sized planets orbiting in the habitable zones of sunlike stars and red dwarf stars within the Milky Way galaxy. With such huge numbers of “possible Earths” in sight, Astrobiology and SETI are becoming research fields more and more attractive to a number of young scientists. Mathematically innovative papers like the Evo-SETI ones, should thus be welcome. But in this book we did more than just in all previous Evo-SETI papers. While just preserving all the advantages of the b-lognormal probability density functions, we kept these b-lognormals good only for the first part of the curve: the one between birth and peak. The second part, between peak and death, was replaced by just a simple descending half-parabola, thus avoiding any inflexion point like the “senility” point typical of b-lognormals that was so difficult to estimate numerically in most cases. Thus LOGPAR curves have greatly simplified the description of any finite phenomenon in time like the lifetime of a cell, or a human, or a civilization or even like an ET civilization. In addition to all that, we abandoned the normalization condition of b-lognormals retaining just their shape, and not their numbers. This transformed the logpars into power curves, both in the popular sense where “power” means “political and military power” and in the strictly physical sense, where “power” means a curve measured in watts. And the area under such a logpar is indeed the ENERGY associated to the logpar phenomenon between birth and death. So, for the first time in the creation of our Evo-SETI Theory, we were able to add ENERGY to the ENTROPY previously considered already. And energy and entropy are the two pillars of classical Thermodynamics thus making Evo-SETI even more neatly applicable. Finally, there is one more crucial step ahead that we made by introducing logpars. Without mentioning it so far, we actually “stumbled” into the PRINCIPLE OF LEAST ACTION. This is of course the #1 mathematical tool of all theoretical physics: just think of all the unified theories of gravitation, where a certain action

278

Life Expectancy and Life Energy According to Evo-SETI Theory

function is postulated, then the Least Action Principle (or Hamilton’s Principle) and then the relevant Euler-Lagrange differential equations are derived, and finally (hopefully) solved, yielding the trajectory of particles. Well, the ACTION has the dimension of an ENERGY MULTIPLED BY THE TIME, and this is precisely what we did when finding the area under the logpar and considering the logpar integral in between birth and death. So we claim that… the logpar is the optimal trajectory of our Evo-SETI Theory, also in regard to the Least Action Principle. An adequate description of that result would require one or more papers giving more profound justifications. But the needs of mundane life, like the IAA SETI Permanent Committee Chairmanship, do not allow this author to pursue these mathematical delights as much as he would love to.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

J. Felsenstein, Inferring Phylogenies (Sinauer Associates Inc., Sunderland, 2004) https://en.wikipedia.org/wiki/Molecular_clock C. Maccone, The statistical Drake equation. Acta Astronaut. 67, 1366–1383 (2010) C. Maccone, A mathematical model for evolution and SETI. Orig. Life Evol. Biosph. (OLEB) 41, 609–619 (2011) C. Maccone, Mathematical SETI, 2012 edn. (Praxis-Springer in the fall of 2012), 724 p. ISBN10: 3642274366 | ISBN-13: 978-3642274367| C. Maccone, SETI, evolution and human history merged into a mathematical model. Int. J. Astrobiol. 12(3), 218–245 (2013) C. Maccone, Evolution and mass extinctions as lognormal stochastic processes. Int. J. Astrobiol. 13(4), 290–309 (2014) C. Maccone, Evo-SETI entropy identifies with molecular clock. Acta Astronaut. 115, 286–290 (2015) T. Maruyama, Stochastic Problems in Population Genetics. Lecture Notes in Biomathematics #17 (Springer, Berlin, 1977) M. Nei, K. Sudhir, Molecular Evolution and Phylogenetics (Oxford University Press, Oxford, 2000) M. Nei, Mutation-Driven Evolution (Oxford University Press, Oxford, 2013)

Evo-SETI: A Mathematical Tool for Cladistics, Evolution, and SETI

Abstract The discovery of new exoplanets makes us wonder where each new exoplanet stands along its way to develop life as we know it on Earth. Our Evo-SETI Theory is a mathematical way to face this problem. We describe cladistics and evolution by virtue of a few statistical equations based on lognormal probability density functions (pdf) in the time. We call b-lognormal a lognormal pdf starting at instant b (birth). Then, the lifetime of any living being becomes a suitable b-lognormal in the time. Next, our “Peak-Locus Theorem” translates cladistics: each species created by evolution is a b-lognormal whose peak lies on the exponentially growing number of living species. This exponential is the mean value of a stochastic process called “Geometric Brownian Motion” (GBM). Past mass extinctions were all-lows of this GBM. In addition, the Shannon Entropy (with a reversed sign) of each b-lognormal is the measure of how evolved that species is, and we call it EvoEntropy. The “molecular clock” is re-interpreted as the EvoEntropy straight line in the time whenever the mean value is exactly the GBM exponential. We were also able to extend the Peak-Locus Theorem to any mean value other than the exponential. For example, we derive in this chapter the EvoEntropy corresponding to the Markov-Korotayev (2007) “cubic” evolution: a curve of logarithmic increase. Keywords Cladistics · Darwinian evolution · Molecular clock · Entropy · SETI

1 Purpose of This Chapter This paper describes the recent developments in a new statistical theory describing Evolution and SETI by mathematical equations. I call this the Evo-SETI mathematical model of Evolution and SETI. The main question which this chapter focuses on is, whenever a new exoplanet is discovered, what is the evolutionary stage of the exoplanet in relation to the life on it, compared to how it is on Earth today? This is the central question for Evo-SETI. In this chapter, it is also shown that the (Shannon) Entropy of b-lognormals addresses this question, thus allowing the creation of an Evo-SETI SCALE that may be applied to exoplanets.

© Springer Nature Switzerland AG 2020 C. Maccone, Evo-SETI, https://doi.org/10.1007/978-3-030-51931-5_7

279

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Evo-SETI: A Mathematical Tool for Cladistics, Evolution, and SETI

An important new result presented in this chapter stresses that the cubic in the work of Markov-Korotayev [5, 3–4, 15–18] can be taken as the mean value curve of a lognormal process, thus reconciling their deterministic work with our probabilistic Evo-SETI theory.

2 During the Last 3.5 Billion Years, Life Forms Increased as in a (Lognormal) Stochastic Process Figure 1 shows the time t on the horizontal axis, with the convention that negative values of t are past times, zero is now, and positive values are future times. The

Fig. 1 Increasing number of living species on Earth between 3.5 billion years ago and now. The red solid curve is the mean value of the GBM stochastic process L GBM (t) given by Eq. (22) (with t replaced by (t-ts)), while the blue dot-dot curves above and below the mean value are the two standard deviation upper and lower curves, given by Eqs. (11) and (12), respectively, with m GBM (t) given by Eq. (22). The “Cambrian Explosion” of life, that on Earth started around 542 million years ago, is evident in the above plot just before the value of −0.5 billion years in time, where all three curves “start leaving the time axis and climbing up”. Notice also that the starting value of living species 3.5 billion years ago is one by definition, but it “looks like” zero in this plot since the vertical scale (which is the true scale here, not a log scale) does not show it. Notice finally that nowadays (i.e. at time t = 0) the two standard deviation curves have exactly the same distance from the middle mean value curve, i.e. 30 million living species more or less the mean value of 50 million species. These are assumed values that we used just to exemplify the GBM mathematics: biologists might assume other numeric values

2 During the Last 3.5 Billion Years, Life Forms Increased …

281

starting point on the time axis is ts = 3.5 billion (109 ) years ago, i.e., the accepted time of the origin of life on Earth. If the origin of life started earlier than that, for example 3.8 billion years ago, the following equations would remain the same and their numerical values would only be slightly changed. On the vertical axis is the number of species living on Earth at time t, denoted L(t) and standing for “life at time t”. We do not know this “function of the time” in detail, and so it must be regarded as a random function, or stochastic process L(t). This paper adopts the convention that capital letters represent random variables, i.e., stochastic processes if they depend on the time, while lower-case letters signify ordinary variables or functions.

3 Mean Value of the Lognormal Process L(t) The most important, ordinary and continuous function of the time associated with a stochastic process like L(t) is its mean value, denoted by: m L (t) ≡ L(t).

(1)

The probability density function (pdf ) of a stochastic process like L(t) is assumed in the Evo-SETI theory to be a b-lognormal, and its equation thus reads: e

ln(n)−M L (t)]2 2σ L2 (t−ts)

−[

L(t)_ pd f (n; M L (t), σ, t) = √ with √ 2π σ L t − tsn  σ ≥ 0, and L M L (t) = arbitrary function of t.



n ≥ 0, t ≥ ts, (2)

This assumption is in line with the extension in time of the statistical Drake equation, namely the foundational and statistical equation of SETI, as shown in [8]. The mean value (Eq. (1)) is related to the pdf Eq. (2) by the relevant integral in the number n of living species on Earth at time t, as follows: ∞ n·√

m L (t) ≡ 0

ln(n)−M L (t)]2 2σ L2 (t−ts)

−[

e dn. √ 2π σ L t − tsn

(3)

The “surprise” is that this integral in Eq. (3) may be exactly computed with the key result, so that the mean value m L (t) is given by: m L (t) = e M L (t) e

σ L2 2

(t−ts)

.

(4)

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Evo-SETI: A Mathematical Tool for Cladistics, Evolution, and SETI

In turn, the last equation has the “surprising” property that it may be exactly inverted, i.e., solved for M L (t): M L (t) = ln(m L (t)) −

σ L2 (t − ts). 2

(5)

4 L(t) Initial Conditions at ts In relation to the initial conditions of the stochastic process L(t), namely concerning the value L(ts), it is assumed that the exact positive number L(ts) = N s

(6)

is always known, i.e., with a probability of one: Pr{L(ts) = N s} = 1.

(7)

In practice, N s will be equal to one in the theories of the evolution of life on Earth or on an exoplanet (i.e., there must have been a time ts in the past when the number of living species was just one, be it RNA or something else), and it is considered as equal to the number of living species just before the asteroid/comet impacted in the theories of mass extinction of life on a planet. The mean value m L (t) of L(t) must also equal the initial number N s at the initial time ts, that is: m L (ts) = N s .

(8)

Replacing t with ts in Eq. (4), one then finds: m L (ts) = e M L (ts) .

(9)

That, checked against Eq. (8), immediately yields: N s = e M L (ts) that is M L (ts) = ln(N s) .

(10)

These are the initial conditions for the mean value. After the initial instant ts, the stochastic process L(t) unfolds, oscillating above or below the mean value in an unpredictable way. Statistically speaking however, it is expected that L(t) does not “depart too much” from m L (t), and this fact is graphically shown in Fig. 1 by the two dot-dot blue curves above and below the mean value solid red curve m L (t). These two curves are the upper standard deviation curve

4 L(t) Initial Conditions at ts

283

   2 upper_ st_ dev_ curve (t) = m L (t) 1 + eσL (t−ts) − 1

(11)

and the lower standard deviation curve    2 lower_ st_ dev_ curve(t) = m L (t) 1 − eσL (t−ts) − 1

(12)

respectively (see [15]). Both Eqs. (11) and Eqs. (12), at the initial time t = ts, equal the mean value m L (ts) = N s. With a probability of one, the initial value N s is the same for all of the three curves shown in Fig. 1. The function of the time  variation_ coefficient(t) =

eσL (t−ts) − 1 2

(13)

is called the variation coefficient, since the standard deviation of L(t) (noting that this is just the standard deviation  L (t) of L(t) and not either of the above two “upper” and “lower” standard deviation curves given by Eqs. (11) and (12), respectively) is:  st_ dev_ curve(t) ≡  L (t) = m L (t)

eσL (t−ts) − 1. 2

(14)

Thus, Eq. (14) shows that the variation coefficient of Eq. (13) is the ratio of  L (t) to m L (t), i.e. it expresses how much the standard deviation “varies” with respect to the mean value. Having understood this fact, the two curves of Eqs. (11) and (12) are obtained:  2 (15) m L (t) ±  L (t) = m L (t) ± m L (t) eσL (t−ts) − 1.

5 L(t) Final Conditions at te > ts With reference to the final conditions for the mean value curve, as well as for the two standard deviation curves, the final instant can be termed te, reflecting the end time of this mathematical analysis. In practice, this te is zero (i.e., now) in the theories of the evolution of life on Earth or exoplanets, but it is the time when the mass extinction ends (and life starts to evolve again) in the theories of mass extinction of life on a planet. First of all, it is clear that, in full analogy to the initial condition Eq. (8) for the mean value, the final condition has the form: m L (te) = N e

(16)

where N e is a positive number denoting the number of species alive at the end time te. However, it is not known what random value L(te) will take, but only that its

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standard deviation curve Eq. (14) will, at time te, have a certain positive value that will differ by a certain amount δ N e from the mean value Eq. (16). In other words, from Eq. (14):  2 δ N e =  L (te) = m L (te) eσL (te−ts) − 1.

(17)

When dividing Eq. (17) by Eq. (16), the common factor m(te) is cancelled out, and one is left with:  δNe 2 = eσL (te−ts) − 1. (18) Ne Solving this for σ L finally yields:   2  ln 1 + δNNee σL = . √ te − ts

(19)

This equation expresses the so far unknown numerical parameter σ L in terms of the initial time ts plus the three final-time parameters (te, N e, δ N e). Therefore, in conclusion, it is shown that once the five parameters (ts, N s, te, N e, δ N e) are assigned numerically, the lognormal stochastic process L(t) is completely determined. Finally, notice that the square of Eq. (19) may be rewritten as:

σ L2 =

 2  ln 1 + δNNee te − ts

= ln

⎧ ⎨ ⎩

 1+

δNe Ne

⎫ 1 2  te−ts ⎬ ⎭

(20)

.

(21)

from which the following formula is inferred: 

e

σ L2

=e

ln

 1  2 te−ts 1+( δNNee )







= 1+

δNe Ne

1 2  te−ts

This Eq. (21) enables one to get rid of eσL , replacing it by virtue of the four boundary parameters: (ts, te, N e, δ N e). It will be later used in Sect. 8 to rewrite the Peak-Locus Theorem in terms of the boundary conditions, rather than in terms 2 of eσL . 2

6 Important Special Cases of m(t)

285

6 Important Special Cases of m(t) (1) The particular case of Eq. (1) when the mean value m(t) is given by the generic exponential: m GBM (t) = N0 eμG B M t = or, alternatively, = Ae B t

(22)

is called the Geometric Brownian Motion (GBM), and is widely used in financial mathematics, where it represents the “underlying process” of the stock values (BlackSholes models). This author used the GBM in his previous models of Evolution and SETI ([8–13]), since it was assumed that the growth of the number of ET civilizations in the Galaxy, or, alternatively, the number of living species on Earth over the last 3.5 billion years, grew exponentially (Malthusian growth). Upon equating the two right-hand-sides of Eqs. (4) and (22) (with t replaced by (t − ts)), we find: e MGBM (t) e

2 σG BM 2

(t−ts)

= N0 eμGBM (t−ts) .

(23)

Solving this equation for MGBM (t) yields: 

σ2 MGBM (t) = ln N0 + μGBM − GBM 2

 (t − ts) .

(24)

This is (with ts = 0) the mean value at the exponent of the well-known GBM pdf, i.e.,: 

GBM(t)_ pd f (n; N0 , μ, σ, t) =

e−

   2 2 ln(n)− ln N0 + μ− σ2 t 2 σ2 t

√ √ 2π σ t n

, (n ≥ 0).

(25)

This short description of the GBM is concluded as the exponential sub-case of the general lognormal process Eq. (2), by warning that GBM is a misleading name, since GBM is a lognormal process and not a Gaussian one, as the Brownian Motion is. (2) As it has been mentioned already, another interesting case of the mean value function m(t) in Eq. (1) is when it equals a generic polynomial in t starting at ts, namely (with ck being the coefficient of the k-th power of the time t − ts in the polynomial) 

polynomial_ degree

m polynomial (t) =

k=0

ck (t − ts)k .

(26)

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The case where Eq. (26) is a second-degree polynomial (i.e., a parabola in t − ts) may be used to describe the Mass Extinctions on Earth over the last 3.5 billion years (see [12]). (3) Having so said, the notion of a b-lognormal must also be introduced, for t > b = birth, representing the lifetime of living entities, as single cells, plants, animals, humans, civilizations of humans, or even extra-terrestrial (ET) civilizations (see [11], in particular pp. 227–245) [ln(t−b)−μ]2

e− 2 σ 2 b-lognormal_ pdf(t; μ, σ, b) = √ . 2π σ (t − b)

(27)

7 Boundary Conditions When m(t) Is a First, Second, or Third Degree Polynomial in the Time (t − ts) In [12], the reader may find a mathematical model of Darwinian Evolution different from the GBM model. That model is the Markov-Korotayev model, for which this author proved the mean value (1) to be a Cubic(t) i.e., a third degree polynomial in t − ts. In summary, the key formulae proven in [12], relating to the case when the assigned mean value m L (t) is a polynomial in t starting at ts, can be shown as: 

polynomial_ degree

m L (t) =

ck (t − ts)k .

(28)

k=0

(1) The mean value is a straight line. This straight line is the line through the two points, (ts, N s) and (te, N e), that, after a few rearrangements, becomes: m straight_ line (t) = (N e − N s)

t − ts + N s. te − ts

(29)

(2) The mean value is a parabola, i.e., a quadratic polynomial in t − ts. Then, the equation of such a parabola reads:   t − ts t − ts m parabola (t) = (N e − N s) 2− + N s. te − ts te − ts

(30)

Equation (30) was actually firstly derived by this author in [12] (pp. 299–301), in relation to Mass Extinctions, i.e., it is a decreasing function of time. (3) The mean value is a cubic. In [12] (pp. 304–307), this author proved, in relation to the Markov-Korotayev model of Evolution, that the cubic mean value of the L(t) lognormal stochastic process is given by the cubic equation in t − ts:

7 Boundary Conditions When m(t) Is a First, Second …

287

m cubic (t) = (N e − N s)   (t − ts) 2(t − ts)2 − 3(tMax + tmin − 2ts)(t − ts) + 6(tMax − ts)(tmin − ts)  + N s.  · (te − ts) 2(te − ts)2 − 3(tMax + tmin − 2ts)(te − ts) + 6(tMax − ts)(tmin − ts)

(31) Notice that, in Eq. (31), one has, in addition to the usual initial and final conditions N s = m L (ts) and N e = m L (te), two more “middle conditions” referring to the two instants (t Max , tmin ) at which the Maximum and the minimum of the cubic Cubic(t) occur, respectively: 

tmin = time_ of_ the_ Cubic_ minimum tMax = time_ of_ the_ Cubic_ Maximum.

(32)

8 Peak-Locus Theorem The Peak-Locus theorem is the new mathematical discovery of ours, playing a central role in Evo-SETI. In its most general formulation, it can be used for any lognormal process L(t) or arbitrary mean value m L (t). In the case of GBM, it is shown in Fig. 2. The Peak-Locus theorem states that the family of b-lognormals, each having its own peak located exactly upon the mean value curve (1), is given by the following three equations, specifying the three parameters μ( p), σ ( p), and b( p) appearing in

Fig. 2 The Darwinian Exponential is used as the geometric locus of the peaks of b-lognormals for the GBM case. Each b-lognormal is a lognormal starting at a time b (birth time) and represents a different species that originated at time b of the Darwinian Evolution. This is cladistics, as seen from the perspective of the Evo-SETI model. It is evident that, when the generic “running b-lognormal” moves to the right, its peak becomes higher and narrower, since the area under the b-lognormal always equals one. Then, the (Shannon) entropy of the running b-lognormal is the degree of evolution reached by the corresponding species (or living being, or a civilization, or an ET civilization) in the course of Evolution (see, for instance, [1, 6, 7, 13, 14, 19])

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Eq. (27) as three functions of the peak abscissa, i.e. the independent variable p. In other words, we were actually pleased to find out that these three equations may be written directly in terms of m L ( p) as follows: ⎧ ⎪ ⎪ ⎪ μ( p) = ⎨

2

eσ L p 4π[m L ( p)]2

−p

σ L2 2

σ L2

2 p √ e 2π m L ( p) μ( p)−[σ ( p)]2

σ ( p) = ⎪ ⎪ ⎪ ⎩ b( p) = p − e

(33) .

The proof of Eq. (33) is lengthy and was given as a special file (written in the language of the Maxima symbolic manipulator) that the reader may freely download from the web site of [12]. An important result is now presented. The Peak-Locus Theorem Eq. (33) is rewritten, not in terms of σ L , but in terms of the four boundary parameters known as: (ts, te, N e, δ N e). To this end, we must insert Eqs. (21) and (20) into Eq. (33), producing the following result: ⎧ ⎪ ⎪ ⎪ ⎪ μ( p) = ⎪ ⎨



1+( δNNee )

2



p te−ts

  p δ N e 2  2(t−ts) − ln 1 + N e

4 π [m L ( p)]2   p 2 2(t−ts) 1+( δNNee ) √ 2π m L ( p) μ( p)−[σ ( p)]2

⎪ ⎪ σ ( p) = ⎪ ⎪ ⎪ ⎩ b( p) = p − e

(34) .

√ In the particular GBM case, the mean value is Eq. (22) with μG B M = B, σ L = 2B and N0 = N s = A. Then, the Peak-Locus theorem Eq. (33) with ts = 0 yields: ⎧ 1 ⎪ ⎨ μ( p) = 4π A2 − Bp, 1 σ = √2π A , ⎪ ⎩ b( p) = p − eμ( p)−σ 2 .

(35)

In this simpler form, the Peak-Locus theorem had already been published by the author in [9–11], while its most general form is Eqs. (33) and (34).

9 EvoEntropy(p) as a Measure of Evolution The (Shannon) Entropy of the b-lognormal Eq. (27) is (for the proof, see [10], p. 686):     √ 1 1 ln 2π σ ( p) + μ( p) + . H ( p) = ln(2) 2

(36)

9 EvoEntropy(p) as a Measure of Evolution

289

This is a function of the peak abscissa p and is measured in bits, as in Shannon’s Information Theory. By virtue of the Peak-Locus Theorem Equation (33), it becomes:   2 eσ L p 1 1 H ( p) = − ln(m L ( p)) + . ln(2) 4π [m L ( p)]2 2

(37)

One may also directly rewrite Eq. (37) in terms of the four boundary parameters (ts, te, N e, δ N e), upon inserting Eq. (21) into Eq. (37), with the result: ⎧ p δ N e 2  te−ts ⎪ ⎨ 1 + Ne 1 − ln(m L ( p)) + H ( p) = 2 ln(2) ⎪ 4π [m L ( p)] ⎩

⎫ ⎪ 1⎬ . 2⎪ ⎭

(38)

Thus, Eq. (37) and Eq. (38) yield the entropy of each member of the family of ∞1 b-lognormals (the family’s parameter is p) peaked upon the mean value curve (1). The b-lognormal Entropy Eq. (36) is thus the measure of the extent of evolution of the b-lognormal: it measures the decreasing disorganization in time of what that b-lognormal represents. Entropy is thus disorganization decreasing in time. However, one would prefer to use a measure of the increasing organization in time. This is what we call the EvoEntropy of p: EvoEntropy( p) = −[H ( p) − H (ts)].

(39)

The Entropy of evolution is a function that has a minus sign in front of Eq. (36), thus changing the decreasing trend of the (Shannon) entropy Eq. (36) into the increasing trend of this EvoEntropy Eq. (39). In addition, this EvoEntropy starts at zero at the initial time ts, as expected. EvoEntropy(ts) = 0.

(40)

By virtue of Eq. (37), the EvoEntropy Eq. (39), invoking also the initial condition Eq. (8), becomes: EvoEntropy( p)_ of_ the_ Lognormal_ Process_ L(t)    2 2 eσL ts eσ L p m L ( p) 1 − + ln . = ln(2) 4π N s 2 Ns 4π [m L ( p)]2

(41)

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Evo-SETI: A Mathematical Tool for Cladistics, Evolution, and SETI

Alternatively, this could be directly rewritten in terms of the five boundary parameters (ts, N s, te, N e, δ N e), upon inserting Eq. (38) into Eq. (39), thus finding: EvoEntropy( p)_ of_ the_ Lognormal_ Process_ L(t) ⎧ ⎫ p ts  δ N e 2  te−ts δ N e 2  te−ts ⎪ ⎪   ⎨ 1 + Ne 1 + Ne m L ( p) ⎬ 1 . − + ln = ⎪ ln(2) ⎪ 4π N s 2 Ns 4π [m L ( p)]2 ⎩ ⎭

(42)

It is worth noting that the standard deviation at the end time, δ N e, is irrelevant for the purpose of computing the simple curve of the EvoEntropy Eq. (39). In fact, the latter is just a continuous curve, and not a stochastic process. Therefore, any numeric arbitrary value may be assigned to δ N e, and the EvoEntropy curve must not change. Keeping this in mind, it can be seen that the true EvoEntropy curve is obtained by “squashing” down Eq. (42) into the mean value curve m L (t) and this only occurs if we let: δ N e = 0.

(43)

Inserting Eq. (43) into Eq. (42), the latter can be simplified into: EvoEntropy( p)_ of_ the_ Lognormal_ Process_ L(t)    1 1 m L ( p) 1 − + ln = ln(2) 4π N s 2 Ns 4π [m L ( p)]2

(44)

which is the final form of the EvoEntropy curve. Equation (44) will be used in the sequel. It can now be clearly seen that the final EvoEntropy Equation (44) is made up of three terms, as follows: (a) The constant term 1 4π N s 2

(45)

whose numeric value in the particularly important case of N s = 1 is: 1 = 0.079577471545948 4π

(46)

that is, it approximates almost zero. (b) The denominator square term in Eq. (44) rapidly approaches zero as m L ( p) increases to infinity. In other words, this inverse-square term −

1 4π [m L ( p)]2

(47)

9 EvoEntropy(p) as a Measure of Evolution

291

may become almost negligible for large values of the time p. (c) Finally, the dominant, natural logarithmic, term, i.e., that which is the major term in this EvoEntropy Eq. (45) for large values of the time p.  m L ( p) . ln Ns 

(48)

In conclusion, the EvoEntropy Equation (44) depends upon its natural logarithmic term Eq. (48), and so its shape in time must be similar to the shape of a logarithm, i.e., nearly vertical at the beginning of the curve and then progressively approaching the horizontal, though never reaching it. This curve has no maxima nor minima, nor any inflexions.

10 Perfectly Linear EvoEntropy When the Mean Value Is Perfectly Exponential (GBM): This Is just the Molecular Clock In the GBM case of Eq. (22) (with t replaced by (t − ts)), when the mean value is given by the exponential m GBM (t) = N se

σ L2 2

(t−ts)

= N se B (t−ts)

(49)

the EvoEntropy Eq. (44) is exactly a linear function of the time p, since the first two terms inside the braces in Eq. (44) cancel each other out, as we now prove. Proof Insert Eq. (49) into Eq. (44) and then simplify: EvoEntropy( p)_ of_ GBM ⎧ ⎫ ⎪ ⎪ ⎛ ⎞ ⎪ ⎪ 2 σL ⎪ σ 2 ts ⎪ ⎨ σ L2 p ( p−ts) ⎬ L 2 e N se e 1 ⎝ ⎠ − + ln = 2  ⎪ ln(2) ⎪ 4π N s 2 Ns σ L2 ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ 4π N se 2 ( p−ts)    2 2 2 σL 1 eσL ts eσ L p p−ts) ( = − + ln e 2 2 ln(2) 4π N s 2 4π N s 2 eσL ( p−ts)   2 eσL ts 1 σ L2 1 − + = ( p − ts) 2 ln(2) 4π N s 2 2 4π N s eσL (−ts)   2 2 eσL ts eσL ·(ts) σ L2 1 − + = ( p − ts) ln(2) 4π N s 2 4π N s 2 2

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Evo-SETI: A Mathematical Tool for Cladistics, Evolution, and SETI

 2  σL 1 1 {B · ( p − ts)}. = ( p − ts) = ln(2) 2 ln(2)

(50)

In other words, the GBM EvoEntropy is given by: GBM_ EvoEntropy( p) =

B · ( p − ts). ln(2)

(51)

This is a straight line in the time p, starting at the time ts of the origin of life on Earth and increasing linearly thereafter. It is measured in bits/individual and is shown in Fig. 3. This is the same linear behaviour in time as the molecular clock, which is the technique in molecular evolution that uses fossil constraints and rates of molecular change to deduce the time in geological history when two species or other taxa diverged. The molecular data used for such calculations are usually nucleotide sequences for DNA or amino acid sequences for proteins (see [1, 6, 7]).

Fig. 3 EvoEntropy (in bits per individual) of the latest species appeared on Earth during the last 3.5 billion years if the mean value is an increasing exponential, i.e. if our lognormal stochastic process is a GBM. This straight line shows that a Man (nowadays) is 25.575 bits more evolved than the first form of life (RNA) 3.5 billion years ago

10 Perfectly Linear EvoEntropy When the Mean Value …

293

In conclusion, we have ascertained that the EvoEntropy in our Evo-SETI theory and the molecular clock are the same linear time function, apart from multiplicative constants (depending on the adopted units, like bits, seconds, etc.). This conclusion appears to be of key importance when assessing the stage at which a newly discovered exoplanet is in the process of its chemical evolution towards life.

11 Markov-Korotayev Alternative to Exponential: A Cubic Growth Figure 3, showing the linear growth of the Evo-Entropy over the last 3.5 billion years of evolution of life on Earth, illustrates the key factor in molecular evolution and allows for an immediate quantitative estimate of how much (in bits per individuals) any two species differ from each other; this being the key to cladistics. However, after 2007, this exponential vision was shaken by the alternative “cubic vision” now outlined. This cubic vision is detailed in the full list of papers published by Andrey Korotayev and Alexander V. Markov et al., since 2007 [5, 2, 3, 15–18]. Another important publication is their mathematical paper [4] relating to the new research field entitled “Big History”. In addition, a synthetic summary of the Markov-Korotayev theory of evolution appears on Wikipedia at http://en.wikipedia.org/wiki/Andrey_Korotayev, for which an adapted excerpt is seen below: According to the above list of published papers, in 2007–2008 the Russian scientists Alexander V. Markov and Andrey Korotayev showed that a ‘hyperbolic’ mathematical model can be developed to describe the macrotrends of biological evolution. These authors demonstrated that changes in biodiversity through the Phanerozoic correlate much better with the hyperbolic model (widely used in demography and macrosociology) than with the exponential and logistic models (traditionally used in population biology and extensively applied to fossil biodiversity as well). The latter models imply that changes in diversity are guided by a first-order positive feedback (more ancestors, more descendants) and/or a negative feedback arising from resource limitation. Hyperbolic model implies a second-order positive feedback. The hyperbolic pattern of the world population growth has been demonstrated by Markov and Korotayev to arise from a second-order positive feedback between the population size and the rate of technological growth. According to Markov and Korotayev, the hyperbolic character of biodiversity growth can be similarly accounted for by a feedback between the diversity and community structure complexity. They suggest that the similarity between the curves of biodiversity and human population probably comes from the fact that both are derived from the interference of the hyperbolic trend with cyclical and stochastic dynamics [5, 2, 3, 15–18].

This author was inspired by Fig. 4 (taken from the Wikipedia site http://en.wikipe dia.org/wiki/Andrey_Korotayev), showing the increase, but not monotonic increase, of the number of Genera (in thousands) during the last 542 million years of life on Earth, making up the Phanerozoic. Thus, it is postulated that the red curve in Fig. 4 could be regarded as the “Cubic mean value curve” of a lognormal stochastic process, just as the exponential mean value curve is typical of GBMs.

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Evo-SETI: A Mathematical Tool for Cladistics, Evolution, and SETI

Fig. 4 According to Markov and Korotayev, during the Phanerozoic, the biodiversity shows a steady, but not monotonic, increase from near zero to several thousands of genera

The Cubic Equation (31) may be used to represent the red line in Fig. 4, thus reconciling the Markov-Korotayev theory with our Evo-SETI theory. This is realized when considering the following numerical inputs to the Cubic Equation (31), that we derive from looking at Fig. 4. The precision of these numerical inputs is relatively unimportant at this early stage of matching the two theories (this one and the MarkovKorotayev’s), as we are just aiming for a “proof of concept”, and better numeric approximations might follow in the future. ⎧ ts = −530 ⎪ ⎪ ⎨ Ns = 1 ⎪ te = 0 ⎪ ⎩ N e = 4000.

(52)

In other words, the first two equations of Eq. (52) mean that the first of the genera appeared on Earth about 530 million years ago, i.e., the number of genera on Earth was zero before 530 million years ago. In addition, the last two equations of Eq. (52) mean that, at the present time t = 0, the number of genera on Earth is approximately 4000, noting that a standard deviation of about ±1000 affects the average value of 4000. This is shown in Fig. 4 by the grey stochastic process referred to as all genera. It is re-phrased mathematically by assigning the fifth numeric input: δ N e = 1000.

(53)

11 Markov-Korotayev Alternative to Exponential: A Cubic Growth

295

Then, as a consequence of the five numeric boundary inputs (ts, N s, te, N e, δ N e), plus the standard deviation σ on the current value of genera, Eq. (19) yields the numeric value of the positive parameter σ :

σ =

#   $ $ ln 1 + δ N e 2 % Ne te − ts

= 0.011.

(54)

Having thus assigned numerical values to the first five conditions, only the conditions on the two abscissae of the Cubic maximum and minimum, respectively, tone to be assigned. Figure 4 establishes them (in millions of years ago) as: 

tMax = − 400 tmin = − 220.

(55)

Finally, inserting these seven numeric inputs into the Cubic Equation (31), as well as into both of the equations of Eq. (15) of the upper and lower standard deviation curves, the final plot shown in Fig. 5 is produced.

Fig. 5 The Cubic mean value curve (thick red solid curve) ± the two standard deviation curves (thin solid blue and green curve, respectively) provide more mathematical information than Fig. 4. One is now able to view the two standard deviation curves of the lognormal stochastic process, Eqs. (11) and (12), that are completely missing in the Markov-Korotayev theory and in their plot shown in Fig. 4. This author claims that his Cubic mathematical theory of the Lognormal stochastic process L(t) is a more profound mathematization than the Markov-Korotayev theory of Evolution, since it is stochastic, rather than simply deterministic

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Evo-SETI: A Mathematical Tool for Cladistics, Evolution, and SETI

12 EvoEntropy of the Markov-Korotayev Cubic Growth What is the EvoEntropy Equation (44) of the Markov-Korotayev Cubic growth Equation (31)? To answer this question, Eq. (31) needs to be inserted into Eq. (44) and the resulting equation can then be plotted:    1 1 m Cubic (t) 1 · . − + ln Cubic_ EvoEntropy(t) = ln(2) 4π N s 2 Ns 4π [m Cubic (t)]2 (56) The plot of this function of t is shown in Fig. 6.

Fig. 6 The EvoEntropy Equation (44) of the Markov-Korotayev Cubic mean value Equation (31) of our lognormal stochastic process L(t) applies to the growing number of Genera during the Phanerozoic. Starting with the left part of the curve, one immediately notices that, in a few million years around the Cambrian Explosion of 542 million years ago, the EvoEntropy had an almost vertical growth, from the initial value of zero, to the value of approximately 10 bits per individual. These were the few million years when the bilateral symmetry became the dominant trait of all primitive creatures inhabiting the Earth during the Cambrian Explosion. Following this, for the next 300 million years, the EvoEntropy did not significantly change. This represents a period when bilaterally-symmetric living creatures, e.g., reptiles, birds, and very early mammals, etc., underwent little or no change in their body structure (roughly up to 310 million years ago). Subsequently, after the “mother” of all mass extinctions at the end of the Paleozoic (about 250 million years ago), the EvoEntropy started growing again in mammals. Today, according to the Markov-Korotayev model, the EvoEntropy is about 12.074 bits/individual for humans, i.e., much less than the 25.575 bits/individual predicted by the GBM exponential growth shown in Fig. 3. Therefore, the question is: which model is correct?

13 Comparing the EvoEntropy of the Markov-Korotayev …

297

13 Comparing the EvoEntropy of the Markov-Korotayev Cubic Growth, to the Hypothetical (1) Linear and (2) Parabolic Growth It is a good idea to consider two more types of growth in the Phanerozoic: (1) The LINEAR (= straight line) growth, given by the mean value of Eq. (29) (2) The PARABOLIC (= quadratic) growth, given by the mean value of Eq. (30). These can be compared with the CUBIC growth Equation (31) typical for the Markov-Korotayev model. The results of this comparison are shown in the two diagrams (upper one and lower one) in Fig. 7. For the sake of simplicity, we omit all detailed mathematical calculations and confine ourselves to writing down the equation of the: (1) LINEAR EvoEntropy: STRAIGHT_ LINE_ EvoEntropy(t)    m straight_ line (t) 1 1 1 − = . (57)  2 + ln ln(2) 4π N s 2 Ns 4π m straight_ line (t) (2) PARABOLIC (quadratic) EvoEntropy: PARABOLA_ EvoEntropy(t)    m parabola (t) 1 1 1 − . =  2 + ln ln(2) 4π N s 2 Ns 4π m parabola (t)

(58)

(3) CUBIC (MARKOV-KOROTAYEV) EVOENTROPY, i.e., Equation (56).

14 Conclusions The evolution of life on Earth over the last 3.5 to 4 billion years has barely been demonstrated in a mathematical form. Since 2012, I have attempted to rectify this deficiency by resorting to lognormal probability distributions in time, starting each at a different time instant b (birth), called b-lognormals [8–14]. My discovery of the Peak-Locus Theorem, which is valid for any enveloping mean value (and not just the exponential one (GBM), for the general proof see [1], in particular supplementary materials over there), has made it possible for the use of the Shannon Entropy of Information Theory as the correct mathematical tool for measuring the evolution of life in bits/individual.

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Fig. 7 Comparing the mean value m L (t) (a) and the EvoEntropy(t) (b) in the event of growth with the CUBIC mean value of Eq. (31) (blue solid curve), with the LINEAR Equation (29) (dash-dash orange curve), or with the PARABOLIC Equation (30) (dash-dot red curve). It can be seen that, for all these three curves, starting with the left part of the curve, in a few million years around the Cambrian Explosion of 542 million years ago, the EvoEntropy had an almost vertical growth from the initial value of zero to the value of approximately 10 bits per individual. Again, as is seen in Fig. 6, these were the few million years where the bilateral symmetry became the dominant trait of all primitive creatures inhabiting the Earth during the Cambrian Explosion

In conclusion, the processes which occurred on Earth during the past 4 billion years can now be summarized by statistical equations, noting that this only relates to the evolution of life on Earth, and not on other exoplanets. The extending of this Evo-SETI theory to life on other exoplanets will only be possible when SETI, the

14 Conclusions

299

current scientific search for extra-terrestrial intelligence, achieves the first contact between humans and an alien civilization. Supplementary Materials The following are available online at www.mdpi.com/ link. Conflicts of Interest The author declares no conflict of interest.

Proof of the CUBIC MEAN VALUE Equation (31)

Author: Claudio Maccone, October 28, 2013, at 22:17. This 2020 "Evo-SETI" book version was made on April 17th, 2020. Clearing the Maxima memory from all previous data. (%i52) kill(all); (%o0) done

Equation of the Cubic(t) having the value Ns at t=ts. (%i1) Cubic(t):=a*(t-ts)^3+b*(t-ts)^2+c*(t-ts)+Ns; (%o1) Cubic( t ) := a ( t - ts ) 3 + b ( t - ts ) 2 + c ( t - ts ) + Ns (%i2) at_start_condition:Cubic(ts); (%o2) Ns

Imposing that the above Cubic(t) equals Ne at t=te. (%i3) at_end_condition:Ne=Cubic(te); (%o3) Ne = c ( te - ts ) + a ( te - ts ) 3 + b ( te - ts ) 2 + Ns

First derivative of Cubic(t), that is a quadratic equation in t. Also, let us set this quadratic equal to zero, ready to be solved for the relevant two roots. (%i4) zero_first_derivative:diff(Cubic(t),t)=0; (%o4) 2 b ( t - ts ) + 3 a ( t - ts ) 2 + c = 0

Every cubic only has a Maximum and a minimum. Let us define the time of the Maximum tM of Cubic(t). (%i5) def_tM:subst(tM,t,zero_first_derivative); (%o5) 3 a ( tM - ts ) 2 + 2 b ( tM - ts ) + c = 0

Let us define the time of the minimum tm of Cubic(t).

300

Evo-SETI: A Mathematical Tool for Cladistics, Evolution, and SETI (%i6) def_tm:subst(tm,t,zero_first_derivative); (%o6) 2 b ( tm - ts ) + 3 a ( tm - ts ) 2 + c = 0

The last two equations may be regarded as a system of two symultaneous equations in the two unknown variables b and c. Taking their difference, the term in c disappears, and b may expressed in terms of a, tM and tm. (%i7) with_a_and_b_only:def_tM-def_tm; (%o7) 3 a ( tM - ts ) 2 + 2 b ( tM - ts ) - 2 b ( tm - ts ) - 3 a ( tm - ts ) 2 = 0

Solving the last equation for b, yields b vs a, tM, tm and ts. (%i8) b_with_tM_and_tm:first(factor(solve(with_a_and_b_only,b))); (%o8) b = -

3 a ( tM - 2 ts + tm ) 2

Similarly, adding the two linear equations we get an equation expressing c in terms of a, tM, tm and ts. (%i9) with_a_and_c_only:def_tM+def_tm; (%o9) 3 a ( tM - ts ) 2 + 2 b ( tM - ts ) + 2 b ( tm - ts ) + 3 a ( tm - ts ) 2 + 2 c = 0 (%i10) c_with_tM_and_tm:first(factor(solve(with_a_and_c_only,c))); (%o10) c = -

3 a tM 2 - 6 a ts tM + 2 b tM + 6 a ts 2 - 6 a tm ts - 4 b ts + 3 a tm 2 + 2 b tm 2

Let us now insert the b equation into the last c equation. (%i11) c_with_tM_and_tm,b_with_tM_and_tm; (%o11) c = - ( 3 a tM 2 - 3 a tM ( tM - 2 ts + tm ) + 6 a ts ( tM - 2 ts + tm ) - 3 a tm ( tM - 2 ts + tm ) - 6 a ts tM + 6 a ts 2 - 6 a tm ts + 3 a tm 2 ) / 2

that, once factorized, yields c vs a, tM, tm and ts: (%i12) c_vs_a_tM_tm:factor(%); (%o12) c = - 3 a ( ts - tm ) ( tM - ts )

Let us now invoke the at_end condition and insert into it both the expressions for b and c. The result is clearly a new equation IN a ONLY vs. tM, tm, ts and te.

Proof of the CUBIC MEAN VALUE Equation (31)

301

(%i13) at_end_condition,b_with_tM_and_tm,c_vs_a_tM_tm; (%o13) Ne = - 3 a ( te - ts ) ( ts - tm ) ( tM - ts ) -

3 a ( te - ts ) 2 ( tM - 2 ts + tm ) 2

+ a ( te - ts ) 3 + Ns

Solving for a, we thus get the full expression of a vs. tM, tm, ts and te. We call this a_OK. (%i14) a_OK:factor(first(solve(%,a))); (%o14) a = -

2 ( Ns - Ne ) ( ts - te ) ( 3 ts tM - 6 tm tM + 3 te tM - 2 ts 2 + 3 tm ts - 2 te ts + 3 te tm - 2 te 2 )

Now it's a child's game to find b_OK and c_OK similarly. b_OK reads: (%i15) b_OK:b_with_tM_and_tm,a_OK; (%o15) b =

3 ( Ns - Ne ) ( tM - 2 ts + tm ) ( ts - te ) ( 3 ts tM - 6 tm tM + 3 te tM - 2 ts 2 + 3 tm ts - 2 te ts + 3 te tm - 2 te 2 )

and c_OK reads: (%i16) c_OK:c_vs_a_tM_tm,a_OK; (%o16) c =

6 ( Ns - Ne ) ( ts - tm ) ( tM - ts ) ( ts - te ) ( 3 ts tM - 6 tm tM + 3 te tM - 2 ts 2 + 3 tm ts - 2 te ts + 3 te tm - 2 te 2 )

Game over! You just replace a_OK, b_OK and c_OK into the Cubic(t) and you get the complete cubic (though here written as a long sum of four terms): (%i17) whole_cubic:Cubic(t),a_OK,b_OK,c_OK; (%o17)

6 ( Ns - Ne ) ( t - ts ) ( ts - tm ) ( tM - ts ) ( ts - te ) ( 3 ts tM - 6 tm tM + 3 te tM - 2 ts 2 + 3 tm ts - 2 te ts + 3 te tm - 2 te 2 ) 2

3 ( Ns - Ne ) ( t - ts ) ( tM - 2 ts + tm ) ( ts - te ) ( 3 ts tM - 6 tm tM + 3 te tM - 2 ts 2 + 3 tm ts - 2 te ts + 3 te tm - 2 te 2 ) 2 ( Ns - Ne ) ( t - ts ) 3 ( ts - te ) ( 3 ts tM - 6 tm tM + 3 te tM - 2 ts 2 + 3 tm ts - 2 te ts + 3 te tm - 2 te 2 )

+ Ns

This already is the CUBIC we were looking for inasmuch as it includes the time t plus the four assigned constants tM, tm, ts and te. For instance, let us check that the value of whole_cubic at ts is Ns.

+

302

Evo-SETI: A Mathematical Tool for Cladistics, Evolution, and SETI (%i18) subst(ts,t,whole_cubic); (%o18) Ns

For instance, let us check that the value of whole_cubic at te is Ne. (%i19) ratsimp(subst(te,t,whole_cubic)); (%o19) Ne

Finally, let us check that the whole_cubic Maximum and minimum fall at tM and tm, respectively: (%i20) solve(diff(whole_cubic,t)=0,t); (%o20) [ t = tM , t = tm ]

and the Cubic inflexion falls at the mid-point in between the Maximum and the minimum: (%i21) solve(diff(whole_cubic,t,2)=0,t); (%o21) [ t =

tM + tm 2

]

Yet, it is rather obvious that whole_cubic may be rewrtten in a much more COMPACT FORM. Here we describe how we finally got it. (%i22) towards_compact_cubic:ratsimp(whole_cubic-Ns); (%o22) - ( ( ( 3 Ns - 3 Ne ) ts 2 + ( 6 Ne - 6 Ns ) tm ts + ( 6 Ns - 6 Ne ) t tm + ( 3 Ne - 3 Ns ) t 2 ) tM + ( 2 Ne - 2 Ns ) ts 3 + ( 3 Ns - 3 Ne ) tm ts 2 + ( 3 Ne - 3 Ns ) t 2 tm + ( 2 Ns - 2 Ne ) t 3 ) / ( ( 3 ts 2 - 6 tm ts + 6 te tm - 3 te 2 ) tM - 2 ts 3 + 3 tm ts 2 - 3 te 2 tm + 2 te 3 )

As for the denominator, one may rewrite it as follows: (%i23) denominator:factor(denom(towards_compact_cubic)); (%o23) ( ts - te ) ( 3 ts tM - 6 tm tM + 3 te tM - 2 ts 2 + 3 tm ts - 2 te ts + 3 te tm - 2 te 2 )

this may be better rewritten (%i24) compact_denominator:((te-ts)*((-3)*(te-ts)*(tM+tm+(-2)*ts)+6*(tm-ts)*(tM-ts)+2*(te-ts)^2)); (%o24) ( te - ts ) ( 6 ( tm - ts ) ( tM - ts ) - 3 ( te - ts ) ( tM - 2 ts + tm ) + 2 ( te - ts ) 2 )

in fact one has

Proof of the CUBIC MEAN VALUE Equation (31)

303

(%i25) ratsimp(denominator-compact_denominator); (%o25) 0

As for the numeror, one has: (%i26) towards_compact_numerator:factor(num(ratsimp(whole_cubic-Ns))); (%o26) - ( Ns - Ne ) ( ts - t ) ( 3 ts tM - 6 tm tM + 3 t tM - 2 ts 2 + 3 tm ts - 2 t ts + 3 t tm - 2 t 2 ) (%i27) compact_numerator:(Ne-Ns)*(t-ts)*((-3)*(t-ts)*(tM+tm+(-2)*ts)+6*(tm-ts)*(tM-ts)+2*(t-ts)^2); (%o27) ( Ne - Ns ) ( t - ts ) ( 6 ( tm - ts ) ( tM - ts ) - 3 ( t - ts ) ( tM - 2 ts + tm ) + 2 ( t - ts ) 2 ) (%i28) ratsimp(towards_compact_numerator-compact_numerator); (%o28) 0

In conclusion, whole_cubic-Ns reads: (%i29) compact_cubic:(compact_numerator/compact_denominator)+Ns; (%o29)

( Ne - Ns ) ( t - ts ) ( 6 ( tm - ts ) ( tM - ts ) - 3 ( t - ts ) ( tM - 2 ts + tm ) + 2 ( t - ts ) 2 ) ( te - ts ) ( 6 ( tm - ts ) ( tM - ts ) - 3 ( te - ts ) ( tM - 2 ts + tm ) + 2 ( te - ts ) 2 )

+ Ns

And so the CUBIC MEAN VALUE reads: (%i30) m(t):=((Ne-Ns)*(t-ts)*(-3*(t-ts)*(tM+tm-2*ts)+ 6*(tm-ts)*(tM-ts)+2*(t-ts)^2))/((te-ts)*(-3*(te-ts)*(tM+tm-2*ts)+ 6*(tm-ts)*(tM-ts)+2*(te-ts)^2))+Ns; (%o30) m( t ) :=

( Ne - Ns ) ( t - ts ) ( ( - 3 ) ( t - ts ) ( tM + tm + ( - 2 ) ts ) + 6 ( tm - ts ) ( tM - ts ) + 2 ( t - ts ) 2 ) ( te - ts ) ( ( - 3 ) ( te - ts ) ( tM + tm + ( - 2 ) ts ) + 6 ( tm - ts ) ( tM - ts ) + 2 ( te - ts ) 2 )

The last equation is equation (31) of the "Life" paper of 2017. With the usual four checks: (%i31) m(ts); (%o31) Ns (%i32) m(te); (%o32) Ne (%i33) solve(diff(m(t),t)=0,t); (%o33) [ t = tM , t = tm ]

+ Ns

304

Evo-SETI: A Mathematical Tool for Cladistics, Evolution, and SETI (%i34) solve(diff(m(t),t,2)=0,t); (%o34) [ t =

tM + tm 2

]

Approx MINIMUM-CURVATURE TIME, i.e. third derivative. (%i35) third_derivative_is_this_constant::diff(m(t),t,3); (%o35)

12 ( Ne - Ns ) ( te - ts ) ( 6 ( tm - ts ) ( tM - ts ) - 3 ( te - ts ) ( tM - 2 ts + tm ) + 2 ( te - ts ) 2 )

(%i36) third_derivative:diff(Cubic(t),t,3); (%o36) 6 a (%i37) a_OK; (%o37) a = -

2 ( Ns - Ne ) ( ts - te ) ( 3 ts tM - 6 tm tM + 3 te tM - 2 ts 2 + 3 tm ts - 2 te ts + 3 te tm - 2 te 2 )

(%i38) new_third_derivative:6*rhs(a_OK); (%o38) -

12 ( Ns - Ne ) ( ts - te ) ( 3 ts tM - 6 tm tM + 3 te tM - 2 ts 2 + 3 tm ts - 2 te ts + 3 te tm - 2 te 2 )

(%i39) ratsimp(third_derivative_is_this_constant-new_third_derivative); (%o39) 0

Markov_Korotayev Cubic(t) EXAMPLE. (%i40) Markov_Korotayev:[ts=-530,Ns=1,te=0,Ne=4000,tM=-400,tm=-220,deltaNe=1000]; (%o40) [ ts = - 530 , Ns = 1 , te = 0 , Ne = 4000 , tM = - 400 , tm = - 220 , deltaNe = 1000 ] (%i41) ev(new_third_derivative,Markov_Korotayev,numer); (%o41) 8.7060957910014517 10 -4 (%i42) numeric_MK_cubic:((Ne-Ns)*(t-ts)*((-3)*(t-ts)*(tM+tm+(-2)*ts)+6*(tm-ts)*(tM-ts)+2*(t-ts)^2))/((te-ts)*( (%o42)

3999 ( t + 530 ) ( 2 ( t + 530 ) 2 - 1320 ( t + 530 ) + 241800 ) 55120000

+1

(%i43) plot2d((3999*(t+530)*(2*(t+530)^2-1320*(t+530)+241800))/55120000+1,[t,-530,0]); (%o43)

Let us compute the numeric sigma for the Markov-Korotayev (MK) case.

Proof of the CUBIC MEAN VALUE Equation (31)

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References 1. J. Felsenstein, Inferring Phylogenies (Sinauer Associates Inc., Sunderland, MA, USA, 2004) 2. L. Grinin, A. Markov, A. Korotayev, On similarities between biological and social evolutionary mechanisms: mathematical modeling. Cliodynamics 4, 185–228 (2013) 3. L.E. Grinin, A.V. Markov, A.V. Korotayev, Mathematical modeling of biological and social evolutionary macrotrends. Hist. Math. 4, 9–48 (2014) 4. A.V. Korotayev, A.V. Markov, Mathematical modeling of biological and social phases of big history, in Teaching and Researching Big History—Exploring a New Scholarly Field, ed. by L. Grinin, D. Baker, E. Quaedackers, A. Korotayev (Uchitel Publishing House, Volgograd, Russia, 2014), pp. 188–219 5. A.V. Korotayev, A.V. Markov, L.E. Grinin, Mathematical modeling of biological and social phases of big history, in Teaching and Researching Big History: Exploring a New Scholarly Field (Uchitel, Volgograd, Russia, 2014), pp. 188–219 6. M. Nei, K. Sudhir, Molecular Evolution and Phylogenetics (Oxford University Press, New York, NY, USA, 2000) 7. M. Nei, Mutation-Driven Evolution (Oxford University Press, New York, NY, USA, 2013) 8. C. Maccone, The statistical drake equation. Acta Astronaut. 67, 1366–1383 (2010) 9. C. Maccone, A mathematical model for evolution and SETI. Orig. Life Evolut. Biosph. 41, 609–619 (2011) 10. C. Maccone, Mathematical SETI (Berlin, Germany, Praxis-Springer, 2012) 11. C. Maccone, SETI, evolution and human history merged into a mathematical model. Int. J. Astrobiol. 12, 218–245 (2013) 12. C. Maccone, Evolution and mass extinctions as lognormal stochastic processes. Int. J. Astrobiol. 13, 290–309 (2014) 13. C. Maccone, New Evo-SETI results about civilizations and molecular clock. Int. J. Astrobiol. 16, 40–59 (2017) 14. C. Maccone, Kurzweil’s singularity as a part of Evo-SETI theory. Acta Astronaut. 132, 312–325 (2017) 15. A.V. Markov, V.A. Anisimov, A.V. Korotayev, Relationship between genome size and organismal complexity in the lineage leading from prokaryotes to mammals. Paleontol. J. 44, 363–373 (2010) 16. A.V. Markov, A.V. Korotayev, Phanerozoic marine biodiversity follows a hyperbolic trend. Palaeoworld 16, 311–318 (2007) 17. A.V. Markov, A.V. Korotayev, The dynamics of phanerozoic marine biodiversity follows a hyperbolic trend. Zhurnal Obschei Biologii 68, 3–18 (2007) 18. A.V. Markov, A.V. Korotayev, Hyperbolic growth of marine and continental biodiversity through the Phanerozoic and community evolution. Zh. Obshch. Biol. 69, 175–194 (2008) 19. T. Maruyama, Stochastic Problems in Population Genetics; Lecture Notes in Biomathematics #17 (Springer, Berlin, Germany, 1977)

The Statistical Drake Equation

Abstract We provide the statistical generalization of the Drake equation. From a simple product of seven positive numbers, the Drake equation is now turned into the product of seven positive random variables. We call this “the Statistical Drake Equation”. The mathematical consequences of this transformation are then derived. The proof of our results is based on the Central Limit Theorem (CLT) of Statistics. In loose terms, the CLT states that the sum of any number of independent random variables, each of which may be ARBITRARILY distributed, approaches a Gaussian (i.e. normal) random variable. This is called the Lyapunov Form of the CLT, or the Lindeberg Form of the CLT, depending on the mathematical constraints assumed on the third moments of the various probability distributions. In conclusion, we show that: (1) The new random variable N, yielding the number of communicating civilizations in the Galaxy, follows the LOGNORMAL distribution. Then, as a consequence, the mean value of this lognormal distribution is the ordinary N in the Drake equation. The standard deviation, mode, and all the moments of this lognormal N are also found. (2) The seven factors in the ordinary Drake equation now become seven positive random variables. The probability distribution of each random variable may be ARBITRARY. The CLT in the so-called Lyapunov or Lindeberg forms (that both do not assume the factors to be identically distributed) allows for that. In other words, the CLT “translates” into our statistical Drake equation by allowing an arbitrary probability distribution for each factor. This is both physically realistic and practically very useful, of course. (3) An application of our statistical Drake equation then follows. The (average) DISTANCE between any two neighboring and communicating civilizations in the Galaxy may be shown to be inversely proportional to the cubic root of N. Then, in our approach, this distance becomes a new random variable. We derive the relevant probability density function, apparently previously unknown and dubbed “Maccone distribution” by Paul Davies. (4) DATA ENRICHMENT PRINCIPLE. It should be noticed that ANY positive number of random variables in the Statistical Drake Equation is compatible with the CLT. So, our generalization allows for many more factors to be added in the future as long as more refined scientific knowledge about each factor will © Springer Nature Switzerland AG 2020 C. Maccone, Evo-SETI, https://doi.org/10.1007/978-3-030-51931-5_8

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The Statistical Drake Equation

be known to the scientists. This capability to make room for more future factors in the statistical Drake equation, we call the “Data Enrichment Principle,” and we regard it as the key to more profound future results in the fields of Astrobiology and SETI. Finally, a practical example is given of how our statistical Drake equation works numerically. We work out in detail the case, where each of the seven random variables is uniformly distributed around its own mean value and has a given standard deviation. For instance, the number of stars in the Galaxy is assumed to be uniformly distributed around (say) 350 billions with a standard deviation of (say) 1 billion. Then, the resulting lognormal distribution of N is computed numerically by virtue of a MathCad file that the author has written. This shows that the mean value of the lognormal random variable N is actually of the same order as the classical N given by the ordinary Drake equation, as one might expect from a good statistical generalization. Keywords Drake equation · Statistics · SETI

1 Introduction The Drake equation is now a famous result (see Ref. [2] for the Wikipedia summary) in the fields of the Search for ExtraTerrestial Intelligence (SETI, see Ref. [3]) and Astrobiology (see Ref. [4]). Devised in 1961, the Drake equation was the first scientific attempt to estimate the number N of ExtraTerrestrial civilizations in the Galaxy, with which we might come in contact. Frank D. Drake (see Ref. [5]) proposed it as the product of seven factors: N = N s · f p · ne · f l · f i · f c · f L

(1)

where (1) Ns is the estimated number of stars in our Galaxy. (2) fp is the fraction (=percentage) of such stars that have planets. (3) ne is the number of “Earth-type” such planets around the given star; in other words, ne is number of planets, in a given stellar system, on which the chemical conditions exist for life to begin its course: they are “ready for life”. (4) fl is fraction (=percentage) of such “ready for life” planets on which life actually starts and grows up (but not yet to the “intelligence” level). (5) fi is the fraction (=percentage) of such “planets with life forms” that actually evolve until some form of “intelligent civilization” emerges (like the first, historic human civilizations on Earth). (6) fc is the fraction (=percentage) of such “planets with civilizations”, where the civilizations evolve to the point of being able to communicate across the interstellar distances with other (at least) similarly evolved civilizations. As far as we know in 2021, this means that they must be aware of the Maxwell equations governing radio waves, as well as of computers and radioastronomy (at least).

1 Introduction

309

(7) fL is the fraction of galactic civilizations alive at the time when we, poor humans, attempt to pick up their radio signals (that they throw out into space just as we have done since 1900, when Marconi started the transatlantic transmissions). In other words, fL is the number of civilizations now transmitting and receiving, and this implies an estimate of “how long will a technological civilization live?” that nobody can make at the moment. Also, are they going to destroy themselves in a nuclear war, and thus live only a few decades of technological civilization? Or are they slowly becoming wiser, reject war, speak a single language (like English today), and merge into a single “nation”, thus living in peace for ages? Or will robots take over one day making “flesh animals” disappear forever (the so-called “post-biological universe”)? No one knows… But let us go back to the Drake Eq. (1). In the seventy years of its existence, a number of suggestions have been put forward about the different numeric values of its seven factors. Of course, every different set of these seven input numbers yields a different value for N, and we can endlessly play that way. But we claim that these are like… children plays! We claim the classical Drake Eq. (1), as we shall call it from now on to distinguish it from our statistical Drake equation to be introduced in the coming sections, well, the classical Drake equation is scientifically inadequate in one regard at least: it just handles sheer numbers and does not associate an error bar to each of its seven factors. At the very least, we want to associate an error bar to each Di . Well, we have thus reached STEP ONE in our improvement of the classical Drake equation: replace each sheer number by a probability distribution! The reader is now asked to look at the flow chart in the next page as a guide to this paper, please.

2 Step 1: Letting Each Factor Become a Random Variable In this paper, we adopt the notations of the great book “Probability, Random Variables and Stochastic Processes” by Athanasios Papoulis (1921–2002), now re-published as Papoulis-Pillai, Ref. [11]. The advantage of this notation is that it makes a neat distinction between probabilistic (or statistical: it is the same thing here) variables, always denoted by capitals, from non-probabilistic (or “deterministic”) variables, always denoted by lower-case letters. Adopting the Papoulis notation also is a tribute to him by this author, who was a Fulbright Grantee in the United States with him at the Polytechnic Institute (now Polytechnic University) of New York in the years 1977–79.

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The Statistical Drake Equation

We thus introduce seven new (positive) random variables Di (“D” from “Drake”) defined as ⎧ D1 = N s ⎪ ⎪ ⎪ ⎪ ⎪ D2 = f p ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ D3 = ne D4 = f l (2) ⎪ ⎪ ⎪ D = f i ⎪ 5 ⎪ ⎪ ⎪ ⎪ D ⎪ 6 = fc ⎪ ⎪ ⎩ D7 = f L so that our STATISTICAL Drake equation may be simply rewritten as N=

7 

Di .

(3)

i=1

Of course, N now becomes a (positive) random variable too, having its own (positive) mean value and standard deviation. Just as each of the Di has its own (positive) mean value and standard deviation… … the natural question then arises: how are the seven mean values on the right related to the mean value on the left? … and how are the seven standard deviations on the right related to the standard deviation on the left? Just take the next step, STEP TWO.

2.1 Step 2: Introducing Logs to Change the Product into a Sum Products of random variables are not easy to handle in probability theory. It is actually much easier to handle sums of random variables, rather than products, because: (1) The probability density of the sum of two or more independent random variables is the convolution of the relevant probability densities (worry not about the equations, right now). (2) The Fourier transform of the convolution simply is the product of the Fourier transforms (again, worry not about the equations, at this point).

2 Step 1: Letting Each Factor Become a Random Variable

311

312

The Statistical Drake Equation

So, let us take the natural logs of both sides of the Statistical Drake Eq. (3) and change it into a sum:  ln(N ) = ln

7 

 Di

=

i=1

7

ln(Di ).

(4)

i=1

It is now convenient to introduce eight new (positive) random variables defined as follows:

Y = ln(N ) (5) Yi = ln(Di ) i = 1, . . . , 7. Upon inversion, the first equation of Eq. (5) yields the important equation, that will be used in the sequel N = eY .

(6)

We are now ready to take STEP THREE.

2.2 Step 3: The Transformation Law of Random Variables So far we did not mention at all the problem: “which probability distribution shall we attach to each of the seven (positive) random variables Di ?” It is not easy to answer this question because we do not have the least scientific clue to what probability distributions fit at best to each of the seven points listed in Sect. 1. Yet, at least one trivial error must be avoided: claiming that each of those seven random variables must have a Gaussian (i.e. normal) distribution. In fact, the Gaussian distribution, having the well-known bell-shaped probability density function f X (x; μ, σ ) = √

1 2πσ

e−

(x−μ)2 2σ 2

(σ ≥ 0)

(7)

has its independent variable x ranging between −∞ and ∞ and so it can apply to a real random variable X only, and never to positive random variables like those in the statistical Drake Eq. (3). Period. Searching again for probability density functions that represent positive random variables, an obvious choice would be the gamma distributions (see, for instance, Ref. [6]). However, we discarded this choice too because of a different reason: please keep in mind that, according to Eq. (5), once we selected a particular type of probability density function (pdf) for the last seven of Eq. (5), then we must compute the (new

2 Step 1: Letting Each Factor Become a Random Variable

313

and different) pdf of the logs of such random variables. And the pdf of these logs certainly is not gamma-type any more. It is high time now to remind the reader of a certain theorem that is proved in probability courses, but, unfortunately, does not seem to have a specific name. It is the transformation law (so we shall call it, see, for instance, Ref. [11]) allowing us to compute the pdf of a certain new random variable Y that is a known function Y = g(X) of another random variable X having a known pdf. In other words, if the pdf f X (x) of a certain random variable X is known, then the pdf f Y (y) of the new random variable Y, related to X by the functional relationship Y = g(X )

(8)

can be calculated according to this rule: (1) First, invert the corresponding non-probabilistic equation y = g(x) and denote by x i (y) the various real roots resulting from this inversion. (2) Second, take notice whether these real roots may be either finitely- or infinitely many, according to the nature of the function y = g(x). (3) Third, the probability density function of Y is then given by the (finite or infinite) sum

f Y (y) =

f X (xi (y)) |g  (xi (y))| i

(9)

  where the summation extends to all roots x i (y) and g  (xi (y)) is the absolute value of the first derivative of g(x), where the i-th root x i (y) has been replaced instead of x. Since we must use this transformation law to transfer from the Di to the Y i = ln(Di ), it is clear that we need to start from a Di pdf that is as simple as possible. The gamma pdf is not responding to this need because the analytic expression of the transformed pdf is very complicated (or, at least, it looked so to this author in the first instance). Also, the gamma distribution has two free parameters in it, and this “complicates” its application to the various meanings of the Drake equation. In conclusion, we discarded the gamma distributions and confined ourselves to the simpler uniform distribution instead, as shown in the next section.

3 Step 4: Assuming the Easiest Input Distribution for Each Di : The Uniform Distribution Let us now suppose that each of the seven Di is distributed UNIFORMLY in the interval ranging from the lower limit ai ≥ 0 to the upper limit bi ≥ ai .

314

The Statistical Drake Equation

This is the same as saying that the probability density function of each of the seven Drake random variables Di has the equation f uniform_ Di (x) =

1 bi − ai

with 0 ≤ ai ≤ x ≤ bi

(10)

as it follows at once from the normalization condition bi f uniform_ Di (x) d x = 1.

(11)

ai

Let us now consider the mean value of such uniform Di defined by bi uniform_ Di  = ai

=

2 bi

x 1 bi − ai 2

1 x f uniform_ Di (x) d x = bi − ai =

ai

bi xd x ai

− ai + bi = . 2(bi − ai ) 2 bi2

ai2

By words (as it is intuitively obvious): the mean value of the uniform distribution simply is the mean of the lower plus upper limit of the variable range uniform_ Di  =

ai + bi . 2

(12)

In order to find the variance of the uniform distribution, we first need finding the second moment 

Di2

uniform_



bi =

x 2 f uniform_ Di (x) d x ai

1 = bi − ai =

bi

3 bi b3 − ai3 x 1 x dx = = i bi − ai 3 ai 3(bi − ai ) 2

ai

a 2 + ai bi + bi2 (bi − ai )(ai2 + ai bi + bi2 ) = i . 3(bi − ai ) 3

The second moment of the uniform distribution is thus 

 a 2 + ai bi + bi2 uniform_ Di2 = i . 3

(13)

3 Step 4: Assuming the Easiest Input Distribution for Each …

315

From Eqs. (12) and (13), we may now derive the variance of the uniform distribution   2 2 2 σuniform_ Di = uniform_ Di − uniform_ Di  =

ai2 + ai bi + bi2 (ai + bi )2 (bi − ai )2 − = . 3 4 12

(14)

Upon taking the square root of both sides of Eq. (14), we finally obtain the standard deviation of the uniform distribution: σuniform_ Di =

bi − ai √ . 2 3

(15)

We now wish to perform a calculation that is mathematically trivial, but rather unexpected from the intuitive point of view, and very important for our applications to the statistical Drake equation. Just consider the two simultaneous Eqs. (12) and (15) 

i uniform_ Di  = ai +b 2 bi √ −ai σuniform_ Di = 2 3 .

(16)

Upon inverting this trivial linear system, one finds

√ ai = uniform_ Di  − √3 σuniform_ Di bi = uniform_ Di  + 3 σuniform_ Di .

(17)

This is of paramount importance for our application the Statistical Drake equation inasmuch as it shows that: If one (scientifically) assigns the mean value and standard deviation of a certain Drake random variable Di , then the lower and upper limits of the relevant uniform distribution are given by the two Eqs. (17), respectively. √ In other words, there is a factor of 3 = 1.732 included in the two Eqs. (17) that is not obvious at all to human intuition, and must indeed be taken into account. The application of this result to the Statistical Drake equation is discussed in the next section.

316

The Statistical Drake Equation

3.1 Step 5: A Numerical Example of the Statistical Drake Equation with Uniform Distributions for the Drake Random Variables Di The first variable Ns in the classical Drake Eq. (1) is the number of stars in our Galaxy. Nobody knows how many they are exactly (!). Only statistical estimates can be made by astronomers, and they oscillate (say) around a mean value of 350 billions (if this value is indeed correct!). This being the situation, we assume that our uniformly distributed random variable Ns has a mean value of 350 billions minus or plus a standard deviation of (say) one billion (we do not care whether this number is scientifically the best estimate as of January 2021: we just want to set up a numerical example of our Statistical Drake equation). In other words, we now assume that one has:

uniform_ Di  = 350 × 109 (18) σuniform_ D1 = 1 × 109 . Therefore, according to Eq. (17), the lower and upper limit of our uniform distribution for the random variable Ns = D1 are, respectively

√ a N s = uniform_ D1  − √3σuniform_ D1 = 348.3 × 109 b N s = uniform_ D1  + 3σuniform_ D1 = 351.7 × 109 .

(19)

Similarly, we proceed for all the other six random variables in the Statistical Drake Eq. (3). For instance, we assume that the fraction of stars that have planets is 50%, i.e. 50/100, and this will be the mean value of the random variable fp = D2 . We also assume that the relevant standard deviation will be 10%, i.e. that σ fp = 10/100. Therefore, the relevant lower and upper limits for the uniform distribution of fp = D2 turn out to be √

a f p = uniform_ D2  − √3σuniform_ D2 = 0.327 (20) b f p = uniform_ D2  + 3σuniform_ D2 = 0.673. The next Drake random variable is the number ne of “Earth-type” planets in a given star system. Taking example from the Solar System, since only the Earth is truly “Earth-type”, the mean value of ne is clearly 1, but the standard deviation is not zero if we assume that Mars also may be regarded as Earth-type. Since there are thus two √Earth-type planets in the Solar√System, we must assume a standard deviation of 1/ 3 = 0.577 to compensate the 3 appearing in Eq. (17) in order to finally yield two “Earth-type” planets (Earth and Mars) for the upper limit of the random variable ne. In other words, we assume that

3 Step 4: Assuming the Easiest Input Distribution for Each …

317

Table 1 Input values (i.e. mean values and standard deviations) for the seven Drake uniform random variables Di Ns := 350 × 109

μNs : = Ns

σ Ns := 1 × 109

fp :=

μfp : = fp

σ fp :=

50 100

ne := 1

μne : = ne

fl :=

μfl : = fl

50 100 20 fi := 100 20 fc := 100 1000 fL := 1010

μf : = fi μfc : = fc μfL : = fL

N : = Ns · fp · ne · fl · fi · fc · fL

10 100 σ ne := √1 3 10 σ fl := 100 10 σ fi := 100 10 σ fc := 100 1000 σ fL := 1010

N = 3500

The first column on the left lists the seven input sheer numbers that also become the mean values (middle column). Finally, the last column on the right lists the seven input standard deviations. The bottom line is the classical Drake Eq. (1)



√ ane = uniform_ D3  − √3 σuniform_ D3 = 0 bne = uniform_ D3  + 3 σuniform_ D3 = 2.

(21)

The next four Drake random variables have even more “arbitrarily” assumed values that we simply assume for the sake of making up a numerical example of our Statistical Drake equation with uniform entry distributions. So, we really make no assumption about the astronomy, or the biology, or the sociology of the Drake equation: we just care about its mathematics. All our assumed entries are given in Table 1. Please notice that, had we assumed all the standard deviations to equal zero in Table 1, then our Statistical Drake Eq. (3) would have obviously reduced to the classical Drake Eq. (1), and the resulting number of civilizations in the Galaxy would have turned out to be 3500: N = 3500.

(22)

This is an important deterministic number that we will use in the sequel of this chapter for comparison with our statistical results on the mean value of N, i.e. N . This will be explained in Sects. 3.3 and 5.

3.2 Step 6: Computing the Logs of the Seven Uniformly Distributed Drake Random Variables Di Intuitively speaking, the natural log of a uniformly distributed random variable may not be an another uniformly distributed random variable! This is obvious from the trivial diagram of y = ln(x) shown in Fig. 1.

The Statistical Drake Equation REAL values of the natural log: y=ln(x)

318 Natural logarithm of x 2

1

0

1

2

0

1 2 3 4 POSITIVE independent variable x

5

Fig. 1 The simple function y = ln(x)

So, if we have a uniformly distributed random variable Di with lower limit ai and upper limit bi , the random variable Yi = ln(Di ) i = 1, . . . , 7

(23)

must have its range limited in between the lower limit ln(ai ) and the upper limit ln(bi ). In other words, these are the lower and upper limits of the relevant probability density function f Yi (y). But what is the actual analytic expression of such a pdf? To find it, we must resort to the general transformation law for random variables, defined by Eq. (9). Here, we obviously have y = g(x) = ln(x).

(24)

That, upon inversion, yields the single root x1 (y) = x(y) = e y .

(25)

On the other hand, differentiating Eq. (24) one gets g  (x) =

1 1 1 and g  (x1 (y)) = = y x x1 (y) e

(26)

where Eq. (25) was already used in the last step. By virtue of the uniform probability density function (10) and of Eq. (26), the general transformation law (9) finally yields f Y (y) =

f X (xi (y)) ey 1 1   = . = 1 |g  (xi (y))| bi − ai  e y  bi − ai i

(27)

3 Step 4: Assuming the Easiest Input Distribution for Each …

319

In other words, the requested pdf of Y i is f Yi (y) =

ey bi − ai

i = 1, . . . , 7 ln(ai ) ≤ y ≤ ln(bi ).

(28)

These are the probability density functions of the natural logs of all the uniformly distributed Drake random variables Di . This is indeed a positive function of y over the interval ln(ai ) ≤ y ≤ ln(bi ), as for every pdf, and it is easy to see that its normalization condition is fulfilled ln(b i)

ln(b i)

f Yi (y) dy = ln(ai )

ln(ai )

ey eln(bi ) − eln(ai ) dy = = 1. bi − ai bi − ai

(29)

Next, we want to find the mean value and standard deviation of Y i , since these play a crucial role for future developments. The mean value Y i  of the random variables Y i = ln(Di ) is given by ln(b i)

Yi  =

ln(b i)

y f Yi (y) dy = ln(ai )

ln(ai )

ye y dy bi − ai

bi [ln(bi ) − 1] − ai [ln(ai ) − 1] = . bi − ai

(30)

This is thus the mean value of the natural log of all the uniformly distributed Drake random variables Di Yi  = ln(Di ) =

bi [ln(bi ) − 1] − ai [ln(ai ) − 1] . bi − ai

(31)

In order to find the variance also, we must first compute the mean value of the square of Y i , i.e. 

Yi2



ln(b i)

=

y f Yi (y) dy = ln(ai )

=

ln(b i) 2

ln(ai )

y2 ey dy bi − ai

bi [ln (bi ) − 2 ln(bi ) + 2] − ai [ln2 (ai ) − 2 ln(ai ) + 2] . bi − ai 2

(32)

The variance of Y i = ln(Di ) is now given by Eq. (32) minus the square of Eq. (31), that, after a few reductions, yield: 2 =1− σY2i = σln(D i)

ai bi [ln(bi ) − ln(ai )]2 . (bi − ai )2

(33)

320

The Statistical Drake Equation

Whence the corresponding standard deviation  σYi = σln(Di ) =

1−

ai bi [ln(bi ) − ln(ai )]2 . (bi − ai )2

(34)

Let us now turn to an another topic: the use of Fourier transforms, that, in probability theory, are called “characteristic functions”. Following again the notations of Papoulis (Ref. [11]) we call “characteristic function,” Φ Yi (ζ), of an assigned probability distribution Y i ,√the Fourier transform of the relevant probability density function, that is (with j = −1) ∞ ΦYi (ζ ) =

e j ζ y f Yi (y) dy.

(35)

−∞

The use of characteristic functions simplifies things greatly. For instance, the calculation of all moments of a known pdf becomes trivial if the relevant characteristic function is known, and greatly simplified also are the proofs of important theorems of statistics, like the Central Limit Theorem that we will use in Sect. 4. Another important result is that the characteristic function of the sum of a finite number of independent random variables is simply given by the product of the corresponding characteristic functions. This is just the case we are facing in the Statistical Drake Eq. (4), and so we are now led to find the characteristic function of the random variable Y i , i.e. ∞ ΦYi (ζ ) =

e

jζ y

ln(b i)

−∞

=

=

1 bi − ai e

ej ζ y

f Yi (y) dy = ln(ai )

ln(b i)

e(1+ j ζ ) y dy =

ln(ai )

ey dy bi − ai

ln(bi )  1 (1/1 + jζ ) e(1+ jζ )y ln(ai ) bi − ai

(1+ jζ ) ln(bi )

bi − ai − e(1+ jζ ) ln(ai ) = . (bi − ai )(1 + jζ ) (bi − ai )(1 + jζ ) 1+ jζ

1+ jζ

(36)

Thus, the characteristic function of the natural log of the Drake uniform random variable Di is given by 1+ jζ

ΦYi (ζ ) =

1+ jζ

− ai bi . (bi − ai )(1 + jζ )

(37)

3 Step 4: Assuming the Easiest Input Distribution for Each …

321

3.3 Step 7: Finding the Probability Density Function of N, but Only Numerically, Not Analytically Having found the characteristic functions ΦYi (ζ ) of the logs of the seven input random variables Di , we can now immediately find the characteristic function of the random variable Y = ln(N) defined by Eq. (5). In fact, by virtue of Eq. (4), of the well-known Fourier transform property stating that “the Fourier transform of a convolution is the product of the Fourier transforms,” and of Eq. (37), it immediately follows that Φ Y (ζ ) equals the product of the seven ΦYi (ζ ) 7 

ΦY (ζ ) =

ΦYi (ζ ) =

i=1

7  i=1

1+ jζ

1+ jζ

bi − ai . (bi − ai )(1 + jζ )

(38)

The next step is to invert this Fourier transform in order to get the probability density function of the random variable Y = ln(N). In other words, we must compute the following inverse Fourier transform (Fig. 1). ∞ f Y (y) = (1/2π )

e− j ζ y ΦY (ζ ) dζ

−∞

∞ = (1/2π )

e

e

−∞

 7 

 ΦYi (ζ ) dζ

i=1

−∞ ∞

= (1/2π )

−j ζ y

−j ζ y

 7  i=1

 1+ jζ 1+ jζ bi − ai dζ. (bi − ai ) (1 + jζ )

(39)

This author regrets that he was unable to compute the last integral analytically. He had to compute it numerically for the particular values of the 14 ai and bi that follow from Table 1 and Eq. 17. The result was the probability density function for Y = ln(N) plotted in the following Fig. 2. We are now just one more step from finding the probability density of N, the number of ExtraTerrestrial Civilizations in the Galaxy predicted by our Statistical Drake Eq. (3). The point here is to transfer from the probability density function of Y to that of N, knowing that Y = ln(N), or alternatively, that N = exp(Y ), as stated by Eq. (6). We must thus resort to the transformation law of random variables Eq. (9) by setting y = g(x) = e x .

(40)

This, upon inversion, yields the single root x1 (y) = x(y) = ln(y).

(41)

The Statistical Drake Equation Probability density function of Y

322

PROB. DENSITY FUNCTION OF Y=ln(N)

0.4 0.3 0.2 0.1 0

0 1 2 3 4 5 6 7 8 9 10 11 12 Independent variable Y = ln(N)

Fig. 2 Probability density function of Y = ln(N) computed numerically by virtue of the integral (39). The two “funny gaps” in curve are due to the numeric limitations in the MathCad numeric solver that the author used for this numeric computation

On the other hand, differentiating Eq. (40) one gets g  (x) = e x and g  (x1 (y)) = eln(y) = y

(42)

where Eq. (41) was already used in the last step. The general transformation law (9) finally yields f N (y) =

f X (xi (y)) 1 = f Y (ln(y)).  (x (y))| |g |y| i i

(43)

Prob. density function of N

This probability density function f N (y) was computed numerically by using Eq. (43) and the numeric curve given by Eq. (39), and the result is shown in Fig. 3. We now want to compute the mean value N  of the probability density Eq. (43). Clearly, it is given by 4 .10

4

3 .10

4

2 .10

4

1 .10

4

0

PROBABILITY DENSITY FUNCTION OF N

0 1000 2000 3000 4000 N = Number of ET Civilizations in Galaxy

Fig. 3 The numeric (and not analytic) probability density function curve f N (y) of the number N of ExtraTerrestrial Civilizations in the Galaxy according to the Statistical Drake Eq. (3). We see that curve peak (i.e. the mode) is very close to low values of N, but the tail on the right is long, meaning that the resulting mean value N  is of the order of thousands

3 Step 4: Assuming the Easiest Input Distribution for Each …

323

∞ N  =

y f N (y) dy.

(44)

0

This integral too was computed numerically, and the result was a perfect match with N = 3500 of Eq. (22), i.e. N  = 3499.99880177509 + 0.000000124914686 j.

(45)

Note that this result was computed numerically in the complex domain because of the Fourier transforms, and that the real part is virtually 3500 (as expected), while the imaginary part is virtually zero because of the rounding errors. So, this result is excellent, and proves that the theory presented so far is mathematically correct. Finally, we want to consider the standard deviation. This also had to be computed numerically, resulting in σ N = 3953.42910143389 + 0.000000032800058i.

(46)

This standard deviation, higher than the mean value, implies that N might range in between 0 and 7453. This completes our study of the probability density function of N if the seven uniform Drake input random variable Di have the mean values and standard deviations listed in Table 1. We conclude that, unfortunately, even under the simplifying assumptions that the Di be uniformly distributed, it is impossible to solve the full problem analytically, since all calculations beyond Eq. (38) had to be performed numerically. This is no good. Shall we thus loose faith, and declare “impossible” the task of finding an analytic expression for the probability density function f N (y)? Rather surprisingly, the answer is “no”, and there is indeed a way out of this dead-end, as we shall see in the next section.

4 The Central Limit Theorem (CLT) of Statistics Indeed there is a good, approximating analytical expression for f N (y), and this is the following lognormal probability density function f N (y, μ, σ ) =

(ln(y)−μ)2 1 1 e− 2σ 2 √ y 2π σ

(y ≥ 0).

(47)

To understand why, we must resort to what is perhaps the most beautiful theorem of Statistics: the Central Limit Theorem (abbreviated CLT). Historically, the CLT was in

324

The Statistical Drake Equation

fact proven first in 1901 by the Russian mathematician Alexandr Lyapunov (1857– 1918), and later (1920) by the Finnish mathematician Jarl Waldemar Lindeberg (1876–1932) under weaker conditions. These conditions are certainly fulfilled in the context of the Drake equation because of the “reality” of the astronomy, biology and sociology involved with it, and we are not going to discuss this point any further here. A good, synthetic description of the Central Limit Theorem of Statistics is found at the Wikipedia site (Ref. [7]) to which the reader is referred for more details, such as the equations for the Lyapunov and the Lindeberg conditions, making the theorem “rigorously” valid. Put in loose terms, the CLT states that, if one has a sum of random variables even NOT identically distributed, this sum tends to a normal distribution when the number of terms making up the sum tends to infinity. Also, the normal distribution mean value is the sum of the mean values of the addend random variables, and the normal distribution variance is the sum of the variances of the addend random variables. Let us now write down the equations of the CLT in the form needed to apply it to our Statistical Drake Eq. (3). The idea is to apply the CLT to the sum of random variables given by Eqs. (4) and (5) whatever their probability distributions can possibly be. In other words, the CLT applied to the Statistical Drake Eq. (3) leads immediately to the following three equations: (1) The sum of the (arbitrarily distributed) independent random variables Y i makes up the new random variable Y. (2) The sum of their mean values makes up the new mean value of Y. (3) The sum of their variances makes up the new variance of Y. In equations ⎧ 7  ⎪ ⎪ Y = Yi ⎪ ⎪ ⎪ i=1 ⎪ ⎨ 7  Y  = Yi  ⎪ i=1 ⎪ ⎪ ⎪ 7  ⎪ ⎪ ⎩ σY2 = σY2i .

(48)

i=1

This completes our synthetic description of the CLT for sums of random variables.

5 The Lognormal Distribution Is the Distribution of the Number N of Extraterrestrial Civilizations in the Galaxy The CLT may of course be extended to products of random variables upon taking the logs of both sides, just as we did in Eq. (3). It then follows that the exponent

5 The Lognormal Distribution Is the Distribution of the Number …

325

random variable, like Y in Eq. (6), tends to a normal random variable, and, as a consequence, it follows that the base random variable, like N in Eq. (6), tends to a lognormal random variable. To understand this fact better in mathematical terms consider again of the transformation law (9) of random variables. The question is: what is the probability density function of the random variable N in Eq. (6), i.e. what is the probability density function of the lognormal distribution? To find it, set y = g(x) = e x .

(49)

This, upon inversion, yields the single root x1 (y) = x(y) = ln(y).

(50)

On the other hand, differentiating (49) one gets g  (x) = e x and g  (x1 (y)) = eln(y) = y

(51)

where Eq. (50) was already used in the last step. The general transformation law Eq. (9) finally yields f N (y) =

f X (xi (y)) 1 f Y (ln(y)). =  |g (xi (y))| |y| i

(52)

Therefore, replacing the probability density on the right by virtue of the wellknown normal (or Gaussian) distribution given by Eq. (7), the lognormal distribution of Eq. (47) is found, and the derivation of the lognormal distribution from the normal distribution is proved. In view of future calculations, it is also useful to point out the so-called “Gaussian integral,” i.e. 

∞ e

−Ax 2 Bx

e

−∞

dx =

π B2 e 4 A , A > 0, B = real. A

(53)

This follows immediately from the normalization condition of the Gaussian Eq. (7), i.e. ∞ −∞

(x−μ)2 1 e− 2σ 2 d x = 1, √ 2π σ

(54)

just upon expanding the square at the exponent and making the two replacements (we skip all steps)

326

The Statistical Drake Equation



A= B=

1 > 0, 2σ 2 μ = real. σ2

(55)

In the sequel of this paper, we shall denote the independent variable of the lognormal distribution (47) by a lower-case letter n to remind the reader that corresponding random variable N is the positive integer number of ExtraTerrestrial Civilizations in the Galaxy. In other words, n will be treated as a positive real number in all calculations to follow, because it is a “large” number (i.e. a continuous variable) compared to the only civilization that we know of, i.e. ourselves. In conclusion, from now on the lognormal probability density function of N will be written as f N (n) =

1 1 2 2 ·√ e−(ln(n)−μ) /(2σ ) n 2π σ

(n ≥ 0).

(56)

Having so said, we now turn to the statistical properties of the lognormal distribution (56), i.e. to the statistical properties that describe the number N of ExtraTerrestrial Civilizations in the Galaxy. Our first goal is to prove an equation yielding all the moments of the lognormal distribution (56), i.e. for every non-negative integer k = 0, 1, 2, …, one has 

 2 σ2 N k = ekμ ek 2 .

(57)

The relevant proof starts with the definition of the k-th moment 

N

k



∞ =

n k f N (n) dn 0

∞ = 0

1 1 2 2 nk √ e−(ln(n)−μ) /(2σ ) dn. n 2π σ

One then transforms the above integral by virtue of the substitution ln[n] = z.

(58)

The new integral in z is then seen to reduce to the Gaussian integral (53) (we skip all steps here) and Eq. (57) follows = ekμ ek

2 σ2 2

.

5 The Lognormal Distribution Is the Distribution of the Number …

327

Upon setting k = 0 into Eq. (57), the normalization condition for f N (n) follows ∞ f N (n) dn = 1.

(59)

0

Upon setting k = 1 into Eq. (57), the important mean value of the random variable N is found N  = eμ e

σ2 2

.

(60)

Upon setting k = 2 into Eq. (57), the mean value of the square of the random variable N is found 

 2 N 2 = e2μ e2σ .

(61)

The variance of N now follows from the last two formulae σ N2 = e2μ eσ (eσ − 1). 2

2

(62)

The square root of this is the important standard deviation formula for the N random variable  2 (63) σ N = eμ eσ /2 eσ 2 − 1. The third moment is obtained upon setting k = 3 into Eq. (57) 

9σ 2  N 3 = e3μ e 2 .

(64)

Finally, upon setting k = 4, the fourth moment of N is found 

 2 N 4 = e4μ e8σ .

(65)

Our next goal is to find the cumulants of N. In principle, we could compute all the cumulants K i from the generic i-th moment μi by virtue of the recursion formula (see Ref. [8]) Ki =

μi

 i−1  i −1 − K k μn−k . k−1

(66)

k=1

In practice, however, here we shall confine ourselves to the computation of the first four cumulants only because they only are required to find the skewness and kurtosis of the distribution. Then, the first four cumulants in terms of the first four

328

The Statistical Drake Equation

moments read ⎧ K1 ⎪ ⎪ ⎨ K2 ⎪ K ⎪ ⎩ 3 K4

= μ1 = μ2 − K 12 = μ3 − 3K 1 K 2 − K 13 = μ4 − 4K 1 K 3 − 3K 22 − 6K 2 K 12 − K 14 .

(67)

These equations yield, respectively K 1 = eμ eσ

2

/2

.

(68)

K 2 = e2μ eσ (eσ − 1). 2

2

K 3 = e3μ e

9σ 2 2

(69)

.

(70)

K 4 = e4μ+2σ (eσ − 1)3 (e3σ + 3e2σ + 6eσ + 6). 2

2

2

2

2

(71)

From these we derive the skewness  K3 σ2 = (e + 2) eσ 2 − 1 (K 2 )3/2

(72)

K4 2 2 2 = e4σ + 2e3σ + 3e2σ − 6. (K 2 )2

(73)

and the kurtosis

Finally, we want to find the mode of the lognormal probability density function, i.e. the abscissa of its peak. To do so, we must first compute the derivative of the probability density function f N (n) of Eq. (56), and then set it equal to zero. This derivative is actually the derivative of the ratio of two functions of n, as it plainly appears from Eq. (56). Thus, let us set for a moment E(n) =

(ln(n) − μ)2 2σ 2

(74)

where “E” stands for “exponent”. Upon differentiating this, one gets E  (n) =

1 1 2(ln(n) − μ) . 2σ 2 n

(75)

5 The Lognormal Distribution Is the Distribution of the Number …

329

But the lognormal probability density function (56), by virtue of Eq. (74), now reads 1 e−E(n) f N (n) = √ · . n 2π σ

(76)

So that its derivative is 1 −e−E(n) E  (n)n − 1e−E(n) d f N (n) =√ dn n2 2π σ    −E(n) E (n)n + 1 1 −e =√ . n2 2π σ

(77)

Setting this derivative equal to zero means setting E  (n)n + 1 = 0.

(78)

That is, upon replacing Eq. (75), 1 (ln(n) − μ) + 1 = 0. σ2

(79)

Rearranging, this becomes ln(n) − μ + σ 2 = 0

(80)

and finally n mode ≡ n peak = eμ e−σ . 2

(81)

This is the most likely number of ExtraTerrestrial Civilizations in the Galaxy. How likely? To find the value of the probability density function f N (n) corresponding to this value of the mode, we must obviously substitute Eq. (81) into Eq. (56). After a few rearrangements, one then gets 1 2 e−μ eσ /2 . f N (n mode ) = √ 2π σ

(82)

This is “how likely” the most likely number of ExtraTerrestrial Civilizations in the Galaxy is, i.e. it is the peak height in the lognormal probability density function f N (n). Next to the mode, the median m (Ref. [9]) is one more statistical number used to characterize any probability distribution. It is defined as the independent variable abscissa m such that a realization of the random variable will take up a value lower than m with 50% probability or a value higher than m with 50% probability again.

330

The Statistical Drake Equation

In other words, the median m splits up our probability density in exactly two equally probable parts. Since the probability of occurrence of the random event equals the area under its density curve (i.e. the definite integral under its density curve), then the median m (of the lognormal distribution, in this case) is defined as the integral upper limit m: m

m f N (n)dn =

0

0

(ln(n)−μ)2 1 1 1 e− 2σ 2 dn = . √ n 2π σ 2

(83)

In order to find m, we may not differentiate Eq. (83) with respect to m, since the “precise” factor ½ on the right would then disappear into a zero. On the contrary, we may try to perform the obvious substitution z2 =

(ln(n) − μ)2 2 σ2

z ≥ 0.

(84)

into the integral (83) to reduce it to the following integral defining the error function erf (z) 2 er f (x) = √ π

x

e−z dz. 2

(85)

0

Then, after a few reductions that we skip for the sake of brevity, the full Eq. (83) is turned into   1 ln(m) − μ 1 = + er f (86) √ 2 2 2σ i.e.  er f

ln(m) − μ √ 2σ

 = 0.

(87)

Since from the definition Eq. (85) one obviously has erf (0) = 0, Eq. (87) becomes ln(m) − μ =0 √ 2σ

(88)

median = m = eμ .

(89)

whence finally

5 The Lognormal Distribution Is the Distribution of the Number …

331

This is the median of the lognormal distribution of N. In other words, this is the number of ExtraTerrestrial civilizations in the Galaxy such that, with 50% probability the actual value of N will be lower than this median, and with 50% probability it will be higher. In conclusion, we feel useful to summarize all the equations that we derived about the random variable N in the following Table 2. We want to complete this section about the lognormal probability density function (56) by finding out its numeric values for the inputs to the Statistical Drake Eq. (3) listed in Table 1. According to the CLT, the mean value μ to be inserted into the lognormal density Eq. (56) is given (according to the second Eq. (48)) by the sum of all the mean values Yi , that is, by virtue of Eq. (31), by μ=

7

Yi  =

i=1

7 bi [ln(bi ) − 1] − ai [ln(ai ) − 1] i=1

bi − ai

.

(90)

Table 2 Summary of the properties of the lognormal distribution that applies to the random variable N = number of ET communicating civilizations in the Galaxy Random variable

N = number of communicating ET civilizations in Galaxy

Probability distribution

Lognormal

Probability density function

f N (n) =

1 n

·

√1 2πσ

e

−

(ln(n)−μ)2 2σ 2

Mean value

N  = eμ e

Variance

σ N2 = e2μ eσ (eσ − 1) 2

2

All the moments, i.e. k-th moment

σ2  2 σ N = eμ e 2 eσ − 1  k 2 σ2 N = ekμ ek 2

Mode (=abscissa of the lognormal peak)

n mode ≡ n peak = eμ e−σ

Value of the mode peak

f N (n mode ) =

Median (=fifty–fifty probability value for N)

Median = m

Skewness

K3 (K 2 )3/2

Kurtosis

K4 (K 2 )2

Expression of μ in terms of the lower (ai ) and upper (bi ) limits of the Drake uniform input random variables Di

μ= 7

Standard deviation

(n ≥ 0)

σ 2 /2

2

2 √ 1 e−μ eσ /2 2πσ = eμ

 2 2 = (eσ + 2) eσ − 1 2

2

2

= e4σ + 2e3σ + 3e2σ − 6 7

i=1

i=1 Yi  = bi [ln(bi )−1]−ai [ln(ai )−1] bi −ai

Expression of σ 2 in terms of the lower (ai ) and upper σ 2 = 7 σ 2 = i=1 Yi  (bi ) limits of the Drake uniform input random 7  ai bi [ln(bi )−ln(ai )]2 2 variables Di i=1 1 − (bi −ai )

332

The Statistical Drake Equation

Upon replacing the 14 ai and bi listed in Table 1 into Eq. (90), the following numeric mean value μ is found μ ≈ 7.462176.

(91)

Similarly, to get the numeric variance σ 2 one must resort to the last of Eq. (48) and to Eq. (33): σ2 =

7 i=1

σY2i =

 7  ai bi [ln(bi ) − ln(ai )]2 1− (bi − ai )2 i=1

(92)

yielding the following numeric variance σ 2 to be inserted into the lognormal pdf Eq. (56) σ 2 ≈ 1.938725

(93)

whence the numeric standard deviation σ σ ≈ 1.392381.

(94)

Upon replacing these two numeric values Eqs. (91) and (94) into the lognormal pdf Eq. (56), the latter is perfectly determined. It is plotted in Fig. 4 as the thin curve. In other words, Fig. 4 shows the lognormal distribution for the number N of ExtraTerrestrial Civilizations in the Galaxy derived from the Central Limit Theorem as applied to the Drake equation (with the input data listed in Table 1).

Fig. 4 Comparing the two probability density functions of the random variable N found: (1) at the end of Sect. 3.3 in a purely numeric way and without resorting to the CLT at all (thick curve) and (2) analytically by using the CLT and the relevant lognormal approximation (thin curve)

5 The Lognormal Distribution Is the Distribution of the Number …

333

We now like to point out the most important statistical properties of this lognormal pdf: (1) Mean Value of N. This is given by Eq. (60) with μ and σ given by Eqs. (91) and (94), respectively: N  = eμ eσ

2

/2

≈ 4589.559.

(95)

In other words, there are 4590 ET Civilizations in the Galaxy according to the Central Limit Theorem of Statistics with the inputs of Table 1. This number 4590 is HIGHER than the 3500 foreseen by the classical Drake equation working with sheer numbers only, rather than with probability distributions. Thus, Eq. (95) IS GOOD FOR NEWS FOR SETI, since it shows that the expected number of ETs is HIGHER with an adequate statistical treatment than just with the too simple Drake sheer numbers of Eq. (1). (2) Variance of N. The variance of the lognormal distribution is given by Eq. (62) and turns out to be a huge number σ N2 = e2μ eσ (eσ − 1) ≈ 125328623. 2

2

(96)

(3) Standard deviation of N. The standard deviation of the lognormal distribution is given by Eq. (63) and turns out to be σ N = eμ e

σ2 2



eσ 2 − 1 = 11195.

(97)

Again, this is GOOD NEWS FOR SETI. In fact, such a high standard deviation means that N may range from very low values (zero, theoretically, and one since humanity exists) up to tens of thousands (4590 + 11,195 = 15,785 is Eq. (95) + Eq. (97)). (4) Mode of N: the mode (=peak abscissa) of the lognormal distribution of N is given by Eq. (81), and has a surprisingly low numeric value n mode ≡ n peak = eμ e−σ ≈ 250. 2

(98)

This is well shown in Fig. 4: the mode peak is very pronounced and close to the origin, but the right tail is long, and this means that the mean value of the distribution is much higher than the mode: 4590  250. (5) Median of N: the median (=fifty–fifty abscissa, splitting the pdf in two exactly equi-probable parts) of the lognormal distribution of N is given by Eq. (89), and has the numeric value n median ≡ eμ ≈ 1740.

(99)

334

The Statistical Drake Equation

In words, assuming the input values listed in Table 1, we have exactly a 50% probability that the actual value of N is lower than 1740, and 50% that it is higher than 1740.

6 Comparing the CLT Results with the Non-CLT Results The time is now ripe to compare the CLT-based results about the lognormal distribution of N, just described in Sect. 5, against the Non-CLT-based results obtained numerically in Sect. 3.3. To do so in a simple, visual way, let us plot on the same diagram two curves (see Fig. 4): (1) The numeric curves appearing in Fig. 2 and obtained after laborious Fourier transform calculations in the complex domain, and (2) The lognormal distribution (56) with numeric μ and σ given by Eqs. (91) and (94), respectively. We see in Fig. 4 that the two curves are virtually coincident for values of N larger than 1500. This is a consequence of the law of large numbers, of which the CLT is just one of the many facets. Similarly, it happens for natural log of N, i.e. the random variable Y of Eq. (5), that is plotted in Fig. 5 both in its normal curve version (thin curve) and in its numeric version, obtained via Fourier transforms and already shown in Fig. 2. The conclusion is simple: from now on we shall discard forever the numeric calculations and we will stick only to the equations derived by virtue of the CLT, i.e. to the lognormal Eq. (56) and its consequences.

7 Distance of the Nearest Extraterrestrial Civilization as a Probability Distribution As an application of the Statistical Drake Equation developed in the previous sections of this chapter, we now want to consider the problem of estimating the distance of the ExtraTerrestrial Civilization nearest to us in the Galaxy. In all Astrobiology textbooks (see for instance, Ref. [1]) and in several web sites, the solution to this problem is reported with only slight differences in the mathematical proofs among the various authors. In the first of the coming two sections (Sect. 7.1), we derive the expression for this “ET_Distance” (as we like to denote it) in the classical, non-probabilistic way: in other words, this is the classical, deterministic derivation. In the second Sect. 7.2, we provide the probabilistic derivation, arising from our Statistical Drake Equation, of the corresponding probability density function f ET_distance (r): here, r is the distance between us and the nearest ET civilization assumed as the independent variable of its own probability density function. The ensuing sections provide more mathematical

7 Distance of the Nearest Extraterrestrial Civilization …

335

PROBABILITY DENSITY FUNCTION OF Y=ln(N)

Probability density function of Y

0.5

0.4

0.3

0.2

0.1

0

0

1

2

3

4 5 6 7 8 Independent variable Y = ln(N)

9

10

11

12

Fig. 5 Comparing the two probability density functions of the random variable Y = ln(N) found: (1) at the end of Sect. 3.3 in a purely numeric way and without resorting to the CLT at all (thick curve) and (2) analytically by using the CLT and the relevant normal (Gaussian) approximation (thin Gaussian curve)

details about this f ET_distance (r) such as its mean value, variance, standard deviation, all central moments, mode, median, cumulants, skewness and kurtosis.

7.1 Classical, Non-probabilistic Derivation of the Distance of the Nearest ET Civilization Consider the Galactic Disk and assume that: (1) The diameter of the Galaxy is (about) 100,000 light years, (abbreviated ly) i.e. its radius, RGalaxy , is about 50,000 ly. (2) The thickness of the Galactic Disk at half-way from its center, hGalaxy , is about 16,000 ly. Then, (3) The volume of the Galaxy may be approximated as the volume of the corresponding cylinder, i.e. 2 VGalax y = π RGalax y h Galax y .

(100)

(4) Now consider the sphere around us having a radius r. The volume of such as sphere is

336

The Statistical Drake Equation

VOur _Spher e

  ET_ Distance 3 4 = π . 3 2

(101)

In the last equation, we had to divide the distance “ET_Distance” between ourselves and the nearest ET Civilization by 2, because we are now going to make the unwarranted assumption that all ET Civilizations are equally spaced from each other in the Galaxy! This is a crazy assumption, clearly, and should be replaced by more scientifically grounded assumptions as soon as we know more about our Galactic Neighbourhood. At the moment, however, this is the best guess that we can make, and so we shall take it for granted, although we are aware that this is weak point in the reasoning. Having thus assumed that ET Civilizations are UNIFORMLY SPACED IN THE GALAXY, we can write down this proportion VOur _Spher e VGalax y = . N 1

(102)

That is, upon replacing both Eq. (100) and Eq. (101) into Eq. (102) 2 π RGalax y h galax y

N

=

3  4 π ET_ Distance 3 2 1

.

(103)

The only unknown in the last equation is ET_Distance, and so we may solve for it, thus getting the (AVERAGE) DISTANCE BETWEEN ANY PAIR OF NEIGHBORING CIVILIZATIONS IN THE GALAXY 3

ET_ Distance =

2 6RGalax y h Galax y C =√ √ 3 3 N N

(104)

where the positive constant C is defined by C=

3

2 6RGalax y h Galax y ≈ 28, 845 light years.

(105)

Equations (104) and (105) are the starting point for our first application of the Statistical Drake equation, that we discuss in detail in the coming sections of this chapter.

7 Distance of the Nearest Extraterrestrial Civilization …

337

7.2 Probabilistic Derivation of the Probability Density Function for ET_Distance The probability density function (pdf) yielding the distance of the ET Civilization nearest to us in the Galaxy and presented in this section, was discovered by this author on September 5th, 2007. He did not disclose it to other scientists until the SETI meeting run by the famous mathematical physicist and popular science author, Paul Davies, at the “Beyond” Center of the University of Arizona at Phoenix, on February 5–8, 2008. This meeting was also attended by SETI Institute experts Jill Tarter, Seth Shostak, Doug Vakoch, Tom Pierson and others. During the author’s talk, Paul Davies suggested to call “the Maccone distribution” the new probability density function that yields the ET_Distance and is derived in this section. Let us go back to Eq. (104). Since N is now a random variable (obeying the lognormal distribution), it follows that the ET_Distance must be a random variable as well. Hence, it must have some unknown probability density function that we denote by f ET_Distance (r )

(106)

where r is the new independent variable of such a probability distribution (it is denoted by r to remind the reader that it expresses the three-dimensional radial distance separating us from the nearest ET civilization in a full spherical symmetry of the space around us). The question then is: what is the unknown probability distribution (106) of the ET_Distance? We can answer this question upon making the two formal substitutions

N→x ET_ distance → y

(107)

into the transformation law (8) for random variables. As a consequence, Eq. (104) takes form C = C x −1/3 . y = g(x) = √ 3 x

(108)

In order to find the unknown probability density f ET_Distance (r), we now apply the rule of Eq. (9) to Eq. (108). First, notice that Eq. (108), when inverted to yield the various roots x i (y), yields a single real root only x1 (y) =

C3 . y3

(109)

338

The Statistical Drake Equation

Then, the summation in Eq. (9) reduces to one term only. Second, differentiating Eq. (108) one finds g  (x) = −

C −4/3 x 3

(110)

Thus, the relevant absolute value reads    C  C |g  (x)| = − x −4/3  = x −4/3 . 3 3

(111)

Upon replacing Eq. (111) into Eq. (9), we then find

−4/3

 C −4/3 C C3 C C −4 y4 = = = . |g (x1 )| = x 3 3 y3 3 y 3C 3 

(112)

This is the denominator of Eq. (9). The numerator simply is the lognormal probability density function (56) where the old independent variable x must now be rewritten in terms of the new independent variable y by virtue of Eq. (109). By doing so, we finally arrive at the new probability density function f Y (y) ⎛ ⎡

f Y (y) =

3

3C 1 1 · C3 · √ · e− 4 y 2πσ y3

⎝ln⎣

⎤

⎞2

C3 ⎦ ⎠ −μ y3

.

2σ 2

Rearranging and replacing y by r, the final form is ⎛ ⎡

f ET_ Distance (r ) =

3 1 · e− √ r 2π σ

⎝ln⎣

⎤

⎞2

C3 ⎦ ⎠ −μ r3 2σ 2

.

(113)

Now, just replace C in Eq. (113) by virtue of Eq. (105). Then: We have discovered the probability density function yielding the probability of finding the nearest ExtraTerrestrial Civilization in the Galaxy in the spherical shell between the distances r and r + dr from Earth: ⎛ ⎡

f ET_ Distance (r ) = holding for r ≥ 0.

1 3 ·√ · e− r 2π σ

⎜ ⎢ ⎝ln⎣

⎤

⎞2

2 6RGalax y h Galax y ⎥ ⎟ ⎦−μ⎠ r3 2 σ2

.

(114)

7 Distance of the Nearest Extraterrestrial Civilization …

339

7.3 Statistical Properties of This Distribution We now want to study this probability distribution in detail. Our next questions are: (1) (2) (3) (4) (5) (6)

What is its mean value? What are its variance and standard deviation? What are its moments to any higher order? What are its cumulants? What are its skewness and kurtosis? What are the coordinates of its peak, i.e. the mode (peak abscissa) and its ordinate? (7) What is its median? The first three points in the list are all covered by the following theorem: all the moments of Eq. (113) are given by (here k is the generic and non-negative integer exponent, i.e. k = 0, 1, 2, 3, … ≥ 0) 



∞ r k f ET_ Distance (r )dr

ET_ Distance = k

0

∞ = 0

3 1 rk √ e− r 2πσ μ

= C k e−k 3 ek

2 σ2 · 18

  2 3 ln C3 −μ y 2σ 2

dr

.

(115)

To prove this result, one first transforms the above integral by virtue of the substitution

3 C (116) ln 3 = z. r Then, the new integral in z is then seen to reduce to the known Gaussian integral (53) and, after several reductions that we skip for the sake of brevity, Eq. (115) follows from Eq. (53). In other words, we have proven that 

 μ 2 σ2 ET_ Distancek = C k e−k 3 ek 18 .

(117)

Upon setting k = 0 into Eq. (117), the normalization condition for f ET_Distance (r) follows ∞ f ET_ Distance (r ) dr = 1. 0

(118)

340

The Statistical Drake Equation

Upon setting k = 1 into Eq. (117), the important mean value of the random variable ET_Distance is found. ET_ Distance = C e

−μ σ 2 3 e 18 .

(119)

Upon setting k = 2 into Eq. (117), the mean value of the square of the random variable ET_Distance is found 2 2 2  ET_ Distance2 = C 2 e− 3 μ e 9 σ .



(120)

The variance of ET_Distance now follows from the last two formulae with a few reductions:   2 2 2 σET_ Distance = ET_ Distance − ET_ Distance   2 σ2 σ2 = C 2 e− 3 μ e 9 e 9 − 1 .

(121)

So, the variance of ET_Distance is 2

2 2 −3μ σET_ e Distance = C e

σ2 9

(e

σ2 9

− 1).

(122)

The square root of this is the important standard deviation of the ET_Distance random variable μ

σ2

σET_ Distance = C e− 3 e 18

e

σ2 9

− 1.

(123)

The third moment is obtained upon setting k = 3 into Eq. (117) 

σ2  ET_ Distance3 = C e−μ e 2 .

(124)

Finally, upon setting k = 4 into Eq. (117), the fourth moment of ET_Distance is found 4 8 2   ET_ Distance4 = C 4 e− 3 μ e 9 σ .

(125)

Our next goal is to find the cumulants of the ET_Distance. In principle, we could compute all the cumulants K i from the generic i-th moment μi by virtue of the recursion formula (see Ref. [8]) Ki =

μi

−

i=1 k=1



 i −1 K k μn−k . k−1

(126)

7 Distance of the Nearest Extraterrestrial Civilization …

341

In practice, however, here we shall confine ourselves to the computation of the first four cumulants, because they only are required to find the skewness and kurtosis of the distribution (113). Then, the first four cumulants in terms of the first four moments read ⎧ K1 ⎪ ⎪ ⎨ K2 ⎪ K ⎪ ⎩ 3 K4

= μ1 = μ2 − K 12 = μ3 − 3K 1 K 2 − K 13 = μ4 − 4K 1 K 3 − 3K 22 − 6K 2 K 12 − K 14 .

(127)

These equations yield, respectively: K 1 = C e−μ/3 eσ K 2 = C 2 e−2μ/3 eσ K 3 = C 3 e−μ (eσ K 4 = C 4 e4μ/3 (e8σ

2

/9

− 4e5σ

2

2

/2

/9

2

/9

/18

(eσ

− 3e5σ

− 3e4σ

2

2

2

/9

2

.

/9

/18

(128) − 1).

+ 2eσ

+ 12eσ

2

2

(129) /6

/3

).

− 6e2σ

(130) 2

/9

).

(131)

From these, we derive the skewness 2σ 2

σ2

K3 e 9 +e 9 −2 = . 3/2 (K 2 ) σ2 3 9 C e −1

(132)

K4 4σ 2 σ2 2σ 2 = e 9 + 2e 3 + 3e 9 − 6. 2 (K 2 )

(133)

and the kurtosis

Next we want to find the mode of this distribution, i.e. the abscissa of its peak. To do so, we must first compute the derivative of the probability density function f ET_Distance (r) of Eq. (113), and then set it equal to zero. This derivative is actually the derivative of the ratio of two functions of r, as it plainly appears from Eq. (113). Thus, let us set for a moment

E(r ) =

2  - 3. ln Cr 3 − μ 2 σ2

.

where “E” stands for “exponent”. Upon differentiating, one gets

(134)

342

The Statistical Drake Equation

  3 1 1 C E (r ) = 2 ln 3 − μ C 3 C 3 (−3)r −4 2σ 2 r r3  3  C 1 1 = 2 ln 3 − μ (−3) . σ r r 

(135)

But the probability density function (113) now reads 3 e−E(r ) · f ET_ Distance (r ) = √ · r 2π σ

(136)

So that its derivative is d f ET_ Distance (r ) 3 −e−E(r ) E  (r )r − 1 e−E(r ) =√ dr r2 2π σ 3 −e−E(r ) [E  (r )r + 1] =√ . r2 2π σ

(137)

Setting this derivative equal to zero means setting E  (r )r + 1 = 0.

(138)

That is, upon replacing Eq. (135) into Eq. (138), we get  3  C 1 1 ln 3 − μ (−3) r + 1 = 0. 2 σ r r

(139)

Rearranging, this becomes   3 C −3 ln 3 − μ + σ 2 = 0 r

(140)

that is

−3 ln

 C3 + 3μ + σ 2 = 0 r3

(141)

whence ln and finally

 μ σ2 C = + r 3 9

(142)

7 Distance of the Nearest Extraterrestrial Civilization …

343

μ

rmode ≡ rpeak = C e− 3 e−

σ2 9

.

(143)

This is the most likely ET_Distance from Earth. How likely? To find the value of the probability density function f ET_Distance (r) corresponding to this value of the mode, we must obviously replace Eq. (143) into Eq. (113). After a few rearrangements, which we skip for the sake of brevity, one gets Peak Value of f ET_ Distance (r ) ≡ f ET_ Distance (rmode ) μ 3 σ2 = √ · e 3 · e 18 . C 2π σ

(144)

This is the peak height in the pdf f ET_Distance (r). Next to the mode, the median m (Ref. [9]) is one more statistical number used to characterize any probability distribution. It is defined as an independent variable abscissa m such that a realization of the random variable will take up a value lower than m with 50% probability or a value higher than m with 50% probability again. In other words, the median m splits up our probability density in exactly two equally probable parts. Since the probability of occurrence of the random event equals the area under its density curve (i.e. the definite integral under its density curve), then the median m (of the Maccone distribution, Eq. (113)) is defined as the integral upper limit m m f ET_ Distance (r )dr =

1 . 2

(145)

0

Upon replacing Eq. (113), this becomes m 0

 - 3 . 2 / (2σ 2 ) ln Cr 3 −μ

− 3 1 e √ r 2πσ

dr =

1 . 2

(146)

In order to find m, we may not differentiate Eq. (146) with respect to m, since the “precise” factor ½ on the right would then disappear into a zero. On the contrary, we may try to perform the obvious substitution

z2 =

2  - 3. ln Cr 3 − μ 2 σ2

z ≥ 0.

(147)

into the integral (146) to reduce it to the integral (85) defining the error function erf(z). Then, after a few reductions that we leave to the reader as an exercise, the full Eq. (145), defining the median, is turned into the corresponding equation involving the error function erf (x) as defined by Eq. (85)

344

The Statistical Drake Equation

⎛ - 3. ⎞ ln mC 3 − μ 1 ⎠= 1 + er f ⎝ √ 2 2 2σ

(148)

i.e. ⎛ - 3. ⎞ ln mC 3 − μ ⎠ = 0. er f ⎝ √ 2σ

(149)

Since from the definition Eq. (85) one obviously has erf (0) = 0, Eq. (149) yields ln

C3 m3

√

.

−μ

2σ

=0

(150)

whence finally median = m = C e−μ/3 .

(151)

This is the median of the Maccone distribution of ET_distance. In other words, this is the distance from the Sun such that, with 50% probability the actual value of ET_distance will be smaller than this median, and with 50% probability it will be higher. In conclusion, we feel useful to summarize all the equations that we derived about the random variable ET_distance in the following Table 3.

7.4 Numerical Example of the ET_Distance Distribution In this section, we provide a numerical example of the analytic calculations carried on so far. Consider the Drake Equation input values reported in Table 1. Then, the graph of the corresponding probability density function of the nearest ET_Distance, f ET_Distance (r), is shown in Fig. 6. From Fig. 6, we see that the probability of finding ExtraTerrestrials is practically zero up to a distance of about 500 light years from Earth. Then, it starts increasing with the increasing distance from Earth, and reaches its maximum at rmode ≡ r peak = Ce−μ/3 e−σ

2

/9

≈ 1933 light years.

(152)

This is the MOST LIKELY VALUE of the distance at which we can expect to find the nearest ExtraTerrestrial civilization.

7 Distance of the Nearest Extraterrestrial Civilization …

345

Table 3 Summary of the properties of the probability distribution that applies to the random variable ET_Distance yielding the (average) distance between any two neighboring communicating civilizations in the Galaxy Random variable

ET_Distance between any two neighboring ET civilizations in Galaxy assuming they are UNIFORMLY distributed throughout the whole Galaxy volume

Probability distribution

Unnamed (Paul Davies suggested “Maccone distribution”) ⎛ ⎡

Probability density function

⎝ln⎣

f ET_ Distance (r ) =

− 3 √1 r 2πσ e

⎤ ⎞2 2 6RGalax y h Galax y ⎦ ⎠ −μ r3 2σ 2

Defining the positive numeric constant C

C=

Mean value

ET_ Distance = C e−μ/3 eσ /18   2 σ2 σ 2 2 2 e− 3 μ e 9 e 9 − 1 σET_ = C Distance

Variance Standard deviation

3

2 6RGalax y h Galax y ≈ 28, 845 light years

2

μ

σ2

σET_ Distance = C e− 3 e 18

All the moments, i.e. k-th moment



Mode (=abscissa of the probability density function peak)

rmode ≡ r peak = C e− 3 e−

Value of the mode peak

 ET_ Distancek =

e

σ2 9

−1

μ 2 σ2 C k e−k 3 ek 18 μ

σ2 9

Peak value of f ET_ Distance (r )= ≡ f ET_ Distance (rmode )=

√3 C 2πσ

μ

σ2

· e 3 · e 18

Median (=fifty–fifty Median = m = C e−μ/3 probability value for ET_Distance) Skewness

Kurtosis

K3 (K 2 )3/2 K4 (K 2 )2

Expression of μ in μ = terms of the lower (ai ) and upper (bi ) limits of the Drake uniform input random variables Di

=

e

2σ 2 9

C3

= 7

4σ 2 e 9

σ2 9 −2 σ2 e 9 −1

+e

σ2

2σ 2

+ 2e 3 + 3e 9 − 6 7 bi [ln(bi )−1]−ai [ln(ai )−1] i=1 Yi  = i=1 bi −ai

(continued)

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The Statistical Drake Equation

Table 3 (continued) Random variable

Probability density function (1/meters)

Expression of σ2 in terms of the lower (ai ) and upper (bi ) limits of the Drake uniform input random variables Di

5.63 .10

20

4.5 .10

20

3.38 .10

20

2.25 .10

20

1.13 .10

20

0

ET_Distance between any two neighboring ET civilizations in Galaxy assuming they are UNIFORMLY distributed throughout the whole Galaxy volume  7 7  2 i )−ln(ai )] σ 2 = i=1 σY2i = i=1 1 − ai bi [ln(b (b −a )2 i

i

DISTANCE OF NEAREST ET_CIVILIZATION

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

ET_Distance from Earth (light years) Fig. 6 This is the probability of finding the nearest ExtraTerrestrial Civilization at the distance r from Earth (in light years) if the values assumed in the Drake Equation are those shown in Table 1. The relevant probability density function f ET_Distance (r) is given by Eq. (113). Its mode (peak abscissa) equals 1933 light years, but its mean value is higher since the curve has a long tail on the right: the mean value equals in fact 2670 light years. Finally, the standard deviation equals 1309 light years: THIS IS GOOD NEWS FOR SETI, inasmuch as the nearest ET Civilization might lie at just 1 sigma = 2670 − 1309 = 1361 light years from us

It is not, however, the mean value of the probability distribution (113) for f ET_Distance (r). In fact, the probability density Eq. (113) has an infinite tail on the right, as clearly shown in Fig. 6, and hence its mean value must be higher than its peak value. As given by Eq. (119), its mean value is rmean_value = Ce−μ/3 eσ

2

/18

≈ 2670 light years.

(153)

7 Distance of the Nearest Extraterrestrial Civilization …

347

This is the MEAN (value of the) DISTANCE at which we can expect to find ExtraTerrestrials. After having found the above two distances (1933 and 2670 light years, respectively), the next natural question that arises is: “what is the range, forth and back around the mean value of the distance, within which we can expect to find ExtraTerrestrials with “the highest hopes?”. The answer to this question is given by the notion of standard deviation, that we already found to be given by Eq. (123) σET_Distance = Ce

−μ σ 2 3 e 18

e

σ2 9

− 1 ≈ 1309 light years.

(154)

More precisely, this is the so-called 1-sigma (distance) level. Probability theory then shows that the nearest ExtraTerrestrial civilization is expected to be located within this range, i.e. within the two distances of (2670 − 1309) = 1361 light years and (2670 + 1309) = 3979 light years, with probability given by the integral of f ET_Distance (r) taken in between these two lower and upper limits, i.e. 3979 light years

f ET_ Distance (r )dr ≈ 0.75 = 75%.

(155)

1361 light years

In plain words: with 75% probability, the nearest ExtraTerrestrial civilization is located in between the distances of 1361 and 3979 light years from us, having assumed the input values to the Drake Equation given by Table 1. If we change those input values, then all the numbers change again.

8 The “DATA ENRICHMENT PRINCIPLE” as the Best CLT Consequence upon the Statistical Drake Equation (Any Number of Factors Allowed) As a fitting climax to all the statistical equations developed so far, let us now state our “DATA ENRICHMENT PRINCIPLE”. It simply states that “The Higher the Number of Factors in the Statistical Drake equation, The Better”. Put in this simple way, it simply looks like a new way of saying that the CLT lets the random variable Y approach the normal distribution when the number of terms in the sum (4) approaches infinity. And this is the case, indeed. However, our “Data Enrichment Principle” has more profound methodological consequences that we cannot explain now, but hope to describe more precisely in one or more coming papers.

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The Statistical Drake Equation

9 Conclusions We have sought to extend the classical Drake equation to let it encompass Statistics and Probability. This approach appears to pave the way to future, more profound investigations intended not only to associate “error bars” to each factor in the Drake equation, but especially to increase the number of factors themselves. In fact, this seems to be the only way to incorporate into the Drake equation more and more new scientific information as soon as it becomes available. In the long run, the Statistical Drake equation might just become a huge computer code, growing up in size and especially in the depth of the scientific information it contained. It would thus be the humanity’s first “Encyclopaedia Galactica”. Unfortunately, to extend the Drake equation to Statistics, it was necessary to use a mathematical apparatus that is more sophisticated than just the simple product of seven numbers. The first IAC presentation of the Statistical Drake Equation was made by the author on October 1st, 2008, at the 59th International Astronautical Congress held in Glasgow, Scotland, UK (Ref. [10]). When this author had the honour and privilege to present his results at the SETI Institute on April 11th, 2008, in front of an audience also including Professor Frank Drake, he felt he had to add these words: “My apologies, Frank, for disrupting the beautiful simplicity of your equation”. Acknowledgements The author is grateful to Drs. Jill Tarter, Paul Davies, Seth Shostak, Doug Vakoch, Tom Pierson, Carol Oliver, Paul Shuch and Kathryn Denning for attending his first presentation ever about these topics at the “Beyond” Center of the University of Arizona at Phoenix on February 8th, 2008. He also would like to thank Dr. Dan Werthimer and his School of SETI young experts for keeping alive the interplay between experimental and theoretical SETI. But the greatest “thanks” goes of course to the Teacher to all of us: Professor Frank Donald Drake, whose equation opened a new way of thinking about the past and the future of Humans in the Galaxy.

References 1. J. Bennett, S. Shostak, in Life in the Universe, 2nd edn (Pearson–Addison-Wesley, San Francisco, 2007). ISBN 0-8053-4753-4. See in particular page 404 2. C. Maccone, The statistical Drake equation. Paper presented on October 1, 2008 at the 59th International Astronautical Congress (IAC) held in Glasgow, Scotland, UK, Sept 29–Oct 3, 2008. Paper #IAC-08-A4.1.4 3. http://en.wikipedia.org/wiki/Drake_equation 4. http://en.wikipedia.org/wiki/SETI 5. http://en.wikipedia.org/wiki/Astrobiology 6. http://en.wikipedia.org/wiki/Frank_Drake 7. http://en.wikipedia.org/wiki/Gamma_distribution 8. http://en.wikipedia.org/wiki/Central_limit_theorem

References

349

9. http://en.wikipedia.org/wiki/Cumulants 10. http://en.wikipedia.org/wiki/Median 11. A. Papoulis, S. Unnikrishna Pillai, in Probability, Random Variables and Stochastic Processes, 4th edn (Tata McGraw-Hill, New Delhi, 2002). ISBN 0-07-048658-1

SETI and SEH (Statistical Equation for Habitables)

Abstract The statistics of habitable planets may be based on a set of ten (and possibly more) astrobiological requirements first pointed out by Stephen H. Dole in his book “Habitable planets for man” (1964). In this chapter, we first provide the statistical generalization of the original and by now too simplistic Dole equation. In other words, a product of ten positive numbers is now turned into the product of ten positive random variables. This we call the SEH, an acronym standing for “Statistical Equation for Habitables”. The mathematical structure of the SEH is then derived. The proof is based on the central limit theorem (CLT) of Statistics. In loose terms, the CLT states that the sum of any number of independent random variables, each of which may be arbitrarily distributed, approaches a Gaussian (i.e. normal) random variable. This is called the Lyapunov form of the CLT, or the Lindeberg form of the CLT, depending on the mathematical constraints assumed on the third moments of the various probability distributions. In conclusion, we show that (1) The new random variable N Hab , yielding the number of habitables (i.e. habitable planets) in the Galaxy, follows the lognormal distribution. By construction, the mean value of this lognormal distribution is the total number of habitable planets as given by the statistical Dole equation. But now we also derive the standard deviation, the mode, the median and all the moments of this new lognormal N Hab random variable. (2) The ten (or more) astrobiological factors are now positive random variables. The probability distribution of each random variable may be arbitrary. The CLT in the so-called Lyapunov or Lindeberg forms (that both do not assume the factors to be identically distributed) allows for that. In other words, the CLT “translates” into our SEH by allowing an arbitrary probability distribution for each factor. This is both astrobiologically realistic and useful for any further investigations. (3) An application of our SEH then follows. The (average) distance between any two nearby habitable planets in the Galaxy may be shown to be inversely proportional to the cubic root of N Hab . Then, in our approach, this distance becomes a new random variable. We derive the relevant probability density function, apparently previously unknown and dubbed “Maccone distribution” by Paul Davies in 2008.

© Springer Nature Switzerland AG 2020 C. Maccone, Evo-SETI, https://doi.org/10.1007/978-3-030-51931-5_9

351

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SETI and SEH (Statistical Equation for Habitables)

(4) Data Enrichment Principle. It should be noticed that ANY positive number of random variables in the SEH is compatible with the CLT. So, our generalization allows for many more factors to be added in the future as long as more refined scientific knowledge about each factor will be known to the scientists. This capability to make room for more future factors in the SEH we call the “Data Enrichment Principle”, and we regard it as the key to more profound future results in the fields of Astrobiology and SETI. (5) A practical example is then given of how our SEH works numerically. We work out in detail the case where each of the ten random variables is uniformly distributed around its own mean value as given by Dole back in 1964 and has an assumed standard deviation of 10%. The conclusion is that the average number of habitable planets in the Galaxy should be around 100 million ± 200 million, and the average distance in between any couple of nearby habitable planets should be about 88 light years ± 40 light years. (6) Finally, we match our SEH results against the results of the Statistical Drake Equation that we introduced in our 2008 IAC presentation. As expected, the number of currently communicating ET civilizations in the Galaxy turns out to be much smaller than the number of habitable planets (about 10,000 against 100 million, i.e. one ET civilization out of 10,000 habitable planets). And the average distance between any two nearby habitable planets turns out to be much smaller than the average distance between any two neighboring ET civilizations: 88 light years versus 2000 light years, respectively. This means an ET average distance about 20 times higher than the average distance between any couple of adjacent habitable planets. Keywords Habitable planets · Statistics · Dole equation · Drake equation

1 Introduction to SETI SETI is an acronym for “Search for Extra Terrestrial Intelligence”. SETI is a comparatively new branch of scientific research that began only in 1959. The goal of SETI is to ascertain whether Alien Civilizations exist in the universe, how far from us they exist, and possibly how much more advanced than us they may be. As of 2021, the only physical tools we know that could help us get in touch with Aliens are the electromagnetic waves that an Alien Civilization could emit and we could detect. This forces us to use the largest radiotelescopes on Earth for SETI research because the higher our collecting area of electromagnetic radiation, the higher our sensitivity, i.e. the further in space we can probe. Yet, even by using the largest radiotelescopes we have on Earth, we cannot search for Aliens beyond, say, a few hundred light years away. This is a very, very small amount of space around us within our Galaxy, the Milky Way, that is about a hundred thousand light years in diameter. Thus, current SETI can cover only a very tiny fraction of the Galaxy,

1 Introduction to SETI

353

and it is not surprising that in the past 50 years of SETI searches NO extraterrestrial civilization was discovered. Quite simply, we did not get far enough! This demands the construction of much more powerful and radically new radiotelescopes. Rather than big and heavy metal dishes, whose mechanical problems hamper SETI research too much, we are now turning to “software radiotelescopes”. These are made up by a large number of small dishes (ATA = Allen telescope array, and ALMA = Atacama large millimeter/submillimeter array) or even just of simple dipoles (LOFAR = low frequency array) using state-of-the-art electronics and very high-speed computing that can outperform the classical radiotelescopes in many regards. The final dream in this field is the SKA (=square kilometer array), currently being designed.

2 The Key Question: How Far Are They? But still, the key question remains: how far are they? Or, more correctly, how far do we expect the NEAREST extraterrestrial civilization to be from the Solar System in the Galaxy? This question was first faced in a scientific manner back in 1961 by the same scientist who also was the first experimental SETI radio astronomer ever: the American, Frank Donald Drake (born 1930). He first considered the shape and size of the Galaxy where we are living: the Milky Way. This is a spiral galaxy measuring some 100,000 light years in diameter and some 1600 light years in thickness of the Galactic Disk at half-way from its center. That is: (1) The diameter of the Galaxy is (about) 100,000 light years (abbreviated ly), i.e., its radius, RGalaxy , is about 50,000 ly. (2) The thickness of the Galactic Disk at half-way from its center, hGalaxy , is about 1600 ly. Then (3) The volume of the Galaxy may then be approximated as the volume of the corresponding cylinder, i.e. 2 VGalax y = π RGalax y h Galax y .

(1)

(4) Now consider the sphere around us having as radius the half distance in between us and the nearest ET Civilization. Its volume is given by VOur _Spher e =

  ET_ Distance 3 4 π 3 2

(2)

In the last equation, we had to divide the distance “ET_Distance” between ourselves and the nearest ET civilization by 2 because we are now going to make the unwarranted assumption that all ET Civilizations are equally spaced from each other in the Galaxy! This is a crazy assumption, clearly, and should be replaced

354

SETI and SEH (Statistical Equation for Habitables)

by more scientifically grounded assumptions as soon as we know more about our Galactic neighborhood. At the moment, however, this is the best guess that we can make, and so we shall take it for granted, although we are aware that this is a weak point in the reasoning. Furthermore, let us denote by N the total number of civilizations now living in the Galaxy, including ourselves. Of course, this number N is unknown. We only know that N ≥ 1 since one civilization does at least exist! Having thus assumed that ET civilizations are uniformly spaced in the Galaxy, we can then write down the proportion: VOur _Spher e VGalax y = . N 1

(3)

That is, upon replacing both (1) and (2) into (3): 2 π RGalax y h Galax y

N

=

4 π 3

  3 ET_ Distance 2 . 1

(4)

The last equation contains two unknowns: N and ET_Distance, and so we do not know which one it is better to solve for. However, we may suppose that, by resorting to the (rather uncertain) knowledge that we have about the Evolution of the Galaxy through the last 10 billion years or so, we might somehow compute an approximate value for N. Then, we may solve (4) for ET_Distance thus obtaining the (average) distance between any pair of neighboring civilizations in the Galaxy (Distance Law)  3

ET_ Distance(N ) =

2 6 RGalax y h Galax y C =√ √ 3 3 N N,

(5)

where the positive constant C is defined by C=

 3 2 6 RGalax y h Galax y ≈ 28, 845 light years.

(6)

Equations (5) and (6) are the starting point to understand the origin of the Drake equation that we discuss in detail in Sect. 3 of this paper (see also Ref. [2]). Let us just complete this section by pointing out three different numerical cases of the distance law (5): (1) We know that we exist, so N may not be smaller than 1, i.e., N ≥ 1. Suppose then that we are alone in the Galaxy, i.e., that N = 1. Then the distance law (5) yields as distance to the nearest civilization from us just the constant C, i.e., 28,845 light years. This is about the distance in between ourselves and the center of the Galaxy (i.e. the Galactic Bulge). Thus, this result seems to suggest that, if we do not find any extraterrestrial civilization around us in these outskirts of

2 The Key Question: How Far Are They?

355

the Galaxy where we live, we should look around the Galactic Center first. And this is indeed what is happening, i.e., many SETI searches are actually pointing the antennas towards the Galactic Center, looking for beacons (see, for instance Ref. [1]). (2) Suppose next that N = 1000, i.e. there are about a thousand extraterrestrial communicating civilizations in the whole Galaxy right now. Then the distance law (5) yields an average distance of 2885 light years. This is a distance that most radiotelescopes in Earth may not reach for SETI searches right now: hence the need to build larger radiotelescopes, like ALMA, FAST and the SKA. (3) Suppose finally that N = 1,000,000, i.e., there are a million communicating civilizations now in the Galaxy. Then the distance law (5) yields an average distance of 288 light years. This is within the (upper) range of distances that our current radiotelescopes may reach for SETI searches, and that justifies all SETI searches that have been done so far in the first sixty years of SETI (1960–2010). In conclusion, interpolating the above three special cases of N, we may say that the distance law (5) yields the following key diagram of the average ET distance versus the assumed number of communicating civilizations, N, in the Galaxy right now (Fig. 1). This we call the Distance Law, i.e., the average distance (plot along the vertical axis in light years) versus the number of communicating civilizations assumed to exist in the Galaxy right now. Average DISTANCE of the nearest ET civilization vs. the ASSUMED NUMBER of ET civilizations in the Galaxy Average DISTANCE of the civilization nearest to us in LIGHT YEARS

2000

1750

1500

1250

1000

750

500

250

0 0

100000 200000 300000 400000 500000 600000 700000 800000 900000 ASSUMED NUMBER of civilizations in the Galaxy (that is, N in the Drake equation)

1000000

Fig. 1 Distance Law, i.e., the average distance (plot along the vertical axis in light years) versus the number of communicating civilizations assumed to exist in the Galaxy right now

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SETI and SEH (Statistical Equation for Habitables)

3 Computing N by Virtue of the Drake Equation (1961) In the previous section, the problem of finding how close the nearest ET civilization may be was “solved” by reducing it to the computation of N, the total number of extraterrestrial civilizations now existing in this Galaxy. In this section the famous Drake equation is described, which was proposed back in 1961 by Frank Donald Drake to estimate the numerical value of N. N can be written as the product or multiplication of a number of factors, each a kind of filter, every one of which must be sizable for there to be a large number of civilizations: Ns fp ne fl fi fc fL

The number of stars in the Milky Way Galaxy The fraction of stars that have planetary systems The number of planets in a given system that are ecologically suitable for life The fraction of otherwise suitable planets on which life actually arises The fraction of inhabited planets on which an intelligent form of life evolves The fraction of planets inhabited by intelligent beings on which a communicative technical civilization develops; and The fraction of planetary lifetime graced by a technical civilization. Written out, the equation reads N = N s · f p · ne · f l · f i · f c · f L

(7)

All of the f s are fractions, having values between 0 and 1; they will pare down the large value of Ns. To derive N we must estimate each of these quantities (see also Ref. [6]).

4 The Drake Equation Is Over-Simplified In the sixty years elapsed since Frank Drake proposed his equation, a number of scientists and writers tried to find out which numerical values of its seven independent variables are more realistic in agreement with our present-day knowledge. Thus there is a considerable amount of literature about the Drake equation nowadays, and, as one can easily imagine, the results obtained by the various authors largely differ from one another. In other words, the value of N, that various authors obtained by different assumptions about the astronomy, the biology and the sociology implied by the Drake equation, may range from a few tens (in the pessimist’s view) to some million or even billion in the optimist’s opinion. A lot of uncertainty is thus affecting our knowledge of N as of 2021. In all cases, however, the final result about N has always been a sheer number, i.e., a positive integer number ranging from 1 to million or billion. This is precisely the aspect of the Drake equation that this author regarded

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357

as “too simplistic” and improved mathematically in his paper #IAC-08-A4.1.4, entitled “The Statistical Drake Equation” and presented on October 1st, 2008, at the 59th International Astronautical Congress (IAC) held in Glasgow, Scotland, UK, September 29th–October 3rd, 2008, that is Refs. [4, 5].

5 The Statistical Drake Equation by Maccone (2008) We start by an example. Consider the first independent variable in the Drake equation (7), i.e., Ns, the number of stars in the Milky Way Galaxy. Astronomers tell us that approximately there should be about 350 billion stars in the Galaxy. Of course, nobody has counted (or even seen in the photographic plates) all the stars in the Galaxy! There are too many practical difficulties preventing us from doing so: just to name one, the dust clouds that do not allow us to see even the Galactic Bulge (i.e. the central region of the Galaxy) in the visible light (although we may “see it” at radio frequencies like the famous neutral hydrogen line at 1420 MHz). So, it does not make any sense to say that Ns = 350 × 109 , or, say (even worse) that the number of stars in the Galaxy is (say) 354, 233, 321, 123, or similar fanciful exact integer numbers. That is just silly and non-scientific. Much more scientific, on the contrary, is to say that the number of stars in the Galaxy is 350 billion plus or minus, say, 50 billion (or whatever values the astronomers may regard as more appropriate, since this is just an example to let the reader understand the difficulty). Thus, it makes sense to replace each of the seven independent variables in the Drake equation (7) by a mean value (350 billion, in the above example) plus or minus a certain standard deviation (50 billion, in the above example). By doing so, we have made a great step ahead: we have abandoned the toosimplistic equation (7) and replaced it by something more sophisticated and scientifically more serious: the statistical Drake equation. In other words, we have transformed the classical and simplistic Drake equation (7) into an advanced statistical tool for the investigation of a host of facts hardly known to us in detail. In other words still: (1) We replace each independent variable in (7) by a positive random variable, labelled Di (from Drake). (2) We assume that the mean value of each Di is the same numerical value previously attributed to the corresponding algebraic independent variable in (7). (3) But now we also add a standard deviation σ Di on each side of the mean value, that is provided by the knowledge gathered by scientists in each discipline encompassed by each Di . Having so done, the next question is: How can we find out the probability distribution for each Di ? For instance, shall that be a Gaussian, or what?

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SETI and SEH (Statistical Equation for Habitables)

This is a difficult question, for nobody knows, for instance, the probability distribution of the number of stars in the Galaxy, not to mention the probability distribution of the other six variables in the Drake equation (7). There is a brilliant way to get around this difficulty, though. We start by excluding the Gaussian because each variable in the Drake equation is a positive (or, more precisely, a non-negative) random variable, while the Gaussian applies to real random variables only. So, the Gaussian is out. Then, one might consider the large class of well-studied and positive probability densities called “the gamma distributions,” but it is then unclear why one should adopt the gamma distributions and not any other one. The solution to this apparent conundrum comes from Shannon’s Information Theory and a theorem that he proved in 1948: “The probability distribution having maximum entropy (=uncertainty) over any finite range of real values is the uniform distribution over that range”. So, at this point, we assume that each of the seven Di in (7) is a uniform random variable, whose mean value and standard deviation is known by the scientists working in the respective field (let it be astronomy, or biology, or sociology). Notice that, for such a uniform distribution, the knowledge of the mean value μDi and of the standard deviation σ Di automatically determines the RANGE of that random variable in between its lower (called ai ) and upper (called bi ) limits: in fact these limits are given by the equations: 

√ ai = μ Di − √3σ Di bi = μ Di + 3σ Di

(8)

(the “surprising” factor 3 in the above equations comes from the definitions of mean value and standard deviation: please see Eqs. (12), (15) and (17) in Refs. [4, 5] for the relevant proof). So the uniform distribution of each random variable Di is perfectly determined by its mean value and standard deviation, and so are all its other properties. The next problem is the following: OK, since we now know everything about each uniformly distributed Di , what is the probability distribution of N, given that N is the product (7) of all the Di ? In other words, not only do we want to find the analytical expression of the probability density function of N, but we also want to relate its mean value μN to all mean values μ Di of the Di , and its standard deviation σ N to all standard deviations σ Di of the Di . This is a difficult problem. It occupied the author’s mind for no less than about ten years (1997–2007). It is actually an analytically unsolvable problem, in that, to the best of this author’s knowledge, it is impossible to find an analytic expression for any finite product of uniform random variables Di . This result is proven in Sects. 2 and 3.3 of Refs. [4, 5] (unfortunately!).

6 Solving the Statistical Drake Equation by Virtue …

359

6 Solving the Statistical Drake Equation by Virtue of the Central Limit Theorem (CLT) of Statistics The solution to the problem of finding the analytical expression for the probability density function of N in the statistical Drake equation was found by this author only in September 2007. The key steps are the following: (1) Take the natural logs of both sides of the statistical Drake equation (7). This changes the product into a sum. (2) The mean values and standard deviations of the logs of the random variables Di may all be expressed analytically in terms of the mean values and standard deviations of the Di . (3) Recall the Central Limit Theorem (CLT) of Statistics, stating that (loosely speaking) if you have a sum of independent random variables, each of which is arbitrarily distributed (hence, also including uniformly distributed), then, when the number of terms in the sum increases indefinitely (i.e. for a sum of random variables infinitely long)y the sum random variable tends to a Gaussian. (4) Thus, the natural log of N tends to a Gaussian. (5) Thus, N tends to the lognormal distribution. (6) The mean value and standard deviations of this lognormal distribution of N may all be expressed analytically in terms of the mean values and standard deviations of the logs of the Di already found previously. This result is fundamental. All the relevant equations are summarized in Table 1. This table is actually the same as Table 2 of the author’s original paper (Ref. [5]) IAC-08-A4.1.4, entitled “The Statistical Drake Equation” and presented by him at the International Astronautical Congress (IAC) held in Glasgow, UK, on October 1st, 2008. This original paper is reproduced just the same in Ref. [4]. To sum up, not only we have found that N approaches the completely known lognormal distribution for an infinity of factors in the statistical Drake equation (7), but we also paved the way to further applications by removing the condition that the number of terms in the product (7) must be finite. This possibility of adding any number of factors in the Drake equation (7) was not envisaged, of course, by Frank Drake back in 1961, when “summarizing” the evolution of life in the Galaxy in seven simple steps. But today, the number of factors in the Drake equation should already be increased: for instance, there is no mention in the original Drake equation of the possibility that asteroidal impacts might destroy the life on Earth at any time, and this is because the demise of the dinosaurs at the K/T impact had not been yet understood by scientists in 1961, and was so only in 1980!

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Table 1 Summary of the properties of the lognormal distribution that applies to the random variable N = number of ET communicating civilizations in the Galaxy Random variable

N = number of communicating ET civilizations in Galaxy

Probability distribution

Lognormal

Probability density function

f N (n) =

Mean value

  1 √1 −(ln(n)−μ)2 / 2σ 2 n 2πσ e

N  = eμ e



All the moments, i.e. kth moment

2 eσ − 1  2 2 σ N = eμ eσ /2 eσ − 1 k 2 2 N = ekμ ek σ /2

Mode (=abscissa of the lognormal peak)

n mode ≡ n peak = eμ e−σ

Value of the Mode Peak

f N (n mode ) =

Median (=fifty–fifty probability value for N)

median =m 2

 2 K3 σ +2 eσ − 1 3 = e

Variance Standard deviation

Skewness

(n ≥ 0)

σ 2 /2

σ N2 = e2μ eσ

2

2

2 √ 1 e−μ eσ /2 2πσ = eμ

(K 2 ) 2

7

Expression of μ in terms of the lower (ai ) and upper (bi ) limits of the Drake uniform input random variables Di

μ= 7

Expression of σ 2 in terms of the lower (ai ) and upper (bi ) limits of the Drake uniform input random variables Di

2 σ 7=

i=1

i=1 Yi  = bi [ln(bi )−1]−ai [ln(ai )−1] bi −ai

2 i=1 σYi

=

7 i=1

1−

ai bi [ln(bi )−ln(ai )]2 (bi −ai )2



In practice, we are suggesting increasing the number of factors as much as necessary in order to get better and better estimates of N as long as our scientific knowledge increases. This we call the “Data Enrichment Principle” and believe should be the next important goal in the study of the statistical Drake equation. Finally, we wish to provide a numerical example explaining how the statistical Drake equation works in the practice. This will be done in the next section.

7 An Example Explaining the Statistical Drake Equation To understand how things work in practice for the statistical Drake equation, please consider Input Table 1. It is made up of three columns: (1) The first column on the left lists the seven input sheer numbers that also become… (2) The mean values (middle column). (3) Finally the last column on the right lists the seven input standard deviations.

7 An Example Explaining the Statistical Drake Equation

361

Table 2 Summary of the properties of the probability distribution that applies to the random variable ET_Distance yielding the (average) distance between any two neighboring communicating civilizations in the Galaxy Random variable

ET_Distance between any two neighboring ET civilizations in Galaxy assuming they are UNIFORMLY distributed throughout the whole Galaxy volume

Probability distribution

Unnamed (Paul Davies suggested “Maccone distribution”)

Probability density function

f ET_ distance (r ) =  

− ln

3 √1 r 2πσ

2 6 RGalax y h Galax y



2 

−μ

r3



2 σ2



e  Numerical constant C related to the Milky Way C = 3 6R 2 Galax y h Galax y ≈ 28, 845 light years size ET_ Distance = Ce−μ/3 eσ

2 /18

All the moments, i.e. kth moment



2 2 − 23 μ eσ 2 /9 eσ 2 /9 − 1 σET_ Distance = C e  2 2 σET_ Distance = Ce−μ/3 eσ /18 eσ /9 − 1  

2 2 ET_ Distancek = C k e−k(μ/3) ek σ /18

Mode (=abscissa of the lognormal peak)

rmode ≡ r peak = Ce−μ/3 e−σ

2 /9

Value of the Mode Peak

Peak Value Of f ET_ Distance (r ) =≡ 2 f ET_ Distance (rmode ) = √3 eμ/3 eσ /18

Median (=fifty–fifty probability value for N)

median =m = Ce−μ/3

Mean value Variance Standard deviation

C 2πσ

Skewness

K3

3 (K 2 ) 2

Kurtosis Expression of μ in terms of the lower (ai ) and upper (bi ) limits of the Drake uniform input random variables Di

K4 (K 2 )2

μ=

=

e

2σ 2 9

C3

= e4σ 7

σ2 9 −2 σ2 e 9 −1

+e

2 /9

i=1 Yi 

+ 2eσ =

2 /3

7 i=1

Expression of σ 2 in terms of the lower (ai ) and σ 2 = 7 7 2 upper (bi ) limits of the Drake uniform input i=1 σYi = i=1 1 − random variables Di

+ 3e2σ

2 /9

−6

bi [ln(bi )−1]−ai [ln(ai )−1] bi −ai

ai bi [ln(bi )−ln(ai )]2 (bi −ai )2



The bottom line is the classical Drake equation (7). We see that, for this particular set of seven inputs, the classical Drake equation (i.e. the product of the seven numbers) yields a total of 3500 communicating extraterrestrial civilizations existing in the Galaxy right now. The statistical Drake equation, however, provides a much more articulated answer than just the above sheer number N = 3500. In fact, a MathCad code written by this author and capable of performing all the numerical calculations required by the statistical Drake equation for a given set of seven input mean values plus seven input

362

SETI and SEH (Statistical Equation for Habitables)

Input Table 2 Input values (i.e. mean values and standard deviations) for the seven Drake uniform random variables Di N s : 350 × 109

μN s = N s

σ N s = 1 × 109

fp =

μfp = f p

σfp =

50 100

ne = 1

μne = ne

fl =

50 100 20 f i = 100 20 f c = 100 f L = 10000 1010

μfl = f l

N : N s · f p · ne · f l · f i · f c · f L

N = 3500

μfi = f i μfc = f c μ f L = Fl

10 100 σne = √1 3 10 σ f l = 100 10 σ f i = 100 10 σ f c = 100 10 σ f L = 100

The first column on the left lists the seven input sheer numbers that also become the mean values (middle column). Finally the last column on the right lists the seven input standard deviations. The bottom line is the classical Drake equation (7)

standard deviations, yields for N the lognormal distribution (thin curve) plotted in Fig. 2. We see immediately that the peak of this thin curve (i.e. the mode) falls at about 2 n mode ≡ n peak = eμ e−σ × 250 (this is Eq. (99) of Refs. [4, 5]), while the median (fifty–fifty value splitting the lognormal density in two parts with equal undergoing areas) falls at about nmedian × eμ ≈ 1740. These seem to be smaller values than N = 3500 provided by the classical Drake equations, but it is a wrong impression due to a poor “intuitive” understanding of what statistics is! In fact, neither the mode nor the PROBABILITY DENSITY FUNCTION OF N

Prob. density function of N

6.10-4 5.10-4 4.10-4 3.10-4 2.10-4 1.10-4 0 0

1000 2000 3000 N = Number of ET Civilizations in Galaxy

4000

Fig. 2 Comparing the two probability density functions of the random variable N found: (1) at the end of Sect. 3.3 in a purely numerical way and without resorting to the CLT at all (thick curve), and (2) analytically using the CLT and the relevant lognormal approximation (thin curve)

7 An Example Explaining the Statistical Drake Equation

363

median are the “really important” values: the really important value for N is the mean value! Now if you look at the thin curve in Fig. 2 (i.e. the lognormal distribution arising from the Central Limit Theorem), you see that this curve has a long tail on the right! In other words, it does not immediately go down to nearly zero beyond the peak of the mode. Thus, when you actually compute the mean value, you should 2 not be too surprised to find out that it equals N  = eμ eσ ≈ 4589.559 ∼ 4590 communicating civilizations now in the Galaxy. This is the important number, and it is higher than the 3500 provided by the classical Drake equation. Thus, in conclusion, the statistical extension of the classical Drake equation that we made increases our hopes to find an extraterrestrial civilization! Even more so our hopes are increased when we go on to consider the standard deviation associated with the mean value 4590. In fact, thestandard deviation is given 2 by equation (97) of Refs. [4, 5]. This yields σ N = eμ eσ /2 eσ 2 − 1 = 11, 195 and so the expected number of N may actually be even much higher than the 4590 provided by the mean value alone! The “upper limit of the 1-sigma confidence interval” (as statisticians call it), i.e. the sum 4590 + 11,195 = 15,785, yields a higher number still! Finally, the reader should not worry about the thick curve depicted in Fig. 2: it is just the numerical solution of the statistical Drake equation for a finite number of 7 input factors. Figure 2 actually shows that this curve “is well interpolated” by the lognormal distribution (thin curve), i.e., by the neat analytical expression provided by the Central Limit Theorem for an infinite number of factors in the Drake equation. That is, in conclusion, Fig. 2 visually shows that taking 7 factors or an infinity of factors “is almost the same thing” already for a value as small as 7.

8 Finding the Probability Distribution of the ET-Distance by Virtue of the Statistical Drake Equation Having solved the statistical Drake equation by finding the lognormal distribution, we are now in a position to solve the ET-distance problem by resorting to statistics again, rather than just to the purely deterministic Distance Law (5), as we did in Sect. 2. This is “scientifically more serious” than just the purely deterministic Distance Law (5) inasmuch as the new statistical Distance Law will yield a probability density for the Distance, with the relevant mean value and standard deviation, of course. In other words, the Distance Law (5), now itself becomes a random variable whose probability distribution, mean value and standard deviation must be computed by “replacing” (so to say) into (5) the fact that N is now known to follow the lognormal distribution. This was done by this author in September 2007 also, and is mathematically described in detail in Sect. 7 of Refs. [4, 5].

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SETI and SEH (Statistical Equation for Habitables)

So, the important new result is the probability density for the distance, that the well-known physicist Paul Davies dubbed “the Maccone distribution” and whose equation reads ⎛ ⎡

f ET_ Distance (r ) =

⎝ln⎣

3 1 e− √ r 2πσ

⎤ ⎞2 2 6 RGalax y h Galax y ⎦ ⎠ −μ 3 r 2 σ2

(9)

and holds for r ≥ 0. This is Eq. (114) of Refs. [4, 5]. Starting from this equation, the author computed the mean value of the random variable ET_Distance ET_ Distance = C e−μ/3 eσ

2

/18

(10)

which is Eq. (119) of Refs. [4, 5], and finally the ET_Distance standard deviation σET_ Distance = C e−μ/3 eσ

2

/18



eσ 2 /9 − 1

(11)

which is Eq. (123) of Refs. [4, 5]. Of course, all other descriptive statistical quantities, such as moments, cumulants, etc. can be computed upon starting from the probability density (9), and the result is Table 2 hereafter, that is Table 3 of Refs. [4, 5]. Finally, we wish to complete this section, as well as this “easy introduction to the statistical Drake equation,” by pointing out the numerical values that Eqs. (10) and (11) yield for the Input Table 1. They are, respectively: rmean_value = C e−μ/3 eσ

2

/18

≈ 2670 light years

(12)

 eσ 2 /9 − 1 ≈ 1309 light years

(13)

which is Eq. (153) of Ref. [5], and σET_ Distance = C e−μ/3 eσ

2

/18

which is Eq. (154) of Refs. [4, 5]. It is actually clarifying to draw the graph of the ET_Distance probability density (9), that is Fig. 3. From Fig. 3 we see that the probability of finding extraterrestrials is practically zero up to a distance of about 500 light years from Earth. Then it starts increasing with the increasing distance from Earth, and reaches its maximum at rmode ≡ r peak = C e−μ/3 e−σ

2

/9

≈ 1933 light years.

(14)

This is the most likely value of the distance at which we can expect to find the nearest extraterrestrial civilization. It is not, as we said, the mean value of the probability distribution (9) for f ET_Distance (r). In fact, the probability density (9) has an infinite tail on the right, as clearly shown

8 Finding the Probability Distribution of the ET-Distance … DISTANCE OF NEAREST ET_CIVILIZATION

5.63·10-20

Probability density function (1/meters)

365

4.5·10-20

3.38·10-20

2.25·10-20

1.13·10-20

0 0

500

1000

1500 2000 2500 3000 3500 4000 ET_Distance from Earth (light years)

4500

5000

Fig. 3 This is the probability of finding the nearest extraterrestrial civilization at the distance r from Earth (in light years) if the values assumed in the Drake Equation are those shown in Input Table 1. The relevant probability density function f ET_Distance (r) is given by Eq. (9). Its mode (peak abscissa) equals 1933 light years, but its mean value is higher since the curve has a long tail on the right: the mean value equals in fact 2670 light years. Finally, the standard deviation equals 1309 light years: this is good news for SETI, inasmuch as the nearest ET civilization might lie at just 1 sigma = 2670 − 1309 = 1361 light years from us

in Fig. 3, and hence its mean value must be higher than its peak value. As given by 2 (10) and (12), its mean value is r mean_value = Ce−μ/3 eσ /18 ≈ 2670 light years. This is the mean (value of the) distance at which we can expect to find extraterrestrials. After having found the above two distances (1933 and 2670 light years, respectively), the next natural question that arises is: “what is the range, back and forth around the mean value of the distance, within which we can expect to find extraterrestrials with “the highest hopes?”. The answer to this question is given by the notion of standard deviation that we already found to be given by (11) and (13), σET_ Distance = C e−μ/3 eσ

2

/18



eσ 2 /9 − 1 ≈ 1309 light years.

More precisely, this is the so-called 1-sigma (distance) level. Probability theory then shows that the nearest extraterrestrial civilization is expected to be located within this range, i.e. within the two distances of (2670 − 1309) = 1361 light years and (2670 + 1309) = 3979 light years, with probability given by the integral of f ET_Distance (r) taken in between these two lower and upper limits, that is:

366

SETI and SEH (Statistical Equation for Habitables) 3979  light years

f ET_ Distance (r )dr ≈ 0.75 = 75%.

(15)

136 light years

In plain words: with 75% probability, the nearest extraterrestrial civilization is located in between the distances of 1361 and 3979 light years from us, having assumed the input values to the Drake Equation given by Input Table 1. If we change those input values, then all the numbers change again, of course.

9 The “Data Enrichment Principle” as the Best CLT Consequence upon the Statistical Drake Equation (Any Number of Factors Allowed) As a fitting climax to all the statistical equations developed so far, let us now state our “Data Enrichment Principle.” It simply states that “The higher the number of factors in the statistical Drake equation, the better.” Put in this simple way, it simply looks like a new way of saying that the CLT lets the random variable Y = ln(N) approach the normal distribution when the number of terms in the product (7) approaches infinity. And this is the case, indeed. However, our “Data Enrichment Principle” has more profound methodological consequences that we cannot explain now, but hope to describe more precisely in one or more coming papers.

10 Habitable Planets for Man Let us now change topics completely! Rather than seeking for ETs in the Galaxy, we now seek for habitable planets for man in the Galaxy. How many are there? And how far from us is the nearest such a habitable planets? These topics seem to have been faced “seriously” for the first time in 1964 by Stephen H. Dole, then with the Rand Corporation. Back in 1964, only three years had elapsed since Frank Drake had made known his now famous Drake equation. Dole learned the lesson of the Drake equation perfectly, and in his now famous book entitled “Habitable Planets for Man” (Ref. [3]) Dole used the same mathematical structure as the Drake equation (7) in order to find the number of habitable planets for man in the Galaxy. In other words, on page 82 of his book, he wrote the same mathematical thing as the Drake equation, but he applied it to habitable planets. Nowadays Dole’s 1964 book can be freely downloaded from the Rand Corporation web site. The equation on page 82 we shall call “the classical Dole equation”.

10 Habitable Planets for Man

367

As we can see from Dole’s book, the classical Dole equation is made up by TEN factors (instead of SEVEN factors as in the Drake equation): N H ab = N s · P p · Pi · P D · P M · Pe · P B · P R · P A · P L .

(16)

Here N Hab is the total number of habitable planets for man in the Galaxy, and it is given by the product of the following TEN input numbers: (1)

Ns is the number of stars in the suitable mass range 0.35–1.43 solar masses (this is Dole’s assumption about to the mass of “habitable stars”). (2) Pp is the probability that a given star has planets in orbit around it. (3) Pi is the probability that the inclination of the planet’s equator is correct for its orbital distance. (4) PD is the probability that at least one planet orbits within an ecosphere. (5) PM is the probability that the planet has a suitable mass, 0.4–2.35 Earth masses (again, this is Dole’s assumption in this regard). (6) Pe is the probability that the planet’s orbital eccentricity is sufficiently low. (7) PB is the probability that the presence of a second star has not rendered the planet uninhabitable. (8) PR is the probability that the planet’s rate of rotation is neither too fast nor too slow. (9) PA is the probability that the planet is of the proper age. (10) PL is the probability that, all astronomical conditions being proper, life has developed on the planet.

11 The Statistical Dole Equation It is now natural to rename the above ten input variables of the classical Dole equation (16) as follows: ⎧ D1 = N s ⎪ ⎪ ⎪ ⎪ D2 = P p ⎪ ⎪ ⎪ ⎪ ⎪ D3 = Pi ⎪ ⎪ ⎪ D = PD ⎪ 4 ⎪ ⎪ ⎨ D5 = P M ⎪ D6 = Pe ⎪ ⎪ ⎪ ⎪ D7 = P B ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ D8 = P R ⎪ ⎪ ⎪ D = PA ⎪ ⎩ 9 D10 = P L so that our classical Dole equation may be simply rewritten as

(17)

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SETI and SEH (Statistical Equation for Habitables)

N H ab =

10 !

Di .

(18)

i=1

We now let (18) undergo exactly the same changes that we applied to the classical Drake equation (7). In other words: (1) All the input variables on the right-hand side of (18) now become positive random variables. (2) All these random variables are supposed to be uniformly distributed with assigned mean values μ Di and standard deviations μ Di . It can then be shown that assigning them actually amounts to assigning the lower and upper limits (ai and bi , respectively) of each uniform random variable Di . (3) As a consequence of these assumptions, the total number of habitable planets in the Galaxy, N Hab , also becomes a random variable, that we already know to be lognormally distributed from our previous similar work about the Drake equation. Thus, we may now call (18) the statistical Dole equation. It is true that the classical Drake equation (7) and the classical Dole equation (16) have a different number of factors (7 and 10, respectively), buty frankly speaking, who cares? This perfectly in line with what we did already for the Drake equation, and so the number of factors in both (7) and (16) is totally irrelevant, thanks to the central limit theorem!

12 The Number of Habitable Planets for Man in the Galaxy Follows the Lognormal Distribution We now just repeat the same arguments developed for the Drake equation to immediately conclude that: the total number of habitable planets in the Galaxy follows the lognormal distribution given in Table 1.

13 The Distance Between Any Two Nearby Habitable Planets Follows the Maccone Distribution Again, we now just repeat the same arguments developed for the Drake equation to immediately conclude that the distance between any two nearby habitable planets follows the Maccone distribution given in Table 2.

14 A Numerical Example: A Some Hundred Million Habitable …

369

14 A Numerical Example: A Some Hundred Million Habitable Planets Exist in the Galaxy! We just need to complete this paper by giving a numerical example of how our Statistical Dole equation (18) works, and this we will do in the present section. Consider Input Table 2. This is in principle comparable to Input Table 1 for the Statistical Drake equation. In fact, the arguments developed by Dole in Chap. 5 of Ref. [3] do provide the mean values of each Di , but only such mean values, and not the relevant standard deviations, of course. To set up a working example of the Statistical Dole Equation, however, we must assign the ten standard deviations also, that were not given by Dole and are unknown to this author from the current scientific literature about these matters. No problem. In order to cut short, this author thus simply assigned the value of 1/10 (i.e. 10%) to each of the ten standard deviations listed in Input Table 2, and Input Table 2 is now complete. Having assumed all the values listed in Input Table 2 as the input values, a new (unpublished) MathCad code was created by this author for the Statistical Dole Equation. For the input values of Input Table 2, this code yielded the results described hereafter. First of all, the lognormal probability density for the random variable N Hab is shown in Fig. 4. We see that the peak (i.e. the mode) corresponds to about ten million planets, but the tail is rather long. Input Table 2 Input values (i.e. mean values and standard deviations) for the ten Dole uniform random variables Di N s = 60448 × 108

μN s : = N s

σ N s = 1 × 107

P p = 1.0

μPp = P p

σPp =

Pi = 0.81

μ Pi = Pi

P D = 0.63

μP D = P D

P M = 0.19

μP M = P M

Pe = 0.94

μ Pe = Pe

P B = 0.95

μP B = P B

P R = 0.9

μP R = P R

P A = 0.7

μP A = P A

PL = 1

μP L = P L

10 100 10 σ Pi = 100 10 σ P D = 100 10 σ P M = 100 10 σ Pe = 100 10 σ P B = 100 10 σ P R = 100 10 σ P A = 100 10 σ P L = 100

N H ab = N s · P p · Pi · P D · P M · Pe · P B · P R · P A · P L N H ab := 3.5171930508624 × 107 The first column on the left lists the ten input sheer numbers that also are the mean values (middle column). The last column on the right lists the ten input standard deviations. The bottom line is the classical Dole equation (16). So, the number of habitable planets in the Galaxy, given by the classical Dole equation just as a sheer number, is 35 million 171 hundred thousand and 930

Lognormal probability density

370

SETI and SEH (Statistical Equation for Habitables) LOGNORMAL DISTRIBUTION OF THE NUMBER OF HAB PLANETS

2.075×10-8

1.038×10-8

0 0

1×107 2×107 3×107 4×107 5×107 6×107 7×107 8×107 9×107 1×108 NUMBER of Hab Planets in the Galaxy

Fig. 4 The lognormal probability density of the overall NUMBER of habitable planets in the Galaxy as described in Stephen H. Dole’s book “Habitable Planets for Man”, first edition published in 1964 (Ref. [3]), and implemented by assigning a 10% standard deviation to all the ten input random variables listed in Input Table 2

To quantify these remarks, let us first point out that the author’s MathCad code yields the following numerical values for the two parameters μ and σ given by the last two rows in both Tables 1 and 2:  μ H ab = 1.76, 268, 289, 631, 314 × 101 (19) σ H ab = 1.27, 010, 132, 908, 265 × 100 . Then, the mean value of the random variable N Hab , given by the fourth row in Table 1, is given by N H ab  = eμ H ab eσ H ab /2 = 1.012 × 108 ≈ 100 million. 2

(20)

In other words, our statistical (and thus more serious, scientifically speaking) treatment of the Dole equation yields 100 million expected habitable planets in the Galaxy. This figure is higher than the 35 million given by the classical Dole equation, and much higher than the value of the mode (10 million) shown by the lognormal curve in Fig. 5. The last result, stating that there are about 100 million habitable planets in the Galaxy, is of course good news for the future “human conquest of the Galaxy” (if there will ever be one!), since it raises to 100 million the expected number of “Earths” to land on! But what about the standard deviation around the mean value given by (20)? Table 1, row 6, shows that such a standard deviation of the random variable N Hab is given by

Probability density function (1/m)

14 A Numerical Example: A Some Hundred Million Habitable …

371

(MACCONE) DISTRIBUTION OF THE DISTANCE OF THE NEAREST HAB PLANET

-15

1.794×10

1.196×10-15

0

0 0

25

50

75 100 125 150 175 200 225 Hab Planet Distance from Earth (light years)

250

275

300

Fig. 5 The Maccone probability distribution of the distance of the nearest habitable planet to us in the Galaxy for the data of the Input Table 2 assumed as inputs to the statistical Dole equation (18). A glance to this plot immediately reveals that it is “hopeless” to expect to detect a habitable planet at distances smaller than 25 light years from us, since the value of the Maccone distribution is practically zero at such distances. Thus, future Interstellar Spacecraft designers should keep this lower bound in mind wished they land on habitable planets, rather than just on “any Planet”. Also, the curve reaches its peak (mode) at about 67 light years from us, its mode (fifty–fifty probability) at about 80 light years and, above all, its mean value at 88 light years from us. The relevant standard deviation turns out to be about 40 light years, since the distribution tail is rather “short”

σ N H ab = e

μ H ab σ H2 ab /2

e

 eσ H ab − 1 = 2.0 × 108 ≈ 200 million. 2

(21)

In other words, the standard deviation of the number of habitable planets is 200 million. And so, with probability 1-sigma, we might expect the actual number of habitable planets to rise up 100 million plus 200 million = 300 million. Finally, the median (fifty–fifty probability of the lognormal distribution shown in Fig. 4) yields a value of median = m = eμ H ab = 4.521 × 107 ≈ 45 million.

(22)

15 Distance (Maccone) Distribution of the Nearest Habitable Planet to Us According to the Previous Numerical Inputs Next comes the distance distribution of the nearest habitable planet to us (of course under the easy hypothesis that the distribution of habitable planets in the Galaxy is uniform). Well, from the third row of Table 2 it follows that the relevant probability density is given by the Maccone distribution, and this is plotted in Fig. 5.

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SETI and SEH (Statistical Equation for Habitables)

The mean value of the Maccone distribution is given by the fifth row in Table 2, that is, for the data given by the Input Table 2: Hab_ Distance = Ce−μ H ab /3 eσ H ab /18 = 8.8 × 101 ly ≈ 88 ly. 2

(23)

The relevant standard deviation is given by the seventh row in Table 2, and reads σHab_ Distance = Ce

−μ H ab /3 σ H2 ab /18

e

 eσ H ab /9 − 1 = 3.9 2

× 101 ly ≈ 40 ly.

(24)

Thus, with probability 1 sigma, it should not be hopeless to expect a detection of a habitable planet even at, say, just 88 − 40 = 48 × 50 light years from us.

16 Comparing the Statistical Dole and Drake Equations: Number of Habitable Planets Versus Number of ET Civilizations in This Galaxy It is now appropriate to make a comparison between the number of habitable planets and the number of expected ET civilizations in the Galaxy. In other words, we want the “get the feeling” of the numbers that we have worked out in this chapter just to see if the comparison among them “makes sense”. This we can do by putting on a same table (1) the mean value and standard deviation of the total number of both habitable planets and ET civilizations, and (2) the mean value and standard deviation of their respective distances from us (of course, under the hypothesis that both of them are uniformly scattered throughout the Galaxy). The result is Table 5, clearly showing that how much “more rare” the ET civilizations are with respect to the habitable planets. Roughly, one has: N H ab  100 million ≈ 20, 202 = N E T  4950

(25)

so that the habitable planets seem to 20,000 more frequent than ET civilizations, or, if you wish, only one ET civilization emerges out of 20,000 habitable planets. As for the distances, the ratio is the other way round: Hab_ Distance 2670 ly = ≈ 30.340 ET_ Distance 88 ly

(26)

meaning that ETs are, on the average, 30 times further out that habitable planets.

16 Comparing the Statistical Dole and Drake Equations …

373

Table 3 Comparing the results of the statistical Dole and Drake equation found by inputting to them the Input Tables 2 and 1, respectively Statistical Dole equation

Statistical Drake equation

Mean value of the total number of

Habitable planets in the Galaxy ~100 million

ET civilizations in the Galaxy ~4590

Standard deviation of the total number of

Habitable planets in the Galaxy ~200 million

ET civilizations in the Galaxy ~11,195

Mean value of the distance of

Nearest habitable planet ~88 light years

Nearest ET civilization ~2670 light years

Standard deviation of the distance of

Nearest habitable planet ~40 light years

Nearest ET civilization ~1309 light years

And all these results, however, are just statistical, of course!

17 SEH, the “Statistical Equation for Habitables” Is just the Statistical Dole Equation So far we have referred to (18) as to the Statistical Dole equation. In view of further improvements in the mathematical analysis of this equation, however, it appears to be suitable to rename it “SEH”, an acronym standing for “Statistical Equation for Habitables”. This will be clear in the future papers by the author, where a number possibly higher than ten will be the new number of independent, uniform random variables describing the equation inputs. These topics have to be deferred to a further paper, though.

18 Conclusions We have sought to extend both the classical Drake and Dole equations to let them encompass statistics and probability. This approach appears to pave the way to future, more profound investigations intended not only to associate “error bars” to each factor in the equation, but especially to increase the number of factors themselves. In fact, this seems to be the only way to incorporate into the equations more and more new scientific information as soon as it becomes available. In the long run, our Statistical equations might just become a huge computer code, growing in size and especially in the depth of the scientific information it contained. It would thus be Humanity’s first “Encyclopaedia Galactica.” Unfortunately, to extend the Drake equation to Statistics, it was necessary to use a mathematical apparatus that is more sophisticated than just the simple product of seven numbers.

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SETI and SEH (Statistical Equation for Habitables)

When this author had the honour and privilege to present his results at the SETI Institute on April 11th, 2008, in front of an audience also including Professor Frank Drake, he felt he had to add these words of apology to him: “My apologies, Frank, for disrupting the beautiful simplicity of your equation.” Acknowledgements The author is grateful to the leader of the Tau Zero Foundation, Marc Millis, and to Drs. Hal Puthoff and Eric Davis of the Institute for Advanced Study at Austin for the reading the author’s paper #IAC-08-A4.1.4 and letting him have their appreciated comments. Also, the author is grateful to Drs. Jill Tarter, Paul Davies, Seth Shostak, Doug Vakoch, Tom Pierson, Carol Oliver, Paul Shuch and Kathryn Denning for attending his first presentation ever about these topics at the “Beyond” Center of the University of Arizona at Phoenix on February 8, 2008. He also would like to thank Dr. Dan Werthimer and his School of SETI experts at the Berkeley Space Sciences Laboratory for keeping alive the interplay between experimental and theoretical SETI. But the greatest “thanks” go of course to the Teacher to all of us: Professor Frank D. Drake, whose equation opened to Humanity a new way of thinking about the past and the future of Humans in the Galaxy.

References 1. G. Benford, J. Benford, D. Benford, Cost optimized interstellar beacons: SETI, arXiv.org web site, 22 Oct 2008 2. J. Bennet, S. Seth, in Life in the Universe, 2nd edn (Pearson—Addison Wesley, San Francisco, 2007). see in particular p. 404 3. S.H. Dole, Habitable planets for man, 1st edn, 1964, © 1964 by the RAND Corporation, Library of Congress Catalogue Card Number 64-15992. See in particular p. 82, i.e. the beginning of Chapter 5, entitled “Probability of Occurrence of habitable planets” 4. C. Maccone, The statistical Drake equation, in press, Acta Astronautica, 2010, in the Proceedings of the First IAA Symposium on “Searching for Life Signatures” held at UNESCO, Paris, 22–26 Sept 2008 5. C. Maccone, The Statistical Drake Equation, paper #IAC-08-A4.1.4 presented on October 1, 2008, at the 59th International Astronautical Congress (IAC) held in Glasgow, Scotland, UK, September 29–October 3 2008 6. C. Sagan, in Cosmos (Random House, New York, 1983). See in particular, p. 298–302

Societal Statistics by Virtue of the Statistical Drake Equation

Abstract The Drake equation, first proposed by Frank D. Drake in 1961, is the foundational equation of SETI. It yields an estimate of the number N of extraterrestrial communicating civilizations in the Galaxy given by the product N = Ns × fp × ne × fl × fi × fc × fL, where: Ns is the number of stars in the Milky Way Galaxy; fp is the fraction of stars that have planetary systems; ne is the number of planets in a given system that are ecologically suitable for life; fl is the fraction of otherwise suitable planets on which life actually arises; fi is the fraction of inhabited planets on which an intelligent form of life evolves; fc is the fraction of planets inhabited by intelligent beings on which a communicative technical civilization develops; and fL is the fraction of planetary lifetime graced by a technical civilization. The first three terms may be called “the astrophysical terms” in the Drake equation since their numerical value is provided by astrophysical considerations. The fourth term, fl, may be called “the origin-of-life term” and entails biology. The last three terms may be called “the societal terms” inasmuch as their respective numerical values are provided by anthropology, telecommunication science and “futuristic science”, respectively. In this chapter, we seek to provide a statistical estimate of the three societal terms in the Drake equation basing our calculations on the Statistical Drake Equation first proposed by this author at the 2008 IAC. In that paper the author extended the simple 7-factor product so as to embody Statistics. He proved that, no matter which probability distribution may be assigned to each factor, if the number of factors tends to infinity, then the random variable N follows the lognormal distribution (central limit theorem of Statistics). This author also proved at the 2009 IAC that the Dole (1964) (Stephen H. Dole, Habitable planets for Man, first edition, 1964, © 1964 by the RAND Corporation, Library of Congress Catalogue Card Number 64–15,992. See in particular page 82, i.e. the beginning of Chapter 5, entitled Probability of Occurrence of Habitable Planets.) equation, yielding the number of Habitable Planets for Man in the Galaxy, has the same mathematical structure as the Drake equation. So the number of Habitable Planets follows the lognormal distribution as well. But the Dole equation is described by the first FOUR factors of the Drake equation. Thus, we may “divide” the 7-factor Drake equation by the 4-factor Dole equation getting the probability distribution of the last-3-factor Drake equation, i.e. the probability distribution of the SOCIETAL TERMS ONLY. These we study in detail in this paper, achieving new statistical results about the SOCIETAL ASPECTS OF SETI. © Springer Nature Switzerland AG 2020 C. Maccone, Evo-SETI, https://doi.org/10.1007/978-3-030-51931-5_10

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Keywords Statistical drake equation · Societal aspects of SETI · Lognormal probability densities

1 Introducing the Drake Equation The Drake equation (see, for instance, Ref. [8]) is the foundational equation of SETI. It was proposed back in 1961 by Frank Donald Drake to estimate the numerical value of N, the number of communicating extraterrestrial civilizations now existing in the Galaxy. To summarize what the Drake equation is, we think that that no better introductory description exists other than the one given by Carl Sagan in his 1983 book “Cosmos” (Ref. [7]), in its turn based on the famous TV series “Cosmos”. So, in this paragraph we report Carl Sagan’s description of the Drake equation unabridged. “But is there anyone out there to talk to? With a third or a half a trillion stars in our Milky Way Galaxy alone, could ours be the only one accompanied by an inhabited planet? How much more likely it is that technical civilizations are a cosmic commonplace, that the Galaxy is pulsing and humming with advanced societies, and, therefore, that the nearest such culture is not so very far away—perhaps transmitting from antennas established on a planet of a naked-eye star just next door. Perhaps when we look up the sky at night, near one of those faint pinpoints of light is a world on which someone quite different from us is then glancing idly at a star we call the Sun and entertaining, for just a moment, an outrageous speculation. It is very hard to be sure. There may be several impediments to the evolution of a technical civilization. Planets may be rarer than we think. Perhaps the origin of life is not so easy as our laboratory experiments suggest. Perhaps the evolution of advanced life forms is improbable. Or it may be that complex life forms evolve more readily, but intelligence and technical societies require an unlikely set of coincidences—just as the evolution of the human species depended on the demise of the dinosaurs and the ice-age recession of the forests in whose trees our ancestors screeched and dimly wondered. Or perhaps civilizations arise repeatedly, inexorably, on innumerable planets in the Milky Way, but are generally unstable; so all but a tiny fraction are unable to survive their technology and succumb to greed and ignorance, pollution and nuclear war. It is possible to explore this great issue further and make a crude estimate of N, the number of advanced civilizations in the Galaxy. We define an advanced civilization as one capable of radio astronomy. This is, of course, a parochial if essential definition. There may be countless worlds on which the inhabitants are accomplished linguists or superb poets but indifferent radio astronomers. We will not hear from them. N can be written as the product or multiplication of a number of factors, each a kind of filter, every one of which must be sizable for there to be a large number of civilizations: Ns fp ne fl

the number of stars in the Milky Way Galaxy; the fraction of stars that have planetary systems; the number of planets in a given system that are ecologically suitable for life; the fraction of otherwise suitable planets on which life actually arises;

1 Introducing the Drake Equation

fi fc

377

the fraction of inhabited planets on which an intelligent form of life evolves; the fraction of planets inhabited by intelligent beings on which a communicative technical civilization develops; and. the fraction of planetary lifetime graced by a technical civilization.

fL

Written out, the equation reads N = N s × f p × ne × f l × f i × f c × f L .

(1)

All of the f ’s are fractions, having values between 0 and 1; they will pare down the large value of Ns. To derive N we must estimate each of these quantities. We know a fair amount about the early factors in the equation, the number of stars and planetary systems. We know very little about the later factors, concerning the evolution of intelligence or the lifetime of technical societies. In these cases our estimates will be little better than guesses. I invite you, if you disagree with my estimates below, to make your own choices and see what implications your alternative suggestions have for the number of advanced civilizations in the Galaxy. One of the great virtues of this equation, due to Frank Drake of Cornell, is that it involves subjects ranging from stellar and planetary astronomy to organic chemistry, evolutionary biology, history, politics and abnormal psychology. Much of the Cosmos is in the span of the Drake equation. We know Ns, the number of stars in the Milky Way Galaxy, fairly well, by careful counts of stars in a small but representative region of the sky. It is a few hundred billion; some recent estimates place it at 4 × 1011 . Very few of these stars are of the massive short-lived variety that squander their reserves of thermonuclear fuel. The great majority have lifetimes of billions or more years in which they are shining stably, providing a suitable energy source for the energy and evolution of life on nearby planets. There is evidence that planets are a frequent accompaniment of star formation: in the satellite systems of Jupiter, Saturn and Uranus, which are like miniature solar systems; in theories of the origin of the planets; in studies of double stars; in observations of accretion disks around stars; and is some preliminary investigations of gravitational perturbations of nearby stars.1 Many, perhaps even most, stars may have planets. We take the fraction of stars that have planets, fp, as roughly equal to 1/3. Then the total number of planetary systems in the Galaxy would be Ns × fp ~ 1.3 × 1011 (the symbol ~ means “approximately equal to”). If each system were to have about ten planets, as ours does, the total number of worlds in the Galaxy would be more than a trillion, a vast arena for the cosmic drama. 1 Carl

Sagan was writings these lines back in the 1970s, when no extrasolar planets had been discovered yet. The first such discovery occurred in 1995, when the two Swiss astronomers from the Geneva Observatory, Michel Mayor and Didier Queloz, working at the “Observatoire de Haute Provence” in France, discovered the first extrasolar planet orbiting the nearby star 51 Peg. This first extrasolar planet was hence named 51 Peg B. Many more extrasolar planets were discovered around nearby stars ever since. As of 28 July 2010, 473 extrasolar planets (exoplanets) are listed in the Extrasolar Planets Encyclopedia [8] maintained by Jean Schneider of the Paris Observatory at Meudon.

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In our own solar system there are several bodies that may be suitable for life of some sort: the Earth certainly, and perhaps Mars, Titan and Jupiter. Once life originates, it tends to be very adaptable and tenacious. There must be many different environments suitable for life in a given planetary system. But conservatively we choose ne = 2. Then the number of planets in the Galaxy suitable for life becomes Ns × fp × ne ~ 3 × 1011 . Experiments show that under the most common cosmic conditions the molecular basis of life is readily made, the building blocks of molecules able to make copies of themselves. We are now on less certain grounds; there may, for example, be impediments in the evolution of the genetic code, although I think this is unlikely over billions of years of primeval chemistry. We choose fl ~ 1/3, implying a total number of planets in the Milky Way on which life has arisen at least once as Ns × fp × ne × fl ~ 1 × 1011 , a hundred billion inhabited worlds. That in itself is a remarkable conclusion. But we are not yet finished. The choices of fi and fc are more difficult. On the one hand, many individually unlikely steps had to occur in biological evolution and human history for our present intelligence and technology to develop. On the other hand, there must be quite different pathways to an advanced civilization of specified capabilities. Considering the apparent difficulty in the evolution of large organisms, represented by the Cambrian explosion, let us choose fi × fc = 1/100, meaning that only 1% of planets on which life arises actually produce a technical civilization. This estimate represents some middle ground among the varying scientific options. Some think that the equivalent of the step from the emergence of trilobites to the domestication of fire goes like a shot in all planetary systems; others think that, even given ten or fifteen billion years, the evolution of a technical civilization is unlikely. This is not a subject on which we can do much experimentation as long as our investigations are limited to a single planet. Multiplying these factors together, we find Ns × fp × ne × fl × fi × fc ~ 1 × 109 , a billion planets on which technical civilizations have arisen at least once. But that is very different from saying that there are a billion planets on which technical civilizations now exist. For this we must also estimate fL. What percentage of the lifetime of a planet is marked by a technical civilization? The Earth has harbored a technical civilization characterized by radio astronomy for only a few decades out of a lifetime of a few billion years. So far, then, for our planet fL is less than 1/108 , a millionth of a percent. And it is hardly out of the question that we might destroy ourselves tomorrow. Suppose this were a typical case, and the destruction so complete that no other technical civilization—of the human or any other species—were able to emerge in the five or so billion years remaining before the Sun dies. Then Ns × fp × ne × fl × fi × fc × fL ~ 10, and, at a given time there would be only a tiny smattering, a handful, a pitiful few technical civilizations in the Galaxy, the steady state number maintained as emerging societies replace those recently self-immolated. The number N might be even as small as 1, if civilizations tend to destroy themselves soon after reaching a technological phase, there might be no one for us to talk with but ourselves. And that we do but poorly. Civilizations would take billions of years of tortuous evolution, and then snuff themselves out in an instant of unforgivable neglect.

1 Introducing the Drake Equation

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But consider the alternative, the prospect that at least some civilizations learn to live with technology; that the contradictions posed by the vagaries of past brain evolution are consciously resolved and do not lead to self-destruction; or that, even if major disturbances occur, they are reveries in the subsequent billions of years of biological evolution. Such societies might live to a prosperous old age, their lifetimes measured perhaps on geological or stellar evolutionary time scales. If 1% of civilizations can survive technological adolescence, take the proper fork at this critical historical branch point and achieve maturity, then fL ~ 1/100, N ~ 107 , and the number of extant civilizations in the Galaxy is in the millions. Thus, for all our concern about the possible unreliability of our estimates of the early factors in the Drake equation, which involve astronomy, organic chemistry and evolutionary biology, the principal uncertainty comes to economics and politics and what, on Earth, we call human nature. It seems fairly clear that if self-destruction is not the overwhelmingly preponderant fate of galactic civilizations, then the sky is softly humming with messages from the stars. These estimates are stirring. They suggest that the receipt of a message from space is, even before we decode it, a profoundly hopeful sign. It means that someone has learned to live with high technology; that it is possible to survive technological adolescence. This alone, quite apart from the contents of the message, provides a powerful justification for the search for other civilizations.”

2 The Drake Equation is Over-Simplified In the sixty years (1961–2021) elapsed since Frank Drake proposed his equation, a number of scientists and writers tried to find out which numerical values of its seven independent variables are more realistic in agreement with our present-day knowledge. Thus there is a considerable amount of literature about the Drake equation nowadays, and, as one can easily imagine, the results obtained by the various authors largely differ from one another. In other words, the value of N, that various authors obtained by different assumptions about the astronomy, the biology and the sociology implied by the Drake equation, may range from a few tens (in the pessimist’s view) to some million or even billion in the optimist’s opinion. A lot of uncertainty is thus affecting our knowledge of N as of 2011. In all cases, however, the final result about N has always been a sheer number, i.e. a positive integer number ranging from 1 to million or billion. This is precisely the aspect of the Drake equation that this author regarded as “too simplistic” and improved mathematically in his paper #IAC-08A4.1.4, entitled “The Statistical Drake Equation” and presented on October 1st, 2008, at the 59th International Astronautical Congress (IAC) held in Glasgow, Scotland, UK, September 29th–October 3rd, 2008, i.e. Ref. [3], later (2010) re-published as Ref. [5], and, in Russian, in Ref. [4]. Newcomers to SETI and to the Drake equation, however, mind find that papers too difficult to be understood mathematically at a first reading. Thus, we shall now explain the content of that paper “by speaking easily”, we beg the reader’s indulgence and draw their attention to these new mathematical

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developments about the Drake equation. Also, it should be pointed out that much later, on November 10th, 2010, the author was allowed gave a talk at the SETI Institute by the title of “Statistical Equation for Habitables (SEH) and Statistical Fermi Paradox”. This talk covers most of the topics described in this paper, lasts about an hour and can be watched at the site: https://www.youtube.com/watch?v=xNe5hVvaXkk.

3 The Statistical Drake Equation by Maccone [3] We start by an example. Consider the first independent variable in the Drake equation (1), i.e. Ns, the number of stars in the Milky Way Galaxy. Astronomers tell us that approximately there should be about 350 billion stars in the Galaxy. Of course, nobody has counted (or even seen in the photographic plates) all the stars in the Galaxy! There are too many practical difficulties preventing us from doing so: just to name one, the dust clouds that do not allow us to see even the Galactic Bulge (i.e. the central region of the Galaxy) in the visible light (although we may “see it” at radio frequencies like the famous neutral hydrogen line at 1420 MHz). So, it does not make any sense to say that Ns = 350 × 109 , or, say (even worse) that the number of stars in the Galaxy is (say) 354,233,321,123, or similar fanciful exact integer numbers. That is just silly and non-scientific. Much more scientific, on the contrary, is to say that the number of stars in the Galaxy is 350 billion plus or minus, say, 50 billion (or whatever values the astronomers may regard as more appropriate, since this is just an example to let the reader understand the difficulty). Thus, it makes sense to replace each of the seven independent variables in the Drake equation (1) by a mean value (350 billion, in the above example) plus or minus a certain standard deviation (50 billion, in the above example). By doing so, we have made a great step ahead: we have abandoned the too simplistic Eq. (1) and replaced it by something more sophisticated and scientifically more serious: the statistical Drake equation. In other words, we have transformed the classical and simplistic Drake equation (1) into an advanced statistical tool for the investigation of a host of facts hardly known to us in detail. In other words still: (1) We replace each independent variable in (1) by a random variable, labeled Di (from Drake). (2) We assume that the mean value of each Di is the same numerical value previously attributed to the corresponding independent variable in (1). (3) But now we also add a standard deviation σ Di on each side of the mean value, that is provided by the knowledge gathered by scientists in each discipline encompassed by each Di . Having so done, the next question is: How can we find out the probability distribution for each Di ? For instance, shall that be a Gaussian, or what?

3 The Statistical Drake Equation by Maccone [3]

381

This is a difficult question, for nobody knows, for instance, the probability distribution of the number of stars in the Galaxy, not to mention the probability distribution of the other six variables in the Drake equation (1). There is a brilliant way to get around this difficulty, though. We start by excluding the Gaussian because each variable in the Drake equation is a positive (or, more precisely, a non-negative) random variable, while the Gaussian applies to real random variables only. So, the Gaussian is out. Then, one might consider the large class of well-studied and positive probability densities called “the gamma distributions”, but it is then unclear why one should adopt the gamma distributions and not any other one. The solution to this apparent conundrum comes from Shannon’s Information Theory and a theorem that he proved in 1948: “The probability distribution having maximum entropy (=uncertainty) over any finite range of real values is the uniform distribution over that range” (Ref. [1]). So, at this point, we assume that each of the seven Di in (1) is a uniform random variable, whose mean value and standard deviation is known by the scientists working in the respective field (let it be astronomy, or biology, or sociology). Notice that, for such a uniform distribution, the knowledge of the mean value μ Di and of the standard deviation σ Di automatically determines the range of that random variable in between its lower (called ai ) and upper (called bi ) limits: in fact these limits are given by the equations 

√ 3σ Di , √ + 3σ Di ,

ai = μ Di − bi = μ Di

(2)

√ (the “surprising” factor 3 in the above equations comes from the definitions of mean value and standard deviation: see Eqs. (12), (15) and (4) in Ref. [3] for the relevant proof). So the uniform distribution of each random variable Di is perfectly determined by its mean value and standard deviation, and so are all its other properties. The next problem is the following: Now that we know everything about each uniformly distributed Di , what is the probability distribution of N, given that N is the product (1) of all the Di ? In other words, not only do we want to find the analytical expression of the probability density function of N, but we also want to relate its mean value μN to all mean values μ Di of the Di , and its standard deviation σ N to all standard deviations σ Di of the Di . This is a difficult problem. It occupied the author’s mind for no less than about ten years (1997–2007). It is actually an analytically unsolvable problem, in that, to the best of this author’s knowledge, it is impossible to find an analytic expression for any finite product of uniform random variables Di . This result is proven in Sect. 2–3.3 of Ref. [3] (unfortunately!).

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4 Solving the Statistical Drake Equation by Virtue of the Central Limit Theorem (CLT) of Statistics The solution to the problem of finding the analytical expression for the probability density function of N in the statistical Drake equation was found by this author only in September 2007. The key steps are the following: (1) Take the natural logs of both sides of the statistical Drake equation (1). This changes the product into a sum. (2) The mean values and standard deviations of the logs of the random variables Di may all be expressed analytically in terms of the mean values and standard deviations of the Di . (3) Recall the Central Limit Theorem (CLT) of statistics, stating that (loosely speaking) if you have a sum of independent random variables, each of which is arbitrarily distributed (hence, also including uniformly distributed), then, when the number of terms in the sum increases indefinitely (i.e. for a sum of random variables infinitely long)… the sum random variable tends to a Gaussian. (4) Thus, the natural log of N tends to a Gaussian. (5) Thus, N tends to the lognormal distribution. (6) The mean value and standard deviations of this lognormal distribution of N may all be expressed analytically in terms of the mean values and standard deviations of the logs of the Di already found previously. This result is fundamental. All the relevant equations are summarized in Table 1. This table is actually the same as Table 2 of the author’s original paper IAC-08-A4.1.4, entitled “The Statistical Drake Equation” and presented by him at the International Astronautical Congress (IAC) held in Glasgow, UK, on October 1st, 2008. This original paper is reproduced just the same in Ref. [3]. Input Table 1 Input values (i.e. mean values and standard deviations) for the seven Drake uniform random variables Di Ns: = 350 × 109

μNs: = Ns

σ Ns: = 1 × 109

fp: = 50/100

μfp: = fp

ne: = 1

μne: = ne

σ fp: = 10/100 √ σ ne: = 1/ 3

fl: = 50/100

μfl: = fl

σ fl: = 10/100

fi: = 20/100

μfi:fi

σ fi: = 10/100

fc: = 20/100

μfc: = fc

σ fc: = 10/100

fL: = 10,000/1010

μfL: = fL

σ fL: = 1000/1010

N: = Ns × fp × ne × fl × fi × fc × fL

N = 3500

The first column on the left lists the seven input sheer numbers that also become the mean values (middle column). Finally the last column on the right lists the seven input standard deviations. The bottom line is the classical Drake equation (1)

4 Solving the Statistical Drake Equation by Virtue …

383

Table 1 Summary of the properties of the lognormal distribution that applies to the random variable N = number of ET communicating civilizations in the Galaxy Random variable

N = number of communicating ET civilizations in Galaxy

Probability distribution

Lognormal

Probability density function

f N (n) =    1 √1 − (ln(n)−μ)2 / 2σ 2 n 2πσ e 2 N  = eμ eσ /2

Mean value

(n ≥ 0)

 2  eσ − 1

Variance

σ N2 = e2μ eσ

Standard deviation All the moments, i.e. kth moment

 σ2 2 σ N = eμ e 2 eσ − 1    k 2 2 N = ekμ ek σ /2

Mode (= abscissa of the lognormal peak)

n mode ≡ n peak = eμ e−σ

Value of the Mode Peak

f N (n mode ) =

Median (= fifty–fifty probability value for N)

median = m  2  2 K3 σ +2 eσ − 1 3/ 2 = e

Skewness Kurtosis

2

2

2 √ 1 e−μ eσ /2 2πσ = eμ

(K 2 )

K4 (K 2 )2

2

2

Expression of μ in terms of the lower (ai ) and upper (bi ) limits of the Drake uniform input random variables Di

μ=

Expression of σ 2 in terms of the lower (ai ) and upper (bi ) limits of the Drake uniform input random variables Di

σ2 = 7 7 



1− σY2i =

7

Yi  =

i=1

i=1

2

= e4σ + 2e3σ + 3e2σ − 6 7

i=1

i=1

bi [ln(bi )−1]−ai [ln(ai )−1] bi −ai

ai bi [ln(bi )]−[ln(ai )]2 (bi −ai )2



To sum up, not only have we found that N approaches the completely known lognormal distribution for an infinity of factors in the statistical Drake equation (1), but we also paved the way to further applications by removing the condition that the number of terms in the product (1) must be finite. This possibility of adding any number of factors in the Drake equation (1) was not envisaged, of course, by Frank Drake back in 1961, when “summarizing” the evolution of life in the Galaxy in seven simple steps. But today, the number of factors in the Drake equation should already be increased: for instance, there is no mention in the original Drake equation of the possibility that asteroidal impacts might destroy the life on Earth at any time, and this is because the demise of the dinosaurs at the K/T impact had not been yet understood by scientists in 1961, and was first suggested in 1980! In practice, we are suggesting increasing the number of factors as much as necessary in order to get better and better estimates of N as long as our scientific knowledge

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Societal Statistics by Virtue of the Statistical Drake Equation

increases. This we call the “Data Enrichment Principle” and we believe this should be the next important goal in the study of the statistical Drake equation. Finally, we wish to provide a numerical example explaining how the statistical Drake equation works in practice. This will be done in the next section.

5 An Example Explaining the Statistical Drake Equation To understand how things work in practice for the statistical Drake equation, please consider the following Input Table 1. It is made up of three columns: (1) The first column on the left lists the seven input sheer numbers that also become the mean values if one lets the values in the third column (input standard deviations) approach zero. (2) The mean values (middle column). (3) Finally the last column on the right lists the seven input standard deviations. The bottom line is the classical Drake equation (1). We see that, for this particular set of seven inputs, the classical Drake equation (i.e. the product of the seven numbers) yields a total of 3500 communicating extraterrestrial civilizations existing in the Galaxy right now. The statistical Drake equation, however, provides a much more articulated answer than just the above sheer number N = 3500. In fact, a MathCad code written by this author and capable of performing all the numerical calculations required by the statistical Drake equation for a given set of seven input mean values plus seven input standard deviations, yields for N the lognormal distribution (thin curve) plotted in Fig. 2. We see immediately that the peak of this thin curve (i.e. the mode) falls at 2 about nmode ≡ npeak = eμ e−σ ≈ 250 (this is Eq. (99) of Ref. [3]), while the median (fifty-fifty value splitting the lognormal density in two parts with equal undergoing areas) falls at about nmedian ≡ eμ ≈ 1740. These seem to be smaller values than N = 3500 provided by the classical Drake equations, but it is a wrong impression due to a poor “intuitive” understanding of what statistics is! In fact, neither the mode nor the median are the “really important” values: the really important value for N is the mean value. Now if you look at the thin curve in Fig. 1 (i.e. the lognormal distribution arising from the Central Limit Theorem), you see that this curve has a long tail on the right. In other words, it does not immediately go down to nearly zero beyond the peak of the mode. Thus, when you actually compute the mean value, you should not be too surprised to find out that it equals = eμ e(σ 2/2) ≈ 4589.559~ 4590 communicating civilizations now in the Galaxy. This is the important number, and it is higher than the 3500 provided by the classical Drake equation. Thus, in conclusion, the statistical extension of the classical Drake equation that we made increases our hopes of finding an extraterrestrial civilization. Our chances are increased even more when we go on to consider the standard deviation associated with the mean value 4590. In fact,the standard deviation is 2 given by Eq. (97) of Ref. [3]. This yields σN = eμ e(σ /2) eσ 2 − 1 = 11, 195 and so

5 An Example Explaining the Statistical Drake Equation PROBABILITY DENSITY FUNCTION OF N

6.10–4

Prob. density function of N

385

5.10–4 4.10–4 3.10–4 2.10–4 1.10–4 0 0

1000 2000 3000 N = Number of ET Civilizations in Galaxy

4000

Fig. 1 Comparing the two probability density functions of the random variable N found: (1) at the end of Sect. 3.3 of Ref. [5] in a purely numerical way and without resorting to the CLT at all (thick curve). (2) Analytically by using the CLT and the relevant lognormal approximation (thin curve)

the expected number of N may actually be even much higher than the 4590 provided by the mean value alone! The “upper limit of the one-sigma confidence interval” (as statisticians call it), i.e. the sum 4590 + 11,195 = 15,785, yields a higher number still! (Note: the “lower limit of the one-sigma confidence interval”, i.e. –6605, “seems” to be negative, but it is actually zero because the lognormal distribution is positive (or, more correctly, non-negative) and so negative values of its support are ruled out from the start!). Finally, the reader should not worry about the thick curve depicted in Fig. 1: it is just the numerical solution of the statistical Drake equation for a finite number of 7 input factors. Figure 1 actually shows that this curve “is well interpolated” by the lognormal distribution (thin curve), i.e. by the neat analytical expression provided by the Central Limit Theorem for an infinite number of factors in the Drake equation. That is, in conclusion, Fig. 1 visually shows that taking 7 factors or an infinity of factors “is almost the same thing” already for a value as small as 7.

6 The “Data Enrichment Principle” as the Best CLT Consequence upon the Statistical Drake Equation (Any Number of Factors Allowed) As a fitting climax to all the statistical equations developed so far, let us now state our “Data Enrichment Principle”. It simply states that “The Higher the Number of Factors in the Statistical Drake equation, The Better”.

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Societal Statistics by Virtue of the Statistical Drake Equation

Fig. 2 Reproduction of page 82 of Stephen H. Dole’s book “Habitable Planets for Man”, first edition published in 1964. It can be freely downloaded from the web site of the Rand Corporation

6 The “Data Enrichment Principle” as the Best CLT Consequence …

387

Put in this simple way, it simply looks like a new way of saying that the CLT lets the random variable Y approach the normal distribution when the number of terms in the product (1) approaches infinity. And this is the case, indeed. However, our “Data Enrichment Principle” has more profound methodological consequences that we cannot explain now, but hope to describe more precisely in one or more coming papers.

7 Habitable Planets for Man Let us now change topics: rather than seeking for ETs in the Galaxy, we now seek for Habitable Planets for Man in the Galaxy. How many are there? And how far from us is the nearest such Habitable Planet? These topics seem to have been faced “seriously” for the first time in 1964 by Stephen H. Dole, then with the Rand Corporation. Back in 1964, only three years had elapsed since Frank Drake had made known his now famous Drake equation. Dole learned the lesson of the Drake equation perfectly, and in his now famous book entitled “Habitable Planets for Man” (Ref. [2]) he used the same mathematical structure as the Drake equation (1) in order to find the number of habitable planets for Man in the Galaxy. In other words, on page 82 of his book, he wrote the same mathematical thing as the Drake equation, but he applied it to Habitable Planets. Figure 2 reproduces this crucial page of Dole’s 1964 book, that nowadays can be freely downloaded from the Rand Corporation web site. This equation we shall now call “the classical Dole equation”. As we can see from Fig. 2, the classical Dole equation is made up by ten factors (instead of seven factors as in the Drake equation): N H ab = N s × P p × Pi × P D × P M × Pe × P B × P R × P A × P L .

(3)

Here N Hab is the total number of Habitable Planets for Man in the Galaxy, and it is given by the product of the following TEN input numbers: (1) (2) (3) (4) (5) (6) (7)

Ns is the number of stars in the suitable mass range 0.35–1.43 solar masses (this is Dole’s assumption about to the mass of “habitable stars”). Pp is the probability that a given star has planets in orbit around it. Pi is the probability that the inclination of the planet’s equator is correct for its orbital distance. PD is the probability that at least one planet orbits within an ecosphere. PM is the probability that the planet has a suitable mass, 0.4–2.35 Earth masses (again, this is Dole’s assumption in this regard). Pe is the probability that the planet’s orbital eccentricity is sufficiently low. PB is the probability that the presence of a second star has not rendered the planet uninhabitable.

388

Societal Statistics by Virtue of the Statistical Drake Equation

(8)

PR is the probability that the planet’s rate of rotation is neither too fast nor too slow. (9) PA is the probability that the planet is of the proper age. (10) PL is the probability that, all astronomical conditions being proper, life has developed on the planet.

8 The Statistical Dole Equation It is now natural to rename the above ten input variables of the classical Dole equation (3) as follows: ⎧ D ⎪ ⎪ 1 ⎪ ⎪ ⎪ D2 ⎪ ⎪ ⎪ ⎪ ⎪ D3 ⎪ ⎪ ⎪ ⎪ ⎪ D4 ⎪ ⎪ ⎪ ⎨D

= Ns = Pp = Pi = PD

= PM ⎪ ⎪ D6 = Pe ⎪ ⎪ ⎪ ⎪ D7 = P B ⎪ ⎪ ⎪ ⎪ ⎪ D8 = P R ⎪ ⎪ ⎪ ⎪ ⎪ D9 = P A ⎪ ⎪ ⎩ D10 = P L , 5

(4)

so that our classical Dole equation may be simply rewritten as N H ab =

10 

Di .

(5)

i=1

We now let (5) undergo exactly the same changes that we applied to the classical Drake equation (1). In other words: (1) All the input variables on the right-hand side of (5) now become positive random variables. (2) All these random variables are supposed to be uniformly distributed with assigned mean values μ Di and standard deviations σ Di . It can then be shown that assigning them actually amounts to assigning the lower and upper limits (ai and bi , respectively) of each uniform random variable Di . (3) As a consequence of these assumptions, the total number of Habitable Planets in the Galaxy, N Hab , also becomes a random variable, that we already know to be lognormally distributed from our previous similar work about the Drake equation.

8 The Statistical Dole Equation

389

Thus, we may now call (5) the statistical Dole equation. The notation Di obviously comes from “Dole”, but the lucky coincidence that both Frank Drake’s and Stephen Dole’s family names both start with a “D” will save us from introducing new notations other than these Di ! It is true that the classical Drake equation (1) and the classical Dole equation (3) have a different number of factors (7 and 10, respectively), but… frankly speaking, who cares? This perfectly in line with what we did already for the Drake equation, and so the number of factors in both (1) and (3) is totally irrelevant, thanks to the central limit theorem.

9 The Number of Habitable Planets for Man in the Galaxy Follows the Lognormal Distribution We now just repeat the same arguments developed for the Drake equation to immediately conclude that: the total number of Habitable planets for Man in the Galaxy follows the lognormal distribution given in Table 1.

10 An Example Explaining the Statistical Dole Equation: Some Hundred Million Habitable Planets Exist in the Galaxy! We now provide a numerical example of how our Statistical Dole equation (3) works. Consider Input Table 2. This is in principle comparable to Input Table 1 for the Statistical Drake equation. In fact, the arguments developed by Dole in Chap. 5 of Ref. [2] do provide the mean values of each Di , but these are only mean values, and not the relevant standard deviations, of course. To set up a working example of the Statistical Dole Equation, however, we must assign the ten standard deviations also, that were not given by Dole and are unknown to this author from the current scientific literature on these matters. As a simple exercise, the author thus simply assigned the value of 1/10 (i.e. 10%) to each of the ten standard deviations listed in Input Table 2, and Input Table 2 is now complete. Having assumed all the values listed in Input Table 2 as the input values, a new (unpublished) MathCad code was created by this author for the Statistical Dole Equation. For the input values of Input Table 2, this code yielded the results described hereafter. First of all, the lognormal probability density for the random variable N Hab is shown in Fig. 3. We see that the peak (i.e. the mode) corresponds to about ten million planets, but the tail is rather long.

390

Societal Statistics by Virtue of the Statistical Drake Equation

Imput Table 2 Input values (i.e. mean values and standard deviations) for the ten Dole uniform random variables Di Ns: = 6.448 × 108

μNs: = Ns

σ Ns: = 1 × 107

Pp: = 1.0

μPp: = Pp

σ Pp: = 10/100

Pi: = 0.81

μPi: = Pi

σ Pi: = 10/100

PD: = 0.63

σ PD: = PD

σ PD: = 10/100

PM: = 0.19

μPM: = PM

σ PM: = 10/100 σ Pe: = 10/100

Pe: = 0.94

μPe: = Pe

PB: = 0.95

μPD: = i

σ PB: = 10/100

PR: = 0.9

μPR: = PR

σ PR: = 10/100

PA: = 0.7

μPA: = Pc

σ PA: = 10/100

PL: = 1

μPL: = PL

σ PL: = 10/100

N Hab : = Ns × Pp × Pi × PD × PM × PB × PR × PR × PA × PL N Hab : = 3.5171930508624 × 107

Lognormal probability density

The first column on the left lists the ten input numbers that also are the mean values (middle column). The last column on the right lists the ten input standard deviations. The bottom line is the classical Dole equation (3). So, the number of Habitable Planets in the Galaxy, given by the classical Dole equation just as a sheer number, is 35,171,930

LOGNORMAL DISTRIBUTION OF THE NUMBER OF HAB PLANETS

2.075×10−8

1.038×10−8

0 0

1×107 2×107 3×107 4×107 5×107 6×107 7×107 8×107 9×107 1×108 NUMBER of Hab Planets in the Galaxy

Fig. 3 The lognormal probability density of the overall NUMBER of Habitable Planets in the Galaxy as described in Stephen H. Dole’s book “Habitable Planets for Man”, first edition published in 1964, and implemented by assigning a 10% standard deviation to all the ten input random variables listed in Input Table 2

To quantify these remarks, let us first point out that the author’s MathCad code yields the following numerical values for the two parameters μ and σ given by the last two rows in both Input Tables 1 and 2:

10 An Example Explaining the Statistical Dole Equation …



μ H ab = 1.76268289631314 × 101 σ H ab = 1.27010132908265 × 100 .

391

(6)

Then, the mean value of the random variable N Hab , given by the fourth row in Table 1, is given by 

N H ab  = eμ H ab e

σ2

H ab/2



= 1.012 × 108 ≈ 100 million.

(7)

In other words, our statistical (and thus more serious, scientifically speaking) treatment of the Dole equation yields 100 million expected Habitable Planets in the Galaxy. This figure is higher than the 35 million given by the classical Dole equation, and much higher than the value of the mode (10 million) shown by the lognormal curve in Fig. 3. The last result, stating that there are about 100 million Habitable Planets in the Galaxy, is of course good news for the future “human conquest of the Galaxy” (if there will ever be one!), since it raises to 100 million the expected number of “Earths” to land on! But what about the standard deviation around the mean value given by (7)? Table 1, row 6, shows that such a standard deviation of the random variable N Hab is given by  2 2 σ N H ab = eμ H ab e(σ H ab /2) eσ H ab − 1 = 2.0 × 108 ≈ 200 million.

(8)

In other words, the standard deviation of the number of Habitable Planets is 200 million. And so, with probability one-sigma, we might expect the actual number of Habitable Planets to rise up 100 million plus 200 million = 300 million. A more profound discussion would be required by find “how many sigmas or parts of sigma” would yield non-negative standard deviations as given by (8), but we have no time to deal with this problem now, with apologies. Finally, the median (fifty-fifty probability of the lognormal distribution shown in Fig. 3) yields a value of median = m = eμ H ab = 4.521 × 107 ≈ 45 million.

(9)

392

Societal Statistics by Virtue of the Statistical Drake Equation

Table 2 Comparing the results of the Statistical Dole and Drake equation found by inputting to them the Input Tables 2 and 1, respectively Statistical Dole equation

Statistical Drake equation

Mean value of the TOTAL NUMBER of

Habitable Planets in the Galaxy ~100 million

ET civilizations in the Galaxy ~4590

Standard Deviation of the TOTAL NUMBER of

Habitable Planets in the Galaxy ~200 million

ET civilizations in the Galaxy ~11,195

11 Comparing the Statistical Dole and Drake Equations: Number of Habitable Planets Versus Number of ET Civilizations in This Galaxy It is now appropriate to make a comparison between the number of Habitable Planets and the number of expected ET Civilizations in the Galaxy. In other words, we want the “get the feeling” of the numbers that we have worked out in this paper just to see if the comparison among them “makes sense”. This we can do by putting on a same table the mean value and standard deviation of the total number of both Habitable Planets and ET Civilizations. The result is in Table 2, clearly showing how much “more rare” the ET Civilizations are with respect to the Habitable Planets. Roughly, one has N H ab  100 million ≈ 20, 202, = N E T  4950

(10)

so that the Habitable Planets seem to 20,000 more frequent than ET Civilizations, or, if you wish, only one ET Civilization emerges out of 20,000 Habitable Planets.

12 The Probability Distribution of the Ratio of Two Lognormally Distributed Random Variables This section is “just mathematical”, meaning that we shall only face and solve the mathematical problem of finding the probability distribution of the ratio of two lognormally distributed random variables. Why we are doing so will become apparent in the next section. So, let us consider a first lognormally distributed random variable X, as usual characterized by its own two parameters μX and σ X , and thus having the pdf: f X (x) =

1 1 2 2 e−((ln(x)−μ X ) /(2σ X )) (x ≥ 0). √ x 2π σ X

(11)

12 The Probability Distribution of the Ratio of Two Lognormally …

393

Table 3 Summary of the properties of the LOL (lognormal-over-lognormal) distribution that applies to the random variable Random variable X = N = number of communicating ET civilizations in the Galaxy = X left-hand-side of the Drake equation Probability distribution

Lognormal

Probability density function

f X (x) =

    1 √ 1 − (ln(x)−μ X )2 / 2σ X2 x 2πσ X e

(x ≥ 0)

Random variable Y = N Hab = number of Habitable Planets for Man in the Galaxy = Y Left-hand-side of the Dole equation Probability distribution

Lognormal

Probability density function

f Y (y) =

Random variable Z

Z=

Probability distribution

No name, but we call it… LOL = Lognormal-Over-Lognormal

Probability density function

fZ (z) =   2 2 z (μ X −μY )/ σ X +σY −1 √

Mean value

Z  = eμ X −μY e

Variance

X Y

    1 √ 1 − (ln(y)−μY )2 / 2σY2 y 2πσY e

=

N N H ab

=

(y ≥ 0)

N s× f p×ne× f l× f i× f c× f L N s× f p×ne× f l

= fi × fc× f L

    2  2 2 2 1 e− (μ X −μY ) +(ln(z)) / 2 σ X +σY 2 2 2π σ X +σY

(z ≥ 0)

 2  σ X +σY2 /2

 2 2  2 2 σ Z2 = e2(μ X −μY )+σ X +σ y eσ X +σ y − 1

   2 2 (μ −μ )+ σ X2 +σ y2 /2 σZ = e X Y eσ X +σ y − 1  2    2 2 All the moments, Z k = ek(μ X −μY ) ek σ X +σY /2 i.e. kth moment

Standard deviation

Mode (= abscissa of the lognormal peak)

z mode ≡ z peak = eμ X −μY e−

Value of the Mode Peak

f Z (z mode ) =

Median (= fifty–fifty probability value for Z)

median = m Z = eμ X −μY

√

 2  σ X +σY2

  −(μ X −μY ) e σ X2 +σY2 /2 1 e 2π σ X2 +σY2

Next to this, we also consider one more arbitrary and different lognormally distributed random variable Y having the pdf: f Y (y) =

1 1 2 2 e−((ln(y)−μY ) /2σY ) (y ≥ 0). √ y 2π σY

(12)

394

Societal Statistics by Virtue of the Statistical Drake Equation

The problem we solve in this section is to answer the question: what is the pdf of the third positive random variable Z given by the ratio of the two previous positive and lognormally distributed X and Y? Z=

a certain log normal distribution X = . Y another log normal distribution

(13)

Standard textbooks about Probability Theory (for instance, see the book by Papoulis and Pillai, “Probability Random Variables and Stochastic Processes”, Ref. [6], in particular pp. 186–187 and Eqs. (6)–(59)) tell us that the random variable Z has its pdf given by the integral: ∞ |y| f X Y (yz, y)dy,

f Z (z) =

(14)

−∞

where the function f XY (…, …) is the joint pdf of the two random variables X and Y. Now, if we assume the two random variables X and Y to be statistically independent of each other, inasmuch as “the physics” tells us this to be the case, then their joint pdf f XY (…, …) simply is the product of the two pdfs, i.e. the two lognormal distributions (11) and (12). That is ln(x)−μ X ) ln(y)−μY ) −( −( 1 1 1 1 2 2σ X 2σY2 e e √ √ x 2π σ X y 2π σY 2

f X,Y (x, y) = f X (x) f Y (y) =  − 1 e = 2π σ X σY x y

(ln(x)−μ X )2 + (ln(y)−μY )2 2 2σ X

2σY2

2



.

(15)

This is the joint pdf that must be introduced into the integral (14). Notice, however, that the integral actually ranges from 0 to infinity only, since both x and y do so. The modulus affecting y in (14) thus disappears also, and we are just left with the computation of the definite integral ∞ f Z (z) = f (X / Y ) (z) =

y f X,Y (yz, y)dy.

(16)

0

That is, replacing the last expression (15) under the integral sign, 1 f Z (z) = 2π σ X σY

∞ y

1 −[((ln(yz)−μ X )2 /2σ X2 )+((ln(y)−μY )2 /2σY2 )] e dy yz

0

1 = 2π σ X σY z

∞ 0

e−[((ln(yz)−μ X )

2

/2σ X2 )+((ln(y)−μY )2 /2σY2 )]

dy.

(17)

12 The Probability Distribution of the Ratio of Two Lognormally …

395

This is a tough integral to compute analytically. Basically, it can be reduced to the Gauss integral, i.e. to the normalization condition of the ordinary Gaussian curve, but many, many steps are required to perform the integration with respect to y. This author, when faced with this analytical computation, turned to Macsyma (concisely described, for instance, at the site: https://en.wikipedia.org/wiki/Mac syma). Macsyma is a wonderful Computer Algebra code that was created at the MIT Artificial Intelligence Laboratory back in the 1960s to let NASA re-compute analytically the orbits requested for the Apollo astronauts to safely reach the Moon and come back. So, Macsyma was able to perform the analytical integration in (17) in a matter of seconds. The outcome is the function of z given by 1 2 2 2 2 2 2 e−(((μ X −μY ) +(ln(z)) )/(2(σ X +σY ))) . f Z (z) = z ((μ X −μY )/(σ X +σY ))−1 √  2 2 2π σ X + σY (18) As one can see, this function of z is a complicated mix of exponentials in z through the natural log of z squared, times a power of z, times many other constants. Also, the pdf (18) does not have any specific name. Thus, this author thought of giving it a name, and called it LOL, and acronym for Lognormal-Over-Lognormal pdf, as indeed it is. N X N s × f p × ne × f l × f i × f c × f L = = Y N H ab N s × f p × ne × f l = f i × f c × f L = Societal Par t o f the Statistical Drake Equation.

Z=

Though the LOL pdf (18) is difficult for humans to handle by hand, it can be easily handled by Macsyma! Thus, one can quickly prove that it indeed fulfills the normalization condition of any pdf ∞ f Z (z)dz = 1.

(19)

0

A similar calculation then shows that the mean value of the LOL random variable Z reads ∞ μ Z = Z  =

z f Z (z)dz = eμ X −μY e(σ X +σY )/2 . 2

2

(20)

0

The corresponding LOL variance was again found by Macsyma through a similar calculation, and reads  2 2  2 2 (21) σ Z2 = e2(μ X −μY )+σ X +σ y eσ X +σ y − 1 .

396

Societal Statistics by Virtue of the Statistical Drake Equation

Its square root is thus the relevant LOL standard deviation: σZ = e

(μ X −μY )+((σ X2 +σY2 )/2)



eσ X +σ y − 1. 2

2

(22)

In order to find the mode of the pdf (18), i.e. the abscissa of its peak, we must find out the first derivative of (18) with respect to z and then set the resulting equation equal to zero. Good old Macsyma did a good job again, and the two results are the two abscissas of the minimum of (18), obviously at z = 0, and of the maximum (i.e. the peak, or mode) at z mode ≡ z peak = eμ X −μY e−(σ X +σY ) . 2

2

(23)

Replacing this into (18) and rearranging, we the find the LOL pdf peak value: f Z (z mode ) = √

1 2 2  e−(μ X −μY ) e(σ X +σY )/2 . 2π σ X2 + σY2

(24)

To summarize, all the statistical results about the LOL random variable Z that we derived in this section are listed in Table 3.

13 Breaking the Drake Equation up into the Dole Equation Times the Drake Equation’s Societal Part Now we are ready to achieve the main result of the present chapter, namely estimating the societal part of the Drake equation by virtue of both the full Drake equation and the Dole equation. The idea is as follows. The Dole equation yields the number of Habitable Planets for Man in the Galaxy. This number we regard as the product of the first four terms of the Drake equation, namely N H ab = N s × f p × ne × f l.

(25)

This “new definition” of the Dole equation we regard as acceptable inasmuch as for Man to live on a certain planet not only must that planet exist, have liquid water and be breathable, but also animals and fishes (= food) as we know them must be there to let Man survive and progress. Thus, the fl term of the Drake equation must be included in the Dole equation also inasmuch as it symbolizes all sorts of living creatures (from trilobites up to monkeys) that we know developed during the Evolution on Earth but may not be regarded as “intelligent” as Homo Sapiens is.

13 Breaking the Drake Equation up into the Dole Equation …

397

Having accepted (25) as the new definition of the Dole equation as a subset of the Drake equation (1), we may then break up the Drake equation (1) into two parts: the Dole equation (25) times the remaining last three terms of the Drake equation (1), that is N = N H ab × f i × f c × f L .

(26)

From this partition it obviously follows that: fi × fc× f L =

Drake lognormal N . = N H ab Dole lognormal

(27)

But then we made a “discovery”, that is: the societal part of the Drake equation, namely the product of the three random variables fi × fc × fL, has a probability distribution given by the ratio of the two lognormal distributions of Drake (at the numerator) over Dole (at the denominator). This is the key new result presented in this chapter. We may now use the mathematics developed in the previous Sect. 11 to immediately conclude that the pdf of the societal part of the Drake equation is given by (18), namely f f i× f c× f L (z) = z ((μ X −μY )/(σ X +σY ))−1 √ 2

2



1

e−(((μ X −μY )

2

+(ln(z))2 )/(2(σ X2 +σY2 )))

2π σ X2 + σY2

. (28)

This is the key result presented in this chapter for the first time, namely we have identified the following three random variables: ⎧ ⎪ ⎨ N=X N H ab = Y ⎪ ⎩ N = Z. N H ab

(29)

This of course implies the identification of the relevant means and standard deviations also, that is it implies the following four equations: ⎧ ⎪ ⎪ ⎨

μ X = μ N = N  σX = σN ⎪ μ = μ H ab = N H ab  ⎪ ⎩ Y σY = σ N H ab . Replacing (30) into (28), the latter is turned into its final form

(30)

398

Societal Statistics by Virtue of the Statistical Drake Equation 

2 2 z ((μ N −μ N H ab ) (σ N +σ H ab ))−1 −(((μ N −μ H ab )2 +(ln(z))2 ) (2(σ N2 +σ H2 ab )))  . f f i× f c× f L (z) = e √ 2π σ N2 + σ H2 ab

(31) This is the probability density function followed by the Societal Statistics of Human-like beings in the Galaxy. Is it a lognormal distribution? No, of course. But it reduces to a lognormal distribution in the special case μ N = μ H ab .

(32)

This is the case when all Habitable planets are indeed inhabited by Humans (statistically speaking), so that the two mean values coincide. In fact, replacing (32) into (31), one finds that (31) reduces in this special case to  z −1 2 2 2 e−(((ln(z)) ) (2(σ N +σ H ab ))) , f f i× f c× f LIμN =μ H ab (z) = √  2π σ N2 + σ H2 ab

(33)

which is indeed a lognormal distribution with zero mean value and variance σ N2 +

σ H2 ab .

14 Conclusions We have written this chapter for the mathematical delight of showing how “playing mathematically” with both the Drake and Dole equations one may find new and unexpected results concerning the Societal Part of the Drake equation, that is notoriously something “nobody knows about”. This paper is just a beginning: for instance how could one modify the Drake equation by inserting a special further term dealing with the “end of life” caused by an asteroid impacting on a living planet? These questions deserve far more mathematical research work. Acknowledgements The author is grateful to the leader of the Tau Zero Foundation, Marc Millis, and to Drs. Hal Puthoff and Eric Davis of the Institute for Advanced Study at Austin for the reading the author’s paper #IAC-08-A4.1.4 and letting him have their appreciated comments. Also, the author is grateful to Drs. Jill Tarter, Paul Davies, Seth Shostak, Doug Vakoch, Tom Pierson, Carol Oliver, Paul Shuch and Kathryn Denning for attending his first presentation ever about these topics at the “Beyond” Center of the University of Arizona at Phoenix on February 8th, 2008. He also would like to thank Dr. Dan Werthimer and his School of SETI experts at the Berkeley Space Sciences Laboratory for keeping alive the interplay between experimental and theoretical SETI. But the greatest “thanks” go of course to the Teacher to all of us: Professor Frank D. Drake, whose equation opened to Humanity a new way of thinking about the past and the future of Humans in the Galaxy.

References

399

References 1. E. Claude, Shannon: a mathematical theory of communication. Bell Syst. Tech. J. 27, 379–423, 623–656 (1948) 2. S.H. Dole, Habitable planets for Man, 1st edn, 1964, © 1964 by the RAND Corporation, Library of Congress Catalogue Card Number 64-15992. See in particular page 82, i.e. the beginning of Chapter 5, entitled Probability of Occurrence of Habitable Planets 3. C. Maccone, The Statistical Drake Equation, paper #IAC-08-A4.1.4 presented on October 1, 2008, at the 59th International Astronautical Congress (IAC) held in Glasgow, Scotland, UK, September 29–October 3, 2008 4. C. Maccone, The statistical Drake equation and A.M. Lyapunov’s theorem—problems of search for extraterrestrial intelligence, Part I, Int. Sci. J.—Actual Probl. Aviat. Aerosp. Syst. 16(1(32)), 38–63 (2011) (This is essentially a Russian translation of Ref. [4]) (in Russian) 5. C. Maccone, The statistical Drake Equation. Acta Astronaut. 67, 1366–1383 (2010). (This was a special Acta Astronautica volume edited by John Elliott and collecting all papers presented at the First IAA Symposium on “Searching for Life Signatures”, held at UNESCO, Paris, September 22–26, 2008) 6. A. Papoulis, S.U. Pillai, Probability, Random Variables and Stochastic Processes, 4th edn (TataMcGraw-Hill, New Delhi, 2002), pp.186–187 7. C. Sagan, Cosmos (Random House, New York, 1983), pp. 298–302 8. Wikipedia 2011 site about the Drake equation: https://en.wikipedia.org/wiki/Drake_equation

Evolution and History in a New “Mathematical SETI” Model

Abstract In a paper (Maccone, Orig. Life Evol. Biosph. (OLEB) 41:609–619, 2011, [15]) and in a book (Maccone, Mathematical SETI, 2012, [17]), this author proposed a new mathematical model capable of merging SETI and Darwinian Evolution into a single mathematical scheme. This model is based on exponentials and lognormal probability distributions, called “b-lognormals” if they start at any positive time b (“birth”) larger than zero. Indeed: (1) Darwinian evolution theory may be regarded as a part of SETI theory in that the factor f l in the Drake equation represents the fraction of planets suitable for life on which life actually arose, as it happened on Earth. (2) In 2008 (Maccone, The Statistical Drake Equation, 2008, [9]) this author firstly provided a statistical generalization of the Drake equation where the number N of communicating ET civilizations in the Galaxy was shown to follow the lognormal probability distribution. This fact is a consequence of the Central Limit Theorem (CLT) of Statistics, stating that the product of a number of independent random variables whose probability densities are unknown and independent of each other approached the lognormal distribution if the number of factors is increased at will, i.e. it approaches infinity. (3) Also, in Maccone (Orig. Life Evol. Biosph. (OLEB) 41:609–619, 2011, [15]), it was shown that the exponential growth of the number of species typical of Darwinian Evolution may be regarded as the geometric locus of the peaks of a one-parameter family of b-lognormal distributions constrained between the time axis and the exponential growth curve. This was a brand-new result. And one more new and far-reaching idea was to define Darwinian Evolution as a particular realization of a stochastic process called Geometric Brownian Motion (GBM) having the above exponential as its own mean value curve. (4) The b-lognormals may be also be interpreted as the lifespan of any living being, let it be a cell, or an animal, a plant, a human, or even the historic lifetime of any civilization. In Maccone (Mathematical SETI, 2012, [17, Chapters 6, 7, 8 and 11]), as well as in the present paper, we give important exact equations yielding the b-lognormal when its birth time, senility-time (descending inflexion point) and death time (where the tangent at senility intercepts the time axis) are known. These also are brand-new results. In particular, the σ = 1 b-lognormals are shown to be related to the golden ratio, so famous in the arts and in architecture, and these special b-lognormals we call “golden b-lognormals”. (5) Applying this new mathematical apparatus to Human History leads to the discovery of the exponential trend of progress between Ancient Greece and the current USA © Springer Nature Switzerland AG 2020 C. Maccone, Evo-SETI, https://doi.org/10.1007/978-3-030-51931-5_11

401

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Evolution and History in a New “Mathematical SETI” Model

Empire as the envelope of the b-lognormals of all Western Civilizations over a period of 2500 years. (6) We then invoke Shannon’s Information Theory. The entropy of the obtained b-lognormals turns out to be the index of “development level” reached by each historic civilization. As a consequence, we get a numerical estimate of the entropy difference (i.e. the difference in the evolution levels) between any two civilizations. In particular, this was the case when Spaniards first met with Aztecs in 1519, and we find the relevant entropy difference between Spaniards an Aztecs to be 3.84 bits/individual over a period of about 50 centuries of technological difference. In a similar calculation, the entropy difference between the first living organism on Earth (RNA?) and Humans turns out to equal 25.57 bits/individual over a period of 3.5 billion years of Darwinian Evolution. (7) Finally, we extrapolate our exponentials into the future, which is of course arbitrary, but is the best Humans can do before they get in touch with any alien civilization. The results are appalling: the entropy difference between aliens 1 million years more advanced than Humans is of the order of 1000 bits/individual, while 10,000 bits/individual would be requested to any Civilization wishing to colonize the whole Galaxy (Fermi Paradox). (8) In conclusion, we have derived a mathematical model capable of estimating how much more advanced than humans an alien civilization will be when SETI succeeds. Keywords Darwinian evolution · Statistical drake equation · Lognormal probability densities · Human history · Entropy

1 Introduction: Interstellar Flight and SETI Are the Two Sides of the Same Coin 1.1 Astronautics and Interstellar Flight Since the Moon Landings When humans first landed on the moon on July 20, 1969, a new era in the history of humanity started: expansion into space became our “manifest destiny”. Driven by these events, some scientists then began studying how to overcome the vast distances of interstellar space in agreement with the laws of known physics (interstellar flight). That task, however, proved to be much more difficult than expected. As a consequence, in the 49 years elapsed since the last moon landing (December 14, 1972), not only no more moon landing occurred, but the very spirit of manned space exploration slowly faded away and was replaced by a spirit of “robotic exploration first”. As of 2021, this situation is of course fully justified by the increasing financial difficulties in the Western World and the changed balance of power: China replaced the Soviet Union as main rival of the USA. But she is hardly technologically as good as the USA yet, and so manned spaceflight has few supporters at the moment, except for new moon landings or asteroid landings. Nevertheless, in January 2011 NASA and DARPA launched a new initiative: the 100 Year Spaceship Study (100YSS, described, for instance, at: http://en.wik ipedia.org/wiki/100_Year_Starship). Two 100YSS conferences were later held at

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Orlando (September 30 and October 1 and 2, 2011) and Houston (September 13– 16, 2012), and they gathered most interstellar flight experts worldwide. The idea of manned interstellar flight was thus revived, in spite of all financial and technological difficulties suffered by today’s astronautics.

1.2 The “FOCAL” Space Missions Enabling Us to Use the Sun as a Gravitational Lens (1992) This author attended both the above 100YSS conferences and suggested, as a first practical step towards interstellar flight, to send a large (12 m) relay antenna spacecraft to at least the distance of 550 AU from the Sun in the direction opposite to the target star system. This mission he dubbed FOCAL (an acronym for “Fast Outgoing Cyclopean Astronomical Lens”) and it would open up to Humanity the use of the Sun as a gravitational lens, thus dramatically reducing the powers necessary to keep the radio communications (link) between Earth and any future truly interstellar probe [10]. FOCAL could also be used to get hugely magnified radio pictures of exoplanets. Figure 1 shows the Sun gravity lens and two positions of the FOCAL spacecraft: the minimal one at 550 AU from the Sun, and another one, much further out.

Fig. 1 Sun gravity lens basic geometry, showing its minimal focal length at 550AU (i.e. 3.17 lightdays (about half a light-week) or about 13 times the distance of Pluto from the Sun). The FOCAL spacecraft, also shown on the right, must at least reach this distance of 550 AU away from the Sun in order to take advantage of the huge radio magnification provided by the focusing effect of the Sun’s gravity lens. In reality, however, it would be preferable for FOCAL to reach the distance of about 1000 AU from the Sun, in order to get rid of the undesired “coronal effects”, namely the divergent-lens effect on incoming radio waves created by the electrons in the lower layers of the Sun Corona. For instance, at the Cosmic Microwave Background (CMB) peak frequency of about 160 GHz, the “true focus” of the Sun’s gravity lens plus its Coronal effects turns out to be at 763 AU. The FOCAL space mission was studied in detail in [10]

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1.3 A “RADIO BRIDGE” Among Two Stars to Enable the Radio Link Among Their Civilizations (2011) The consideration of the Sun as a gravitational lens for use in interstellar communications also inevitably led to one more new idea: the “radio bridge” among any two stars in the Galaxy. In fact, any pair of stars (rather than just one star only) could be used as gravitational lenses at the same time, thus enabling the radio link between their two hosted civilizations to become feasible at low power cost. Of course, two FOCAL space missions, either on the opposite side of each star, would be required to complete this radio bridge, which also implies either an agreement between the two civilizations or a single civilization expanding up to another nearby star. For example, suppose for a moment that a human spacecraft was already able to reach the nearest star system (Alpha Centauri, 4.37 light years away). The question would then be: “what shall we do with that spacecraft after it got there?” This author’s answer is “let us put it on the other side of Alpha Centauri B with respect to the Sun, and thus create a radio bridge between the Sun and the Alpha Centauri B system. That would enable cheap (in terms of the required powers) radio communications between the two stellar systems thus enabling the further exploration of the Alpha Centauri B system, where a small planet was recently discovered” [16]. Figure 2 shows the Radio Bridge between any two stars, with the two relevant FOCAL spacecrafts on opposite sides.

Fig. 2 Radio bridge between any two stars. The electromagnetic waves path (red, solid lines) is deflected by the gravity field of each star, and made to focus on the two FOCAL spacecrafts placed on opposite sides with respect to the two stars. These two FOCAL spacecrafts must be strictly positioned along the axis passing through the two stars’ centers. Then, the transmission powers between the #1 FOCAL spacecraft and the #2 FOCAL spacecraft are greatly reduced. In fact, the huge (antenna) gains of the two stars, plus the modest (antenna) gains of the two FOCAL spacecrafts combine to yield a huge total (antenna) gain of the whole system, meaning that the transmissions powers between the two stellar systems become quite affordable. This is the key to build a Galactic Internet, that might already have been created in the Galaxy by aliens capable of sending FOCAL spacecrafts around. But we, humans, still are cut off from all that since we did not yet send any FOCAL spacecraft 550 AU away from the Sun, namely we have not yet learned about how to use the Sun as a gravitational magnifying lens. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article)

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1.4 A Galactic Internet Might Be in Use Already, but by Aliens, Not by Humans yet (2021) Having understood what a radio bridge among two stars is, one may now take next, bolder and final step: the Galactic Internet. By this we mean that one or more alien civilizations harboring in the Galaxy might have understood the use of radio bridges among stars long ago already. Thus, they might have created a Galactic Internet among the colonized stellar systems according to their needs. This is no science fiction: it is just the physics of star gravitational lensing applied to more than one civilization in the Galaxy. In [18] this author computed how much information could be handled by radio bridges among the Sun and Alpha Centauri A, the Sun and Barnard’s star, the Sun and Sirius A, and even between the Sun and a Sun-like star located at the center of the Galaxy and even in M31 (Andromeda galaxy). Also, use of the KLT (Karhunen-Loève Transform, a signal-extraction algorithm much more powerful than the Fourier Transform) would improve things significantly in this regard (see Refs. [13, 23]). Now, radio communications between a pair of (not too far apart) stars in the Galaxy is exactly the goal of SETI, the Search for Extraterrestrial Intelligence, and so to SETI we now turn.

2 SETI and Darwinian Evolution Merged Mathematically 2.1 Introduction: The Drake Equation (1961) as the Foundation of SETI In 1961 the American astronomer Frank D. Drake tried to estimate the number N of communicating civilizations in the Milky Way galaxy by virtue of a simple equation now called the Drake equation. N was written as the product of seven factors, each a kind of filter, every one of which must be sizable for there to be a large number of civilizations: Ns fp ne fl fi fc fL

the number of stars in the Milky Way Galaxy; the fraction of stars that have planetary systems; the number of planets in a given system that are ecologically suitable for life; the fraction of otherwise suitable planets on which life actually arises; the fraction of inhabited planets on which an intelligent form of life evolves (as in human history); the fraction of planets inhabited by intelligent beings on which a communicative technical civilization develops (as we have here now); the fraction of planetary lifetime graced by a technical civilization (a totally unknown factor).

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Written out, the equation reads N = N s · f p · ne · f l · f i · f c · f L .

(1)

All the f ’s are fractions, having values between 0 and 1; they will pare down the large value of Ns. To derive N we must estimate each of these quantities. We know a fair amount about the early factors in the equation, the number of stars and planetary systems. We know very little about the later factors, concerning the evolution of life, the evolution of intelligence or the lifetime of technical societies. In these cases our estimates will be little better than guesses. In the 50 years elapsed since Drake proposed his equation, a number of scientists and writers tried either to improve it or criticize it in many ways. For instance, in 1980, Walters et al. [25] suggested to insert a new parameter in the equation taking interstellar colonization into account. In 1981, Wallenhorst [24] tried to prove that there should be an upper limit of about 100 to the number N. In 2004, Ksanfomality [7] again asked for more new factors to be inserted into the Drake equation, this time in order to make it compatible with the peculiarities of planets of Sun-like stars. Also ´ the temporal aspect of the Drake equation was stressed by Cirkovi´ c [3]. But while these authors were concerned with improving the Drake equation, other simply did not consider it useful and preferred to forget about it, like Burchell [2]. Also, it has been correctly pointed out that the habitable part of the Galaxy is probably much smaller than the entire volume of the Galaxy itself (the important relevant references are [5, 6, 8]). For instance, it might be a sort of a torus centered around the so called “corotation circle”, i.e. a circle around the Galactic Bulge such that stars orbiting around the Bulge and within such a torus never fall inside the dangerous spiral arms of the Galaxy, where supernova explosions would probably fry any living organism before it could develop to the human level or beyond. Fortunately for Humans, the orbit of the Sun around the Bulge is just a circle staying within this torus for 5 billion years or more [1, 19]. In all cases the final result about N has always been a sheer number, i.e., a positive integer number ranging from 1 to thousands or millions. This “integer or real number” aspect of all variables making up the Drake equation is what this author regarded as “too simplistic”. He extended the Drake equation so as to embrace Statistics in his 2008 paper [9]. This paper was later published in Acta Astronautica [11], and more mathematical consequences were derived in Maccone [12, 14].

2.2 Statistical Drake Equation (2008) Consider Ns, the number of stars in the Milky Way Galaxy, i.e. the first independent variable in the Drake Equation (1). Astronomers tell us that approximately there should be about 350 billions stars in the Galaxy. Of course, nobody has counted all stars in the Galaxy! There are too many practical difficulties preventing us from doing so: just to name one, the dust clouds that do not allow us to see even the Galactic

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Bulge (central region of the Galaxy) in visible light, although we may “see it” at radio frequencies like the famous neutral hydrogen line at 1420 MHz. So, it does not really make much sense to say that Ns = 350 × 109 , or similar fanciful exact integer numbers. More scientific is saying that the number of stars in the Galaxy is 350 billion plus or minus, say, 50 billions (or whatever values the astronomers may regard as more appropriate). It makes thus sense to replace each of the seven independent variables in the Drake Equation (1) by a mean value (350 billions, in the above example) plus or minus a certain standard deviation (50 billions, in the above example). By doing so, we made a step ahead: we have abandoned the too-simplistic Eq. (1) and replaced it by something more sophisticated and scientifically serious: the statistical Drake equation. In other words, we have transformed the simplistic classical Drake Equation (1) into a statistical tool capable of investigating of a host of facts hardly known to us in detail. In other words still: 1. We replace each independent variable in (1) by a random variable, labeled Di (from Drake). 2. We assume the mean value of each Di to be the same numerical value previously attributed to the corresponding input variable in (1). 3. But now we also add a standard deviation σ Di on each side of this mean value, as provided by the knowledge obtained by scientists in the discipline covered by each Di . Having so done, we wonder: how can we find out the probability distribution for each Di ? For instance, shall that be a Gaussian, or what? This is a difficult question, for nobody knows, for instance, the probability distribution of the number of stars in the Galaxy, not to mention the probability distribution of the other six variables in the Drake Equation (1). In 2008, however, this author found a way to get around this difficulty, as explained in the next section.

2.3 The Statistical Distribution of N Is Lognormal The solution to the problem of finding the analytical expression for the probability density function of the positive random variable N is as follows: 1. Take the natural logs of both sides of the statistical Drake Equation (1). This changes the product into a sum. 2. The mean values and standard deviations of the logs of the random variables Di may all be expressed analytically in terms of the mean values and standard deviations of the Di [9]. 3. The central limit theorem (CLT) of statistics, states that (loosely speaking) if you have a sum of independent random variables, each of which is arbitrarily distributed (hence, also including uniformly distributed), then, when the number of terms in the sum increases indefinitely (i.e. for a sum of random variables infinitely long)…, the sum random variable approaches a Gaussian.

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Table 1 Summary of the statistical properties of the lognormal distribution that applies to the random variable N = number of ET communicating civilizations in the Galaxy Random variable

N = number of communicating ET civilizations in Galaxy

Probability distribution

Lognormal

Probability density function

f N (n) =

Mean value

1 n

N  = eμ e

2 2 √ 1 e−(ln(n)−μ) /2σ 2πσ

·



All the moments, i.e. kth moment

 2 eσ − 1  2 2 σ N = eμ eσ /2 eσ − 1  k 2 2 N = ekμ ek ·(σ /2)

Mode (= abscissa of the lognormal peak)

n mode ≡ n peak = eμ e−σ

Value of the mode peak

f N (n mode ) =

Median (= fifty–fifty probability value for N)

Median = m

Skewness

K3 (K 2 )3/ 2

Kurtosis

K4 (K 2 )2

Expression of μ in terms of the lower (ai ) and upper (bi ) limits of the Drake uniform input random variables Di

μ= 7

Expression of σ 2 in terms of the lower (ai ) and upper (bi ) limits of the Drake uniform input random variables Di

2 σ 7=

Variance Standard deviation

(n ≥ 0)

σ 2 /2

σ N2 = e2μ eσ

2

√1 2πσ = eμ

2

· e−μ · eσ

2 /2

 2 2 = (eσ + 2) eσ − 1 2

2

2

= e4σ + 2e3σ + 3e2σ − 6

i=1 Yi 

2 i=1 σYi

=

=

7 i=1

7 i=1

bi [ln(bi )−1]−ai [ln(ai )−1] bi −ai

 1−

ai bi [ln(bi )−ln(ai )]2 (bi −ai )2



4. Thus, the ln(N) approaches a Gaussian. 5. Namely, N approaches the lognormal distribution. Table 1 shows the most important statistical properties of a lognormal. 6. The mean value and standard deviations of this lognormal distribution of N may be expressed analytically in terms of the mean values and standard deviations of the logs of the Di already found previously, as shown in Table 1. For all the relevant mathematical proofs, more mathematical details and a few numerical examples of how the statistical Drake equation works, see Maccone [11].

2.4 Darwinian Evolution as (Overall) Exponential Increase in the Number of Living Species Consider now Darwinian evolution. To assume that the number of species increased exponentially over the 3.5 billion years of evolutionary time span is certainly a gross oversimplification of the real situation, as proven, for instance, by Rohde and Muller

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[21]. However, we will now temporarily assume this exponential increase of the number of living species in time just in order to cast the theory into a mathematically simple form. We will do much better in Sect. 2.5 by re-defining the exponential not as the actual number of living species at a certain instant, but rather as the mean value of an important stochastic process called geometric Brownian motion (GBM). In other words, we now assume that 3.5 billion years ago there was on Earth only one living species, whereas now there may be (say) 50 million living species or more (see, for instance, the site http://en.wikipedia.org/wiki/Species). Note that the actual number of species currently living on earth does not really matter as a number for us: we just want to stress the exponential character of the growth of species. Thus, we shall assume that the number of living species on Earth increases in time as E(t) (standing for “exponential in time”): E(t) = Ae Bt

(2)

where A and B are two positive constants that we will soon determine numerically. Let us now adopt the convention that the current epoch corresponds to the origin of the time axis, i.e. to the instant t =0. This means that all the past epochs of Darwinian evolution correspond to negative times, whereas the future ahead of us (including finding ETs) corresponds to positive times. Setting t = 0 in (2), we immediately find E(0) = A

(3)

proving that the constant A equals the number of living species on earth right now. We shall assume A = 50 million species = 5 × 107 species.

(4)

To also determine the constant B numerically, consider the two values of the exponential (2) at two different instants t 1 and t 2 , with t 1 < t 2 , that is 

E(t1 ) = A e B t1 E(t2 ) = A e B t2 .

(5)

Dividing the lower equation by the upper one, A disappears and we are left with an equation in B only E(t2 ) = e B(t2 −t1 ) . E(t1 )

(6)

ln(E(t2 )) − ln(E(t1 )) . t2 − t1

(7)

Solving this for B yields B=

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Fig. 3 Darwinian exponential curve representing the growing number of species on Earth up to now

We may now impose the initial condition stating that 3.5 billion years ago there was just one species on Earth, the first one (whether this was RNA is unimportant in the present simple mathematical formulation)

t1 = −3.5 × 109 years E(t1 ) = 1 whence ln(E(t1 )) = ln(1) = 0.

(8)

The final condition is of course that today (t 2 = 0) the number of species equals A given by (4). Upon replacing both (4) and (8) into (7), the latter becomes B=−

1.605 × 10−16 ln(5 × 107 ) ln(E(t2 )) = . =− t1 −3.5 × 109 year s

(9)

Having thus determined the numerical values of both A and B, the exponential in (2) is fully specified. This curve is plot in Fig. 3 just over the last billion years, rather than over the full range between −3.5 billion years and now.

2.5 Darwinian Evolution Is Just a Particular Realization of Geometric Brownian Motion in the Number of Living Species Consider again the exponential curve described in the previous section. The most frequent question that non-mathematically minded persons ask this author is: “then you do not take the mass extinctions into account”. The answer to this objection is that our exponential curve is just the mean value of a certain stochastic process that may run above and below that exponential in a totally unpredictable way. Such a

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Fig. 4 Geometric Brownian motion: Two particular realizations of the stochastic process called geometric Brownian motion (GBM) taken from the Wikipedia site http://en.wikipedia.org/wiki/ Geometric_Brownian_motion. Their mean values is the exponential (2) with different values of A and B for each shown stochastic process

stochastic process is called geometric Brownian motion (abbreviated GBM) and is described, for instance, at the web site: http://en.wikipedia.org/wiki/Geometric_Bro wnian_motion, from which Fig. 4 is taken. In other words, mass extinctions that occurred in the past are indeed taken into account as unpredictable fluctuations in the number of living species occurred in the particular realization of the GBM between −3.5 billion years and now. So, extinctions are “unpredictable vertical downfalls” in that GBM plot that may indeed happen from time to time. Also notice that: 1. The particular realization of GBM occurred over the last 3.5 billion years is very much unknown to us in its numeric details, but… 2. We would not care either, inasmuch as the theory of stochastic processes only cares about such statistical quantities like the mean value and the standard deviation curves, that are deterministic curves in time with known equations.

3 Geometric Brownian Motion (GBM) Is the Key to Stochastic Evolution of All Kinds 3.1 The N(t) GBM as Stochastic Evolution On January 8, 2012, this author come to realize that his statistical Drake equation, previously described in Sects. 2.2 and 2.3, is the special static case (i.e. “the picture”, so as to say) of a more general time-dependent statistical Drake equation (i.e. “the movie”, so as to say) that we study in this section. In other words, this result is the powerful generalization in time of all results described in Sects. 1 and 2. This section is thus an introduction to a new, exciting mathematical model that one may call “exponential evolution in time of the statistical Drake equation”.

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To be precise, the number N in the statistical Drake Equation (1), yielding the number of extraterrestrial civilizations now existing and communicating in the Galaxy, is replaced in this section by a stochastic process N(t), jumping up and down in time like the number e raised to a Brownian motion, but actually in such a way that its mean value keeps increasing exponentially in time as N (t) = N0 eμt .

(10)

In (10), N 0 and μ are two constants with respect to the time variable t. Their meaning is, respectively: 1. N 0 is the number of ET communicating civilizations at the time t = 0, namely “now”, if one decides to regard the positive times (t > 0) as the future history of the Galaxy ahead of us, and the negative times (t < 0) as the past history of the Galaxy. 2. μ is a positive (if the number of ET civilizations increases in time) or negative (if the number of ET civilizations decreases in time) parameter that we may call “the drift”. To fix the ideas, and to be optimistic, we shall suppose μ > 0. This evolution in time of N(t) is just what we expect to happen in the Galaxy, where the overall number N(t) of ET civilizations does probably increase (rather than decrease) in time because of the obvious technological evolution of each civilization. But this N(t) scenario is a stochastic one, rather than a deterministic one, and certainly does not exclude temporary setbacks, like the end of civilizations due to causes as diverse as: (a) asteroid and comet impacts, (b) rogue planets or stars arriving from somewhere and disrupting the gravitational stability of the planetary system, (c) supernova explosions that would “fry” entire nearby ET civilizations (think of AGN, the Active Nucleus Galaxies and ask: how many ET civilizations are dying in those galaxies right now?), (d) ET nuclear wars, (e) and possibly more causes of civilization end that we do not know about yet. Mathematically, we came to define the probability density function (pdf) of this exponentially increasing stochastic process N(t) as the lognormal 1 2 2 2 N (t)_pdf(n, N0 , μ, σ, t)= √ √ e−([ln(n)−(ln N0 +μt−(σ t/2))] )/(2σ t)) 2π σ tn for 0 ≤ n ≤ ∞. (11)

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It is easy to prove that this lognormal pdf obviously fulfills the normalization condition

∞ N (t)_pdf(n, N0 , μ, σ, t) dn 0

∞ = 0

1 2 2 2 √ √ e−([ln(n)−(ln N0 +μt−(σ t/2))] ))/(2σ t) dn = 1. 2π σ tn

(12)

Also, the mean value of (11) yields indeed the exponential curve (10)

∞ n · N (t)_pdf(n, N0 , μ, σ, t) dn 0

∞ =

n√ 0

1

√

2πσ tn

e−([ln(n)−(ln N0 +μt−(σ

2

t/2))]2 )/(2σ 2 t))

dn = N0 eμt .

(13)

The proof of (12) and (13) is given in as the Maxima file “GBM_as_N_of_t_v47” of [17]. Table 2 summarizes the main properties of GBM, namely of this N(t) stochastic process.

3.2 Our Statistical Drake Equation Is the Static Special Case of N(t) In this section we prove the crucial fact that the lognormal pdf of our statistical Drake equation given in Table 1 is just “the picture” case of the more general exponentially growing stochastic process N(t) (“the movie”) having the lognormal pdf (11) as given in Table 2. To make things neat, let us denote by the subscript “GBM” both the μ and σ appearing in (11). The latter thus takes the form N (t)_pdf(n, N0 , μG B M , σG B M , t) 1 2 2 2 =√ √ e(−([ln(n)−(ln N0 +μG B M t−(σG B M t/2))] )/(2σG B M t)) 2π σG B M tn for 0 ≤ n ≤ ∞.

(14)

Similarly, the let us denote by the subscript “Drake” both the μ and σ appearing in the lognormal pdf given in the third line of Table 1 (this is also Eq. (1.B.56) of [17]), namely the pdf of our statistical Drake equation:

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Table 2 Summary of the properties of the lognormal distribution that applies to the stochastic process N(t) = exponentially increasing number of ET communicating civilizations in the Galaxy, as well as the number of living species on earth over the last 3.5 billion years. Clearly, these two different GBM stochastic processes have different numerical values of N 0 , μ and σ, but the equations are the same for both processes Stochastic process

N (t) =  (1) N umber o f E T civili zations (in S E T I ) (2) N umber o f living species (in evolution)

Probability distribution

Lognormal distribution of the geometric Brownian motion (GBM)

Probability density function

N (t)_ pd f (n, N0 , μ, σ, t) = 2 2 2 √ 1 √ e−([ln(n)−(ln N0 +μt−(σ t/2))] /2σ t) for n ≥ 0 2πσ tn

Mean value

N (t) = N0 eμt

Variance

σ N2 (t) = N02 e2μt (eσ t − 1)  2 σ N (t) = N0 eμt eσ t − 1  k  2 2 N (t) = N0k ekμt e(k −k)σ t/2 2

Standard deviation All the moments, i.e. kth moment Mode (= abscissa of the lognormal peak)

n mode ≡ n peak = N0 eμt e−(3σ

Value of the mode peak

f N (t) (n mode ) =

Median (= fifty–fifty probability value for N(t))

Median = m = N0 eμt e−(σ

Skewness

K3 (K 2 )3/ 2

Kurtosis

K4 (K 2 )2

= (eσ

= e4σ

2t

2t

√1 √ N0 2πσ t

2 t/2)

· e−μt .eσ

2t

2 t/2)

 2 + 2) eσ t − 1

+ 2e3σ

2t

+ 3e2σ

2t

−6

lognormal_ pdf_ of_ statistical_ Drake_ Eq(n, μ Drake , σ Drake ) 1 2 2 =√ e−([ln(n)−μ Drake ] )/(2σ Drake ) for 0 ≤ n ≤ ∞. 2π σ Drake n

(15)

Now, a glance at (14) and (15) reveals that they can be made coincide if, and only if, the two simultaneous equations hold ⎧ ⎨

2 σG2 B M t = σ Drake

2 ⎩ ln N0 + μG B M t − σG B M t = μ Drake . 2

(16)

On the other hand, when we pass (so as to say) “from the movie to the picture”, the two σ must be the same thing, and so must be the two μ, that is, one must have

3 Geometric Brownian Motion (GBM) Is the Key …



σG B M = σ Drake = σ μG B M = μ Drake = μ.

415

(17)

Checking thus the upper Eq. (17) against the upper Eq. (16), we are just left with t = 1.

(18)

So, t = 1 is the correct numeric value of the time leading “from the movie to the picture”. Replacing this into the lower Eq. (16), and keeping in mind the upper Eq. (17), the lower Eq. (16) becomes ln N0 + μG B M −

σ2 = μ Drake . 2

(19)

Since the two μ also must be the same because of the lower Eq. (17), then (19) further reduces to ln N0 −

σ2 =0 2

(20)

that is N0 = e(σ

2

/2)

(21)

and the problem of “passing from the movie to the picture” is completely solved. In conclusion, we have proven the following “movie to picture” theorem: The stochastic process N(t) reduces to the random variable N if, and only if, one inserts ⎧ ⎪ ⎪t = 1 ⎨ σG B M = σ Drake = σ ⎪ =μ =μ μ ⎪ ⎩ G B M (σ 2 /2)Drake N0 = e

(22)

into the lognormal probability density (11) of the stochastic process N(t).

3.3 The N(t) Stochastic Process Is a Geometric Brownian Motion But what is this N(t) stochastic process reducing to the lognormal random variable N in the static case? Well, N(t) is no less than the famous geometric Brownian motion (abbreviated “GBM”), of paramount importance in the mathematics of finance. In fact, in the

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so-called Black–Sholes models, N(t) is related to the log return of the stock price. Huge amounts of money all over the world are handled at stock exchanges according to the mathematics of the stochastic process N(t), that is differently denoted S t there (“S” from Stock). A concise summary about the GBM is given at the Wikipedia site http://en.wikipedia.org/wiki/Geometric_Brownian_motion. But we would not touch these topics in this paper, since this paper is about SETI, rather than about stocks. We just content ourselves to have proven that the GBM used in the mathematics of finance is the same thing as the exponentially increasing process N(t) yielding the number of communicating ET civilizations in the Galaxy!

4 Darwinian Evolution Re-defined as a GBM in the Number of Living Species 4.1 Introducing the “DARWIN” (D) Unit, Measuring the Amount of Evolution that a Given Species Reached In all sciences “to measure is to understand”. In physics and chemistry this is done by virtue of units such as the meter, second, kilogram, coulomb, etc. So, it appears useful to introduce a new unit measuring the degree of evolution that a certain species has reached at a certain time t of Darwinian evolution, and the obvious name for such a new unit is the “Darwin”, denoted by a lower case “d”. For instance, if we adopt the average exponential evolution curve described in the previous sections, we might say that the dominant species on earth right now (humans) have reached an evolution level of 50 million darwins, since there are nowadays 50 million living species on Earth. How many darwins may an alien civilization have reached already? Probably more than 50 millions, i.e. more than 50 Md, but we will not check out until SETI succeeds for the first time. We are not going to discuss further this notion of measuring the “amount of evolution” since we are aware that endless discussions might come out of it. But it is clear to us that such a new measuring unit (and ways to measure it for different species) will sooner or later have to be introduced to make evolution a fully quantitative science.

4 Darwinian Evolution Re-Defined as a GBM …

417

4.2 Cladistics, Namely the GBM Mean Exponential as the Locus of the Peaks of b-Lognormals Representing Each a Different Species Started by Evolution at the Time t=b How is it possible to “match” the GBM mean exponential curve with the lognormals appearing in the statistical Drake equation? Our answer to such a question is “by letting the GBM mean exponential become the envelope of the b-lognormals representing the cladistic branches, i.e. the new species that were produced by evolution at different times as long as evolution unfolded”. Let us now have a look at Fig. 5 hereafter. The envelope shown in Fig. 5 is not really an envelope in the strictly mathematical sense explained in calculus textbooks. However, it is “nearly the same thing in the practice” because it actually is the geometric locus of the peaks of all b-lognormals. We shall now explain this in detail. First of all, let us write down the equation of the b-lognormal, i.e. of the lognormal starting at any instant t = b (while ordinary lognormals all start just at zero); in other words, (t − b) replaces n in the first equation in Table 1 

b_lognormal(t, μ, σ, b) =

√

2 2 1 e−((ln(t−b)−μ) /2σ ) 2πσ ·(t−b)

holding f or t >b and up to t = ∞.

(23)

Then, notice that its peak falls at the abscissa p and ordinate P given by, respectively (as given by the eighth and ninth line in Table 1) ⎧ μ−σ 2 ⎪ = b_ lognormal_ peak_ abscissa, ⎨ p =b+e (σ 2 /2)−μ e ⎪ ⎩ P= √ = b_ lognormal_ peak_ ordinate. 2π σ

(24)

Fig. 5 Darwinian exponential as the envelope of b-lognormals. Each b-lognormal is a lognormal starting at a time (t = b=birth time) larger than zero and represents a different species “born” at time b of the Darwinian evolution

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Evolution and History in a New “Mathematical SETI” Model

Let us now wonder: can we match the second Eq. (24) with the Darwinian exponential (2)? Yes, is the answer, if we set at time t = p ⎧ 1 ⎪ ⎨ A= √ 2π σ ⎪ σ2 ⎩ Bp e = e 2 −μ

⎧ ⎪ ⎪ ⎨ that is

A= √

1 2π σ

2 ⎪ ⎪ ⎩ Bp = σ − μ. 2

(25)

The last system of two equations may now be inverted, i.e. exactly solved with respect to μ and σ ⎧ ⎪ ⎪ ⎨

σ =√

1

2π A ⎪ 1 ⎪ ⎩ μ = −Bp + 4π A2

(26)

showing that each b-lognormal in Fig. 5 (i.e. its μ and σ ) is perfectly determined by the Darwinian exponential (namely by A and B) plus a precise value of the blognormal’s peak time p. In other words, this is a one-parameter (the parameter is p) family of curves that are all constrained between the time axis and the Darwinian Exponential. Clearly, as long as one moves to higher values of p, the peaks of these curves become narrower and narrower and higher and higher, since the area under each b-lognormal always equals 1 (normalization condition).

4.3 Cladogram Branches Are Made up by Increasing, Decreasing or Stable (Horizontal) Exponential Arches It is now possible to understand how cladograms shape up in our mathematical theory of evolution: they depart from the time axis at the birth time (b) of the new species and then either: 1. Increase if the b-lognormal of the i-th new species has (keeping in mind the convention pi < 0 for past events, i.e. events prior to now) 

Ai = Bi =

√1 2π σi (σ12 /2)−μi pi

> 0 that is μi >

σi2 . 2

(27)

4 Darwinian Evolution Re-Defined as a GBM …

419

2. Decrease if the same b-lognormal has (keeping in mind the convention pi < 0 for past events) 

Ai = Bi =

√1 2π σi (σi2 /2)−μi pi

< 0 that is μi
b. On the other hand, b-lognormals like (23) are infinite in time, i.e. spanning from t = b to t = + ∞, so one might immediately wonder how (23) might possibly represent a finite lifespan. Well, the answer to such a question will be given later in Sect. 5.3, when we will introduce the notion of “death instant” t = d as the intersection point between the tangent to (23) in its descending inflexion point and the time axis. At the moment, we content ourselves to start studying some mathematical properties of the infinite b-lognormal pdf (23). This was done in a highly innovative editorial way in the author’s book entitled “Mathematical SETI” [17]. In fact, the mathematical proof of each of the theorems proven there was hardly demonstrated line-by-line in the text. On the contrary, the hardest calculations were performed by aid of Maxima, the powerful computer algebra code (also called Macsyma) created by NASA and MIT in the 1960s and now freely downloadable from the web site http://maxima.sourceforge.net/. So, the reader may find them in the Maxima file “b_lognormals_inflexion_points_and_DEATH_time.wmx” that is reprinted in the author’s 2012 book. From now on, we shall simply state the equation numbers in that Maxima file proving a certain result about b-lognormals, and the interested reader will then find the relevant proof by reading the corresponding Maxima command lines (“i” = input lines) and output lines (“o” = output lines). This way of proving “electronically” the mathematical results simplifies things greatly, if compared to the “ordinary” lengthy proofs of traditional books, and students and researchers will be able to download for free the corresponding Maxima symbolic manipulator from the site: http://maxima.sourceforge.net/.

5 Life-Spans of Living Beings as b-Lognormals

421

5.2 Infinite b-Lognormals Again, a b-lognormal simply is a lognormal probability density function (pdf) starting at any positive value b > 0 (called “birth”) rather than at the origin. As such, a b-lognormal has the following equation in the independent variable t (time) and with the three independent parameters μ, σ and b, of which μ is a real number, while both σ and b are positive numbers: 

b_lognormal(t, μ, σ, b) √2π σ1(t−b) e−((ln(t−b)−μ)

2

/2σ 2 )

holding f or t > b and up to t = ∞.

(31)

This we call the infinite b-lognormal, meaning that it extends to the right up to infinity. Its main mathematical properties are basically the same as those of the ordinary lognormals starting at zero and given in Table 1, with just one exception: all formulas representing an abscissa have the same expression as for ordinary lognormals with a + b term added because of the right-shift of magnitude b. In other words, all infinite b-lognormals have the formulas given in the following Table 3 (a formal, analytical proof of all results in Table 3 can be found in of the author’s book “Mathematical SETI” [17]).

5.3 From Infinite to Finite b-Lognormals: Defining the Death Time, d, as the Time Axis Intercept of the Tangent at Senility The b-lognormal extends up to t = + ∞ and this is in sharp contrast with the fact that every living being sooner or later dies at the finite time d (“death”) such that 0 < b < d < ∞. We thus must somehow define this finite death time d in order to let the b-lognormals became a realistic mathematical model for the life-and-death of every living being. We solved this problem by defining the death time t = d as the intercept point between the time axis and the straight line tangent to the b-lognormal at its descending inflexion point t = s, i.e. the tangent line to the lognormal curve at senility. And, from now on, we shall call finite b-lognormal any such a truncated b-lognormal, ending just at t = d. This section is devoted to the calculation of the equation yielding the d point in terms of the b-lognormal’s μ and σ, and the whole procedure is described at the lines %i45 thru %o56 of the file “blognormals_inflexion_points_and_DEATH_time.wxm” in of [17].

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Table 3 Properties of the b-lognormal distribution, namely the INFINITE b-lognormal distribution given by (31). These are both statistical and geometric properties of the pdf (31), whose importance will become evident later Probability distribution

b-Lognormal, namely the infinite b-lognormal

Probability density function

f b−lognormal (t, μ, σ, b) = 2 2 1 · (t−b) e−((ln(t−b)−μ) /2σ ) (t ≥ b ≥ 0)

√1 2πσ

Abscissa of the ascending inflexion point Ordinate of the ascending inflexion point

adolescence √ ≡a = 2 2 b + e−(((σ σ +4)/2)−(3σ /2)+μ) f b−lognormal (adolescence) ≡ A = e−((σ

Abscissa of the descending inflexion point Ordinate of the descending inflexion point

√

σ 2 +4)/4) e−μ+(σ 2 /4)−(1/2)

√

2πσ

senility ≡ s = b + e(((σ

√ σ 2 +4)/2)−(3σ 2 /2))+μ

f b−lognormal (senility) ≡ S = e(σ

√

σ 2 +4)/4 e−μ+(σ 2 /4)−(1/ 2)

√

2πσ

b − lognormal = b + eμ eσ

Mean value

2 /2

2 σb−lognormal = e2μ eσ (eσ − 1)  2 2 σb−lognormal = eμ eσ /2 eσ − 1 2

Variance Standard deviation Peak abscissa = mode

2

b − lognormal peak ≡ b − lognormalmode ≡ p = b + eμ e−σ = b + eμ−σ 2

Peak ordinate = value of the mode peak

f b−lognormal (b − lognormalmode ) = 2 2 · e−μ · e(σ /2) = √ 1 · e(σ /2)−μ

√1 2πσ

Median (= fifty–fifty probability abscissa)

2

2πσ

Median = m = b + eμ

Let us start by recalling the simple formula yielding the equation of the straight line having an angular coefficient m and tangent to the curve y(t) at the point having the coordinates (t 0 , y0 ) y − y0 = m(t − t0 ).

(32)

Then, the value of y0 clearly is the value of the b-lognormal at its senility time, given by the sixth line in Table 3, that is, rearranging: y0 =

e

 √   σ σ 2 +4 /4 −(μ−(σ 2 /4)+(1/2))

e √ 2π σ

.

(33)

5 Life-Spans of Living Beings as b-Lognormals

423

On the other hand, the abscissa of the senility time t = s is given by the fifth line in Table 2, that is t0 = b + e

  √ σ σ 2 +4 /2−(3σ 2 /2)+μ

.

(34)

Finally, we must find the expression of the angular coefficient m at the senility time, and this involves finding the b-lognormal’s derivative at senility. Maxima has no problem to find this, and the lines %i48 and %o48 show that one gets, after some rearranging  √ 7σ 2 /4 √ ) σ 2 + 4 − σ e− 2e( m=− √ 4 πσ2

 √   σ σ 2 +4+8μ+2 /4

.

(35)

Inserting then (33), (34) and (35) into (32) one gets the equation of the desired straight line tangent to the b-lognormal at senility y−

e((σ

√

σ 2 +4/4) (−μ−(σ 2 /4)+(1/2))

e √

2π σ     √ √ 7σ 2 /4 √ ) σ 2 + 4 − σ e− σ σ 2 +4+8μ+2 /4 2e( =− √ 4 πσ2      √ σ σ 2 +4 /2 −(3σ 2 /2)+μ t −e −b

(36)

In order to find the abscissa of the death point t = d, we just need insert y = 0 into the above Eq. (36) and solve for the resulting t. Maxima yields at first a rather complicated result (%o52). However, keeping in mind that the term in b must obviously appear “alone” in the final equation since the b-lognormal is just an ordinary lognormal shifted to make it start at b, the way to further simplify (36) becomes obvious, and the final result simply is √ d =b+

σ2 + 4 + σ

2 e

 √   σ σ 2 +4 /2 −(3σ 2 /2)+μ

4

This is the “death time” of all living beings born at any time b.

.

(37)

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5.4 Terminology of Various Time Instants Related to a Lifetime The reader is now asked to look carefully at Fig. 6 to familiarize with mathematical notations and their meaning describing the lifetime of all living beings: Obvious are the definitions of the instants of: 1. 2. 3. 4. 5.

birth (b = starting point on the time axis), adolescence (a = ascending inflexion abscissa, with ordinate A), peak (p = maximum point abscissa, with ordinate P), senility (s = descending inflexion abscissa, with ordinate S), and death (d = death abscissa = intercept between the time axis and the straight line tangent to the b-lognormal at the descending inflexion point).

Fig. 6 Lifetime of all living beings, i.e. finite b-lognormal: definitions of the basic instants of birth (b = starting point on the time axis), adolescence (a = ascending inflexion abscissa, with ordinate A), peak (p = maximum point abscissa, with ordinate P), senility (s = descending inflexion abscissa, with ordinate S) and death (d = death abscissa = intercept between the time axis and the straight line tangent to the b-lognormal at the descending inflexion point). Also defined are the obvious single-time-step-spanning segments called childhood (C = a−b), youth (Y = p−a), maturity (M = s−p), decline (D = d−s). In addition, also defined are the multiple-time-step-spanning segments of the all-covering lifetime (L = d−b), vitality (V = s−b) (i.e. lifetime minus decline), and fertility (F = s−a) (i.e. adolescence to senility)

5 Life-Spans of Living Beings as b-Lognormals

425

5.5 Terminology of Various Time Spans Related to a Lifetime Also defined in Fig. 6 are the obvious time segments called: 1. 2. 3. 4. 5. 6. 7.

Childhood (C = a−b). Youth (Y = p−a). Maturity (M = s−p). Decline (D = d−s). Fertility (F = s−a). Vitality (V = s−b). Lifetime (L = d−b).

Then, from all these definitions and from the mathematical properties of the b-lognormals listed in Table 3, one gets immediately the following equations: Childhood ≡ C = a − b = e−((σ

√

σ 2 +4)/2)−(3σ 2 /2)+μ

Youth ≡ Y = p − a = eμ−σ − e−((σ 2

Maturity ≡ M = s − p = e √ Decline ≡ D=d − s = −e

σ2

σ 2 +4)/2)−(3σ 2 /2)+μ

 √   σ σ 2 +4 /2 − (3σ 2 /2)+μ

+4+σ

 √   σ σ 2 +4 /2 −

=e

√

   √ σ σ 2 +4 /2 −



2

(38)

σ

√

  σ 2 +4 /2 −

e

(39)

− eμ−σ

2

(40)

(3σ 2 /2)+μ

4 (3σ 2 /2)+μ (3σ 2 /2)+μ σ ·

√

σ2 + 4 + σ



2

      σ σ 2 + 4)/2 − 3σ 2 /2 + μ  √  − σ σ 2 + 4)/2 −(3σ 2 /2)+μ −e  √  2+4 σ σ 2 = 2 · e−(3σ /2)+μ · sinh 2

(41)

Fertility ≡ F = s − a =

Vitality ≡ V = s − b = e

Lifetime = L = d − b =

 √   σ σ 2 + 4 /2 − (3σ 2 /2)+μ

(42)

(43)

√ 2  √ 2   σ σ + 4 /2 − (3σ 2 /2)+μ σ2 + 4 + σ e 4

.

(44)

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Obviously one also has Lifetime = vitality + decline = (s − b) + (d − s) =d −b  √ 2  √ 2   σ σ +4 /2 −(3σ 2 /2)+μ σ2 + 4 + σ e = . 4

(45)

as one may check analytically by adding (43) and (41), and checking the result against (44). In addition, dividing (41) by (43), all exponentials disappear and one gets the important new equation Decline = vitality

σ

√  σ2 + 4 + σ 2

.

(46)

This we shall use later in Sect. 6 in connection with the “golden ratio” and “golden b-lognormals”.

5.6 Normalizing to One All Finite b-Lognormals Finite b-lognormals are positive functions of the time, as requested for any probability density function, but they are not normalized to one yet, as it is also demanded for any probability density function. This is because: 1. If one computes the integral of the b-lognormal (31) between birth b and senility s one gets

s b_lognormal(t, μ, σ, b)dt b e(σ

√

σ 2 +4)/2−(3σ 2 /2) + μ



1 2 2 e−(((ln(t−b)−μ) )/2σ ) dt √ 2π σ (t − b) b √ √ 1 er f (( 2/4) · ( σ 2 + 4 − 3σ )) = + 2 2

=

(47)

where erf (x) is the well-known error function of probability and statistics, defined by the integral

5 Life-Spans of Living Beings as b-Lognormals

2 er f (x) = √ π

427

x

e−z dz. 2

(48)

0

Notice that, during the integration in (47), the independent variable μ disappeared, leaving a result depending on σ only. We shall not prove (47) here: the proof can be found in Appendix 6B of [17], lines % i78 through %o79. 2. If we add to (47) the integral of the descending straight line tangent to the blognormal at s, taken between s (given by the fifth line in Table 3) and d (given by (37)), we get

d y_from_ Eq._(36)dt s

√ √ σ 2 +4)/2) − (3σ 2 /2)+μ b+((( σ 2 +4+σ )2 e((σ )/4)

=

y_from_ Eq._(36)dt b+e((σ

√

σ 2 +4)/2)−(3σ 2 /2) + μ

√ √ 2 2 ( σ 2 + 4 + σ )e((3σ σ +4)/4) − (5σ /4)−(1/2) = . √ 25/2 π

(49)

Once again μ disappeared, leaving a result depending on σ only. Again, we shall not prove (49) here: the proof can be found in Appendix 6B of [17], lines (%i85) trough (%087). 3. In conclusion, adding (47) and (49), one gets the area under the FINITE blognormal (from b to d)

Area_ under_ finite_ b-lognormal

d =

finite_ b_ lognormal dt = K (σ )

(50)

b

with: √ √ 2 2 1 ( σ 2 + 4 + σ )e((3σ σ +4)/4)−(5σ /4)−(1/2) K (σ ) = + √ 2 25/2 π √ √ er f (( 2/4) · ( σ 2 + 4 − 3σ )) . 2

(51)

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In practice, it will be sufficient to compute the numeric value of K(σ ) for a given σ and divide the corresponding finite b-lognormal by this value to have it normalized to one.

5.7 Finding the b-Lognormals Given b and Two Out of the Four a, p, s, d The question is now: having introduced the five points b, a, p, s, d, do some equations exist enabling one to determine the b-lognormal’s μ and σ in terms of the birth time b (supposed to be always known) and any two more points out of the remaining four (a, p, s and d)? This author was able to discover several such pair of equations, yielding μ and σ exactly (and not as numeric approximations) and they are all listed in Table 4. The mathematical proofs are given in [17], and will not be repeated here. The most important out of all these equations are our brand-new history formulas, given by the two equations 

σ =

√ d−s √ d−b s−b

μ = ln(s − b) +

2s 2 −(3d+b)s+d 2 +bd . (d−b)s−bd+b2

(52)

Essentially, these two equations allow us to find a b-lognormal when its birth, senility and death times are given. This is precisely what happens in the study of human history, since we certainly know when a past civilization was born (for instance when a new town was founded and later became the capital of a new empire), and when it died (because of war, usually). Less precisely we may know the time when its decline began (after reaching its peak), which is the s appearing in the fifth line of Table 3. But if one manages to find that out in history books, then the b-lognormal (31) is fully determined by our history formulas (52).

6 Golden Ratios and Golden b-Lognormals 6.1 Is σ Always Smaller Than 1? So far so good: we derived a number of properties of the b-lognormals given by (31) and representing the life of a living being. But one question remains: is there any specific reason why should σ be smaller or larger than one? More precisely, while we know σ to be necessarily positive, no “plausible” reason seems to exist for it to be smaller than one, as it appears to be numerically in majority of life forms. To explore this topic a little more, consider a trivial rectangular triangle having catheti equal to 2 and σ, respectively. Because of the well-known Pythagorean

p

d

s

⎧      ⎪ p−b p−b ⎪ ⎪ ⎪ ⎨ σ = ln a − b / 1 + ln a − b

a

No exact formula exists, only numeric approximations

√ ⎪ ⎪ 2 2 ⎪ ⎪ ⎩ μ = ln(a − b) + σ σ + 4 + 3σ 2 2  ⎧   2 ⎪ √  ⎪ ⎪ √  s−b ⎪ ⎪ ⎨ σ = 2 ln √ +1−1 a−b ⎪ ⎪ ⎪ ⎪ ⎪ ln[(a − b)(s − b)] 3σ 2 ⎩ μ=+ + 2 2

a

–

Given

No exact formula exists, only numeric approximations

√ ⎪ ⎪ 2 2 ⎪ ⎪ ⎩ μ = ln(s − b) − σ σ + 4 + 3σ 2 2

⎧      ⎪ s−b s−b ⎪ ⎪ ⎪ ⎨ σ = ln p − b / 1 − ln p − b

–

√ ⎪ ⎪ 2 2 ⎪ ⎪ ⎩ μ = ln(a − b) + σ σ + 4 + 3σ 2 2

p ⎧      ⎪ p−b p−b ⎪ ⎪ ⎪ σ = ln / 1 + ln ⎨ a−b a−b

d −s σ = √ √ d −b s−b

⎪ 2s 2 + (−3d − b)s + d 2 + bd ⎪ ⎪ ⎩ μ = ln(s − b) + (d − b)s − bd + b2

History formulas ⎧ ⎪ ⎪ ⎪ ⎨

–

s  ⎧   2 ⎪ √  ⎪ ⎪ √  s−b ⎪ ⎪ ⎨ σ = 2 ln √ +1−1 a−b ⎪ ⎪ ⎪ ⎪ ⎪ ln[(a − b)(s − b)] 3σ 2 ⎩ μ= + 2 2 ⎧      ⎪ s − b s − b ⎪ ⎪ ⎪ ⎨ σ = ln p − b / 1 − ln p − b √ ⎪ ⎪ 2 2 ⎪ ⎪ ⎩ μ = ln(s − b) − σ σ + 4 + 3σ 2 2

d −s σ = √ √ d −b s−b

–

⎪ 2s 2 + (−3d − b)s + d 2 + bd ⎪ ⎪ ⎩ μ = ln(s − b) + (d − b)s − bd + b2

History formulas ⎧ ⎪ ⎪ ⎪ ⎨

No exact formula exists, only Numeric approximations

No exact formula exists, only numeric approximations

d

Table 4 Finding the b-lognormal (i.e. finding both its μ and σ ) given the birth time, b, and any two out of the four instants a = adolescence, p = peak, s = senility, d = death

6 Golden Ratios and Golden b-Lognormals 429

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Evolution and History in a New “Mathematical SETI” Model

√ theorem, the hypotenuse obviously equals σ 2 + 4. Since the hypotenuse always is longer than anyone of the catheti, we conclude that  σ 2 + 4 > σ.

(53)

Now insert (53) into (46). The result is Decline = Vitality

σ

√  σ2 + 4 + σ

>

2 2σ 2 σ (2σ ) = = σ 2. = 2 2

σ (σ + σ ) 2 (54)

Since all variables in this inequality are positive, we may rewrite it as Decline > σ 2 · Vitality.

(55)

Now, in the majority of known life forms, it appears that the vitality time (i.e. the time between birth b and senility s), i.e. (s − b) is longer, or much longer than the decline time, (i.e. the time between senility s and death d, i.e. (d − s)). Thus, the only way to let (55) apply to biological reality is to conclude that it must be σ 2 < 1 or σ 2 1 and so their decline would be larger or much larger than their vitality: just think of some science fiction movies like Star Wars, where some living being declares to be 900 years old or more… Anyway, the dividing line between “good” and “bad” values of σ seems to be the σ = 1 case. Is this case significant? Yes, very much, as we discover in the next section.

6 Golden Ratios and Golden b-Lognormals

431

6.2 Golden Ratios and Golden b-Lognormals If one lets σ = 1 into (46) one gets Decline = Vitality =

σ

 √ σ2 + 4 + σ

2 √  1 5+1

√2 1+ 5 = golden ratio = 2 = 1.6180339887 . . . = φ.

(58)

But this is the famous “golden ratio”, celebrated by artists, architects and mathematicians as esthetically very pleasing for over 2000 years (see, for instance, the site http://en.wikipedia.org/wiki/Golden_ratio). In the Renaissance (actually in 1509) the Italian author Luca Pacioli (1445–1517) wrote a book about it by the Latin title of “De Divina Proportione” (The Divine Proportion), with illustrations by Leonardo Da Vinci! (http://en.wikipedia.org/wiki/Luca_Pacioli). So, let us go back to (58). We now wish to prove that the following “divine proportion” holds among lifetime, vitality and decline (but only for the life forms having σ = 1, of course) Decline Lifetime = Decline Vitality = golden_ratio ≡ φ ≡

√ 1+ 5 2

= 1.618 . . . .

(59)

For the proof, admit for a moment that (59) holds good. Then, because of (45), the supposed (59) may be rewritten as φ= = = = =

Lifetime Decline Vitality + Decline Decline Vitality +1 Decline 1 +1 (Decline/Vitality) 1 + 1. φ

(60)

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Evolution and History in a New “Mathematical SETI” Model

Thus, if this correct, we reach the conclusion that φ must fulfill the equation φ=

1 + 1 that is φ 2 − φ − 1 = 0. φ

(61)

Solving this quadratic equation in φ yields φ=

−(−1) ±

√ √ 1± 5 1 − 4 · (−1) = . 2 2

(62)

Discarding the negative root in (62) (since the ratio of positive quantities may only yield a new positive quantity) leaves the positive root only, and this is just the golden ratio appearing in (58). Thus, we met with no contradiction in assuming the “divine proportion” (59) to be true, and so it is true indeed. As a consequence of this, it appears quite natural to call golden b-lognormal the particular case σ = 1 of (31), that is 

golden_b_lognormal(t, μ, b) =

√

2 1 e−((ln(t−b)−μ) /2) 2π (t−b)

holding f or t >b and up to t = ∞.

(63)

This a “new” statistical distribution, whose main statistical properties are listed in Table 5, of course derived by setting σ = 1 into the corresponding entries of Table 3. Actually, rather than being just a single curve, (63) is a one-parameter family of curves in the (t, golden_b-lognormal) plane, the parameter being μ. One is thus led to wonder what properties might this family of golden b-lognormals possibly have. Then, this author discovered a simple theorem: all the golden b-lognormals (63) have their peaks lying on the equilateral hyperbola of equation Golden_ b − lognormal_ peak_ locus (t, b) = √

1 . √ 2π e(t − b)

(64)

The proof is easy: just solve for μ the equation (line 11 in Table 5) yielding the peak abscissa of (63). The result is μ = ln( p − b) + 1.

(65)

Inserting (65) into the expression for the peak height P given in line 12 of Table 5, (64) is found, and the theorem is thus proven. Figure 7 shows immediately the equilateral hyperbola (64) for the case b = 2. But all these considerations about the golden ratio and the golden b-lognormals appear to be just an iceberg’s tip if one thinks of the many known results relating the golden ratio to the Fibonacci numbers, Lucas numbers, and so on. So, much more work is needed for sure is this new field we have uncovered.

7 Mathematical History of Human Civilizations

433

Table 5 Golden b-lognormal distribution, i.e. the b-lognormal having σ = 1, and its statistical properties Probability distribution

Golden b-lognormal

Probability density function

f golden_b−lognormal (t; μ, b) = 2 √1 · 1 e−(ln(t−b)−μ) /2 (t ≥ b ≥ 2π (t−b)  √  μ− 3+ 5 /2

0)

Abscissa of the ascending inflexion point

adolescence ≡ a = b + e

Ordinate of the ascending inflexion point

f golden_b−lognormal (adolescence) ≡ A =

Abscissa of the descending inflexion point

senility ≡ s = b + e

Ordinate of the descending inflexion point

f golden_b−lognormal (senility) ≡ S =

Abscissa of the death point Mean value Variance Standard deviation

√

μ+

√

1+ e−μ−(( √ 2π

  5−3 /2 √

√

d =b+

) )

5 /4

1− e−μ−(( √ 2π

) )

5 /4

√ 2 5+1 eμ+(( 5−3)/2)

4

Golden_ b -lognormal = b + eμ+(1/2) 2 σgolden_b−lognormal = e2μ+1 (e − 1) √ σgolden_b−lognormal = eμ+(1/2) e − 1

Peak abscissa = mode

Golden_ b -lognormalpeak ≡ Golden_ b -lognormalmode ≡ p = b + eμ−1   Peak ordinate = value of the f golden_b−lognormal golden_ b − lognormalmode = √1 · e(1/2)−μ 2π mode peak Median (= fifty–fifty probability value)

Median = m = b + eμ

Skewness

K3 (K 2 )3/2

Kurtosis

K4 (K 2 )2

√ = (e + 2) e − 1 = 6.185 . . .

= e4 + 2e3 + 3e2 − 6 = 110.936 . . .

7 Mathematical History of Human Civilizations 7.1 Civilizations Unfolding in Time as b-Lognormals Centuries of human history on Earth should have taught us something… Basically, civilizations are born, fight against each other, and “die”, merging, however, with the newer civilizations. To cast all these in terms of mathematical equations is hard. The reason nobody has done so is because the task is so daunting. Indeed, no course on “Mathematical History” is taught at any university in the world. In this paper, however, as well as in Chap. 7 of [17], we will have a stab at this.

434

Evolution and History in a New “Mathematical SETI” Model

Fig. 7 The geometric locus of the peaks of all golden b-lognormals (in the above diagram starting all at b = 2) as the parameter μ takes on all positive values (0 ≤ μ ≤ ∞) is the equilateral hyperbola given by (64)

Our idea is simple: any civilization is born, reaches a peak, then declines… just like a b-lognormal!

7.2 Eight Examples of Western Civilizations as Finite b-Lognormals We now offer eight examples of such a view: the historic development of the civilizations of: 1. 2. 3. 4. 5. 6. 7. 8.

Ancient Greece, Ancient Rome, Renaissance Italy, Portuguese Empire, Spanish Empire, French Empire, British Empire American (USA) Empire.

Other historic empires (for instance the Dutch, German, Russian, Chinese, and Japanese ones, not to mention the Aztec and Incas Empires, or the Ancient ones, like the Egyptian, Persian, Parthian, or the medieval Mongol Empire) should certainly be added to such a picture, but we regret we do not have the time to carry on those studies in this paper. Those historic-mathematical studies will be made at a later

7 Mathematical History of Human Civilizations

435

stage of development of this new research field that, in our view, is “Mathematical History”: the mathematical vision of human history based upon b-lognormal probability distributions. To summarize this section’s content, for each one of the eight civilizations listed above, we define: (a) Birth b, namely the year when that civilization was supposed to be “born”, even if only approximately in time. (b) Senility s, namely the year of an historic event that marked the beginning of the decline of that civilization. (c) Death d, namely the year when an historic event marked the “official passing away” of that civilization from history. Then, consider the two history Eqs. (52). For each civilization, these two equations allow us to compute both μ and σ in terms of the three assigned numbers (b, s, (d). As a consequence, the time of the given civilization peak is found immediately from the upper Eq. (24), that is peak_ time = abscissa_ of_ the_ maximum = p = b + eμ−σ . 2

(66)

Also, we then can write down the equation of the corresponding b-lognormal immediately. The plot of this function of the time gives a clear picture of the historic development of that civilization, though, to save space, we prefer not to reproduce here the above eight b-lognormals separately (as it was indeed done in Chap. 7 of [17]). Inserting the peak time (66) into (23), the peak ordinate of the civilization is found, namely “how civilized that civilization was at its peak”, and this is explicitly given by the lower Eq. (24), namely e((σ /2)−μ) . √ 2π σ 2

peak_ ordinate = P =

(67)

Table 6 summarizes the three input data (b, s, d) drawn by this author from history textbooks, and then the two output data (p, P) of that Civilization’s peak, namely its best legacy to other, subsequent civilizations.

7.3 Plotting All b-Lognormals Together and Finding Exponentials Having determined the b-lognormal for each civilization we wish to study, the time is ripe to plot all of them together and “see what the trends are”. This is done in Fig. 8. We immediately notice some trends:

1250 Frederick II dies Middle Ages end. Free Italian towns

1419 Madeira island discovered

1492 Columbus discovers America

1524 Verrazano first in New York bay

1588 Spanish Armada defeated

Renaissance Italy

Portuguese Empire

Spanish Empire

French Empire

British Empire

1660. 1660 Bruno burned. 1642 Galileo dies 1667 Cimento Academy ended

476 AD Western Roman Empire ends. Dark Ages start

1914 World War One won at a high cost

1815 Napoleon defeated at Waterloo

1973 The UK joins European EEC

1962 Algeria lost, as most colonies

1805 Spanish fleet lost at 1898 Last colonies lost to Trafalgar the USA

1822 Brazil independent, 1999 Last colony Macau colonies retained lost

1564 Council of Trent. Tough Catholic and Spanish rule

753 BC Rome founded. Italy 235 AD Military seized by Romans by 270 Anarchy starts. Rome BC not capital

Ancient Rome

d = death time

323 BC Alexander the 30 BC Cleopatra’s death: Great’s death. Hellenism last Hellenistic queen starts

s = senility time

600 BC Mediterranean Greek coastal expansion

Ancient Greece

b = birth time

1868 Victorian Age. Science: Faraday, Maxwell, Darwin.

1732 French Canada and India conquest tried

1741 California to be settled by Spain, 1759–1776

1716 Black slave trade to Brazil at its peak

1497 Renaissance art and architecture. Science. Copernican revolution

59 AD Christianity preached in Rome by Saints Peter and Paul against slavery

434 BC Pericles’ Age. Democracy peak. Arts and science peak

p = peak time

(continued)

8.447 × 10−3

4.279 × 10−3

5.938 × 10−3

3.431 × 10−3

5.749 × 10−3

2.193 × 10−3

2.488 × 10−3

P = peak ordinate

Table 6 Finding the b-lognormals of eight among the most important civilizations of the Western world: Ancient Greece, Ancient Rome, Renaissance Italy, Portugal, Spain, France, Britain and the USA. For each such civilization three input dates are assigned on the basis of historic facts: (1) the birth time, b; (2) the senility time, s, i.e. the time when the decline began, and (3) the death time, d, when the civilization reached a formal end. From these three inputs and the two Eqs. (52) the b-lognormal of each civilization may be computed. As a result, that civilization’s peak is found, as shown in the last two columns. In general, this peak time turns out to be in good agreement with the main historical facts, thus “proving that our theory is correct”

436 Evolution and History in a New “Mathematical SETI” Model

USA Empire

Table 6 (continued)

b = birth time

1898 Philippines, Cuba, Puerto Rico seized

2001 9/11 terrorist attacks

s = senility time 2050? Will the USA yield to China?

d = death time 1973 Moon landings, 1969–1972

p = peak time 0.013

P = peak ordinate

7 Mathematical History of Human Civilizations 437

438

Evolution and History in a New “Mathematical SETI” Model

Two ENVELOPES for ALL CIVILIZATIONS (800 B.C. - 2200 A.D.). 0.0169

0.0135

0.0101

0.0067

0.0034

0 −800

−600

−400

−200

0

200

400

600

800

1000

1200

1400

1600

1800

2000

2200

Years (B.C. = negative years, A.D. = positive years) Greece 600 B.C. - 30 B.C. Rome 753 B.C. - 476 A.D. Renaissance Italy 1250-1660 Portuguese Empire 1419-1974 Spanish Empire 1492-1898 French Empire 1524-1962 British Empire 1588-1974 USA Empire 1898-2050 (?) Greece-to-Britain EXPONENTIAL ENVELOPE Greece-to-USA EXPONENTIAL ENVELOPE

Fig. 8 Showing the b-lognormals of eight civilizations in Western history, with two exponential envelopes for them

1. The first two civilizations in time (Greece and Rome) are separated from the six modern ones by a large, 1000 years gap. This is of course the Middle Ages, i.e. the Dark Ages that hampered the development of the Western Civilization for about 1000 years. Carl Sagan said, “the millennium gap in the middle of the diagram represents a poignant lost opportunity for the human species” [22]. 2. While the first two civilizations of Greece and Rome lasted more than 600 years each, all modern civilizations lasted much less: 500 years at most, but really less, or much less indeed. 3. Since b-lognormals are probability density functions, the area under each blognormal must be the same, i.e. just 1 (normalization condition). Thus, the shorter a civilization lives, the highest its peak must be! This is obvious from Fig. 8: Greece and Rome lasted so long, and their peak was so much smaller than the British and the American peaks! 4. In other words, our theory accounts for the “higher level of the more recent historic civilizations” in a natural fashion, with no need to introduce further free parameters. Not a small result, we think.

7 Mathematical History of Human Civilizations

439

5. All these remarks appeared already in Chap. 7 in [17] and the figures therewith, starting with Fig. 8 hereafter.

8 Extrapolating History into the Past: Aztecs 8.1 Aztec–Spaniards as an Example of Two Suddenly Clashing Civilizations The only example we know for sure about two suddenly clashing civilizations with very different technological levels comes from human history. That was in 1519, when the Spaniard Hernán Cortés, with 600 men, 15 horsemen, 15 cannons, and hundreds of indigenous carriers and warriors, was able to subdue the Aztec empire of Moctezuma II, numbering some 20 million people. How was that possible? Well, we claim that basically there was a psychological breakdown in the Aztecs due to their obvious technological inferiority to the Spaniards, causing the Aztecs to regard the Spaniards as “Semi-Gods”, or “Gods”. We also claim that this is precisely what might happen to humans when they meet for the first time with a much more technologically advanced alien civilization in the Galaxy: humans might be shocked and paralyzed by the alien superiority, thus simply surrendering to alien will. We also claim, however, that this human–alien sudden clash might be somehow softened were humans able to make a mathematical estimate of how much more advanced than us aliens will be. This mathematical theory of the technological civilization level is now developed in this section with a reference to the Aztecs–Spaniards example (Fig. 9).

8.2 “Virtual Aztecs” Method to Find the “True Aztecs” B-Lognormal First of all, this author has developed a mathematical procedure to correctly locate the b-lognormal of past human civilizations in time. Consider the Aztec–Spaniard case: how much were the Spaniards more technologically developed than the Aztecs? Well, we claim that the answer to this question comes from the consideration of wheels. The use of wheels was unknown to the Aztecs. However, although they did not develop the wheel proper, the Olmec and certain other western hemisphere cultures seem to have approached it, as wheel-like worked stones have been found on objects identified as children’s toys (http://en.wik ipedia.org/wiki/Wheel). This is just the point: we assume that the Aztecs “were on the verge” of discovering wheels when the Spaniards arrived in 1519. But then, when had wheels been discovered by the Asian–European civilizations?

440

Evolution and History in a New “Mathematical SETI” Model

Three ENVELOPES for ALL CIVILIZATIONS (800 B.C. - 2200 A.D.). 0.0169

0.0135

0.0101

0.0067

0.0034

0 −800

−600

−400

−200

0

200

400

600

800

1000

1200

1400

1600

1800

2000

2200

Years (B.C. = negative years, A.D. = positive years) Greece 600 B.C. - 30 B.C. Rome 753 B.C. - 476 A.D. Renaissance Italy 1250-1660 Portuguese Empire 1419-1974 Spanish Empire 1492-1898 French Empire 1524-1962 British Empire 1588-1974 USA Empire 1898-2050 (?) Greece-to-Britain EXPONENTIAL ENVELOPE Greece-to-USA EXPONENTIAL ENVELOPE Greece-to-Spain EXPONENTIAL ENVELOPE

Fig. 9 Showing the b-lognormals of eight Western civilizations as in Fig. 8, but here three exponential “envelopes” (or, more precisely, three “loci of the maxima”) are shown: (1) The AncientGreece-peak (434 BC) to Britain’s peak (1868) exponential, namely the dash–dot black curve. (2) The Ancient-Greece-peak (434 BC) to USA peak (1973) exponential, namely the solid black curve. (3) The Ancient-Greece-peak (434 BC) to Spain peak (1741) exponential, namely the dot– dot black curve. The Greece-to-Spain exponential was introduced since it is needed to understand the clash between the Aztecs and the Spaniards (1519–1521), as described by the “Virtual Aztec” b-lognormal, going back 50 centuries before 1519 (see Fig. 10)

Evidence of wheeled vehicles appears from the mid-fourth millennium BC, near-simultaneously in Mesopotamia, the Northern Caucasus (Maykop culture) and Central Europe, so that the question of which culture originally invented the wheeled vehicle remains unresolved and under debate. The earliest well-dated depiction of a wheeled vehicle (here a wagon—four wheels, two axles) is on the Bronocic pot, a ca. 3500–3350 BC clay pot excavated in a Funnelbeaker culture settlement in southern Poland. The wheeled vehicle spread from the area of its first occurrence (Mesopotamia, Caucasus, Balkans, Central Europe) across Eurasia, reaching the Indus Valley by the third millennium BC. During the second millennium BC, the spoke-wheeled chariot spread at an increased pace,

8 Extrapolating History into the Past: Aztecs

441

reaching both China and Scandinavia by 1200 BC. In China, the wheel was certainly present with the adoption of the chariot in ca. 1200 BC. To fix the number, we shall thus assume that the wheels had been discovered by the Asian–Europeans about 3500 BC. So, summing 3500 plus 1519 (when the wheelless Aztecs clashed against the wheel-aware Spaniards), we get about 5000 years of technological difference of level among these two civilizations. And 5000 years means 50 centuries, and not just “a few centuries” of Aztecs inferiority, as historians having no mathematical background have superficially claimed in the past: our blognormal theory is quantitatively much more precise than just “words”! But let us now extend into the past, up to 3800 BC, the diagram shown in Fig. 9. The newer, resulting diagram is shown in Fig. 10. In Fig. 10, the virtual Aztec b-lognormal is the b-lognormal peaked at the time in the past when the western civilizations discovered the wheel, i.e. about 3500 BC in Mesopotamia, Southern Caucasus and Central Europe. This b-lognormal is the dash–dash black curve in Fig. 10. The Aztecs started their expansion in central Mexico in 1325, so when Cortes arrived in 1519 they were a civilization 1519–1325 = 194 years old. Reporting this 194 years lapse before the year 3500, we find that the virtual Aztecs had been “born” 194 years earlier, namely in 3694 BC, which is thus the b value of the virtual Aztec b-lognormal bV A = −3694.

(68)

Then we have to find the b-lognornal itself, i.e. its μVA and σ VA . In other words, we have to find μVA and σ VA knowing only the two peak coordinates, pVA = − 3500 and PVA (the numeric value of the peak height PVA is obviously known, since it equals the value of the Greece-to-Spain exponential, the dot–dot curve in Fig. 10) 

−3500 = pV A = bV A + eμV A −σV A 2

Gr eece_to_Spain_exponential = 7.305 × 10−4 = PV A =

V A ) V A) e(( √ . 2πσV A σ 2 /2 −μ

(69) One may let μVA disappear from the above two equations by multiplying them side-by-side and then finding the following new equation in σ VA only, that must thus be solved for σ VA e−(σV A /2) =√ . 2π σV A 2

N umerically_ known = ( pV A − bV A )PV A

(70)

Unfortunately, it is not possible to solve this equation for σ VA exactly. The best we can do is to expand its right-hand side into a MacLaurin power series for σ VA (which is acceptable since we know that 0 < σ 0) as the future history of the Galaxy ahead of us, and the negative times (t < 0) as the past. 2. μ is a positive (if the number of ET civilizations increases in time) or negative (if the number of ET civilizations decreases in time) parameter that we may call “the drift”. To fix the ideas, and to be optimistic, we shall suppose μ > 0. This evolution in time of N(t) is just what we expect to happen in the Galaxy, where the overall number N(t) of ET civilizations does probably increase (rather than decrease) in time because of the obvious technological evolution of each civilization. But this N(t) scenario is a stochastic one, rather than a deterministic one, and certainly does not exclude temporary setbacks, like the end of civilizations due to causes as diverse as: (a) asteroid and comet impacts, (b) rogue planets or stars, arriving from somewhere and disrupting the gravitational stability of the planetary system, (c) supernova explosions that would “fry” entire nearby ET civilizations (think of AGN, the Active Nucleus Galaxies and ask: how many ET civilizations are dying in those galaxies right now?), (d) ET nuclear wars, (e) and possibly more causes of civilization end that we do not know about yet. Mathematically, we came to define the probability density function (pdf) of this exponentially-increasing stochastic process N(t) as the lognormal N (t)_pdf(n, N0 , μ, σ, t) = √

1

√ e 2π σ tn

  2 − [ln(n)−(ln N0 +μt−(σ 2 t/2))] (2σ 2 t )

. (16)

It is easy to prove that the mean value of (16) yields indeed the exponential curve (15) ∞ 0

n · N (t)_pdf(n, N0 , μ, σ, t)dn = N0 eμt .

(17)

476

SETI as a Part of Big History

Table 2 Summary of the properties of the lognormal distribution that applies to the stochastic process N(t) = exponentially increasing number of ET communicating civilizations in the Galaxy, as well as the number of living species on Earth over the last 3.5 billion years Stochastic process

N (t) =

(1) N umber o f E T Civili zations (in S E T I ). (2) N umber o f Living Species(in Dar winian Evolution).

Probability distribution

Lognormal distribution of the Geometric Brownian Motion (GBM)

Probability density function N (t)_ pd f (n, N , μ, σ, t) = 0  √

Mean value Variance Standard deviation All the moments, i.e. k-th moment





2  2  − ln(n)− ln N0 +μt− σ 2 t/2 2σ t 1√ e for n 2πσ tn

≥0

N (t) = N0 eμt   2 σ N2 (t) = N02 e2μt eσ t − 1  2 σ N (t) = N0 eμt eσ t − 1

2  2   k  N (t) = N0k ekμt e k −k σ t/2

Mode (=abscissa of the lognormal peak)

n mode ≡ n peak = N0 eμt e−

Value of the mode peak

f N (t) (n mode ) =

Median [= fifty-fifty probability value for N(t)]

median = m =

Skewness

K3 (K 2 )(3/2)

Kurtosis

K4 (K 2 )2

2  3σ t/2

√ 1 √ · e−μt N0 2πσ t

2  N0 eμt e− σ t/2

· eσ

2t

 2  2 = eσ t + 2 eσ t − 1

= e4σ

2t

+ 2e3σ

2t

+ 3e2σ

2t

−6

Clearly, these two different GBM stochastic processes have different numerical values of N 0 , μ and σ, but the equations are the same for both processes

The proof of (17) is given in Appendix 11.A as the Maxima file “GBM_as_N_of_t_v47” of Ref. [16]. Table 2 summarizes the main properties of GBM, i.e. of the N(t) stochastic process.

3.2 Statistical Drake Equation as the Static Special Case of N(t) In this section we prove the crucial fact that the lognormal pdf of our Statistical Drake Equation given in Table 1 is just “the picture” case of the more general exponentially growing stochastic process N(t) (“the movie”) having the lognormal pdf (16) as given in Table 2. To make things neat, let us denote by the subscript “GBM” both the μ

3 Geometric Brownian Motion (GBM) as ...

477

and σ appearing in (16). The latter thus takes the form: N (t)_pdf(n, N0 , μG B M , σG B M , t)   2 1 − [ln(n)−(ln N0 +μG B M t−(σG2 B M t/2))] (2σG2 B M t ) . =√ e √ 2π σG B M tn

(18)

Similarly, let us denote by the subscript “Drake” both the μ and σ appearing in the lognormal pdf given in the third line of Table 1 (this is also Eq. 1.B.56) of Ref. [16]), namely the pdf of our Statistical Drake Equation: lognormal_ pdf_ of_ Statistical_ Drake_ Eq(n, μ Drake , σ Drake )  1 2 2 =√ e−([ln(n)−μ Drake ] ) (2 σ Drake ) 2π σ Drake n for 0 ≤ n ≤ ∞.

(19)

Now, a glance at Eqs. (18) and (19) reveals that they can be made to coincide if, and only if, the two simultaneous equations hold

2 σG2 B M t = σ Drake ln N0 + μG B M t −

σG2 B M t 2

= μ Drake .

(20)

On the other hand, when we pass (so as to say) “from the movie to the picture”, the two σ must be the same thing, and so must be the two μ, that is, one must have:

σG B M = σ Drake = σ μG B M = μ Drake = μ.

(21)

Checking thus the upper Eq. (21) against the upper Eq. (20), we are just left with t = 1.

(22)

So, t = 1 is the correct numeric value of the time leading “from the movie to the picture”. Replacing this into the lower Eq. (20), and keeping in mind the upper Eq. (21), the lower Eq. (20) becomes ln N0 + μG B M −

σ2 = μ Drake . 2

(23)

Since the two μ also must be the same because of the lower Eq. (21), then (23) further reduces to ln N0 −

σ2 =0 2

(24)

478

SETI as a Part of Big History

that is N0 = eσ

2

/2

(25)

and the problem of “passing from the movie to the picture” is completely solved. In conclusion, we have proven the following “movie to picture” theorem: the stochastic process N(t) reduces to the random variable N if, and only if, one inserts ⎧ t =1 ⎪ ⎪ ⎨ σG B M = σ Drake = σ ⎪ μG B M = μ Drake = μ ⎪ ⎩ 2 N0 = eσ /2

(26)

into the lognormal probability density (16) of the stochastic process N(t).

3.3 GBM as the Key to the Mathematics of Finance But what is this N(t) stochastic process reducing to the lognormal random variable N in the static case? Well, N(t) is no less than the famous Geometric Brownian Motion (abbreviated “GBM”), of paramount importance in the mathematics of finance. In fact, in the socalled Black-Sholes models, N(t) is related to the log return of the stock price. Huge amounts of money all over the world are handled at Stock Exchanges according to the mathematics of the stochastic process N(t), that is differently denoted S t there (“S” from Stock). But we will not touch these topics here, since this paper is about Big History, Evolution and SETI, rather than about stocks. We content ourselves to have proven that the GBM used in the mathematics of finance is the same thing as the exponentially increasing process N(t) just described! In particular, the interested reader is advised to get familiar with the recently three published papers related to this field: Refs. [15, 17, 18]. These papers are detailed mathematical descriptions of the GBM used in Darwinian Evolution and Human History, that we now extend to Big History also, by “adjusting” the GBM, as described in the next section.

3.4 Adjusting the GBM: Letting It Take the Value of One at Its Start (Time t = tSTART ) and Deriving Its Current Mean Value and Standard Deviation This section deals with a problem of paramount importance for all applications of the GBMs to Big History, Darwinian Evolution, Human History, and so on: adjusting

3 Geometric Brownian Motion (GBM) as ...

479

the GBM, that is finding the GBM (i.e. its μ and σ ) if we know three “easy-to-find” data: 1. The time in the past when the GBM first reached the value of ONE, i.e. the time of the origin of life on Earth if the GBM represents the number of living species, or (nearly) the Big Bang time if the GBM represents the number of galaxies in time, and so on. This instant we call the START-TIME and denote by t START . In general, t START is a negative number, meaning that t START occurred in the past, while the origin of the time axis (t = 0) is right now, and the future corresponds to positive times (t > 0). 2. The GBM’s current mean value N 0 . 3. The GBM’s current standard deviation, δN 0 , around (plus or minus) its current mean value N 0 . The key idea is that the new INCREASING GBM simply is a GBM starting at START-TIME, namely it is a GBM shifted back in time as follows NINCREASING (t) = N (t − t ST A RT ).

(27)

Thus, because of (15), its mean value must have the form NINCREASING (t) = K eμ(t−tST A RT )

(28)

where K is a constant that we nowfind. Actually, on the one hand, one has by definition of t START NINCREASING (t ST A RT ) = 1.

(29)

On the other hand, letting t = t START into (28), one gets NINCREASING (t ST A RT ) = K eμ(tST A RT −tST A RT ) = K .

(30)

So, checking (29) and (30) against each other, we find K = 1.

(31)

This, replaced into (28), yields the correct mean value formula for our GBM ever increasing since t = t START : NINCREASING (t) = eμ(t−tST A RT ) .

(32)

Let us now find out what happens at out current time t = 0. If we set t = 0 into (32), it yields NINCREASING (0) = eμ(−tST A RT ) .

(33)

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SETI as a Part of Big History

On the other hand, we know that the current mean value of the process has the value N 0 . Thus we have N0 = eμ(−tST A RT ) .

(34)

Solving (34) for μ, yields μ for the INCREASING GBM: μ=

ln(N0 ) . −t ST A RT

(35)

This is the same as (14), but now we have generalized it to any value for N 0 and t START . And this generalization prompts us to consider the total differential of (35) δμ =

log(N0 )δt ST A RT δ N0 − 2 N t t ST 0 ST A RT A RT

(36)

and then the ratio of (36) to (35) δμ 1 δ N0 δt ST A RT = . − . μ ln(N0 ) N0 t ST A RT

(37)

This is the relative error affecting μ expressed in terms of the relative errors affecting both N 0 and t START . Let us next turn to finding σ out of the standard deviation expression, Δ(t), for the N INCREASING (t) given by the 6th line in Table 3:  Δ(t) = eμ(t−tST A RT ) eσ 2 (t−tST A RT ) − 1.

(38)

Let us first notice that, if we replace t = t START into (38) we get:  Δ(t ST A RT ) = eμ(tST A RT −tST A RT ) eσ 2 (tST A RT −tST A RT ) − 1 = 0.

(39)

In other words, at the initial instant t = t START there is no “error” affecting the N INCREASING (t START ) = 1, which is thus the exact value of 1. Next, consider the current (t = 0) value of (38). Using (34), it becomes   Δ(0) = eμ(−tST A RT ) eσ 2 (−tST A RT ) − 1 = N0 eσ 2 (−tST A RT ) − 1.

(40)

This (0) value is actually the standard deviation δN 0 currently affecting N 0 and is a known quantity from the experimental results got by the scientists in the relevant science branches, that is one has Δ(0) = δ N0

(41)

3 Geometric Brownian Motion (GBM) as ...

481

Table 3 Summary of the properties of the lognormal distribution that applies to the stochastic process N INCREASING (t) = exponentially increasing number of living species on Earth over the last 3.5 billion years or, as shown in the next section, increasing number of galaxies in the universe or, as shown in SETI by the statistical Drake Eq. (1), number of evolving ET civilizations in the Milky Way Stochastic process

N I N C R E AS I N G (t) ≡ N I N C (t) = ⎧ ⎪ ⎪ ⎨ (1)N umber o f galaxies in the evolving U niver se(in Big H istor y).

(2)N umber o f Living Species(in Dar winian Evolution). ⎪ ⎪ ⎩ (3)N umber o f evolving E T Civili zations in the sole Milky W ay(in S E T I ). Probability Lognormal distribution of the ADJUSTED and INCREASING GBM starting at distribution t START Probability density function

NINC (t)_ pd f (n; μ, σ, t ST A RT , t) =

Mean value

NINC (t) = eμ(t−t ST A RT )   2 σ N2 INC (t) = e2μ(t−t ST A RT ) eσ (t−t ST A RT ) − 1  2 σ NINC (t) = eμ(t−t ST A RT ) eσ (t−t ST A RT ) − 1

Variance Standard deviation All the moments, i.e. k-th moment

2  2  

− ln(n)− μ(t−t ST A RT )− σ 2 (t−t ST A RT )/2 2σ (t−t ST A RT ) 1 √ √ e 2π σ t−t ST A RT n



for n ≥ 0



2  2  k NINC (t) = ekμ(t−t ST A RT ) e k −k σ (t−t ST A RT )/2



Mode n mode ≡ n peak = eμ(t−t ST A RT ) e− (=abscissa of the lognormal peak) Value of the mode peak

f NINC (t) (n mode ) =

Median [= fifty-fifty probability value for N FIX (t)]

median = m = eμ(t−t ST A RT ) e−

3σ 2 (t−t ST A RT )/2

1 √ √ 2π σ t−t ST A RT

· e−μ(t−t ST A RT ) · eσ



Skewness

K3 (K 2 )(3/2)

Kurtosis

K4 (K 2 )2



σ 2 (t−t ST A RT )/2

2 (t−t

ST A RT )



 2  2 = eσ (t−t ST A RT ) + 2 eσ (t−t ST A RT ) − 1

= e4σ

2 (t−t

ST A RT )

+ 2e3σ

2 (t−t

ST A RT )

+ 3e2σ

2 (t−t

ST A RT )

−6

Clearly, these three different GBM stochastic processes have different numerical values of t START , μ and σ, but the equations are the same for all processes

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SETI as a Part of Big History

Equating (40) and (41), one thus gets  N0 eσ 2 (−tST A RT ) − 1 = δ N0 .

(42)

This is an equation where all quantities are known except for σ. Solving thus (42) for σ 2 one finds 



ln 1 + δ N02 /N02 2 . (43) σ = −t ST A RT Note that, in the limit of no current standard deviation, i.e. δN 0 = 0, then (43) yields σ = 0, as expected. Finally, taking the root of (43), one finds 

 ln 1 + δ N02 /N02 σ = . −t ST A RT

(44)

which is σ for OUR INCREASING GBM process. In conclusion, Table 3 holds.

3.5 Example: Darwinian Evolution as a GBM Taking the Value of One at Its Start (Time t = tSTART ) with Known Current Mean Value and Standard Deviation As an example of the mathematical apparatus developed in the last section, we now consider: 1. The mean value curve given by (32), that is mean_ value(t) = eμ(t−tST A RT ) .

(45)

2. The upper standard deviation curve given by the mean value curve (32) plus (38), that is    (46) upper_ st_ dev(t) = eμ(t−tST A RT ) 1 + eσ 2 (t−tST A RT ) − 1 . 3. The lower standard deviation curve given by the mean value curve (32) minus (38), that is    lower_ st_ dev(t) = eμ(t−tST A RT ) 1 − eσ 2 (t−tST A RT ) − 1 .

(47)

These three curves are plotted in Fig. 3 for the Darwinian Evolution case with the following assumed values: 1. Beginning time of life on Earth (RNA?) t START = –3.5 × 109 years.

3 Geometric Brownian Motion (GBM) as ...

483

2. The current mean value of the number of living species on Earth is 50 million, i.e. N 0 = 50 × 106 . 3. The current standard deviation of the number of living species on Earth is 30 million, i.e. ΔN 0 = 30 × 106 (rather arbitrary, probably). Of course, the stochastic process itself is not shown, since it would change at every different realization (i.e. simulation) of it. Only its mean value and standard deviation always stay the same at every new run! (Figs. 3 and 4).

Fig. 3 DARWINIAN EVOLUTION as the increasing number of living species on Earth between 3.5 billion years ago and now. The red solid curve is the mean value of the GBM stochastic process N INCREASING (t) given by (45), while the blue dot-dot curves above and below the mean value are the two standard deviation upper and lower curves, given by (46) and (47), respectively. The “Cambrian Explosion” of life on Earth started around 542 million years ago, as it is evident in the above plot just before the value of –0.5 billion years in time, and shows that our GBM model of Darwinian Evolution is basically correct. Notice that the starting value of living species 3.5 billion years ago is ONE by definition, but it “looks like” zero in this plot since the vertical scale (which is the true scale, not a log scale) does not show it. To see it better, please look at the following plot, where the vertical scale is logarithmic to base 10. Notice also that nowadays (i.e. at time t = 0) the two standard deviation curves have exactly the same distance from the middle mean value curve, i.e. 30 million living species more or less the mean value of 50 million species. These are assumed values that we used just to exemplify the GBM mathematics: biologists might assume other numeric values

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SETI as a Part of Big History

Fig. 4 The same as Fig. 3 but with the vertical scale in logs to base 10. One may now see that the starting value of all three curves (now become straight lines) (45–47) is indeed ONE, and not zero, as it might erroneously “look like” from the true vertical scale shown in Fig. 3. Also, notice that the “distance” of the two upper and lower standard deviation (dot–dot blue) lines from the central, solid and red mean value line appears to be “different” because of the deformation introduced by the logarithmic scale on the vertical axis (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article)

4 Big History as the Statistical Drake Equation Extended by Adding the “Missing Initial Part” 4.1 Big Bang to Current Stars: The “Missing Initial Part” of the Drake Equation At last we may come back to Big History, since we now possess the full GBM apparatus described in the previous sections in order to study Big History stochastically, and not just “by words”. Big History is the history of the Universe since the Big Bang, known by cosmologists to have occurred 13.798 ± 0.037 billion years ago. Thus, we have for our Big History N INCREASING (t) GBM the initial value tBIG_ BANG = −13.798 × 109 years.

(48)

The next obvious question is: “What” is actually increasing during the 13.8 billion years of Big History? A plausible answer to such a question seems to be provided by the Drake Eq. (1) that covers already the time segment from the creation of the Milky Way galaxy, about 10 billion years ago, up to now (time t = 0). So, we can say that, in order

4 Big History as the Statistical Drake Equation Extended...

485

to describe Big History mathematically, we just have to add one (or possibly more) multiplicative factor(s) in front of the Drake Eq. (1). Let us call this new factor Ng and let it represent the number of galaxies in the universe that evolved since the Big Bang. The equation so created thus reads CU = N g N s f p ne f l f i f c f L .

(49)

Here, the new variable CU on the left is of course the number of Civilizations existing in the whole Universe right now, and not just in the Milky Way galaxy alone, as assumed by the classical Drake Eq. (1). We call (49) the Big History equation. So, our answer to the question “What” is actually increasing during the 13.8 billion years of Big History?” is “THE NUMBER OF CIVILIZATIONS IN THE WHOLE UNIVERSE”. Thus, we are now ready to take the next steps in our “mathematization of Big History”. These are: 1. The replacement of all multiplicative real variables in (49) by random variables, just as we did in jumping from the classical Drake equation of 1961 to our statistical Drake equation of 2008 (readers might wish to read the original papers [11, 12]). 2. The assumption that the set of eight real positive random variables appearing in the Statistical Big History Eq. (49), however distributed, are in a sufficient number to let the Central Limit Theorem (CLT) of Statistics to be “reasonably applied”. This is the same assumption that we made already in the transition from the Classical Drake Equation into the Statistical Drake Equation, as completely described in the author’s 2012 book on “Mathematical SETI” [16]. 3. But now we are ready to take the boldest step also: namely the replacement of all lognormal random variables in the Big History Eq. (49) by stochastic processes of the GBM type given by (27), namely we have the Stochastic Big History Equation CU ≡ N umber o f Civili zation in the U niver se(t) = N g(t)N s(t) f p(t)ne(t) f l(t) f c(t) f L(t).

(50)

Mathematically, this “final” step is acceptable since the probability density functions in (50) are lognormals, as we have seen in all previous sections, and so there is no change in our statistical (or, better, stochastical) assumptions. In conclusion, the Evolution of the Universe, now understood as the GBMincreasing number of Civilizations over the last 10 billion years, may be plotted as a function of the time, as in Fig. 5 hereafter. The plot in Figs. 5 and 6 is based upon the following assumed data: 1. The Start time of the Universe is the Big Bang time, t BIG_BANG = –13.798 × 109 years. However, nearly four billion years were necessary for matter to separate from radiation, and then form quasars, proto-galaxies and galaxies. Thus, we

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SETI as a Part of Big History

shall tentatively assume that the first civilizations in the Universe might possibly have been formed only after the start time of about 10 billion years ago: t ST A RT = −10 × 109 years.

(51)

Fig. 5 Evolution of the universe, from the 10 billion years ago to nowadays regarded as the increasing GBM stochastic process in the number of civilizations populating the universe

Fig. 6 The same as Fig. 5 but with the vertical axis in logarithmic scale to base 10

4 Big History as the Statistical Drake Equation Extended...

487

2. The current number of galaxies in the Universe is assumed to be 170 billion, namely we assumed N0 = 1.7 × 1011

(52)

as provided by [6]. However, this numeric value is probably much smaller than the “true” unknown value, inasmuch as (52) is an estimate of the number of visible galaxies only. 3. The assumed “error” on N 0 is about 58%, namely we assumed (arbitrarily) ΔN0 = 1011 .

(53)

Let us finally make some consideration about Fig. 5, showing our “GBM for Civilizations in the whole Universe” model: 1. As we said, the adopted value of N 0 in (52) is likely to be smaller than the “true” unknown value. 2. Truly arbitrary (and so very possibly “wrong”) also is the ΔN 0 value assumed in (53). 3. But there may be other causes of “error” also. For example, we should make a distinction between quasars, proto-galaxies and true galaxies. Thus, we might introduce three (or more) factors (rather than just one) in front of Ns in the Stochastic Big History Eq. (50). 4. Also, we might consider clusters and super-clusters of galaxies instead of just galaxies in (50), and so on and so forth. Thus, the actual numerical values (51)–(53) that we used in this paper really are conjectural. The true step ahead appears to be in the GBM methodology. The use of GBMs in Big History and its subsets, like Darwinian Evolution and Human History truly opens up a new stochastic vision of the universe where we live, and we content ourselves in having broken this new ground by virtue of introducing the GBM stochastic process with exponential mean, apparently not done before by other authors.

5 Mass Extinctions in the Course of Darwinian Evolution Understood by Virtue of a Decreasing GBM 5.1 A Brand-New Discovery: GBMs to Understand Mass Extinctions of the Past SETI as a part of Big History is the main topic of this research paper, and we already provided examples of how GBMs adapt themselves as mathematical models to SETI and Big History. But there is one more example of GBM application that we would

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SETI as a Part of Big History

like to describe: the use of GBMs to understand the mass extinctions that occurred on Earth several times in its geological past. The most notable example probably is the mass extinction of dinosaurs 64 million years ago, now widely recognized by scientists as caused by the impact of a ~10 km asteroid where is now the Chicxulub crater in Yucatan, Mexico [1, 2]. Clearly, by using the GBMs as our model for mass extinctions, we are automatically ruling out the possibility that mass extinctions are a cyclical (in time) phenomenon, as suggested for instance by Rhodes and Muller [20]. The mathematics of cyclic mass extinctions is different from GBMs, and is concisely described in the Appendix to this paper. This author plans to study that thoroughly in a further coming paper. Then, going back to GBMs, consider the GBM mean value (45) again. That is the mean value of a GBM increasing in time to simulate the rise of more and more species in the course of Darwinian Evolution, so μ > 0 for it. But in modeling mass extinctions we clearly must have a decreasing GBM, i.e. μ < 0, and over a much shorter time lapse, just years or some century instead of billions of years as in Darwinian Evolution. So, the starting time now is the asteroid impact time, t Impact , and (45) becomes mean_ value(t) = Ceμ(t−tImpact )

(54)

where C is a constant that we now determine. Just think that, at the impact time, (54) yields 

mean_ value tImpact = C.

(55)

On the other hand, at the same impact time, one has 

mean_ value tImpact = NImpact

(56)

where N Impact is the number of living species on Earth just seconds before the asteroid impact time. Thus, (55) and (56) immediately yield C = NImpact

(57)

This, inserted into (54), yields the final mean value curve as a function of the time mean_ value(t) = NImpact eμ(t−tImpact )

(58)

Let us now consider what happens after the impact, namely the death of many living species over a period of time called “nuclear winter” and caused by the debris thrown into the Earth atmosphere by the asteroid ejecta. Nobody seem to know exactly how long did the nuclear winter last after the impact that actually killed all dinosaurs and other species, but not the mammals, who, being much smaller and so

5 Mass Extinctions in the Course of Darwinian Evolution Understood ...

489

much more easy-fed, could survive the nuclear winter. Mathematically, let us call t End the time when the nuclear winter ended, so that the overall time span of the mass extinction is given by tEnd − tImpact .

(59)

At time t End , a certain number of living species, say N End , survived. Replacing this into (58) yields NEnd = mean_ value(tEnd ) = NImpact eμ(tEnd −tImpact )

(60)

Solving (60) for μ yields the first basic formula for our GBM model of mass extinctions: 

ln NImpact /NEnd μ=− . (61) tEnd − tImpact This is the (negative) μ for mass extinctions, and corresponds to the μ of (35) for Darwinian Evolution, apart from the opposite sign of μ. Notice that in (61) are four input variables 

tImpact , NImpact , tEnd , NEnd

(62)

that we must assign numerically in order determine μ for a mass extinction. Let us also remark that it is convenient to introduce two new variables, Time_Lapse and t Extinction , respectively defined as the overall amount of time during which the extinction occurs, and the middle instant in this overall time lapse, namely:

T ime_Lapse = tEnd − tImpact t +t tExtinction = Impact2 End .

(63)

Clearly, (61), by virtue of the upper (63), becomes 

ln NImpact /NEnd . μ=− T ime_Lapse

(64)

This version of (61) is easier to differentiate than (61) itself inasmuch as it only has three independent variables instead of four. Thus, the total differential of (64) is found (but we will not write all the steps here), and, once divided by (64), yields the relative error on μ expressed in terms of the relative errors on N Impact , N End and Time_Lapse: δ NImpact δμ δT ime_Lapse 1   =− +

μ T ime_Lapse ln NImpact NEnd NImpact

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SETI as a Part of Big History

−



1



ln NImpact NEnd



δ NEnd . NEnd

(65)

This, of course, corresponds to (37) for the case of Darwinian Evolution, but here we have three independent variables instead of just two, as in (37). Let us now find σ. To this end, we must introduce a fifth input [besides the four ones given by (62)], denoted δN End and representing the standard deviation affecting the number of living species on Earth after the end of the nuclear winter, i.e. when life starts growing up again. This means that we must now consider the GBM standard deviation function of the time, (t), already given by (38), that, in this case, takes the form:  2 Δ(t) = NImpact eμ(t−tImpact ) eσ (t−tImpact ) − 1.

(66)

At the End time, t End , (66) takes the form Δ(tEnd ) = NImpact e

μ(tEnd −tImpact )



2 eσ (tEnd −tImpact ) − 1.

(67)

But this equals δN End by the very definition of δN End , and so we get the new equation δ NEnd = NImpact e

μ(tEnd −tImpact )

 2 eσ (tEnd −tImpact ) − 1.

(68)

This is basically the equation in σ we were seeking. We only have to replace μ into (68) by virtue of (64), and then solve the resulting equation for σ. By doing so (and we will omit the relevant steps for the sake of brevity), we finally get the sought expression of σ:   

    ln 1 + δ NEnd NEnd 2  . σ = T ime_Lapse

(69)

This is the GBM σ for the mass extinctions, and corresponds to (44) for the Darwinian Evolution GBM. Please note the special δN END = 0 case of (69), then immediately yielding σ = 0. This is the special case where the GBM “converges” (so as to say) into a single point at t = t END , namely with probability one there will be exactly N End species that survived the nuclear winter after the impact. If you prefer, this is just like the initial condition of ordinary Brownian motion, B(0) = 0, that is always fulfilled with probability one. But in this case it is a final condition, rather than an initial condition. As such, this particular case of (69) is hardly realistic in the true world of an after-impact. Nevertheless we wanted to point it out just to show how subtle the mathematics of stochastic processes can be.

5 Mass Extinctions in the Course of Darwinian Evolution Understood ...

491

Another remark following from (69) is about the expression of the relative error on σ, namely δσ /σ, expressed in terms of the four inputs (62) plus δN End . The relevant expression is long and complicated, and we will not rewrite it here. Finally, it must to be mentioned that the upper standard deviation curve is given by (58) plus (66), that is upper_ st_ dev(t) = NImpact e

μ(t−tImpact )

   σ 2 (t−tImpact ) 1+ e −1

(70)

and it has a maximum at the just-after-impact time  ⎡  ⎤ 2μ μ2 − 2μσ 2 − σ 4 + σ 2 + 3μ 1 ⎦ tImpact + 2 · ln⎣

2 σ σ 2 + 2μ

(71)

Again, we will not rewrite here all the steps leading to (71), and just confine ourselves to mentioning that one gets a quadratic in eσ 2t that, solved for t, yields (71). Having so given the mathematical theory of mass extinctions provided by GBMs, we now proceed to showing a numerical example. Naturally, the chosen example is about the K–Pg impact and the ensuing nuclear winter, that here we suppose to have lasted a thousand years after the impact itself, though other shorter time lapses could be considered as well.

5.2 Example: The K–Pg Mass Extinction Extending Ten Centuries After Impact Warning: the numeric example and graph we now present is just an exercise, and we do not claim that it really shows what happened 64 million years ago during the K–Pg impact and consequent mass extinction. Yet it provides useful hints about how the GBMs work in the simulations of true mass extinctions, and not just those of the past, but also those of the future, would an asteroid hit the Earth again and cause millions or billions of Human casualties: Planetary Defense is a “must” for us! So, let us assume that: 1. The K–Pg impact occurred exactly 64 million years ago (this is just to simplify the calculations a little): tImpact = −64 × 106 year.

(72)

2. At impact, there were 100 living species on Earth NImpact = 100.

(73)

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SETI as a Part of Big History

Again, this is likely to be very roughly underestimated, but we use 100 so as to immediately draw the percentage of surviving species, namely. 3. At the end of the impact effects, there were only 30 living species, namely only the 30% survived NEnd = 30.

(74)

4. We also assume that the error on the value of (74) is about 33.3%. In other words, we assume δ NEnd = 10.

(75)

5. Finally, we assume that the impact effects lasted for a thousand years, that is tEnd − tImpact = 1000 year

(76)

from which, by virtue of (72), we infer tEnd = −63.999 × 106 year.

(77)

These are our five input data. The two outputs then are: μ = −3.815 × 10−11

1 1 = −1.204 × 10−3 s year

(78)

σ 2 = 3.339 × 10−12

1 1 = 1.054 × 10−4 . s year

(79)

and

Figure 7 shows the mean value curve (solid red curve), and the two upper and lower standard deviation curves (dot–dot blue curves) for the corresponding GBM in the decreasing number of living species on Earth as the consequence of the impact. The K–Pg mass extinction is a decreasing GBM in the number of living species over 1000 years after impact. The maximum of the upper standard deviation curve has the numeric value –6.3999983 × 107 years given by (71). In conclusion, Table 4 summarizes all results about the decreasing GBM representing the decreasing number of species on Earth during a mass extinction. In the future, these ideas should be extended not just to the analysis of all mass extinctions occurred in the geological past of Earth, but also to crucial events in Human History such as wars, famines, epidemics and so on when mass extinctions of Humans occurred. An excellent topic to describe mathematically large sections of History that, so far, were mostly described by means of words only. By doing so, we would significantly contribute the studies on Mathematical History.

6 Conclusion

493

Fig. 7 Decreasing number of species during K–Pg Mass extinction

6 Conclusion Let us finally reach to the conclusions of this paper: 1. We developed here a new mathematical model embracing all of Big History, including Darwinian Evolution (RNA to Humans), Human History (Aztecs to USA) (see [17]) and then we extrapolated even that into the future up to 10 million years (see [18]), the minimum time requested for a civilization to expand to the whole Milky Way (Fermi paradox). 2. Our mathematical model is based on the properties of lognormal probability distributions. It also is fully compatible with the Statistical Drake Equation, i.e. the foundational equation of SETI, the Search for ExtraTerrestrial Intelligence. 3. Merging all these apparently different topics into the larger but single topic called Big History is the achievement of this paper. As such, our statistical theory would be crucial to estimate how much more advanced than Humans the Aliens would be when SETI scientists will succeed in finding the first ET Civilization.

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Table 4 Summary of the properties of the lognormal distribution that applies to the stochastic process N DECREASING (t) = exponentially decreasing number of living species on Earth during a mass extinction Stochastic process

N D EC R E AS I N G (t) ≡ N D EC (t) = N umber o f Living Species (in a Mass E xtinction)

Probability distribution

Lognormal distribution of the ADJUSTED and DECREASING GBM starting at t Impact

Probability density function

NDEC (t)_ pd f (n; μ, σ, tImpact , t) =

Mean value Variance Standard deviation All the moments, i.e. k-th moment Mode (=abscissa of the lognormal peak) Value of the mode peak

e

−(([ln(n)−(ln(NImpact )+μ(t−tImpact )−((σ 2 (t−tImpact ))/2))]2 )/2σ 2 (t−tImpact ))

√

√

t−tImpact n

f or n ≥ 0.

N D EC (t) = N I mpact eμ(t−tImpact )  #   $ 2μ t−tImpact σ 2 t−tImpact 2 2 σDEC(t) = NImpact e −1 e     μ t−tImpact σ 2 t−tImpact

σ N D EC (t) = NImpact e e −1 

2  2  k  k −k σ (t−t I mpact )/2 k kμ t−t ( ) I mpact N D EC (t) = N I mpact e e n mode ≡ n peak = N I mpact eμ(t−t I mpact ) e− f N D EC (t) (n mode ) = √ 1√

N I mpact 2πσ

Median [=fifty-fifty probability value for N FIX (t)]

2πσ

t−t I mpact

2 · e−μ(t−t I mpact ) · eσ (t−t I mpact )

median = m = N I mpact eμ(t−tImpact ) e−

Skewness

K3 (K 2 )(3/2)

Kurtosis

K4 (K 2 )2

2  3σ (t−t I mpact )/2

2  σ (t−tImpact )/2

  2 2 = eσ (t−tImpact ) + 2 eσ (t−tImpact ) − 1

2 2 2 = e4σ (t−t I mpact ) + 2e3σ (t−t I mpact ) + 3e2σ (t−t I mpact ) − 6

Appendix: Cyclic Phenomena as Lognormal Stochastic Processes In Sect. 5.1 of this paper we pointed out that cyclic phenomena cannot be represented by GBMs. This author, however, recently found a way to represent all cyclic phenomena as lognormal stochastic processes, and even more general phenomena whose mean value is a know function m(t). But he is not going to disclose these recent mathematical results in this paper since he feels that a new, adequately extended mathematical paper is necessary to explain the details of his procedure. Just to give the feeling of these new results, let us look at Fig. 8. On the horizontal axis are the years between 1725 and 2025 (the period 2014–2025 is of course a guess about the future). On the vertical axis are three curves:

Appendix: Cyclic Phenomena as Lognormal Stochastic Processes

495

Fig. 8 Kondratiev waves as a lognormal stochastic process

1. The sinusoid (solid red) of the Kondratiev waves (i.e. the “waves of technological evolution” that the Russian economist Nikolay D. Kondratiev (1892–1938) assumed or “proved” (according to some) to describe the modern technological world. This solid red sinusoid is for us the mean value of a lognormal stochastic process that generalizes the Kondratiev theory into the world of statistics. The period of our red sinusoid is assumed to be 58 years. 2. The upper and lower standard deviation curves (dot–dot blue curves) have a sinusoidal behavior “similar” to that of the mean value solid red curve, but they depart more and more from it after the year 1771, that, in the Kondratiev theory, is assumed to be the year when the first Industrial Revolution began [in England, then elsewhere (1771–1829)]. 3. After the first Industrial Revolution, the age of Steam and Railways began in 1829 and continued until about 1887 (second Kondratiev wave). Then the age of steel and heavy engineering began (electricity, oil, big ships, airplanes, bridges, skyscrapers, automobiles, etc.): this is the third Kondratiev wave, about 1887–1945. After 1945 computers and information technology took over (fourth Kondratiev wave, 1945–2003), up to the Internet (~2000). Now (2014) we are living in the fifth Kondratiev wave [2003–2061(?)], which will possibly see the advent of Artificial Intelligence and Robotics, with the possible demise of Humans in front of “creatures” created by Humans, but far superior to them in intelligence (“Technological Singularity” to occur around 2050?). Kondratiev waves of technological evolution in the modern world (for a simpleminded summary, see, for instance, the site https://en.wikipedia.org/wiki/Kondra tieff_Waves) are both economic and technological cycles that this author was able to generalize into statistics as a lognormal stochastic process. Details will be given

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SETI as a Part of Big History

in a coming paper. Similarly to the Kondratiev waves are also the “Cycles in Fossil Diversity” as described by Robert A. Rhode and Richard A. Muller in 2005 in [20]. But the period there is 62 ± 3 million years and the standard deviation curves of this lognormal stochastic process are to be determined yet. This author intends to do so in the future.

References 1. L.W. Alvarez, W. Alvarez, F. Asaro, H.V. Michel, Extraterrestrial cause for the cretaceous– tertiary extinction. Science 208(4448), 1095–1108 (1980) 2. W. Alvarez, In the mountains of Saint Francis: discovering the geologic events that shaped our earth. A popular book available in Kindle edition 3. B. Balazs, The galactic belt of intelligent life, in Biostronomy—The Next Steps, ed. by G. Marx (Kluwer Academic Publishers, The Netherlands, 1988), pp. 61–66 4. M.J. Burchell, W(h)ither the Drake equation? Int. J. Astrobiol. 5, 243–250 (2006) ´ 5. M.M. Cirkovi´ c, On the temporal aspect of the Drake equation and SETI. Astrobiology 4, 225–231 (2004) 6. D. Deutsch, The Fabric of Reality (Penguin Books Limited, UK, 2011), p. 234. ISBN 978-014-196961-9 7. G. Gonzalez, D. Brownlee, P. Ward, The galactic habitable zone: galactic chemical evolution. Icarus 152, 185–200 (2001) 8. G. Gonzalez, Habitable zones in the universe. Orig. Life Evol. Biospheres 35, 555–606 (2005) 9. L.V. Ksanfomality, The Drake equation may need new factors based on peculiarities of planets of sun-like stars, in Proceedings of IAU Symposium #202, Planetary Systems in the Universe (2004), p. 458 10. C.H. Lineweaver, Y. Fenner, B.K. Gibson, The galactic habitable zone and the age distribution of complex life in the Milky Way. Science 303, 59–62 (2004) 11. C. Maccone, The statistical Drake equation. Paper #IAC-08-A4.1.4 Presented on October 1st, 2008, at the 59th International Astronautical Congress (IAC) held in Glasgow, Scotland, UK, Sept 29–Oct 3, 2008 12. C. Maccone, The statistical Drake equation. Acta Astronaut. 67, 1366–1383 (2010) 13. C. Maccone, The statistical fermi paradox. J. Br. Interplanet. Soc. 63, 222–239 (2010) 14. C. Maccone, SETI and SEH (Statistical Equation for Habitables). Acta Astronaut. 68, 63–75 (2011) 15. C. Maccone, A mathematical model for evolution and SETI. Orig. Life Evol. Biospheres (OLEB) 41, 609–619 (2011) (available online 03.12.11) 16. C. Maccone, Mathematical SETI (Praxis-Springer, Zürich, Fall of 2012), 724 p. ISBN, ISBN10: 3642274366|ISBN-13: 978-3642274367|Edition 17. C. Maccone, SETI, evolution and human history merged into a mathematical model. Int. J. Astrobiol. 12(3), 218–245 (2013) (Available online 23.04.13) 18. C. Maccone, Evolution and history merged in a new “Mathematical SETI” model. Acta Astronaut. 93, 317–344 (2014). Available online 13.08.13 19. L.S. Marochnik, L.M. Mukhin, Belt of life in the galaxy, in Biostronomy—The Next Steps, ed. by G. Marx (Kluwer Academic Publishers, UK, 1988), pp. 49–59 20. M. Rohde, Cycles in fossil diversity. Nature 434, 208–210 (2005) 21. S.G. Wallenhorst, The Drake equation reexamined. QJRAS 22, 380 (1981) 22. C. Walters, R.A. Hoover, R.K. Kotra, Interstellar colonization: a new parameter for the Drake Equation? Icarus 41, 193–197 (1980)

Lognormals for SETI, Evolution and Mass Extinctions

Abstract In a series of papers (Ref. Maccone, The Statistical Drake Equation, 2008) through (Maccone, A Mathematical Model for Evolution and SETI, 2011.) and (Maccone, Int J Astrobiol 12(3):218–245, 2013) through (Maccone, Acta Astronautica, 317–344, 2014) and in a book (Ref. Maccone, Mathematical SETI, 2012), this author suggested a new mathematical theory capable of merging Darwinian Evolution and SETI into a unified statistical framework. In this new vision, Darwinian Evolution, as it unfolded on Earth over the last 3.5 billion years, is defined as just one particular realization of a certain lognormal stochastic process in the number of living species on Earth, whose mean value increased in time exponentially. SETI also may be brought into this vision since the number of communicating civilizations in the Galaxy is given by a lognormal distribution (Statistical Drake Equation). Now, in this paper we further elaborate on all that particularly with regard to two important topics: (1) The introduction of the general lognormal stochastic process L(t) whose mean value may be an arbitrary continuous function of the time, m(t), rather than just the exponential m GBM (t) = N0 eμ t typical of the Geometric Brownian Motion (GBM). This is a considerable generalization of the GBM-based theory used in ref. (Maccone, The Statistical Drake Equation, 2008) through (Maccone, Acta Astronautica, 317–344, 2014). (2) The particular application of the general stochastic process L(t) to the understanding of Mass Extinctions like the K/Pg one that marked the dinosaurs’ end 65 million years ago. We first model this Mass Extinction as a decreasing Geometric Brownian Motion (GBM) extending from the asteroid’s impact time all through the ensuing “nuclear winter”. However, this model has a flaw: the “final value” of the GBM cannot have a horizontal tangent, as requested to enable the recovery of life again after this “final extinction value”. (3) That flaw, however, is removed if the rapidly decreasing mean value function of L(t) is the left branch of a parabola extending from the asteroid’s impact time all through the ensuing “nuclear winter” and up to the time when the number of living species on Earth started growing up again, as we show mathematically in Sect. 3. In conclusion, we have uncovered an important generalization of the GBM into the general lognormal stochastic process L(t), paving the way to a better, future understanding the evolution of life on Exoplanets on the basis of what Evolution unfolded on Earth in the last 3.5 billion years. That will be the goal of further research papers in the future. © Springer Nature Switzerland AG 2020 C. Maccone, Evo-SETI, https://doi.org/10.1007/978-3-030-51931-5_13

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Keywords Darwinian evolution · Mass extinctions · Statistical drake equation · Lognormal stochastic processes

1 Introduction: New Statistical Mechanisms This paper describes a new statistical theory casting Evolution and SETI into mathematical terms, rather than just by using words only. The basic statistical tools used in this paper are: (1) The general stochastic process called Lognormal process L(t), embodying an arbitrary function of the time, M(t) and an arbitrary positive numeric parameter σ . Thus, the probability density function of this Lognormal process L(t) is denoted by [ln(n)−M(t)]2

e− 2 σ 2 t L(t)_ pd f (n; M(t), σ, t) = √ √ ,(n ≥ 0). 2π σ t n

(1)

(2) The mean value of the Lognormal process (1) is then given by ∞ m L (t) ≡ 0

(ln(n)−M L (t))2

e− 2 σ 2 t σ2 √ √ dn = e M L (t) e 2 t . 2π σ tn

(2)

When solved for M(t), the last equation yields M L (t) = ln(m L (t)) −

σ2 t. 2

(3)

Table 1 shows the main statistical properties of the lognormal stochastic process (1): we skip all the proofs, since those proofs would take many pages, and would also… deprive the reader from the “mathematical delight” of checking our results by virtue of some symbolic manipulator like Maxima, Maple, Mathematica, and the like. (3) The particular case of (2) when that mean value is given by the generic exponential m GBM (t) = N0 eμ t

(4)

is called Geometric Brownian Motion (GBM), and is widely used in financial mathematics, where it represents the “underlying process” of the stock values (Black-Sholes models). This author also widely used the GBM in his previous mathematical models of Evolution and SETI (Refs. [1–3]), since it was assumed that the growth of the number of ET civilizations in the Galaxy, or, alternatively, the number of living species on Earth over the last 3.5 billion years, grew

1 Introduction: New Statistical Mechanisms

499

Table 1 Summary of the properties of the lognormal distribution that applies to the stochastic process L(t) = lognormally changing number of ET communicating civilizations in the Galaxy, as well as the number of living species on Earth over the last 3.5 billion years. Clearly, these two different L(t) lognormal stochastic processes may have two different time functions for M L (t) and two different numerical values for σ , but the equations are the same for both processes, i.e. for the number of ET civilizations in the Galaxy and for the number of living species in the past of Earth Stochastic Process

L(t) =  1)N umber o f E T Civili zations (in S E T I ). 2)N umber o f Living Species(in Evolution).

Probability distribution

Lognormal distribution of all LOGNORMAL stochastic processes, i.e. the lognormal stochastic processes with ARBITRARY MEAN m L (t)

Probability density function

L(t)_ pd f (n; M L (t), σ, t) = √

[ln(n)−M L (t)]2

− 1√ e 2π σ t n

for n ≥ 0

2 σ2 t

σ2

L(t) ≡ m L(t) = e M L (t) e 2 t  2  2 2 σ L(t) = e2M L (t) eσ t eσ t − 1

Mean value Variance

 2 eσ t − 1    σ2 2 = e M L (t) e 2 t 1 + eσ t − 1 σ2 2

Standard Deviation

σ L(t) = e M L (t) e

Upper Standard Deviation Curve

m L(t) + σ L(t)

Lower Standard Deviation Curve

m L(t) − σ L(t) = e M L (t) e

All the moments, i.e. k-th moment

t

2 L k (t) = ek M L (t) e k

σ2 2

σ2 2

t

t

Mode (=abscissa of the lognormal peak)

n mode ≡ n peak = e M L (t) e− σ

Value of the Mode Peak

f L(t) (n mode ) =

Median (=fifty-fifty probability value for N (t))

median = m = e M L (t)

Skewness Kurtosis

2t

· e−M L (t) · e

σ2 2

t

 2  2 = eσ t + 2 eσ t − 1

K3

3 (K 2 ) 2

K4 (K 2 )2

√ 1 √ 2π σ t

   2 1 − eσ t − 1

= e4 σ

2t

+ 2 e3 σ

2t

+ 3 e2 σ

2t

−6

exponentially (Malthusian growth). However, this author now realizes that this assumption, symbolized by the exponential mean value (4), was too restrictive. The replacement of (4) by (2) in all models for Evolution and SETI is the main achievement of this paper. Notice also that, upon equating the two right-hand-sides of (2) and (4), we find that e MGBM (t) e

σ2 2

t

= N0 eμ t .

(5)

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Solving this equation for MGBM (t) yields  σ2 t MGBM (t) = ln N0 + μ − 2

(6)

which is just the “mean value at the exponent” of the well-known pdf of the GBM, that is GBM(t)_ pd f (n; N0 , μ, σ, t) =

=

e

−

  2 2 ln(n)− ln N0 + μ− σ2 t 2 σ2 t

(7) ,(n ≥ 0).

√ √ 2π σ t n

We conclude this short description of the GBM as the particular exponential (“Malthusian”) case of the general lognormal process (1) by warning the reader that the denomination “Geometric Brownian Motion” is a rather misleading one, since it lets the readers think that GBM are Gaussian (or normal) processes, whereas they are lognormal processes instead. (4) Another interesting particular case of the mean value function m(t) in (2) is when it equals a generic polynomial in t, namely 

polynomial_ degree

m polynomial (t) =

ck t k

(8)

k=0

where the ck is the coefficient of the k-th power of the time t in the polynomial (8). The relevant function Mpolynomial (t) is then found at once by virtue of (8) and (3), and reads: Mpolynomial (t) = ln

polynomial_ degree 

 ck t

k=0

k

−

σ2 t. 2

(9)

It is unfortunately impossible to expand this equation any further, since the logarithm of a sum may not be expanded. (5) Thus, more important than (9), is the time-power case for which the mean value (2) grows like the time t raised to a real exponent α, multiplied by a constant C, that is m time_ power (t) = C t α .

(10)

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501

Then, (3) yields immediately Mtime_ power (t) = ln(C) + α ln(t) −

σ2 t. 2

(11)

(6) In particular, we wish to consider the parabolic case m parabolic (t) = C t 2

(12)

for which (3) yields Mparabolic (t) = ln(C) + 2 ln(t) −

σ2 t. 2

(13)

This case we will use later in this paper to model the Mass Extinctions of living species that occurred on Earth millions of years ago. (7) Finally, the linear case is obviously given by m linear (t) = C t

(14)

with Mlinear (t) = ln(C) + ln(t) −

σ2 t. 2

(15)

2 Mass Extinctions of Darwinian Evolution Described by a Decreasing Geometric Brownian Motion 2.1 GBMs to Understand Mass Extinctions of the Past In this section we describe the use of GBMs to model the mass extinctions that occurred on Earth several times in its geological past. The most notable example probably is the mass extinction of dinosaurs 64 million years ago, now widely recognized by scientists as caused by the impact of a ~10 km asteroid where is now the Chicxulub crater in Yucatan, Mexico (Refs. [8, 9]). Incidentally, in 2007 this author was part of a NASA team in charge of studying a space mission capable of deflecting an asteroid off its collision course against the Earth, should this event unfortunately occur again in the future: so he got a background in Planetary Defense. But let us now go straight to the GBMs and consider the mean value given in the fifth line of Table 2 again. That is the mean value of a GBM increasing in time to simulate the rise of more and more species in the course of Evolution, so μ > 0 for it. But in modeling mass extinctions we clearly must have a decreasing GBM, i.e.

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Table 2 Summary of the properties of the lognormal distribution that applies to the stochastic process N (t) = exponentially increasing number of ET communicating civilizations in the Galaxy, as well as the number of living species on Earth over the last 3.5 billion years. Clearly, these two different GBM stochastic processes have different numerical values of N0 ,μ and σ , but the equations are the same for both processes Stochastic Process

N (t) =  (1) N umber o f E T Civili zations (in S E T I ). (2) N umber o f Living Species (in Evolution).

Probability distribution

Lognormal distribution of the Geometric Brownian Motion (GBM), i.e. the lognormal stochastic process with EXPONENTIAL MEAN

Probability density function

N (t)_ pd f (n, N0 , μ, σ, t) = 2

  2 t ln(n)− ln N0 + μ− σ2 − 2 σ2 t e

for n ≥ 0   2 MGBM (t) = ln N0 + μ − σ2 t √

Particular MGBM (t) function Mean value Variance Standard Deviation Upper Standard Deviation Curve Lower Standard Deviation Curve All the moments, i.e. k-th moment

√ 2π σ t n

N (t) ≡ m N (t) = N0 eμ t  2  σ N2 (t) = N02 e2μ t eσ t − 1  2 σ N (t) = N0 eμ t eσ t − 1    2 m N (t) + σ N (t) = N0 eμ t 1 + eσ t − 1    2 m N (t) − σ N (t) = N0 eμ t 1 − eσ t − 1

 2 

N k (t) = N0k ekμ t e k −k

σ2 t 2

Mode (=abscissa of the lognormal peak)

n mode ≡ n peak = N0 eμ t e−

Value of the Mode Peak

f N (t) (n mode ) =

Median (=fifty-fifty probability value for N (t))

median = m = N0 eμ t e−

Skewness Kurtosis

K3

3 (K 2 ) 2

K4 (K 2 )2

√1 √ N0 2π σ t

3 σ2t 2

· e−μ t · eσ

2t

σ2t 2

 2  2 = eσ t + 2 eσ t − 1 = e4 σ

2t

+ 2 e3 σ

2t

+ 3 e2 σ

2t

−6

μ < 0, over a much shorter time lapse, just years or some century instead of billions of years as in Darwinian Evolution. So, the starting time now is the impact time, tImpact , and our GMB mean value becomes mean_ value(t) = C eμ (t−tImpact )

(16)

2 Mass Extinctions of Darwinian Evolution Described …

503

where C is a constant that we now determine. Just think that, at the impact time, (16) yields   mean_ value tImpact = C .

(17)

On the other hand, at the same impact time, one has   mean_ value tImpact = NImpact

(18)

where NImpact is the number of living species on Earth just seconds before the asteroid impact time. Thus, (17) and (18) immediately yield C = NImpact .

(19)

This, inserted into (16), yields the final mean value curve as a function of the time mean_ value(t) = NImpact eμ (t−tImpact )

(20)

Let us now consider what happens after the impact, namely the death of many living species over a period of time called “nuclear winter” and caused by the debris thrown into the Earth atmosphere by the asteroid ejecta. Nobody seem to know exactly how long did the nuclear winter last after the impact that actually killed all dinosaurs and other species, but not the mammals, who, being much smaller and so much more easy-fed, could survive the nuclear winter. Mathematically, let us call tEnd the time when the nuclear winter ended, so that the overall time span of the mass extinction is given by tEnd − tImpact .

(21)

At time tEnd , a certain number of living species, say NEnd , survived. Replacing this into (20) yields NEnd = mean_ value(tEnd ) = NImpact eμ (tEnd −tImpact )

(22)

Solving (22) for μ yields the first basic formula for our GBM model of Mass Extinctions:   N ln NImpact End μ=− . (23) tEnd − tImpact Notice that in (23) are four input variables 

tImpact , NImpact , tEnd , NEnd



(24)

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Lognormals for SETI, Evolution and Mass Extinctions

that we must assign numerically in order determine μ for a mass extinction. Let us also remark that it is convenient to introduce two new variables, T ime_Lapse and t E xtinction , respectively defined as the overall amount of time during which the extinction occurs, and the middle instant in this overall time lapse, namely: 

T ime_Lapse = tEnd − tImpact t +t t E xtinction = Impact2 End .

(25)

Clearly, (23), by virtue of the upper (25), becomes  μ=−

ln

NImpact NEnd



T ime_Lapse

.

(26)

This version of (23) is easier to differentiate than (23) itself inasmuch as it only has three independent variables instead of four. Thus, the total differential of (26) is found (but we will not write all the steps here), and, once divided by (26), yields the relative error on μ expressed in terms of the relative errors on NImpact , NEnd and T ime_Lapse: δT ime_Lapse δμ =− μ T ime_Lapse δ NImpact 1 1 δ NEnd · · +  −  . NImpact N Impact N NEnd Impact ln NEnd ln NEnd

(27)

Let us now find σ . To this end, we must introduce a fifth input (besides the four ones given by (24)), denoted δ NEnd and representing the standard deviation affecting the number of living species on Earth at the end of the nuclear winter, i.e. when life starts growing up again. This means that we must now consider the GBM standard deviation function of the time, (t), given by the 7th line in Table 2, that, in this case, takes the form: (t) = NImpact eμ(t−tImpact )



2 eσ (t−tImpact ) − 1 .

(28)

At the End time, tEnd , (28) becomes (tEnd ) = NImpact e

μ(tEnd −tImpact )



2 eσ (tEnd −tImpact ) − 1 .

(29)

But this equals δ NEnd by the very definition of δ NEnd , and so we get the new equation δ NEnd = NImpact e

μ(tEnd −tImpact )



2 eσ (tEnd −tImpact ) − 1 .

(30)

2 Mass Extinctions of Darwinian Evolution Described …

505

This is basically the equation in σ we were seeking. We only have to replace μ into (30) by virtue of (26), and then solve the resulting equation for σ . By doing so (we omit the relevant steps for the sake of brevity), we finally get the sought expression of σ :   2    δ NEnd  ln 1 + NEnd  . (31) σ = T ime_Lapse This is the GBM σ for the Mass Extinctions. Notice that the special δ NEND = 0 case of (31), immediately yields σ = 0. This is the special case where the GBM “converges” (so as to say) into a single point at t = tEND , namely with probability one there will be exactly NEnd species that survived the nuclear winter after the impact. If you prefer, this is just like the initial condition of ordinary Brownian motion, B(0) = 0, that is always fulfilled with probability one. But in this case it is a final condition, rather than an initial condition. As such, this particular case of (31) is hardly realistic in the true world of an after-impact. Nevertheless we wanted to point it out just to show how subtle the mathematics of stochastic processes can be. Another remark following from (31) is about the expression of the relative error , expressed in terms of the inputs (24) plus δ NEnd . The relevant on σ , namely δσ σ expression is long and complicated, and we will not rewrite it here. Finally, it must to be mentioned that the upper standard deviation curve given by (20) plus (28), i.e.

  2 upper_ st_ dev(t) = NImpact eμ(t−tImpact ) 1 + eσ (t−tImpact ) − 1

(32)

has its maximum at the just-after-impact time   ⎤ 2 − 2μσ 2 − σ 4 + σ 2 + 3μ 2μ μ 1 ⎦ tImpact + 2 · ln⎣  2 σ σ 2 + 2μ ⎡

(33)

Again, we will not rewrite here all the steps leading to (33), and just confine 2 ourselves to mentioning that one gets a quadratic in eσ t that, solved for t, yields (33). Having so given the mathematical theory of mass extinctions provided by GBMs, we now proceed to showing a numerical example. Naturally, the chosen example is about the K–Pg impact and the ensuing nuclear winter, that here we suppose to have lasted a thousand years after the impact itself, though other shorter time lapses could be considered as well.

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Lognormals for SETI, Evolution and Mass Extinctions

2.2 Example: The K–Pg Mass Extinction Extending Ten Centuries After Impact Warning: the numeric example and graph we now present is just an exercise, and we do not claim that it really shows what happened 64 million years ago during the K–Pg impact and consequent mass extinction. Yet it provides useful hints about how the GBMs work in the simulations of true mass extinctions, and not just those of the past, but also those of the future, would an asteroid hit the Earth again and cause millions or billions of human casualties: Planetary Defense is a “must” for us! So, let us assume that: (1) The K–Pg impact occurred exactly 64 million years ago (this is just to simplify the calculations a little): tImpact = − 64 × 106 year .

(34)

(2) At impact, there were 100 living species on Earth NImpact = 100 .

(35)

Again, this is likely to be very roughly underestimated, but we use 100 so as to immediately draw the percentage of surviving species, namely… (3) At the end of the impact effects, there were only 30 living species, namely only the 30% survived NEnd = 30 .

(36)

(4) We also assume that the error on the value of (36) is about 33.3%. In other words, we assume δ NEnd = 10 .

(37)

(5) Finally, we assume that the impact effects lasted for a thousand years, that is tend − tImpact = 1000 year

(38)

from which, by virtue of (34), we infer tEnd = − 63.999 × 106 year .

(39)

These are our five input data. The two outputs then are: μ = −3.815 × 10−11

1 1 = −1.204 × 10−3 sec year

(40)

2 Mass Extinctions of Darwinian Evolution Described …

507

and σ 2 = 3.339 × 10−12

1 1 = 1.054 × 10−4 . sec year

(41)

Figure 7 shows the mean value curve (solid blue curve), and the two upper and lower standard deviation curves (dash-dash blue curves) for the corresponding GBM in the decreasing number of living species on Earth as the consequence of the impact. In conclusion, Table 4 summarizes all results about the decreasing GBM representing the decreasing number of species on Earth during a mass extinction. In the future, these ideas should be extended not just to the analysis of all mass extinctions occurred in the geological past of Earth, but also to crucial events in Human History such as wars, famines, epidemics and so on when mass extinctions of Humans occurred. An excellent topic to describe mathematically large sections of History that, so far, were mostly described by means of words only. By doing so, we would significantly contribute the studies on Mathematical History.

3 Mass Extinctions Described by an Adjusted Parabola Branch 3.1 Adjusting the Parabola to the Mass Extinctions of the Past The Mass Extinction model described in the previous section and based on an adjusted and decreasing GBM has a flaw: the tangent straight line to its mean value curve at the end time is not horizontal. Thus, it is not a realistic model inasmuch as its end time cannot correctly represent the starting point after which the number of living species on Earth started growing up again. On the contrary, the Mass Extinction model the we build up in this section does not have any such flaw: its end time is both the end time of the decreasing number of living species on Earth and its starting time also for increasing living species numbers. Namely, its tangent straight line there is indeed horizontal, as requested. Getting now over to the mathematics, consider the adjusted mean value curve of  by the parabola (i.e. second-order polynomial in the adjusted time  L(t) given t − tImpact  2   m parabola (t) = c2 t − tImpact + c1 t − tImpact + c0

(42)

In order to find its three unknown coefficients c1 , c2 , c3 we must resort to the initial and final conditions (i.e. the two boundary conditions of the problem):

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Lognormals for SETI, Evolution and Mass Extinctions



  m parabola tImpact = NImpact m parabola (tend ) = Nend

(43)

Inserting (42) into (43), the latter takes on the form 

  NImpact = m parabola tImpact = c0  2   Nend = c2 tend − tImpact + c1 tend − tImpact + c0

(44)

The last two equations reduce to the single one  2   Nend − NImpact = c2 tend − timpact + c1 tend − timpact

(45)

On the other hand, the time derivative of the mean value (42) is   dm parabola (t) = 2 c2 t − tImpact + c1 dt

(46)

Equating this to zero, and replacing the time by the end time, we impose that the tangent straight line at the end time must be horizontal. Thus, from (46) one gets:   2 c2 tend − tImpact + c1 = 0

(47)

that, solved for c1 and matched to (45), yields 

    2 2 Nimpact − Nend = c2 timpact + c1 timpact − tend − tend   c1 = −2 c2 tend − tImpact

(48)

These two linear equations in c1 and c2 may immediately solved for them, with the result ⎧ ⎨ c2 = NImpact −Nend2 (tend −tImpact ) (49) ⎩ c1 = −2 ( NImpact −Nend ) tend −tImpact Finally, inserting both (49) and the upper Eq. (44) into the mean value parabola (42), the latter takes its final form m parabola (t) 

= NImpact − Nend



!  2 t − tImpact t − tImpact + NImpact .  2 − 2 tend − tImpact tend − tImpact

(50)

One may immediately check that the two boundary conditions (43) are indeed fulfilled by (50). Also, the minimum of the parabola (50) (i.e. the zero of its first time

3 Mass Extinctions Described by an Adjusted Parabola Branch

509

derivative) falls at the end time tend , obviously by construction. i.e. because of (46). So, the parabola (50) is indeed the right curve with horizontal tangent line at the end that we were seeking. As for the standard deviation, it is given by the 7th row in Table 3, of course “adjusted” by replacing the time t appearing in Table 3 by the new time difference  t − tImpact appearing in the mean value curve (50) already. Thus, the standard deviation for the Parabolic Mass Extinction model is given by  σparabola (t) = m parabola (t) ·

2 eσ (t−tImpact ) − 1

(51)

Consequently, the upper standard deviation curve is

  2 m parabola (t) + σparabola (t) = m parabola (t) 1 + eσ (t−tImpact ) − 1

(52)

and the lower standard deviation curve is

  2 m parabola (t) − σparabola (t) = m parabola (t) 1 − eσ (t−tImpact ) − 1

(53)

Table 5 shows the statistical properties of our Parabolic Mass Extinction model.

3.2 Example: The Parabola of the K–Pg Mass Extinction Extending Ten Centuries After Impact At this point it is natural to check our Parabolic Mass Extinction model against the corresponding Exponential (i.e. GBM-based) Mass Extinction model. In order to allow for the perfect match between the two relevant plots, we shall assume that the five numeric input values given in Sect. 2.2 for the GBM model are numerically kept just the same for the parabolic model also. Thus, the following Fig. 2 is obtained for the Parabolic K–Pg Mass Extinction. Actually, we may now superimpose the two plots given by Figs. 1 and 2, respectively, thus obtaining Fig. 3.

4 Conclusions Let us finally reach the conclusions of this chapter: (1) In Sect. 1 we described how to “construct” a lognormal stochastic process L(t) whose mean is an assigned and “arbitrary” function of the time m L (t). This is of paramount importance for all future applications.

510

Lognormals for SETI, Evolution and Mass Extinctions

Table 3 Summary of the properties of the lognormal distribution that applies to the stochastic process L(t) = lognormally changing number of ET communicating civilizations in the Galaxy, as well as the number of living species on Earth over the last 3.5 billion years. Clearly, these two different L(t) lognormal stochastic processes may have two different time functions for M(t) and two different numerical values for σ , but the equations are the same for both processes, i.e. for the number of ET civilizations in the Galaxy and for the number of living species in the past of Earth Stochastic Process

P(t) =  (1) N umber o f E T Civili zations (in S E T I ). (2) N umber o f Living Species (in Evolution).

Probability distribution Probability density function

Lognormal distribution of stochastic processes with POLYNOMIAL MEAN   P(t)_ pd f n; Mpolynomial (t), σ, t = √

Particular Mpolynomial (t) function

− 1√ e 2π σ t n

 2 ln(n)−Mpolynomial (t) 2 σ2 t

for n ≥ 0



Mpolynomial (t) = ln

polynomial_ " degree

 ck t k

−

k=0

Mean Value Curve

P(t) ≡ m P(t) = e Mpolynomial (t) e polynomial_ " degree ck t k

Variance

2 σ P(t)

k=0

Standard Deviation

= e2Mpolynomial (t) eσ

2t



eσ

2t

σ2 2

t

σ2 2

t

=

 −1

2  Mpolynomial (t) e σ2 t eσ 2 t − 1 = σ = e P(t)   polynomial_  " degree 2 ck t k eσ t − 1

k=0

Upper Standard Deviation Curve

   σ2 2 m P(t) + σ P(t) = e Mpolynomial (t) e 2 t 1 + eσ t − 1 =     polynomial_  " degree 2 k ck t · 1 + e σ t − 1 k=0

Lower Standard Deviation Curve

   σ2 2 m P(t) + σ P(t) = e Mpolynomial (t) e 2 t 1 + eσ t − 1 =     polynomial_  " degree 2 k ck t · 1 + e σ t − 1 k=0

All the moments, i.e. k-th moment



2 P k (t) = ek Mpolynomial (t) e k

σ2 2

t

Mode (=abscissa of the lognormal peak)

n mode ≡ n peak = e Mpolynomial (t) e− σ

Value of the Mode Peak

f P(t) (n mode ) =

√ 1 √ 2π σ t

2t

· e−Mpolynomial (t) · e

σ2 2

t

(continued)

4 Conclusions

511

Table 3 (continued) Stochastic Process

P(t) =  (1) N umber o f E T Civili zations (in S E T I ). (2) N umber o f Living Species (in Evolution).

Median (=fifty-fifty probability value for P(t)) Skewness

median = m = e Mpolynomial (t) K3

3 (K 2 ) 2

Kurtosis

K4 (K 2 )2

 2  2 = eσ t + 2 eσ t − 1 = e4 σ

2t

+ 2 e3 σ

2t

+ 3 e2 σ

2t

−6

(2) In the practice, this “arbitrary” mean time m L (t) may be either an exponential N0 eμ t , and then the corresponding lognormal process L(t) is the well-known Geometric Brownian Motion (GBM), or it may be a polynomial function of polynomial_ " degree the time, ck t k , and then we have shown how to compute all the k=0

statistical properties of the corresponding lognormal process L(t). (3) In particular, we have given an important application of this duality (either exponential or polynomial assumed as mean value) in the case of the Mass Extinctions that plagued the development of life on Earth several times over the last 3.5 billion years. Our result is that the parabolic model is preferable to the GBM model for Mass Extinctions, inasmuch as the possibility of the recovery of life (as indeed it always happened on Earth) is in the parabolic model, but not in the GBM one. (4) Finally, we did not consider in this paper more applications of the lognormal processes L(t) to SETI and Darwinian evolution. This is because these applications are potentially enormous, and we did not want to write an “encyclopedic” paper here as we did already in refs. [ ] and [ ] for the study of SETI, Darwinian Evolution and Human History by using the GBMs (exponential mean value) only.

512

Lognormals for SETI, Evolution and Mass Extinctions

Table 4 Summary of the properties of the lognormal distribution that applies to the stochastic process NDECREASING (t) = exponentially decreasing number of living species on Earth during a Mass Extinction Stochastic Process

NDECREASING (t) ≡ NDEC (t) = N umber o f Living Species (in a Mass E xtinction)

Probability distribution

Lognormal distribution of the ADJUSTED and DECREASING GBM starting at tImpact   NDEC (t)_ pd f n; μ, σ, tImpact , t =  !2 2 t−t σ ln(n)− ln( N )+μ(t−t ) − ( Impact )

Probability density function

Impact

−

e

Particular MGBM (t) function Mean value curve Variance Standard deviation Upper Standard Deviation Curve

√

Impact

( √

2

2 σ 2 t−tImpact

2π σ

)

f or n ≥ 0 .

t−tImpact n



MGBM (t) = ln N0 + μ −

 2

σ 2

t − tImpact



NDEC (t) ≡ m DEC (t) = NImpact eμ (t−tImpact )  2  2 2 σDEC(t) = NImpact e2μ (t−tImpact ) eσ (t−tImpact ) − 1  2 σ NDEC (t) = NImpact eμ (t−tImpact ) eσ (t−tImpact ) − 1 m NDEC (t) + σ NDEC (t)=   2 NImpact eμ (t−tImpact ) 1 + eσ (t−tImpact ) − 1

Lower Standard Deviation Curve m NDEC (t) − σ NDEC (t)=   2 NImpact eμ (t−tImpact ) 1 − eσ (t−tImpact ) − 1  2 

k k ekμ(t−tImpact ) e k −k NDEC (t) = NImpact

All the moments, i.e. k-th moment



Mode (=abscissa of the lognormal peak)

n mode ≡ n peak = NImpact eμ(t−tImpact ) e−

Value of the Mode Peak

f NDEC (t) (n mode ) = √ 1√ NImpact 2π σ

Median (=fifty-fifty probability value for NFIX (t)) Skewness

t−tImpact

(

)

(

)

σ 2 t−tImpact 2

3 σ 2 t−tImpact 2

2 e−μ (t−tImpact ) · eσ (t−tImpact )

σ 2 (t−tImpact ) 2 median = m = NImpact eμ (t−tImpact ) e−   2 2 K3 σ (t−tImpact ) + 2 eσ (t−tImpact ) − 1 3 = e

(K 2 ) 2

Kurtosis

K4 = (K 2 )2 2 t−t 4 σ ( Impact ) e

2 2 + 2 e3 σ (t−tImpact ) + 3 e2 σ (t−tImpact ) − 6

Value of the Mode Peak

Mode (=abscissa of the lognormal peak)

All the moments, i.e. k-th moment

Lower Standard Deviation Curve

Upper Standard Deviation Curve

Standard Deviation Curve

Variance

Mean value curve (i.e. the parabola)

Particular Mparabola (t) function

f NDEC (t) (n mode ) =

√

2π σ

t−tImpact

√



e

2  ln(n)−Mparabola (t) 2 2σ t

σ 2 t−tImpact

(

− √ 1√ e 2π σ t n

) !  2 t−tImpact ) t−tImpact ( ( NImpact −Nend ) 2 −2 tend −tImpact +NImpact (tend −tImpact )



(continued)

Pparabola (t)_ pd f n; Mparabola (t), σ, t = for n ≥ 0 . 

   (t−tImpact )2 2 t−tImpact Mparabola (t) = ln NImpact − Nend + NImpact − σ2 t 2 − 2 tend −tImpact (tend −tImpact ) 

 (t−tImpact )2  t−tImpact + NImpact m parabola (t) = NImpact − Nend 2 − 2 tend −tImpact (tend −tImpact )  2  2 σparabola(t) = m 2parabola (t) · eσ (t−tImpact ) − 1  2 σparabola (t) = m parabola (t) · eσ (t−tImpact ) − 1    2 m parabola (t) + σparabola (t) = m parabola (t) 1 + eσ (t−tImpact ) − 1    2 m parabola (t) − σparabola (t) = m parabola (t) 1 − eσ (t−tImpact ) − 1 

  $ #  σ2   (t−tImpact )2 2 t−tImpact k Pparabola + NImpact · e k −k 2 (t−tImpact ) (t) = k NImpact − Nend 2 − 2 tend −tImpact (tend −tImpact )  

3 σ 2 (t−tImpact )   (t−tImpact )2 t−tImpact 2 n mode ≡ n peak = NImpact − Nend + NImpact · e− 2 − 2 tend −tImpact (tend −tImpact )



Lognormal distribution of the ADJUSTED and PARABOLIC process starting at tImpact

Probability distribution

Probability density function

Pparabolic (t) = N umber o f Living Species (in a Parabolic Mass E xtinction)

Stochastic Process

Table 5 Summary of the properties of the lognormal distribution that applies to the stochastic process Pparabola (t) = decreasing number of living species on Earth during a Mass Extinction whose mean value decreases like the left-branch of a parabola between tImpact and tend (the parabola minimum, thus having a horizontal line tangent at t = tend )

4 Conclusions 513

Kurtosis

Skewness

Median (=fifty-fifty probability value for NFIX (t))

Stochastic Process

Table 5 (continued)

K4 (K 2 )2

(K 2 ) 2

2 2 2 = e4 σ (t−tImpact ) + 2 e3 σ (t−tImpact ) + 3 e2 σ (t−tImpact ) − 6

Pparabolic (t) = N umber o f Living Species (in a Parabolic Mass E xtinction) 

 σ 2 t−t ( Impact )   (t−tImpact )2 t−tImpact − 2 median = m = NImpact − Nend − 2 + N Impact e tend −tImpact (tend −tImpact )2   2 2 K3 σ (t−tImpact ) + 2 eσ (t−tImpact ) − 1 3 = e

514 Lognormals for SETI, Evolution and Mass Extinctions

4 Conclusions

515

Fig. 1 The K–Pg mass extinction as a decreasing GBM in the number of living species over 1000 years after impact. The maximum of the upper standard deviation curve has the numeric value −6.3999983 × 107 years given by (33)

Fig. 2 The K–Pg mass extinction as a decreasing PARABOLA in the number of living species over 1000 years after impact. The five numeric input values for this plot are just the same as those used for the construction of Fig. 1 in order to allow a perfect comparison between the two models, exponential (i.e. GBM-based) and parabolic

516

Lognormals for SETI, Evolution and Mass Extinctions

Fig. 3 Figs. 1 and 2 superimposed in order to allow the perfect comparison between the two models (Exponential (i.e. GBM-based) and parabolic) of the K–Pg mass extinction as a decreasing lognormal stochastic process in the number of living species over 1000 years after impact

Fig. 4 If we double the horizontal axis time window of Fig. 3, then the result is the current Fig. 4. It clearly shows that the parabolic model (in red) allows for the recovery of life on Earth after the nuclear winter of the Mass Extinction, while the exponential (GBM, in blue) does not so. The superior representation of the parabolic model for Mass Extinctions over the GBM one is thus fully evident

References

517

References 1. C. Maccone, The Statistical Drake Equation, paper #IAC-08-A4.1.4 presented on October 1st, 2008, at the 59th International Astronautical Congress (IAC) held in Glasgow, Scotland, UK, September 29th thru October 3rd, 2008 2. C. Maccone, The statistical Drake equation. Acta Astronaut. 67(2010), 1366–1383 (2010) 3. C. Maccone, The statistical Fermi paradox. J. Br. Interplanetary Soc. 63(2010), 222–239 (2010) 4. C. Maccone, A mathematical model for evolution and SETI. Origins Life Evol. Biospheres (OLEB) 41, 609–619 (2011b) (Available online December 3rd, 2011) 5. C. Maccone, Mathematical SETI, A 724-pages book published by Praxis-Springer in the fall of 2012. ISBN, ISBN-10: 3642274366|ISBN-13: 978-3642274367|Edition: 2012 6. C. Maccone, SETI, evolution and human history merged into a mathematical model. Int. J. Astrobiol. 12(3), 218–245 (2013) (Available online since April 23, 2013) 7. C. Maccone, Evolution and history in a new “Mathematical SETI” model. Acta Astronautica, 317–344 (2014) (currently in press. But available online since August 13, 2013) 8. L.W. Alvarez, W. Alvarez, F. Asaro, H.V. Michel, Extraterrestrial cause for the Cretaceous– Tertiary extinction. Science 208(4448), 1095–1108 (1980) 9. W. Alvarez, In the Mountains of Saint Francis: Discovering the Geologic Events that Shaped Our Earth, A popular book available in Kindle edition (2008)

Statistical Drake–Seager Equation for Exoplanet and SETI Searches

Abstract In 2013, MIT astrophysicist Sara Seager introduced what is now called the Seager Equation (Refs. https://en.wikipedia.org/wiki/Sara_Seager#Seager_equ ation; P. Gilster, Astrobiology: Enter the Seager Equation, Centauri Dreams, 11 September 2013, https://www.centauri-dreams.org/?p=28976): it expresses the number N of exoplanets with detectable signs of life as the product of six factors: Ns = the number of stars observed, fQ = the fraction of stars that are quiet, fHZ = the fraction of stars with rocky planets in the Habitable Zone, fO = the fraction of those planets that can be observed, fL = the fraction that have life, fS = the fraction on which life produces a detectable signature gas. This we call the “classical Seager equation”. Now suppose that each input of that equation is a positive random variable, rather than a sheer positive number. As such, each input random variable has a positive mean value and a positive variance that we assume to be numerically known by scientists. This we call the “Statistical Seager Equation”. Taking the logs of both sides of the Statistical Seager Equation, the latter is converted into an equation of the type log(N) = SUM of independent random variables. Let us now consider the possibility that, in the future, the number of physical inputs considered by Seager when she proposed her equation will actually increase, since scientists will know more and more details about the astrophysics of exoplanets. In the limit for an infinite number of inputs, i.e. an infinite number of independent input random variables, the Central Limit Theorem (CLT) of Statistics applies to the Statistical Seager Equation. Thus, the probability density function (pdf) of the output random variable log(N) will approach a Gaussian (normal) distribution in the limit, whatever the distribution of the input random variables might possibly be. But if log(N) approaches the normal distribution, then N approaches the lognormal distribution, whose mean value is the sum of the input mean values and whose variance is the sum of the input variances. This is just what this author realized back in 2008 when he transformed the Classical Drake Equation into the Statistical Drake Equation (Refs. C. Maccone. 2008, The Statistical Drake Equation, Paper #IAC-08-A4.1.4 presented on 1st October, 2008, at the 59th International Astronautical Congress (IAC), Glasgow, Scotland, UK, 29 September–3 October, 2008; Maccone in Acta Astronaut. 67:1366–1383, 2010). This discovery

This paper was presented during the 65th IAC in Toronto. © Springer Nature Switzerland AG 2020 C. Maccone, Evo-SETI, https://doi.org/10.1007/978-3-030-51931-5_14

519

520

Statistical Drake–Seager Equation for Exoplanet and SETI Searches

led to much more related work in the following years (Refs. Maccone in J. Br. Interplanet. Soc. 63:222–239, 2010; Maccone in Acta Astronaut. 68:63–75, 2011; C. Maccone, A mathematical model for evolution and SETI, Orig. Life Evolut. Biosph. (OLEB) 41 (2011) 609–619. Available online 3rd December, 2011; C. Maccone, 2012, Mathematical SETI, A 724-pages book published by Praxis–Springer in the fall of 2012. ISBN-10: 3642274366; ISBN-13: 978–3,642,274,367, edition: 2012; Maccone in Int. J. Astrobiol. 12:218–245, 2013; C. Maccone, Evolution and history in a new “Mathematical SETI” model, Acta Astronaut. 93 (2014) 317–344. Available online since 13 August 2013; Maccone in Acta Astronaut. 101:67–80, 2014; Maccone in Int. J. Astrobiol. 13:290–309, 2014). In this paper we study the lognormal properties of the Statistical Seager Equation relating them to the present and future knowledge for exoplanets searches from both the ground and space. Keywords Statistical Drake equation · Statistical Seager equation · Lognormal probability densities

1 Introduction As we stated in the Abstract, the Seager equation is mathematically equivalent to the Drake equation well known in SETI, the Search for ExtraTerrestrial Intelligence. Actually, all equations simply made up by an output equal to the multiplication of some independent inputs are equivalent to the Drake equation. For instance, the Dole equation that applies to the number of habitable planets for man in the Galaxy, was studied by this author in Ref. [13] and in Chap. 3 of Ref. [15] exactly in the same mathematical way the Drake equation was studied earlier by him in Refs. [10, 11], and the Seager equation is studied in this paper. More prosaically, all these equations simply are the Law of Compound Probability that everyone learns about in every course on elementary probability theory. Sara Seager herself modestly called her equation “an extended Drake equation”. Therefore, we prefer to describe the following important transition from the classical equation to the statistical one with the language of SETI, and so we now introduce first the classical, and later the Statistical Drake equation.

2 The Classical Drake Equation (1961) The Drake equation is a now famous result (see Ref. [1] for the Wikipedia summary) in the fields of SETI (the Search for ExtraTerrestial Intelligence, see Ref. [2]) and Astrobiology (see Ref. [3]). Devised in 1961, the Drake equation was the first scientific attempt to estimate the number N of ExtraTerrestrial civilizations in the Galaxy with which we might come in contact. Frank D. Drake (see Ref. [4]) proposed it as the product of seven factors:

2 The Classical Drake Equation (1961)

N = N s · f p · ne · f l · f i · f c · f L

521

(1)

where (1) Ns is the estimated number of stars in our Galaxy. (2) fp is the fraction (=percentage) of such stars that have planets. (3) ne is the number “Earth-type” such planets around the given star; in other words, ne is number of planets, in a given stellar system, on which the chemical conditions exist for life to begin its course: they are “ready for life”. (4) fl is fraction (=percentage) of such “ready for life” planets on which life actually starts and grows up (but not yet to the “intelligence” level). (5) fi is the fraction (=percentage) of such “planets with life forms” that actually evolve until some form of “intelligent civilization” emerges (like the first, historic human civilizations on Earth). (6) fc is the fraction (=percentage) of such “planets with civilizations” where the civilizations evolve to the point of being able to communicate across the interstellar distances with other (at least) similarly evolved civilizations. As far as we know in 2015, this means that they must be aware of the Maxwell equations governing radio waves, as well as of computers and radio astronomy (at least). (7) fL is the fraction of galactic civilizations alive at the time when we, poor humans, attempt to pick up their radio signals (that they throw out into space just as we have done since 1900, when Marconi started the transatlantic transmissions). In other words, fL is the number of civilizations now transmitting and receiving, and this implies an estimate of “how long will a technological civilization live?” that nobody can make at the moment. Also, are they going to destroy themselves in a nuclear war, and thus live only a few decades of technological civilization? Or are they slowly becoming wiser, reject war, speak a single language (like English today), and merge into a single “nation”; thus living in peace for ages? Or will robots take over one day making “flesh animals” disappear forever (the so-called “post-biological universe”)? No one knows… But let us go back to the Drake equation (1). In the sixty years of its existence, a number of suggestions have been put forward about the different numeric values of its seven factors. Of course, every different set of these seven input numbers yields a different value for N, and we can endlessly play that way. But we claim that these are like… children plays!

3 Transition from the Classical to the Statistical Drake Equation We claim the classical Drake equation (1), as we shall call it from now on to distinguish it from our statistical Drake equation to be introduced in the coming sections, well, the classical Drake equation is scientifically inadequate in one regard at least:

522

Statistical Drake–Seager Equation for Exoplanet and SETI Searches

it just handles sheer numbers and does not associate an error bar to each of its seven factors. At the very least, we want to associate an error bar to each input variable appearing on the right-hand side of (1). Well, we have thus reached STEP ONE in our improvement of the classical Drake equation: replace each sheer number by a probability distribution!

4 Step 1: Letting Each Factor Become a Random Variable In this book we adopt the notations of the great book “Probability, Random Variables and Stochastic Processes” by Athanasios Papoulis (1921–2002), now re-published as Papoulis-Pillai, Ref. [5]. The advantage of this notation is that it makes a neat distinction between probabilistic (or statistical: it is the same thing here) variables, always denoted by capitals, from non-probabilistic (or “deterministic”) variables, always denoted by lower-case letters. Adopting the Papoulis notation also is a tribute to him by this author, who was a Fulbright Grantee in the United States with him at the Polytechnic Institute (now Polytechnic University) of New York in the years 1977–1979. We thus introduce seven new (positive) random variables Di (“D” from “Drake”) defined as ⎧ D1 = N s ⎪ ⎪ ⎪ ⎪ ⎪ D2 = f p ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ D3 = ne D4 = f l (2) ⎪ ⎪ ⎪ D5 = f ⎪ ⎪ ⎪ ⎪ ⎪ D6 = f c ⎪ ⎪ ⎪ ⎩ D7 = f L so that our Statistical Drake equation may be simply rewritten as N=

7 

Di .

(3)

i =1

Of course, N now becomes a (positive) random variable too, having its own (positive) mean value and standard deviation. Just as each of the Di has its own (positive) mean value and standard deviation… … the natural question then arises: how are the seven mean values on the right related to the mean value on the left? … and how are the seven standard deviations on the right related to the standard deviation on the left? Just take the next step…

5 Step 2: Introducing Logs to Change the Product into a Sum

523

5 Step 2: Introducing Logs to Change the Product into a Sum Products of random variables are not easy to handle in probability theory. It is actually much easier to handle sums of random variables, rather than products, because: (1) The probability density of the sum of two or more independent random variables is the convolution of the relevant probability densities (worry not about the equations, right now). (2) The Fourier transform of the convolution simply is the product of the Fourier transforms (again, worry not about the equations, at this point). So, let us take the natural logs of both sides of the Statistical Drake Equation (3) and change it into a sum:  ln(N ) = ln

7 

 Di

=

i =1

7

ln(Di ).

(4)

i =1

It is now convenient to introduce eight new (positive) random variables defined as follows:

Y = ln(N ) (5) Yi = ln(Di ) i = 1, . . . , 7. Upon inversion, the first equation of (5) yields an important equation that will be used in the sequel: N = eY .

(6)

We are now ready to take Step 3.

6 Step 3: The Transformation Law of Random Variables So far we did not mention at all the problem: “which probability distribution shall we attach to each of the seven (positive) random variables Di ?”. It is not easy to answer this question because we do not have the least scientific clue to what probability distributions fit at best to each of the seven points listed in Sect. 2. Yet, at least one trivial error must be avoided: claiming that each of those seven random variables must have a Gaussian (i.e. normal) distribution. In fact, the Gaussian distribution, having the well-known bell-shaped probability density function

524

Statistical Drake–Seager Equation for Exoplanet and SETI Searches

f X (x; μ, σ ) = √

1 2πσ

e−

(x−μ)2 2 σ2

(σ ≥ 0)

(7)

has its independent variable x ranging between −∞ and ∞ and so it can apply to a real random variable X only, and never to positive random variables like those in the statistical Drake equation (3). Searching again for probability density functions that represent positive random variables, an obvious choice would be the gamma distributions (see, for instance, Ref. [6]). However, we discarded this choice too because of a different reason: keep in mind that, according to (5), once we selected a particular type of probability density function (pdf) for the last seven Di of Eq. (5), then we must compute the (new and different) pdf of the logs of such random variables. And the pdf of these logs certainly is not gamma-type any more. It is high time now to remind the reader of an important theorem that is proved in probability courses, but, unfortunately, does not seem to have a specific name. It is the transformation law (so we shall call it, see, for instance, Ref. [5], pages 130–131) allowing us to compute the pdf of a certain new random variable Y that is a known function Y = g(X) of another random variable X having a known pdf. In other words, if the pdf f X (x) of a certain random variable X is known, then the pdf f Y (y) of the new random variable Y, related to X by the functional relationship Y = g(X )

(8)

can be calculated according to the following rule: (1) First invert the corresponding non-probabilistic equation y = g(x) and denote by x i (y) the various real roots resulting from this inversion. (2) Second, take notice whether these real roots may be either finitely- or infinitelymany, according to the nature of the function y = g(x). (3) Third, the probability density function of Y is then given by the (finite or infinite) sum f Y (y) =

f X (xi (y)) |g  (xi (y))| i

(9)

where the summation extends to all roots x i (y) and |g (x i (y))| is the absolute value of the first derivative of g(x) where the ith root x i (y) has been replaced instead of x. Since we must use this transformation law to transfer from the Di to the Y i = ln(Di ), it is clear that we need to start from a Di pdf that is as simple as possible. The gamma pdf is not responding to this need because the analytic expression of the transformed pdf is very complicated. Also, the gamma distribution has two free parameters in it, and this “complicates” its application to the various meanings of the Drake equation. In conclusion, we discarded the gamma distributions and confined ourselves to the much simpler and much more practical uniform distribution instead, as shown in Sect. 7.

7 Step 4: Assuming the Easiest Input Distribution …

525

7 Step 4: Assuming the Easiest Input Distribution for Each Di : The Uniform Distribution Let us now suppose that each of the seven Di is distributed UNIFORMLY in the interval ranging from the lower limit ai ≥ 0 to the upper limit bi ≥ ai . This is the same as saying that the probability density function of each of the seven Drake random variables Di has the equation f uniform_ Di =

1 with 0 ≤ ai ≤ x ≤ bi bi − ai

(10)

that follows at once from the normalization condition bi f uniform_ Di (x) d x = 1.

(11)

ai

Let us now consider the mean value of such uniform Di , defined by bi uniform_ Di  = ai

=

1 x f uniform_ Di (x) d x = bi − ai

bi x dx ai

2 bi b2 − ai2 1 x ai + bi = . = i bi − ai 2 ai 2(bi − ai ) 2

By words (as it is intuitively obvious): the mean value of the uniform distribution simply is the mean of the lower plus upper limit of the variable range uniform_ Di  =

ai + bi . 2

(12)

In order to find the variance of the uniform distribution, we first need finding the second moment 

uniform_

Di2



bi =

x 2 f uniform_ Di (x) d x ai

3 bi b3 − ai3 x 1 x dx = = i bi − ai 3 ai 3(bi − ai ) ai   2 (bi − ai ) ai + ai bi + bi2 a 2 + ai bi + bi2 = i . = 3(bi − ai ) 3 1 = bi − ai

bi

2

526

Statistical Drake–Seager Equation for Exoplanet and SETI Searches

The second moment of the uniform distribution is thus 

 a 2 + ai bi + bi2 . uniform_ Di2 = i 3

(13)

From (12) and (13) we may now derive the variance of the uniform distribution   2 2 2 σuniform_ Di = uniform_ Di − uniform_ Di  =

ai2 + ai bi + bi2 (ai + bi )2 (bi − ai )2 − = . 3 4 12

(14)

Upon taking the square root of both sides of (14), we finally obtain the standard deviation of the uniform distribution: σuniform_ Di =

bi − ai √ . 2 3

(15)

We now wish to perform a calculation that is mathematically trivial, but rather unexpected from the intuitive point of view, and very important for our applications to the Statistical Drake equation. Just consider the two simultaneous Eqs. (12) and (15)

ai +bi 2

uniform_ Di  = σuniform_ Di =

bi √ −ai 2 3

.

(16)

Upon inverting this trivial linear system, one finds

ai = uniform_ Di  − bi = uniform_ Di  +

√ √

3 σuniform_ Di 3 σuniform_ Di .

(17)

This is of paramount importance for our application the Statistical Drake equation inasmuch as it shows that: if one (scientifically) assigns the mean value and standard deviation of a certain Drake random variable Di , then the lower and upper limits of the relevant uniform distribution are given √ by the two Eq. (17), respectively. In other words, there is a factor of 3 = 1.732 included in the two Eq. (17) that is not obvious at all to human intuition, but must indeed be taken into account. The application of this result to the Statistical Drake equation is discussed in the next section.

8 Step 5: Computing the Logs of the 7 Uniformly Distributed …

527

8 Step 5: Computing the Logs of the 7 Uniformly Distributed Drake Random Variables Di Intuitively speaking, the natural log of a uniformly distributed random variable may not be another uniformly distributed random variable! This is obvious from the trivial diagram of y = ln(x) shown in Fig. 1. So, if we have a uniformly distributed random variable Di with lower limit ai and upper limit bi , the random variable Yi = ln(Di ) i = 1, . . . , 7

(18)

must have its range limited in between the lower limit ln (ai ) and the upper limit ln(bi ). In other words, these are the lower and upper limits of the relevant probability density function f Yi (y). But what is the actual analytic expression of such a pdf? To find it, we must resort to the general transformation law for random variables, defined by Eq. (9). Here we obviously have y = g(x) = ln(x).

(19)

That, upon inversion, yields the single root x1 (y) = x(y) = e y .

(20)

On the other hand, differentiating (19) one gets g  (x) =

1 1 1 and g  (x1 (y)) = = y x x1 (y) e

(21)

where (20) was already used in the last step. By virtue of the uniform probability density function (10) and of (21), the general transformation law (9) finally yields Fig. 1 The simple function y = ln(x)

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Statistical Drake–Seager Equation for Exoplanet and SETI Searches

f Y (y) =

f X (xi (y)) ey 1 1 = ·1 = .    |g (xi (y))| bi − ai bi − ai ey i

(22)

In other words, the requested pdf of Y i is f Y (y) =

eY i = 1, . . . , 7 with ln(ai ) ≤ y ≤ ln(bi ). bi − ai

(23)

Probability density functions of the natural logs of all the uniformly distributed Drake random variables Di . This is indeed a positive function of y over the interval ln(ai ) ≤ y ≤ ln(bi ), as for every pdf, and it is easy to see that its normalization condition is fulfilled: ln(b  i)

ln(b  i)

f Y (y)dy = ln(ai )

ln(ai )

ey eln(bi ) − eln(ai ) dy = = 1. bi − ai bi − ai

(24)

Next we want to find the mean value and standard deviation of Y i , since these play a crucial role for future developments. The mean value Yi  is given by ln(b  i)

Yi  =

ln(b  i)

y · f Y (y)dy = ln(ai )

=

ln(ai )

y · ey dy bi − ai

bi [ln(bi ) − 1] − ai [ln(ai ) − 1] . bi − ai

(25)

This is thus the mean value of the natural log of all the uniformly distributed Drake random variables Di Yi  = ln(Di ) =

bi [ln(bi ) − 1] − ai [ln(ai ) − 1] . bi − ai

(26)

In order to find the variance also, we must first compute the mean value of the square of Y i , that is 

Yi2



ln(b  i)

ln(b  i)

y2 · ey dy bi − ai ln(ai ) ln(ai )     bi ln2 (bi ) − 2 ln(bi ) + 2 − ai ln2 (ai ) − 2 ln(ai ) + 2 . = bi − ai

=

y · f Y (y)dy = 2

(27)

8 Step 5: Computing the Logs of the 7 Uniformly Distributed …

529

The variance of Y i = ln(Di ) is now given by (27) minus the square of (26), that, after a few reductions, yield: 2 =1− σY2i = σln(D i)

ai bi [ln(bi ) − ln(ai )]2 . (bi − ai )2

(28)

Whence the corresponding standard deviation  σYi = σln(Di ) =

1−

ai bi [ln(bi ) − ln(ai )]2 . (bi − ai )

(29)

Let us now turn to another topic: the use of Fourier transforms, that in probability theory, are called “characteristic functions”. Following again the notations of Papoulis (Ref. [5]) we call “characteristic function”, Yi (ζ ), of an assigned the Fourier transform of the relevant probability density probability distribution Y i , √ function, that is (with j = −1) ∞ ΦYi (ζ ) =

e jζ y f Yi (y) dy.

(30)

−∞

The use of characteristic functions simplifies things greatly. For instance, the calculation of all moments of a known pdf becomes trivial if the relevant characteristic function is known, and greatly simplified also are the proofs of important theorems of statistics, like the Central Limit Theorem that we will use in Sect. 9. Another important result is that the characteristic function of the sum of a finite number of independent random variables is simply given by the product of the corresponding characteristic functions. This is just the case we are facing in the Statistical Drake Equation (3) and so we are now led to find the characteristic function of the random variable Y i , i.e. ∞ ΦYi (ζ ) =

ln(b  i)

e

jζ y

f Yi (y) dy =

−∞

=

1 bi − ai

e jζ y ln(ai )

ln(b  i)

e(1+ jζ )y dy =

ln(ai )

ey dy bi − ai

1 1  (1+ jζ )y ln(bi ) e · n(ai ) bi − ai 1 + jζ

bi − a 1+ jζ e(1+ jζ ) ln(bi ) − e(1+ jζ ) ln(ai ) = . (bi − ai )(1 + jζ ) (bi − ai )(1 + jζ ) 1+ jζ

=

(31)

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Statistical Drake–Seager Equation for Exoplanet and SETI Searches

Thus, the characteristic function of the natural log of the Drake uniform random variable Di is given by 1+ jζ

ΦYi (ζ ) =

− a 1+ jζ bi . (bi − ai )(1 + jζ )

(32)

9 The Central Limit Theorem (CLT) of Statistics Indeed there is a good, approximating analytical expression for f N (y), and this is the following lognormal probability density function f N (y; μ, σ ) =

(ln(y)−μ)2 1 1 ·√ e− 2 σ 2 (y ≥ 0, σ ≥ 0) y 2π σ

(33)

To understand why, we must resort to what is perhaps the most beautiful theorem of Statistics: the Central Limit Theorem (abbreviated CLT). Historically, the CLT was in fact proven first in 1901 by the Russian mathematician Alexandr Lyapunov (1857– 1918), and later (1920) by the Finnish mathematician Jarl Waldemar Lindeberg (1876–1932) under weaker conditions. These conditions are certainly fulfilled in the context of the Drake equation because of the “reality” of the astronomy, biology and sociology involved with it, and we are not going to discuss this point any further here. A good, synthetic description of the Central Limit Theorem (CLT) of Statistics is found at the Wikipedia site (Ref. [7]) to which the reader is referred for more details, such as the equations for the Lyapunov and the Lindeberg conditions, making the theorem “rigorously” valid. Put in loose terms, the CLT states that, if one has a sum of random variables even NOT identically distributed, this sum tends to a normal distribution when the number of terms making up the sum tends to infinity. Also, the normal distribution mean value is the sum of the mean values of the addend random variables, and the normal distribution variance is the sum of the variances of the addend random variables. Let us now write down the equations of the CLT in the form needed to apply it to our Statistical Drake Equation (3). The idea is to apply the CLT to the sum of random variables given by (4) and (5) whatever their probability distributions can possibly be. In other words, the CLT applied to the Statistical Drake Equation (3) leads immediately to the following three equations: (1) The sum of the (arbitrarily distributed) independent random variables Y i makes up the new random variable Y. (2) The sum of their mean values makes up the new mean value of Y. (3) The sum of their variances makes up the new variance of Y.

9 The Central Limit Theorem (CLT) of Statistics

531

In equations ⎧ 7 ⎪ ⎪ ⎪ ⎪ Y = Yi ⎪ ⎪ ⎪ ⎪ i =1 ⎪ ⎪ ⎪ 7 ⎨ Y  = Yi  ⎪ ⎪ i = 1 ⎪ ⎪ ⎪ ⎪ 7 ⎪ ⎪ ⎪ 2 ⎪ σ = σY2i . ⎪ ⎩ Y

(34)

i =1

This completes our synthetic description of the CLT for sums of random variables.

10 The Lognormal Distribution is the Distribution of the Number N of Extraterrestrial Civilizations in the Galaxy The CLT may of course be extended to products of random variables upon taking the logs of both sides, just as we did in Eq. (3). It then follows that the exponent random variable, like Y in (6), tends to a normal random variable, and, as a consequence, it follows that the base random variable, like N in (6), tends to a lognormal random variable. To understand this fact better in mathematical terms consider again of the transformation law (9) of random variables. The question is: what is the probability density function of the random variable N in Eq. (6), that is, what is the probability density function of the lognormal distribution? To find it, set y = g(x) = e x .

(35)

This, upon inversion, yields the single root x1 (y) = x(y) = ln(y).

(36)

On the other hand, differentiating (35) one gets g  (x) = e x and g  (x1 (y)) = eln(y) = y.

(37)

where (21) was already used in the last step. The general transformation law (9) finally yields f N (y) =

f X (xi (y)) 1 = f Y (ln(y)).  (x (y))| |g |y| i i

(38)

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Statistical Drake–Seager Equation for Exoplanet and SETI Searches

Therefore, replacing the probability density on the right by virtue of the wellknown normal (or Gaussian) distribution given by Eq. (7), the lognormal distribution of Eq. (33) is found, and the derivation of the lognormal distribution from the normal distribution is proved. Table 1 summarizes all the properties of the lognormal distribution, whose demonstrations may be found in statistical textbooks (for instance see Refs. [7–9]). The last two lines in Table 1, however, are about our own discovery of Eqs. (26) and (28) yielding μ and σ 2 of the lognormal distribution of the output of the statistical Drake (and Seager) equation when all inputs are supposed to be uniformly distributed and both the mean value and standard deviation of each input is assigned. Remember that this mean value and standard deviation may be immediately converted into the uniform distribution lower and upper values thanks to the two Eq. (17). Table 1 Summary of the properties of the lognormal distribution that applies to the random variable N = number of ET communicating civilizations in the Galaxy Random variable

N = number of communicating ET civilizations in Galaxy

Probability distribution

Lognormal

Probability density function

f N (n; μ, σ ) = 0, σ ≥ 0) σ2 2

1 n

·

√1 2πσ

e

Mean value

N  = eμ e

Variance

σ N2 = e2μ eσ

Standard deviation All the moments, i.e. kth moment

 σ2 2 σ N = eμ e 2 eσ − 1 2  k 2σ N = ekμ ek 2

Mode (= abscissa of the lognormal peak)

n mode = n peak = eμ e−σ

Value of the mode peak Median (= fifty–fifty probability value for N) Skewness

2



f N (n mode ) =

eσ − 1 2

√1 2π σ

− (ln(n)−μ) 2

2

2 σ

(n ≥



2

· e−μ · e

σ2 2

median = eμ  2   2 K3 eσ − 1 eσ + 2 3 = (K 2 ) 2

= e4 σ + 2e3 σ + 3 e2 σ − 6 2

2

2

Kurtosis

K4 (K 2 )2

Expression of μ in terms of the lower (ai ) and upper (bi ) limits of the Drake uniform input random variables Di

μ=  7 7    bi [ln(bi )−1]−ai [ln(ai )−1] Yi  = bi −ai

Expression of σ 2 in terms of the lower (ai ) and upper (bi ) limits of the Drake uniform input random variables Di

i=1

σ2 7  i=1

i=1

= σY2i =

7   i=1

1−

ai bi [ln(bi )−ln(ai )]2 (bi −ai )2



11 Data Enrichment Principle

533

11 Data Enrichment Principle It should be noticed that any positive number of random variables in the statistical Drake equation is compatible with the CLT. So, our generalization allows for many more factors to be added in the future as long as more refined scientific knowledge about each factor becomes known to scientists. This capability to make room for more future factors in the statistical Drake equation we call “Data Enrichment Principle”, and we regard it as the key to more profound future results in the fields of Astrobiology and SETI.

12 The Statistical Seager Equation We now come to consider the Statistical Seager Equation. As described in the Abstract of this chapter, in 2013, MIT astrophysicist Sara Seager introduced what is now called the Seager equation (see Refs. [20, 21]): it expresses the number N of exoplanets with detectable signs of life as the product of six factors: (1) (2) (3) (4) (5) (6)

Ns = the number of stars observed, fQ = the fraction of stars that are quiet, fHZ = the fraction of stars with rocky planets in the Habitable Zone, fO = the fraction of those planets that can be observed, fL = the fraction that have life, fS = the fraction on which life produces a detectable signature gas. That is N = Ns · f Q · f H Z · f O · f L · f S

(39)

This we call the “classical Seager equation”. Mathematically speaking, Eq. (39) is exactly the same thing as Eq. (1): only the words change, and, of course, so does its scientific meaning. Mathematically, one may thus immediately apply to (39) the full string of mathematical theorems that we described in all Eqs. (2)–(32) of this paper. The first such step is clearly the transformation of the classical Seager Equation (39) into the Statistical Seager Equation (looking the same, mathematically), having all numeric inputs replaced by positive random variables. No more comments are necessary. The second step is asking the two questions: (1) What is the probability distribution of each of the six input positive random variables in (39)? (2) And what is the probability distribution of the resulting output N?

534

Statistical Drake–Seager Equation for Exoplanet and SETI Searches

Our answers to these two questions are: (1) No analytical solution exists as long as the number of Input random variables is finite. Only numeric calculations may be done, but this requires writing a numeric code, that this author could not and would not write down because is he a mathematical physicist and not a computer programmer. (2) However, if we let the number of input random variables approach infinity (i.e. if we consider a high number of input random variables, like five to ten or even more (“how many” might be discussed later) then the solution of both the statistical Drake Equation (1) and the Statistical Seager Equation (3) is immediate. In both cases the probability distribution of the output random variable N is a Lognormal Distribution: (a) Its real parameter μ is given by the sum of the mean values of all input random variables, and (b) Its positive parameter σ 2 is given by the sum of the variances of all input random variables, whatever their probability distribution might possibly be. This is the result of applying the Central Limit Theorem of Statistics to both the Drake and Seager equations.

13 The Extremely Important Particular Case When the Input Random Variables Are Uniform Now we turn to the particular case of our theory when all the input random variables are uniform random variables. This case we regard as “extremely important” for all practical applications of both the Drake and the Seager statistical equations inasmuch as it is the only case when analytic formulae do exist expressing the two lognormal parameters μ and σ directly in terms of the lower and upper limits of all the uniform input random variables. In fact, define by. (1) (2) (3) (4)

ai the real and positive number representing the LOWER LIMIT of the range of the i-th uniform input random variable. bi the real and positive number representing the UPPER LIMIT of the range of the ith uniform input random variable. Clearly, it is assumed bi > ai . Then. The mean value of the ith uniform input random variable is (obviously) given by (12), that is Uniform_ Di  =

ai + bi 2

(40)

13 The Extremely Important Particular Case When …

(5)

The standard deviation of the ith uniform input random variable is (less obviously) given by (15), that is σUniform_ Di =

(6)

(ln(n)−μ)2 1 1 (n ≥ 0) e− 2 σ 2 ·√ n 2πσ

(41)

(42)

The real and positive parameter μ appearing in the lognormal pdf (33) is expressed directly in terms of all the known ai and bi by Eq. (26), that is number_ of_ inputs 

μ=

i=1

(8)

bi − ai √ . 2 3

Above all, one has. The probability density function (pdf) of the output random variable N is the lognormal pdf (33), that is: f N (n) =

(7)

535

 bi [ln(bi ) − 1] − ai [ln(ai ) − 1] . bi − ai

(43)

The real and positive parameter σ 2 appearing in the lognormal pdf (33) is expressed directly in terms of all the known ai and bi by Eq. (28), that is number_ of_ inputs 

σ = 2

i=1

 ai bi [ln(bi ) − ln(ai )]2 1− . (bi − ai )2

(44)

(9)

In addition to providing the pdf of N explicitly by virtue of the three Eqs. (42)– (44), this case of the all-uniform input random variables also is “realistic” in that the uniform probability distribution is “the most uncertain one” in the sense of Shannon’s Information Theory. This is intuitively obvious (“when you do not know where to go, you look around and all directions are equal to each other, i.e. your probability distribution in the azimuth is uniform between 0 and 2 π ). But it may also be rigorously proven by applying the method of the Lagrange multipliers to the definition of Shannon’s Entropy, as shown in the Appendix to this paper. (10) All the statistical properties of the lognormal output random variable N may be found analytically and are listed in Table 1. This completes our summary of the discoveries that this author made back in 2008 about the Statistical Drake Equation and so necessarily also about the 2013 Statistical Seager Equation also. Especially useful is the simple case when all input variables are Uniformly distributed: then, the lognormal pdf of N is given by Eqs. (42)–(44).

536

Statistical Drake–Seager Equation for Exoplanet and SETI Searches

14 Conclusion The conclusion of this chapter is that this author would respectfully advise Professor Sara Seager and her MIT Team to make use of Eqs. (42)–(44) in order to find the lognormal pdf of the number N of exoplanets with detectable signs of life, having previously estimated all the ai and bi numerically. This research work could be part of the Phase A or B studies for the NASA planned TESS space mission, described at the web sites (https://tess.gsfc.nasa.gov/) and (https://en.wikipedia.org/wiki/Transiting_Exoplanet_Survey_Satellite). Acknowledgements The author wishes to acknowledge the cooperation and support of Professor Sara Seager and her Team about the two scientific meetings they had in 2014: (1) At the Istituto Nazionale di Astrofisica (INAF) in Milan, Italy, when they met on April 1st, 2014, and (2) At MIT, when they met on May 5th, 2014.

Appendix Proof of Shannon’s 1948 Theorem stating that the Uniform distribution is the “most uncertain” one over any Finite range of values. As it is well known, the Shannon entropy of any probability density function p(x) is given by the integral ∞ Shannon_ Entropy_ of_ p(x) = −

p(x) log p(x) d x.

(45)

−∞

In modern textbooks this is also called Shannon differential entropy. Now consider the case when a probability density function p(x) is limited to a finite interval a ≤ x ≤ b. This is obviously the case with any physical positive random variable, such as the number N of extraterrestrial communicating civilizations in the Galaxy. We now wish to prove that for any such finite random variable the maximum entropy distribution is the UNIFORM distribution over a ≤ x ≤ b. Shannon did not bother to prove this simple theorem in his 1948 papers since he probably regarded it as just too trivial. But we prefer to point out this theorem since, in the language of the statistical Drake equation, it sounds like: “Since we don’t know what the probability distribution of any one of the Drake random variables Di is, it is safer to assume that each of them has the maximum possible entropy over a ≤ x ≤ b, i.e., that Di is UNIFORMLY distributed there”.

Appendix

537

The proof of this theorem is as follows: (1) Start by assuming ai ≤ x ≤ bi . (2) Then form the linear combination of the entropy integral plus the normalization condition for Di bi bi δ

[− p(x) log p(x) + λp(x)] d x = 0

(46)

ai

where λ is a Lagrange multiplier. Performing the variation, i.e. differentiating with respect to p(x), one finds − log p(x) − 1 + λ = 0

(47)

p(x) = eλ−1 .

(48)

that is

Applying the normalization condition (constraint) to the last expression for p(x) yields bi

bi p(x) d x =

1= ai

e

λ−1

dx = e

λ−1

ai

bi

d x = eλ−1 (bi − ai )

(49)

ai

that is eλ−1 =

1 bi − ai

(50)

and finally, from (48) and (50) p(x) =

1 bi − ai

with ai ≤ x ≤ bi .

(51)

showing that the maximum-entropy probability distribution over any FINITE interval ai ≤ x ≤ bi is just the UNIFORM distribution.

References 1. https://en.wikipedia.org/wiki/Drake_equation 2. https://en.wikipedia.org/wiki/Search_for_Extra-Terrestrial_Intelligence 3. https://en.wikipedia.org/wiki/Astrobiology

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4. https://en.wikipedia.org/wiki/Frank_Drake 5. A. Papoulis, S.U. Pillai, Probability, Random Variables and Stochastic Processes, 4th edn. (Tata McGraw-Hill, New Delhi, 2002). ISBN 0-07-048658-1 6. https://en.wikipedia.org/wiki/Gamma_distribution 7. https://en.wikipedia.org/wiki/Central_limit_theorem 8. https://en.wikipedia.org/wiki/Cumulant 9. https://en.wikipedia.org/wiki/Median 10. C. Maccone, The Statistical Drake Equation, Paper #IAC-08-A4.1.4 presented on 1st October, 2008, at the 59th International Astronautical Congress (IAC), Glasgow, Scotland, UK, 29 September–3 October, 2008 11. C. Maccone, The Statistical Drake Equation. Acta Astronaut. 67(2010), 1366–1383 (2010) 12. C. Maccone, The statistical Fermi paradox. J. Br. Interplanet. Soc. 63(2010), 222–239 (2010) 13. C. Maccone, SETI and SEH (statistical equation for habitables). Acta Astronaut. 68(2011), 63–75 (2011) 14. C. Maccone, A mathematical model for evolution and SETI. Orig. Life Evolut. Biosph. (OLEB) 41, 609–619. Available online 3rd December, 2011 15. C. Maccone, Mathematical SETI, A 724-pages book published by Praxis–Springer in the fall of 2012. ISBN-10: 3642274366; ISBN-13: 978-3642274367, edition: 2012 16. C. Maccone, SETI, evolution and human history merged into a mathematical model. Available online since April 23, 2013. Int. J. Astrobiol. 12(3), 218–245 (2013) 17. C. Maccone, Evolution and history in a new “Mathematical SETI” model. Acta Astronaut. 93, 317–344 (2014). Available online since 13 August 2013 18. C. Maccone, SETI as a part of big history. Acta Astronaut. 101, 67–80 (2014) 19. C. Maccone, Evolution and mass extinctions as lognormal stochastic processes. Int. J. Astrobiol. 13(4), 290–309 (2014) 20. https://en.wikipedia.org/wiki/Sara_Seager#Seager_equation 21. P. Gilster, Astrobiology: Enter the Seager Equation, Centauri Dreams, 11 September 2013, https://www.centauri-dreams.org/?p=28976

Evo-SETI Entropy Identifies with Molecular Clock

Abstract Darwinian evolution over the last 3.5 billion years was an increase in the number of living species from 1 (RNA?) to the current 50 million. This increasing trend in time looks like being exponential, but one may not assume an exactly exponential curve since many species went extinct in the past, even in mass extinctions. Thus, the simple exponential curve must be replaced by a stochastic process having an exponential mean value. Borrowing from financial mathematics (“Black-Sholes models”), this “exponential” stochastic process is called Geometric Brownian Motion (GBM), and its probability density function (pdf) is a lognormal (not a Gaussian) (Proof: see Ref. C., Maccone, “Mathematical SETI”, A 724-pages Book Published by Praxis-Springer in the Fall of 2012. ISBN, ISBN-10: 3,642,274,366 | ISBN13: 978–3,642,274,367 | Edition: 2012), Chapter 30, and Ref. (Maccone in Int. J. Astrobiol. 12:218–245, 2013). Lognormal also is the pdf of the statistical number of communicating ExtraTerrestrial (ET) civilizations in the Galaxy at a certain fixed time, like a snapshot: this result was found in 2008 by this author as his solution to the Statistical Drake Equation of SETI (Proof: see Ref. (Maccone in Acta Astronaut. 67:1366–1383, 2010). Thus, the GBM of Darwinian evolution may also be regarded as the extension in time of the Statistical Drake equation (Proof: see Ref. Maccone in Int. J. Astrobiol. 12:218–245, 2013). But the key step ahead made by this author in his Evo-SETI (Evolution and SETI) mathematical model was to realize that LIFE also is just a b-lognormal in time: every living organism (a cell, a human, a civilization, even an ET civilization) is born at a certain time b (“birth”), grows up to a peak p (with an ascending inflexion point in between, a for adolescence), then declines from p to s (senility, i.e. descending inflexion point) and finally declines linearly and dies at a final instant d (death). In other words, the infinite tail of the b-lognormal was cut away and replaced by just a straight line between s and d, leading to simple mathematical formulae (“History Formulae”) allowing one to find this “finite b-lognormal” when the three instants b, s, and d are assigned. Next the crucial Peak-Locus Theorem comes. It means that the GBM exponential may be regarded as the geometric locus of all the peaks of a one-parameter (i.e. the peak time p) family of b-lognormals. Since b-lognormals are pdf-s, the area under each of them always equals 1 (normalization condition) and so, going from left to right on the time axis, the b-lognormals become more and more “peaky”, and so they last less and less in time. This is precisely what happened in Human History: civilizations that © Springer Nature Switzerland AG 2020 C. Maccone, Evo-SETI, https://doi.org/10.1007/978-3-030-51931-5_15

539

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Evo-SETI Entropy Identifies with Molecular Clock

lasted millennia (like Ancient Greece and Rome) lasted just centuries (like the Italian Renaissance and Portuguese, Spanish, French, British and USA Empires) but they were more and more advanced in the “level of civilization”. This “level of civilization” is what physicists call ENTROPY. In Refs. (C. Maccone, “Mathematical SETI”, A 724-pages Book Published by Praxis-Springer in the Fall of 2012. ISBN, ISBN-10: 3,642,274,366 | ISBN-13: 978–3,642,274,367 | Edition: 2012.) and (Maccone in Int. J. Astrobiol. 12:218–245, 2013), this author proved that, for all GBMs, the (Shannon) Entropy of the b-lognormals in his Peak-Locus Theorem grows LINEARLY in time. At last, we reach the new, original result justifying the publication of this paper. The Molecular Clock, well known to geneticists since 50 years, shows that the DNA base-substitutions occur LINEARLY in time since they are neutral with respect to Darwinian selection. In simple words: DNA evolved by obeying the laws of quantum physics only (microscopic laws) and not by obeying assumed “Darwinian selection laws” (macroscopic laws). This is Kimura’s neutral theory of molecular evolution. The conclusion of this paper is that the Molecular Clock and the b-lognormal Entropy are the same thing. And, on exoplanets, molecular evolution is proceeding at about the same rate as it did proceed on Earth: rather independently of the physical conditions of the exoplanet, if the DNA had the possibility to evolve in water initially. Keywords Darwinian evolution · Molecular clock · Entropy · SETI

1 Purpose of This Chapter This chapter describes recent developments in a new statistical theory describing Evolution and SETI by mathematical equations. This we call the Evo-SETI model of Evolution and SETI. The goal of this paper is to prove that the Evo-SETI model and the well-known Molecular Clock of Evolution are in agreement with each other. The reason of this agreement is that the (Shannon) Entropy of the b-lognormals in the Evo-SETI model increases linearly with time, just as the Molecular Clock increases linearly with time. In other words, apart from constants with respect to the time, the Molecular Clock and b-lognormal Entropy are the same thing.

2 During the Last 3.5 Billion Years Life Forms Increased Like a (Lognormal) Stochastic Process Let us look at Fig. 1 on the horizontal axis is the time t, with the convention that negative values of t are past times, zero is now, and positive times are future times. The starting point on the time axis is ts = − 3.5 × 109 years i.e. 3.5 billion years ago, the time of the origin of life on Earth that we assume to be correct. If the origin of life started earlier than that, say 3.8 billion years ago, the coming equations would

2 During the Last 3.5 Billion years Life Forms Increased …

541

Fig. 1 Darwinian Evolution as the increasing number of living species on Earth between 3.5 billion years ago and now. The red solid curve is the mean value of the GBM stochastic process L GBM (t) given by (20), while the blue dot–dot curves above and below the mean value are the two standard deviation upper and lower curves, given by (11) and (12), respectively, with mGBM (t) given by (21). The “Cambrian Explosion” of life, that on Earth started around 542 million years ago, is evident in the above plot just before the value of −0.5 billion years in time, where all three curves “start leaving the time axis and climbing up”. Notice also that the starting value of living species 3.5 billion years ago is ONE by definition, but it “looks like” zero in this plot since the vertical scale (which is the true scale here, not a log scale) does not show it. Notice finally that nowadays (i.e. at time t = 0) the two standard deviation curves have exactly the same distance from the middle mean value curve, i.e. 30 million living species more or less the mean value of 50 million species. These are assumed values that we used just to exemplify the GBM mathematics: biologists might assume other numeric values. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

still be the same and their numerical values will only be slightly changed. On the vertical axis is the number of Species living on Earth at time t, denoted L(t). This “function of the time” we don’t know in detail, and so it must be regarded as a random function, or stochastic process, with the notation L(t) standing for “life at time t”. In this paper we adopt the convention that capital letters represent random variables, i.e. stochastic processes if they depend on the time, while lower-case letters mean ordinary variables or functions. The most important ordinary, continuous function of the time associated with a stochastic process like L(t) is its mean value, denoted by m L (t) ≡ L(t).

(1)

The probability density function (pdf) of a stochastic process like L(t) is assumed in Evo-SETI theory to be lognormal, that is, its equation reads

542

Evo-SETI Entropy Identifies with Molecular Clock

e

−

[ln(n)−M L (t)]2 2σ L2 (t−ts)

L(t)_ pd f (n; M L (t), σ, t) = √ √ 2πσ L t − tsn   σ L ≥ 0, n ≥ 0, and with t ≥ ts, M L (t) = arbitrary function of t.

(2)

This assumption is in line with the extension in time of the statistical Drake equation, namely foundational and statistical equation of SETI, as shown in Ref. [1]. The mean value (1) is of course related to the pdf (2) by the relevant integral in the number n of living Species on Earth at time t, that is ∞ m L (t) ≡

n·√ 0

e

−

[ln(n)−M L (t)]2 2σ L2 (t−ts)

dn. √ 2π σ L t − tsn

(3)

The “surprise” is that this integral (3) may be computed exactly with the key result that the mean value mL (t) is given by m L (t) = e M L (t) e

σ L2 2

(t−ts)

.

(4)

In turn, the last equation has the “surprising” property that it may be inverted exactly, i.e. solved for M(t): M L (t) = ln(m L (t)) −

σ L2 (t − ts). 2

(5)

Now about the initial conditions of the stochastic process L(t), namely about the value L(ts). We shall assume that the positive number L(ts) = N s

(6)

is always exactly known, i.e. with probability one: Pr{L(ts) = N s} = 1.

(7)

In the practice, Ns will be equal to 1 in the theories of evolution of life on Earth or on an exoplanet (i.e., there must have been a time ts in the past when the number of living species was just one, let it be RNA or something else), and it will be equal to the number of living species just before the asteroid/comet impact in the theories of mass extinction of life on a planet as done in Ref. [5]. The mean value mL (t) of L(t) also must equal the initial number Ns at the initial time ts, that is m L (ts) = N s.

(8)

2 During the Last 3.5 Billion years Life Forms Increased …

543

Replacing t by ts in (4), one then finds m L (ts) = e M(ts)

(9)

that, checked against (8), immediately yields N s = e M L (ts) that is M L (ts) = ln(N s).

(10)

These are the initial conditions for the mean value. After the initial instant ts, the stochastic process L(t) unfolds oscillating above or below the mean value in an unpredictable way. Statistically speaking, however, we expect L(t) “not to depart too much” from m(t) and this fact is graphically shown in Fig. 1 by the two dot–dot blue curves above and below the mean value solid red curve m(t). These two curves are the upper standard deviation curve    2 upper_ st_ dev_ curve(t) = m L (t) 1 + eσL (t−ts) − 1

(11)

and the lower standard deviation curve    2 σ (t−ts) L −1 lower_ st_ dev_ curve(t) = m L (t) 1 − e

(12)

respectively (Proof: see Table 2 of Ref. [4]). Notice that both (11) and (12), at the initial time t = ts, equal the mean value m(ts) = Ns, that is, with probability one again, the initial value Ns is the same for all the three curves shown in Fig. 1. The function of the time  2 (13) variation_ coefficient(t) = eσL (t−ts) − 1 is called “variation coefficient” by statisticians since the standard deviation of L(t) (be careful: this is just the standard deviation L (t) of L(t) and not either of the above two “upper” and “lower” standard deviation curves given by (11) and (12), respectively) is  2 st_ dev_ curve(t) ≡  L (t) = m L (t) eσL (t−ts) − 1

(14)

Indeed, (14) shows that the variation coefficient (13) is the ratio of (t) to mL (t), i.e. it expresses how much the standard deviation “varies” with respect to the mean value. Having understood this fact, it is then obvious that the two curves (11) and (12) are obtained as respectively.

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Evo-SETI Entropy Identifies with Molecular Clock

 2 m L (t) ±  L (t) = m L (t) ± m L (t) eσL (t−ts) − 1

(15)

Now about the final conditions for the mean value curve as well as for the two standard deviation curves. Let us call te the ending time of our mathematical analysis, namely the time beyond which we don’t care anymore about the values assumed by the stochastic process L(t). In the practice, this te is zero (i.e. now) in the theories of evolution of life on Earth or on an exoplanet, or the time when the mass extinction ends (and life starts growing up again) in the theories of mass extinction of life on a planet. First of all, it is clear that, in full analogy to the initial condition (8) for the mean value, also the final condition has the form m L (te) = N e

(16)

where Ne is a positive number denoting the number of species alive at the end time te. But we don’t know what random value will L(te) take. We only know that its standard deviation curve (14) will take at time te a certain positive value that will differ by a certain amount δNe from the mean value (16). In other words, we only know from (14) that one has  2 δ N e =  L (te) = m L (te) eσL (te−ts) − 1

(17)

Dividing (17) by (16) the common factor m(te) disappears, and one is left with δNe = Ne

 eσL (te−ts) − 1. 2

(18)

Solving this for σ L finally yields   2  ln 1 + δNNee σL = . √ te − ts

(19)

This equation expresses the so far unknow numerical parameter σ L in terms of the initial time ts plus the three final-time parameters (te, Ne, δNe). Thus, in conclusion, we have shown that, once the five parameters (ts, Ns, te, Ne, δNe) are assigned numerically, the lognormal stochastic process L(t) is determined completely. In Ref. [5] the reader will find a mathematical model of Darwinian Evolution different from the GBM model described here. That is the Markov–Korotayev model, for which this author proved the mean value (1) to be a Cubic(t) i.e. a third degree polynomial in t.

3 Important Special Cases of m(t)

545

3 Important Special Cases of m(t) (1) The particular case of (1) when the mean value m(t) is given by the generic exponential m GBM (t) = N0 eμG B M t = or, alternatively, = Ae Bt

(20)

is called Geometric Brownian Motion (GBM), and is widely used in financial mathematics, where it represents the “underlying process” of the stock values (Black-Sholes models). This author used the GBM in his previous models of Evolution and SETI (Refs. [1] through [4]), since it was assumed that the growth of the number of ET civilizations in the Galaxy, or, alternatively, the number of living species on Earth over the last 3.5 billion years, grew exponentially (Malthusian growth). Notice that, upon equating the two right-hand-sides of (4) and (20), we find e MGBM (t) e

2 σG BM 2

(t−ts)

= N0 eμG B M (t−ts) .

(21)

Solving this equation for M GBM (t) yields

σG2 B M (t − ts) MGBM (t) = ln N0 + μG B M − 2

(22)

This is (with ts = 0) just the “mean value showing at the exponent” of the well-known GBM pdf, i.e. GBM(t)_pdf (n; N 0 , μ, σ, t) ⎡

⎛

⎛

⎞ ⎞⎤2

⎣ln(n)− ⎝ln N0 +⎝μ−

=

e−

√

σ 2 ⎠ ⎠⎦ t 2

2σ 2 t

, (n ≥ 0).

√ 2π σ tn

(23)

We conclude this short description of the GBM as the exponential sub-case of the general lognormal process (2) by warning that “GBM” is a misleading name, since GBM is a lognormal process and not Gaussian one, as the Brownian Motion is indeed. (2) Another interesting particular case of the mean value function m(t) in (2) is when it equals a generic polynomial in t, namely 

polynomial_ degree

m polynomial (t) =

ck t k

(24)

k=0

with ck being the coefficient of the k-th power of the time t in the polynomial (24). We just confine ourselves to mention that the case where (24) is a second-degree

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Evo-SETI Entropy Identifies with Molecular Clock

polynomial (i.e. a parabola in t) may be used to describe the Mass Extinctions on Earth over the last 3.5 billion years (see Ref. [5]). (3) We must also introduce the notion of b-lognormal [ln(t−b)−μ]2

e− 2σ 2 b - lognormal_ pdf(t; μ, σ, b) = √ 2π σ (t − b)

(25)

holding for t ≥ b = birth, and meaning the lifetime of a living being, let it be a cell, a plant, a human, a civilization of humans, or even an ET civilization (Ref. [4], in particular pages 227–245).

4 Peak-Locus Theorem The Peak-Locus theorem is a new mathematical discovery of ours playing a central role in Evo-SETI. In its most general formulation, it holds good for any lognormal process L(t) and any arbitrary mean value mL (t). In the GBM case, it is shown in Fig. 2. The Peak-Locus theorem states that the family of b-lognormals each having its peak exactly located upon the mean value curve (1), is given by the following three equations, specifying the parameters μ(p), σ (p) and b(p), appearing in (25) as three functions of the independent variable p, the b-lognormal’s peak: that is, if rewritten directly in terms of mL (p):

Fig. 2 Darwinian Exponential as the geometric LOCUS OF THE PEAKS of b-lognormals for the GBM case. Each b-lognormal is a lognormal starting at a time (t = b = birth time) and represents a different SPECIES that originated at time b of the Darwinian Evolution. This is CLADISTICS, as seen through the glasses of our Evo-SETI model. It is evident that, when the generic “Running blognormal” moves to the right, its peak becomes higher and higher and narrower and narrower, since the area under the b-lognormal always equals 1. Then, the (Shannon) ENTROPY of the running b-lognormal is the DEGREE OF EVOLUTION reached by the corresponding SPECIES (or living being, or a civilization, or an ET civilization) in the course of Evolution

4 Peak-Locus Theorem

547

⎧ 2 σ L2 ⎪ eσ L p ⎪ 2 ⎪ μ( p) = − p ⎪ ⎪ 4π [m L ( p)]2 ⎪ ⎪ ⎨ σ L2 2 p e ⎪ σ ( p) = √ ⎪ ⎪ ⎪ 2π m L ( p) ⎪ ⎪ ⎪ 2 ⎩ b( p) = p − eμ( p)−[σ ( p)] . In the particular GBM case, the mean value is (20) with μGBM = B, σ L = and N 0 = Ns = A. Then, the Peak-Locus theorem (26) with ts = 0 yields: ⎧ μ( p) = 4π1A2 − Bp, ⎪ ⎪ ⎨ 1 σ = √2π , A ⎪ ⎪ 2 ⎩ b( p) = p − eμ( p)−σ .

(26)

√

2B

(27)

In this simpler form, the Peak-Locus theorem was already published by the author in Refs. [2–4], while its most general form (26) is new for this paper.

5 Entropy as Measure of Evolution The (Shannon) Entropy of the b-lognormal (25) is     √ 1 1 ln 2π σ ( p) + μ( p) + . H ( p) = ln(2) 2

(28)

This is a function of the peak abscissa p and is measured in bits, as in Shannon’s Information Theory. By virtue of the Peak-Locus Theorem (26), it becomes   2 1 eσ L p 1 H ( p) = − ln(m L ( p)) + . ln(2) 4π [m L ( p)]2 2

(29)

Thus, (29) is the Entropy of each member of the family of ∞1 b-lognormals (the family’s parameter is p) peaked upon the mean value curve (1). The b-lognormal Entropy (29) is thus the Measure of the Amount of Evolution of that b-lognormal: it measures “the decreasing disorganization in time of what that b-lognormal represents”, let it be a cell, a plant, a human or even a civilization. Entropy is thus disorganization decreasing in time. However, one would prefer to use a measure of the “increasing organization” in time. The EvoEntropy of p EvoEntropy( p) = −[H ( p) − H (ts)]

(30)

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Evo-SETI Entropy Identifies with Molecular Clock

(Entropy of Evolution) is a function of p that has a minus sign in front, thus changing the decreasing trend of the (Shannon) Entropy (28) into the increasing trend of our EvoEntropy (30). In addition, our EvoEntropy starts at zero at the initial time ts, as expected: EvoEntropy(ts) = 0.

(31)

By virtue of (29), the EvoEntropy (30) becomes EvoEntropy( p)_ of_ the_ Lognormal_ Process_ L(t) 

 2 2 eσL ts eσ L p m L ( p) 1 − + ln . = ln(2) 4π [m L (ts)]2 4π [m L ( p)]2 m L (ts)

(32)

In the GBM case, the EvoEntropy (32) becomes just an exact linear function of the time p GBM_ EvoEntropy( p) =

B · ( p − ts). ln(2)

(33)

This is, of course, a straight line in the time p starting at the time ts of the Origin of Life on Earth and increasing linearly thereafter. It is measured in bits/individual and is shown in Fig. 3. But… THIS IS THE SAME LINEAR BEHAVIOUR IN TIME AS THE MOLECULAR CLOCK! (see refs. [6–9]).

Fig. 3 EvoEntropy (in bits per individual) of the latest species appeared on Earth during the last 3.5 billion years. This shows that a Man (nowadays) is 25.575 bits more evolved than the first form of life (RNA?) 3.5 billion years ago

5 Entropy as Measure of Evolution

549

So, we have discovered that the Entropy in our Evo-SETI model and the Molecular Clock are the same linear time function, apart for multiplicative constants (depending on the adopted units, like bits, seconds, etc.). This conclusion appears to be of key importance to understand “where a newly discovered exoplanet stands on its way to develop LIFE”.

6 Conclusions More and more exoplanets are now being discovered by astronomers either by observations from the ground or by virtue of space missions, like “CoRot”, “Kepler”, “Gaia”, and other future space missions. As a consequence, a recent estimate sets at 40 billion the number of Earth-sized planets orbiting in the habitable zones of sun-like stars and red dwarf stars within the Milky Way galaxy. With such huge numbers of “possible Earths” in sight, Astrobiology and SETI are becoming research fields more and more attractive to a number of scientists. Mathematically innovative papers like this one, revealing an unsuspected relationship between the Molecular Clock and the Entropy of b-lognormals in Evo-SETI Theory, should thus be welcome.

References 1. J. Felsenstein, Inferring Phylogenies (Sinauer Associates Inc., Sunderland) 2. C. Maccone, The statistical Drake equation. Acta Astronaut. 67, 1366–1383 (2010) 3. C. Maccone, A mathematical model for evolution and SETI. Orig. Life Evol. Biospheres (OLEB) 41, 609–619 (2011) 4. C. Maccone, Mathematical SETI, A 724-pages Book Published by Praxis-Springer in the Fall of 2012. ISBN, ISBN-10: 3642274366|ISBN-13: 978-3642274367|Edition: 2012 5. C. Maccone, SETI, evolution and human history merged into a mathematical model. Int. J. Astrobiol. 12(3), 218–245 (2013) 6. C. Maccone, Evolution and mass exctinctions as lognormal stochastic processes. Int. J. Astrobiol. 13(4), 290–309 (2014) 7. T. Maruyama, Stochastic Problems in Population Genetics. Lecture Notes in Biomathematics 17 (Springer, Heidelberg New York, 1977) 8. M. Nei, K. Sudhir, Molecular Evolution and Phylogenetics (Oxford University Press, 2000) 9. M. Nei, Mutation-Driven Evolution (Oxford University Press, 2013)

Evo-SETI SCALE to Measure Life on Exoplanets

Abstract Darwinian Evolution over the last 3.5 billion years was an increase in the number of living species from 1 (RNA?) to the current 50 million. This increasing trend in time looks like being exponential, but one may not assume an exactly exponential curve since many species went extinct in the past, even in mass extinctions. Thus, the simple exponential curve must be replaced by a stochastic process having an exponential mean value. Borrowing from financial mathematics (“Black–Scholes models”), this “exponential” stochastic process is called Geometric Brownian Motion (GBM), and its probability density function (pdf) is a lognormal (not a Gaussian) (Proof: see Ref. Maccone [4], Chapter 30, and Ref. Maccone [5]). Lognormal also is the pdf of the statistical number of communicating Extra Terrestrial (ET) civilizations in the Galaxy at a certain fixed time, like a snapshot: this result was found in 2008 by this author as his solution to the Statistical Drake Equation of SETI (Proof: see Ref. Maccone [2]). Thus, the GBM of Darwinian Evolution may also be regarded as the extension in time of the Statistical Drake equation (Proof: see Ref. Maccone [5]). But the key step ahead made by this author in his Evo-SETI (Evolution and SETI) mathematical model was to realize that LIFE also is just a b-lognormal in time: every living organism (a cell, a human, a civilization, even an ET civilization) is born at a certain time b (“birth”), grows up to a peak p (with an ascending inflexion point in between, a for adolescence), then declines from p to s (senility, i.e. descending inflexion point) and finally declines linearly and dies at a final instant d (death). In other words, the infinite tail of the b-lognormal was cut away and replaced by just a straight line between s and d, leading to simple mathematical formulae (“History Formulae”) allowing one to find this “finite blognormal” when the three instants b, s, and d are assigned. Next the crucial Peak-Locus Theorem comes. It means that the GBM exponential may be regarded as the geometric locus of all the peaks of a one-parameter (i.e. the peak time p) family of b-lognormals. Since b-lognormals are pdf-s, the area under each of them always equals 1 (normalization condition) and so, going from left to right on the time axis, the b-lognormals become more and more “peaky”, and so they last less and less in time. This is precisely what happened in human history: civilizations that lasted millennia (like Ancient Greece and Rome) lasted just centuries (like the Italian Renaissance and Portuguese, Spanish, French, British and USA Empires) but they were more and more advanced in the “level of civilization”. This “level of civilization” is what physicists call ENTROPY. Also, in © Springer Nature Switzerland AG 2020 C. Maccone, Evo-SETI, https://doi.org/10.1007/978-3-030-51931-5_16

551

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Evo-SETI SCALE to Measure Life on Exoplanets

Refs. Maccone [4, 5], this author proved that, for all GBMs, the (Shannon) Entropy of the b-lognormals in his Peak-Locus Theorem grows LINEARLY in time. The Molecular Clock, well known to geneticists since 50 years, shows that the DNA base-substitutions occur LINEARLY in time since they are neutral with respect to Darwinian selection. In simple words: DNA evolved by obeying the laws of quantum physics only (microscopic laws) and not by obeying assumed “Darwinian selection laws” (macroscopic laws). This is Kimura’s neutral theory of molecular evolution. The conclusion is that the Molecular Clock and the b-lognormal Entropy are the same thing. At last, we reach the new, original result justifying the publication of this paper. On exoplanets, molecular evolution is proceeding at about the same rate as it did proceed on Earth: rather independently of the physical conditions of the exoplanet, if the DNA had the possibility to evolve in water initially. Thus, Evo-Entropy, i.e. the (Shannon) Entropy of the generic b-lognormal of the Peak-Locus Theorem, provides the Evo-SETI SCALE to measure the evolution of life on exoplanets. © 2015 IAA. Published by Elsevier Ltd. All rights reserved. Keywords Darwinian evolution · Molecular clock · Entropy · SETI

1 Purpose of This Chapter This chapter describes recent developments in a new statistical theory describing Evolution and SETI by mathematical equations. This we call the Evo-SETI mathematical model of Evolution and SETI. Now the question is: whenever a new exoplanet is discovered, where does that exoplanet stand in its evolution towards life as we have it on Earth nowadays, or beyond? This is the central question of Evo-SETI. In this chapter we show that the (Shannon) Entropy of b-lognormals answers such a question, thus allowing the creation of an EVO-SETI SCALE.

2 During the Last 3.5 Billion Years Life Forms Increased Like a (Lognormal) Stochastic Process Let us look at Fig. 1: on the horizontal axis is the time t, with the convention that negative values of t are past times, zero is now, and positive times are future times. The starting point on the time axis is ts = −3:5 × 109 years i.e. 3.5 billion years ago, the time of the origin of life on Earth that we assume to be correct. If the origin of life started earlier than that, say 3.8 billion years ago, the coming equations would still be the same and their numerical values will only be slightly changed. On the vertical axis is the number of species living on Earth at time t, denoted L(t). This “function of the time” we do not know in detail, and so it must be regarded as a random function, or stochastic process, with the notation L(t) standing for “life at

2 During the Last 3.5 Billion Years Life Forms Increased Like …

553

Evolution as INCREASING NUMBER OF SPECIES 8

1×10

Number of LIVING SPECIES on Earth

7

9×10

7

8×10

7

7×10

7

6×10

7

5×10

7

4×10

7

3×10

7

2×10

7

1×10

0 − 3.5

−3

− 2.5

−2

− 1.5

−1

− 0.5

0

Time in billions of years Fig. 1 DARWINIAN EVOLUTION as the increasing number of living species on Earth between 3.5 billion years ago and now. The red solid curve is the mean value of the GBM stochastic process L GBM (t) given by (24), while the blue dot–dot curves above and below the mean value are the two standard deviation upper and lower curves, given by (11) and (12), respectively, with mGBM (t) given by (23). The “Cambrian Explosion” of life, that on Earth started around 542 million years ago, is evident in the above plot just before the value of −0.5 billion years in time, where all three curves “start leaving the time axis and climbing up”. Notice also that the starting value of living species 3.5 billion years ago is ONE by definition, but it “looks like” zero in this plot since the vertical scale (which is the true scale here, not a log scale) does not show it. Notice finally that nowadays (i.e. at time t = 0) the two standard deviation curves have exactly the same distance from the middle mean value curve, i.e. 30 million living species more or less the mean value of 50 million species. These are assumed values that we used just to exemplify the GBM mathematics: biologists might assume other numeric values. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

time t”. In this book we adopt the convention that capital letters represent random variables, i.e. stochastic processes if they depend on the time, while lower-case letters mean ordinary variables or functions.

3 Mean Value of the Lognormal Process L(t) The most important ordinary, continuous function of the time associated with a stochastic process like L(t) is its mean value, denoted by

554

Evo-SETI SCALE to Measure Life on Exoplanets

m L (t) ≡ L(t).

(1)

The probability density function (pdf) of a stochastic process like L(t) is assumed in Evo-SETI Theory to be a lognormal, that is, its equation reads e

−

[ln(n)−M L (t)]2 2σ L2 (t−ts)

L(t)_pdf(n; M L (t), σ, t) = √ √ 2π σ L t − ts n   n ≥ 0, σ L ≥ 0, with and M L (t) = arbitrary function of t. t ≥ ts,

(2)

This assumption is in line with the extension in time of the statistical Drake equation, namely the foundational and statistical equation of SETI, as shown in Ref. [2]. The mean value (1) is of course related to the pdf (2) by the relevant integral in the number n of living species on Earth at time t, that is ∞ m L (t) ≡

n·√ 0

e

−

[ln(n)−M L (t)]2 2σ L2 (t−ts)

dn. √ 2π σ L t − tsn

(3)

The “surprise” is that this integral (3) may be computed exactly with the key result that the mean value mL (t) is given by m L (t) = e M L (t) e

σ L2 2

(t−ts)

.

(4)

In turn, the last equation has the “surprising” property that it may be inverted exactly, i.e. solved for M(t): M L (t) = ln(m L (t) −

σ L2 (t − ts). 2

(5)

4 L(t) Initial Conditions at ts Now about the initial conditions of the stochastic process L(t), namely about the value L(ts). We shall assume that the positive number L(ts) = N s is always exactly known, i.e. with probability one:

(6)

4 L(t) Initial Conditions at ts

555

Pr{L(ts) = N s} = 1.

(7)

In the practice, Ns will be equal to 1 in the theories of evolution of life on Earth or on an exoplanet (i.e., there must have been a time ts in the past when the number of living species was just one, let it be RNA or something else). On the contrary, Ns will be equal to the number of living species just before the asteroid/comet impact in the theories of mass extinction of life on a planet. The mean value mL (t) of L(t) thus must equal the initial number Ns at the initial time ts, that is m L (ts) = N s.

(8)

Replacing t by ts in (4), one then finds m L (ts) = e M(ts)

(9)

that, checked against (8), immediately yields N s = e M L (ts) that is M L (ts) = ln(N s).

(10)

These are the initial conditions for the mean value. After the initial instant ts, the stochastic process L(t) unfolds oscillating above or below the mean value in an unpredictable way. Statistically speaking, however, we expect L(t) “not to depart too much” from m(t) and this fact is graphically shown in Fig. 1 by the two dot–dot blue curves above and below the mean value solid red curve m(t). These two curves are the upper standard deviation curve    2 σ (t−ts) −1 upper_ st_ dev_ curve(t) = m L (t) 1+ e L

(11)

and the lower standard deviation curve    2 lower_ st_ dev_ curve(t) = m L (t) 1 − eσL (t−ts) − 1

(12)

respectively (Proof: see Table 2 of Ref. [5]). Notice that both (11) and (12), at the initial time t = ts, equal the mean value m(ts) = Ns, that is, with probability one again, the initial value Ns is the same for all the three curves shown in Fig. 1. The function of the time  2 (13) variation_ coefficient(t) = eσL (t−ts) − 1

556

Evo-SETI SCALE to Measure Life on Exoplanets

is called “variation coefficient” by statisticians since the standard deviation of L(t) (be careful: this is just the standard deviation L (t) of L(t) and not either of the above two “upper” and “lower” standard deviation curves given by (11) and (12), respectively) is  2 st_ dev_ curve(t) ≡ Δ L (t) = m L (t) eσL (t−ts) − 1

(14)

Indeed, (14) shows that the variation coefficient (13) is the ratio of (t) to mL (t), i.e. it expresses how much the standard deviation “varies” with respect to the mean value. Having understood this fact, it is then obvious that the two curves (11) and (12) are obtained as  2 m L (t) ± Δ L (t) = m L (t) ± m L (t) eeL (t−ts) − 1

(15)

respectively.

5 L(t) Final Conditions at te > ts Now about the final conditions for the mean value curve as well as for the two standard deviation curves. Let us call te the ending time of our mathematical analysis, namely the time beyond which we do not care any more about the values assumed by the stochastic process L(t). In the practice, this te is zero (i.e. now) in the theories of evolution of life on Earth or on an exoplanet, or the time when the mass extinction ends (and life starts growing up again) in the theories of mass extinction of life on a planet. First of all, it is clear that, in full analogy to the initial condition (8) for the mean value, also the final condition has the form m L (te) = N e

(16)

where Ne is a positive number denoting the number of species alive at the end time te. But we do not know what random value will L(te) take. We only know that its standard deviation curve (14) will take at time te a certain positive value that will differ by a certain amount δNe from the mean value (16). In other words, we only know from (14) that one has  δ N e =  L (te) = m L (te) eσL (te−ts) − 1 2

(17)

5 L(t) Final Conditions at te > ts

557

Dividing (17) by (16) the common factor m(te) disappears, and one is left with δNe = Ne

 eσL (te−ts) − 1. 2

(18)

Solving this for σL finally yields   2  ln 1 + δNNee σL = . √ te − ts

(19)

This equation expresses the so far unknown numerical parameter σL in terms of the initial time ts plus the three final-time parameters (te, Ne, δNe). Thus, in conclusion, we have shown that, once the five parameters (ts, Ns, te, Ne, δNe) are assigned numerically, the lognormal stochastic process L(t) is determined completely. Finally notice that the square of (19) may be rewritten in the following different form: ⎧ ⎫  2   ⎪  2  te −1 ts ⎪ ⎨ ⎬ ln 1 + δNNee δNe (20) = ln 1 + σ L2 = ⎪ ⎪ te − ts Ne ⎩ ⎭ from which we infer the formula ⎧ ⎪ ⎨

e

σ L2

=e

1    δ N e 2 te−ts ln 1+ Ne ⎪ ⎩

⎫ ⎪ ⎬ ⎪ ⎭

 = 1+



δNe Ne

1 2  te−ts

.

(21)

This Eq. (21) enables us to get rid of eσL replacing it by virtue of the four boundary parameters supposed to be known: (ts, te, Ne, δNe). It will be later used in Sect. 8 in order to rewrite the Peak-Locus Theorem in terms of the boundary conditions, rather 2 than in terms of eσL . 2

6 Important Special Cases of mL (t) 1. The particular case of (1) when the mean value mL (t) is given by the generic exponential m GBM (t) = N0 eμGBM t = or, alternatively, = Ae Bt

(22)

558

Evo-SETI SCALE to Measure Life on Exoplanets

is called Geometric Brownian Motion (GBM), and is widely used in financial mathematics, where it represents the “underlying process” of the stock values (Black–Scholes models). This author used the GBM in his previous models of Evolution and SETI (Refs. [2–5]), since it was assumed that the growth of the number of ET civilizations in the Galaxy, or, alternatively, the number of living species on Earth over the last 3.5 billion years, grew exponentially (Malthusian growth). Notice that, upon equating the two right-hand-sides of (4) and (22), we find e

MGBM (t)

σ 2GBM

e

2

= N0 eμGBM (t−ts) .

(t−ts)

(23)

Solving this equation for M GBM (t) yields   σ2 MGBM (t) = ln N0 + μGBM − GBM (t − ts). 2

(24)

This is (with ts = 0) just the “mean value showing at the exponent” of the well-known GBM pdf, i.e. ⎡

⎛

⎛

⎞ ⎞⎤2

⎣ln(n)−⎝ln N0 +⎝μ−

=

e−

√

σ 2 ⎠ ⎠⎦ t 2

2σ 2 t

√ 2πσ tn

, (n ≥ 0).

(25)

We conclude this short description of the GBM as the exponential sub-case of the general lognormal process (2) by warning that “GBM” is a misleading name, since GBM is a lognormal process and not Gaussian one, as the Brownian Motion is indeed. 2. Another interesting particular case of the mean value function m(t) in (2) is when it equals a generic polynomial in t starting at ts, namely !

polynomial_ degree

m polynomial (t) =

ck (t − ts)k .

(26)

k=0

with ck being the coefficient of the k-th power of the time t in the polynomial (26). We just confine ourselves to mention that the case where (26) is a second-degree polynomial (i.e. a parabola in t) may be used to describe the Mass Extinctions on Earth over the last 3.5 billion years (see Ref. [6]). 3. We must also introduce the notion of b-lognormal [ln(t−b)−μ]2

2σ 2 e− b - lognormal_ pdf (t; μ, σ, b) = √ 2π σ (t − b)

(27)

6 Important Special Cases of mL (t)

559

holding for t ≥ b = birth, and meaning the lifetime of a living being, let it be a cell, a plant, a human, a civilization of humans, or even an ET civilization (Ref. [5], in particular pages 227–245).

7 Boundary Conditions When mL (t) is a First, Second or Third Degree Polynomial in the Time (t–ts) In Ref. [6] the reader may find a mathematical model of Darwinian Evolution different from the GBM model described in terms of GBMs. That is the Markov–Korotayev model, for which this author proved the mean value (1) to be a Cubic (t) i.e. a third degree polynomial in t. We summarize hereafter the key formulae proven in Ref. [6] about the case when the assigned mean value mL (t) is a polynomial in t starting at ts and up to degree 3, that is: m L (t) =

3 !

ck (t − ts)k .

(28)

k=0

1. The mean value is a straight line. Then this straight line simply is the line through the two points (ts, Ns) and (te, Ne), that, after a few rearrangements, turns out to be: m straight_ line (t) = (N e − N s)

t − ts + N s. te − ts

(29)

2. The mean value is a parabola, i.e. a quadratic polynomial in t. Then, the equation of such a parabola reads m parabola (t) = (N e − N s)

  t − ts t − ts 2− + N s. te − ts te − ts

(30)

Equation (30) was actually firstly derived by this author in Ref. [6], pages 299– 301, in relationship to Mass Extinctions (i.e. it is a decreasing function of the time). 3. The mean value is a cubic. Then, in Ref. [6], pages 304–307, this author proved, in relation to the Markov–Korotayev model of Evolution, that the cubic mean value of the L(t) lognormal stochastic process is given by the cubic equation in t m cubic (t) = (N e − N s) # " (t − ts) 2(t − ts)2 − 3(tMax + tmin − 2ts)(t − ts) + 6(tMax − ts)(tmin − ts) " # + N s. · 2 (te − ts) 2(te − ts) − 3(tMax + tmin − 2ts)(te − ts) + 6(tMax − ts)(tmin − ts)

(31)

560

Evo-SETI SCALE to Measure Life on Exoplanets

Notice that, in (31) one has, in addition to the usual initial and final conditions Ns = mL (ts) and Ne = mL (te), two more “middle conditions” referring to the two instants (t M , t m ) of at which the Maximum and the minimum of the cubic Cubic(t) occur, respectively: 

tmin = time_ of_ the_ Cubic_ minimum tMax = time_ of_ the_ Cubic_ Maximum.

(32)

8 Peak-Locus Theorem The Peak-Locus Theorem is a new mathematical discovery of ours playing a central role in Evo-SETI. In its most general formulation, it holds good for any lognormal process L(t) and any arbitrary mean value mL (t). In the GBM case, it is shown in Fig. 2. The Peak-Locus Theorem states that the family of b-lognormals each having its peak exactly located upon the mean value curve (1), is given by the following three equations, specifying the parameters μ(p), σ (p) and b(p), appearing in (27) as three functions of the independent variable p, the b-lognormal’s peak: that is, if rewritten directly in terms of mL (p):

Fig. 2 Darwinian Exponential as the geometric LOCUS OF THE PEAKS of b-lognormals for the GBM case. Each b-lognormal is a lognormal starting at a time (t = b = birth time) and represents a different SPECIES that originated at time b of the Darwinian Evolution. This is CLADISTICS, as seen through the glasses of our Evo-SETI model. It is evident that, when the generic “Running b-lognormal” moves to the right, its peak becomes higher and higher and narrower and narrower, since the area under the b-lognormal always equals 1. Then, the (Shannon) Entropy of the running b-lognormal is the DEGREE OF EVOLUTION reached by the corresponding SPECIES (or living being, or a civilization, or an ET civilization) in the course of Evolution

8 Peak-Locus Theorem

561

⎧ ⎪ μ( p) = ⎪ ⎪ ⎨

2

σ2 eσ L p − p 2L 4π[m L ( p)]2 σ2 L p 2 e √ 2π m L ( p) μ( p)−[σ ( p)]2

⎪ σ ( p) = ⎪ ⎪ ⎩ b( p) = p − e

(33)

.

The proof of (33) is lengthy and was given as a special pdf file (written in the language of the Maxima symbolic manipulator) that the reader may freely download in the web site of Ref. [6]. But we now present an important new result: the PeakLocus Theorem (33) rewritten not in terms of σL any more, but rather in terms of the four boundary parameters supposed to be known: (ts, te, Ne, δNe). To this end, we must insert (21) and (20) into (33), with the result ⎧ p     ⎪ $ % ⎪ δ N e 2 te−ts p ⎪  1+ ⎪ δ N e 2  2(t−ts) Ne ⎪ ⎪ μ( p) = − ln 1 + N e ⎪ 4π[m L ( p)]2 ⎪ ⎨ p (34)     δ N e 2 2(t−ts) ⎪ ⎪ 1+ ⎪ N e ⎪ ⎪ √ σ ( p) = ⎪ ⎪ 2π m L ( p) ⎪ 2 ⎩ b( p) = p − eμ( p)−[σ ( p)] . In the particular GBM case, the mean value is (22) with μGBM = B, σL = and N 0 = Ns = A. Then, the Peak-Locus Theorem (33) with ts = 0 yields: ⎧ 1 ⎪ ⎨ μ( p) = 4π A2 − Bp, 1 , σ = √2π A ⎪ ⎩ b( p) = p − eμ( p)−σ 2 .

√

2B

(35)

In this simpler form, the Peak-Locus Theorem was already published by the author in Refs. [3–5], while its most general form is (33) and (34).

9 Entropy as Measure of Evolution The (Shannon) Entropy of the b-lognormal (27) is     √ 1 1 ln 2π σ ( p) + μ( p) + . H ( p) = ln(2) 2

(36)

This is a function of the peak abscissa p and is measured in bits, as in Shannon’s Information Theory. By virtue of the Peak-Locus Theorem (33), it becomes

562

Evo-SETI SCALE to Measure Life on Exoplanets

$ % 2 eσ L p 1 1 H ( p) = − ln(m L ( p)) + . ln(2) 4π [m L ( p)]2 2

(37)

One may also rewrite (37) directly in terms of the four boundary parameters (ts, te, Ne, δNe) upon inserting (21) into (37), with the result: ⎧ p δ N e 2  te − ts ⎪ ⎪ ⎨ 1 + Ne 1 − ln(m L ( p)) + H ( p) = 2 ln(2) ⎪ 4π [m L ( p)] ⎪ ⎩

⎫ ⎪ ⎪ 1⎬ 2⎪ ⎪ ⎭

.

(38)

Thus, (37) or (38) yield the Entropy of each member of the family of ∞1 blognormals (the family’s parameter is p) peaked upon the mean value curve (1). The b-lognormal Entropy (37) is thus the Measure of the Amount of Evolution of that b-lognormal: it measures “the decreasing disorganization in time of what that b-lognormal represents”, let it be a cell, a plant, a human or even a civilization. Entropy is thus disorganization decreasing in time. However, one would prefer to use a measure of the “increasing organization” in time. The EvoEntropy of p EvoEntropy( p) = −[H ( p) − H (ts)]

(39)

(Entropy of Evolution) is a function of p that has a minus sign in front, thus changing the decreasing trend of the (Shannon) Entropy (36) into the increasing trend of our EvoEntropy (39). In addition, our EvoEntropy starts at zero at the initial time ts, as expected: EvoEntropy(ts) = 0.

(40)

By virtue of (37), the EvoEntropy (39) becomes EvoEntropy( p)_ of_ the_ Lognormal_ Process_ L(t) $ %  2 2 eσL ts eσ L p m L ( p) 1 − + ln . = ln(2) 4π [m L (ts)]2 4π [m L ( p)]2 m L (ts)

(41)

Alternatively, we may rewrite (41) directly in terms of the four boundary parameters (ts, te, Ne, δNe) upon inserting (21) into (41), thus finding: EvoEntropy( p)_ of_ the_ Lognormal_ Process_ L(t) ⎧ ⎫ p ts  δ N e 2  te−ts δ N e 2  te−ts ⎪ ⎪ ⎪ ⎪   1 + Ne m L ( p) ⎬ 1 ⎨ 1 + Ne . − + ln = ⎪ 4π [m L (ts)]2 ln(2) ⎪ 4π [m L ( p)]2 m L (ts) ⎪ ⎪ ⎩ ⎭

(42)

9 Entropy as Measure of Evolution

563

In the GBM case, the EvoEntropy (41) becomes just an exact linear function of the time p GBM_ EvoEntropy ( p) =

B · ( p − ts). ln(2)

(43)

This is, of course, a straight line in the time p starting at the time ts of the Origin of Life on Earth and increasing linearly thereafter. It is measured in bits/individual and is shown in Fig. 3. But… THIS IS THE SAME LINEAR BEHAVIOR IN TIME AS THE MOLECULAR CLOCK, that is the technique in molecular evolution that uses fossil constraints and rates of molecular change to deduce the time in geologic history when two species or other taxa diverged. The molecular data used for such calculations are usually nucleotide sequences for DNA or amino acid sequences for proteins (see Refs. [1, 7–10]). So, we have discovered that the Entropy in our Evo-SETI model and the Molecular Clock are the same linear time function, apart for multiplicative constants (depending on the adopted units, like bits, seconds, etc.). This conclusion appears to be of key

EvoEntropy of the LATEST SPECIES in bits/individual

EvoEntropy of the LATEST SPECIES in bits/individual 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 −3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

Time in billions of years before present (t=0) Fig. 3 EvoEntropy (in bits per individual) of the latest species appeared on Earth during the last 3.5 billion years. This shows that a Man (nowadays) is 25.575 bits more evolved than the first form of life (RNA?) 3.5 billion years ago

564

Evo-SETI SCALE to Measure Life on Exoplanets

importance to understand “where a newly discovered exoplanet stands on its way to develop LIFE”.

10 Kullback–Leibler Divergence (or, Better, “Distance”) Among Any Two Living Species The Molecular Clock, shown in Fig. 3 as the linear growth of Evo-Entropy over the last 3.5 billion years of evolution of life on Earth, is the key fact in molecular evolution and allows for an immediate quantitative estimate of “how much” (in bits per individuals) two Species “differ” from each other. For instance, Fig. 3 reveals that the difference between modern man (at time zero) and the RNA (regarded as the first Species appeared on Earth at time −3.5 billion years) equals 25.575 bits per individual. But mathematicians and information experts also have another quantitative way of measuring the difference in evolution among any two species. This is the so-called Kullback–Leibler divergence (Ref. [11]) among any two probability distributions, introduced in 1951 by Solomon Kullback (1907–1994) and Richard Leibler (1914– 2003). Given any two probability distributions P and Q, whose continuous probability density functions are p(x) and q(x), respectively, the Kullback–Leibler divergence “among P and Q” (measured in bits) is given by the integral 

∞ D K L (P||Q) =

p(x) · ln −∞

 p(x) d x. q(x)

(44)

We now wish to apply the definition (44) to Evo-SETI Theory, but how shall we do so? The answer is to consider two Running b-lognormals in the Peak-Locus Theorem summarized in Sect. 8 and visualized in Fig. 2. Suppose that: 1. The “enveloping curve” (i.e. the red curve in Fig. 2) is the GBM exponential mean value m GBM (t) = Ae Bt with A > 0 and B > 0.

(45)

This curve is actually the particular case of the more general ts-GBM exponential, i.e. GBM starting at ts, with mean value m tsGBM (t) = e B(t − ts) with B > 0 and A = m tsGBM (0).

(46)

But the Peak-Locus Theorem (35) applies to (45) rather than to (25), and so we will use (45) as enveloping GBM exponential, rather than (25). 2. Then the peak of the first b-lognormal, i.e. “the left one” occurs at the time p.

10 Kullback–Leibler Divergence (or, Better, “Distance”) Among …

565

3. The distance (in time) between the peak of the first and the second b-lognormal in Fig. 2 we assume to be a known amount of time, call it Δ. 4. Then the peak of the second b-lognormal, i.e. “the right one” is at the time p + Δ. 5. The Peak-Locus Theorem (35) for the GBM “enveloping exponential” (45) then applies. 6. Therefore, we insert the first two equations (35) into the Running b-lognormal Eq. (27), thus obtaining the left-and right-b-lognormal, respectively: left_ blognormal_ (t)_ with_ peak_ at_ p =

Ae

 −π A2 p B−

1 +ln(t−b) 4π A2

2

t −b

= Q(t)

(47)

right_blognormal_(t)_with_peak_at_( p + Δ) Ae

=

 −π A2 ( p+Δ)B−

1 +ln(t−b) 4π A2

t −b

2

= P(t).

(48)

Let us now write down the integrand (with respect to t, running from b to ∞), of the Kullback–Leibler divergence (44). Taking both (47) and (48) into account, this rather lengthy integrand reads: 

2 1 +ln(t−b) 4π A2 P(t) t−b = · ln P(t) · ln  2 Q(t) 2 Ae−π A ( p B− 1 2 +ln(t−b) 4π A t−b  2   2  2  2 1 1 Ae−π A ( p + Δ)B − + ln(t − b) · −π A2 ( p + Δ)B − + ln(t − b) + π A2 p B − 1 2 + ln(t − b) 4π A2 4π A2 4π A . = t −b Ae−π A

2



1 + ln(t − b) 4π A2 t −b

( p + Δ)B −

2

Ae−π A

2

( p+Δ)B−

(49)

We must next integrate this integrand (49) with respect to t between b and infinity. To this end, let us notice that (49) “looks very much like” the integrand of the classical Gauss integral 

∞ e

−C x 2 +Dx

dx =

−∞

π D2 e 4C with C > 0. C

(50)

Looking at the two exponents in the integrands (49) and (50), we notice that we may let them coincide if we perform the usual change of variable for a lognormalto-Gaussian pdf change: ln(t − b) = x +

1 − ( p + )B. 4π A2

(51)

566

Evo-SETI SCALE to Measure Life on Exoplanets

dt yielding = d x with (t − b) with the two additional substitutions 



b≤t ≤∞ −∞ ≤ x ≤ ∞

C = π A2 D = 0.

(52)

(53)

Thus, the integral of the integrand (49) now reads: ∞

# " 2 Ae−C x · −π A2 x 2 + π A2 (x − BΔ)2 d x

−∞

∞ = πA

2 −∞ ∞

= π A3

# " 2 Ae−C x · −x 2 + (x − BΔ)2 d x " # 2 e−C x · B 2 Δ2 − 2x BΔ d x

−∞

∞ = πA B Δ 3

2

2

e

−C x 2

∞ d x − 2BΔ

−∞



= π A 3 B 2 Δ2

xe−C x d x 2

−∞

π −0 π A2

= π A 2 B 2 Δ2 .

(54)

In conclusion, we have proven the important new GBM Kullback–Leibler divergence where we now divide the final result by ln(2) in order to measure in bits the natural logarithm appearing in (44) Kullback_Leibler_divergence_between_a_Species and_another_Species_born_at_time_Δ_later =

π A 2 B 2 · Δ2 ln 2

(55)

where A and B are the Darwinian Evolution parameters for Earth, that were derived many times by this author in his previous papers, like Refs. [3–6]: $

A = 50 million Species nowadays on Earth −16 ln A B = − −3.5 billion = 1.605×10 . years ago sec

(56)

11 Conclusions

567

11 Conclusions More and more exoplanets are now being discovered by astronomers either by observations from the ground or by virtue of space missions, like “CoRot”, “Kepler”, “Gaia”, and other future space missions. As a consequence, a recent estimate sets at 40 billion the number of Earth-sized planets orbiting in the habitable zones of sun-like stars and red dwarf stars within the Milky Way galaxy. With such huge numbers of “possible Earths” in sight, Astrobiology and SETI are becoming research fields more and more attractive to a number of scientists. Mathematically innovative papers like this one, revealing an unsuspected relationship between the Molecular Clock and the Entropy of b-lognormals in Evo-SETI Theory, should thus be welcome.

References 1. J. Felsenstein, Inferring Phylogenies (Sinauer Associates Inc., Sunderland, 2004). 2. C. Maccone, Mathematical SETI, a 724-pages book published by Praxis-Springer in the fall of 2012. ISBN, ISBN-10: 3642274366, ISBN-13: 978-3642274367. Edition: 2012 3. C. Maccone, SETI, evolution and human history merged into a mathematical model. Int. J. Astrobiol. 12(3), 218–245 (2013) 4. C. Maccone, The statistical Drake equation. Acta Astronaut. 67, 1366–1383 (2010) 5. C. Maccone, A mathematical model for evolution and SETI. Orig. Life Evolut. Biosp. (OLEB) 41, 609–619 (2011) 6. C. Maccone, Evolution and mass extinctions as lognormal stochastic processes. Int. J. Astrobiol. 13(4), 290–309 (2014) 7. T. Maruyama, Stochastic Problems in Population Genetics (Lecture Notes in Biomathematics 17) (Springer, Berlin, 1977) 8. M. Nei, S. Kumar, Molecular Evolution and Phylogenetics (Oxford University Press, New York, 2000). 9. M. Nei, Mutation-Driven Evolution (Oxford University Press, Oxford, 2013). 10. https://en.wikipedia.org/wiki/Molecular_clock 11. https://en.wikipedia.org/wiki/Kullback%E2%80%93Leibler_divergence

Evo-SETI Theory and Information Gap Among Civilizations

Abstract In a series of recent papers (Refs. [1–9]) this author gave the equations of his mathematical model of Evolution and SETI, simply called “Evo-SETI”. Key features of Evo-SETI are: 1. The Statistical Drake Equation is the extension of the classical Drake equation into Statistics. Probability distributions of the number of ET civilizations in the Galaxy (lognormals) were given, and so is the probable distribution of the distance of ETs from us. 2. Darwinian Evolution is re-defined as a Geometric Brownian Motion (GBM) in the number of living species on Earth over the last 3.5 billion years. Its mean value grew exponentially in time and Mass Extinctions of the past are accounted for as unpredictable low GBM values. 3. The exponential growth of the number of species during Evolution is the geometric locus of the peaks of a one-parameter family of lognormal distributions (b-lognormals, starting each at a different time b = birth) constrained between the time axis and the exponential mean value. This accounts for cladistics (i.e. Evolution Lineages). The above key features of Evo-SETI Theory were already discussed by this author in Refs. [1–9]. Now about this paper’s “novelties”. 4. The lifespan of a living being, let it be a cell, an animal, a human, a historic human society, or even an ET society, is mathematically described as a finite blognormal. This author then described mathematically the historical development of eight human historic civilizations, from Ancient Greece to the USA, by virtue of b-lognormals. 5. Finally, the b-lognormal’s entropy is the measure of a civilization’s advancement level. By measuring the entropy difference between Aztecs and Spaniards in 1519, this author was able to account mathematically for the 20-million-Aztecs defeat by a few thousand Spaniards, due to the latter’s technological (i.e. entropic) superiority. The same might unfortunately happen to Humans when they will face an ET superior civilization for the first time. Now the question is: whenever a new exoplanet is discovered, where does that exoplanet stand in its evolution towards life as we have it on Earth nowadays, or beyond? This is the central question of SETI. This author hopes that his Evo-SETI © Springer Nature Switzerland AG 2020 C. Maccone, Evo-SETI, https://doi.org/10.1007/978-3-030-51931-5_17

569

570

Evo-SETI Theory and Information Gap Among Civilizations

Theory will help addressing this question when SETI astronomers will succeed in finding the first “life signatures” or even ET Civilizations. © 2016 IAA. Published by Elsevier Ltd. All rights reserved. Keywords SETI · Darwinian evolution · Central limit theorem of statistics · Shannon entropy · Lognormal probability distributions

1 Introduction During the last nine years (2012–2021) this author devoted himself to creating a mathematical theory capable of mathematizing the History of past Civilizations on Earth and compare that with the first ET Civilization when SETI astronomers will succeed for the first time in finding one. Our use of mathematics is profoundly innovative, and may hardly be summarized in a short paper like this one. Thus, the reader is referred to the many papers published recently by the author in several journals. We are sorry about “cutting short” this way, but we just did not have the time to write a comprehensive paper. Just to give a glimpse of our Evo-SETI mathematical theory, Fig. 1 here below (that is Fig. 5 of Ref. [7]) shows the fundamental definition of LIFE of all living

Fig. 1 ‘Life’ in the Evo-SET1 Theory is a ‘finite b-lognormal’ made up by a lognormal pdf between birth b and senility (descending inflexion point) s, plus the straight tangent line at s and leading to death d

1 Introduction

571

Probability Density Values (nine b-logormals)

Nine Western Civilizations: 3100 BC to 2035 AD, plus two ENVELOPING EXPONENTIALS 0.02 0.019 0.018 0.017 0.016 0.015 0.014 0.013 0.012 0.011 0.01 0.009 0.008 0.007 0.006 0.005 0.004 0.003 0.002 0.001 0 − 3100 − 2900 − 2700 − 2500 − 2300 − 2100 − 1900 − 1700 − 1500 − 1300 − 1100 − 900 − 700 − 500 − 300 − 100

100

300

500

700

900

1100

1300 1500

1700 1900

2100

Time in years (negative = BC, positive = AD) EGYPT. Birth 3100 BC. Peak 1154 BC. Decline 689 BC. End 30 BC. GREECE. Birth 776 BC. Peak 438 BC. Decline 293 BC. End 30 BC. ROME. Birth 753 BC. Peak 117 AD. Decline 273 AD. End 476 AD. ITALY. Birth 1250. Peak 1497. Decline 1562. End 1660. PORTUGAL. Birth 1419. Peak 1716. Decline 1822. End 1999. SPAIN. Birth 1402. Peak 1798. Decline 1844. End 1898. FRANCE. Birth 1525. Peak 1812. Decline 1870. End 1962. BRITAIN. Birth 1588. Peak 1904. Decline 1947. End 1974. USA. Birth 1898. Peak 1972. Decline 2001. End 2035 SINGULARITY. EXPONENTIAL between Greek and American PEAKS. EXPONENTIAL between Italian and American PEAKS.

Fig. 2 The b-lognormals of nine Historic Western Civilizations computed thanks to the History Formulae with the three numeric inputs for b, p and d of each Civilization given by the corresponding line in Table 1. The corresponding s is derived from b, p and d by virtue of the second-order approximation provided by the solution of the quadratic equation in the birth-peak-death theorem

beings mathematically understood as a finite b-lognormal. Please carefully read the caption to this Fig. 1. Then, Fig. 2 of this paper shows the b-lognormals of nine Historic Western Civilizations computed thanks to our History Formulae with the three numeric inputs for b, p and d of each Civilization given by the corresponding line in Table 1. The corresponding s is derived from b, p and d by virtue of the second-order approximation provided by the solution of the quadratic equation in the birth-peak-death theorem, described in detail in Sect. 5 of this paper.

2 A Simple Proof of the b-Lognormal’s Pdf This paper is based on the notion of a b-lognormal, just as are Refs. [6–8]. To let this paper be self-contained in this regard, we now provide an easy proof of the b-lognormal equation as a probability density function (pdf). Just start from the well-known Gaussian or normal pdf e_ Gaussian_or_normal(x; μ, σ ) = √

⎧ ⎨ −∞ ≤ x ≤ +∞, with −∞ ≤ μ ≤ +∞, ⎩ 2πσ σ ≥ 0. (x−μ)2 2σ 2

(1)

3100 BC Lower and Upper Egypt unified. First Dynasty

776 BC First Olympic Games, from which Greeks compute years

753 BC Rome founded Italy seized by Romans by 270 BC, Carthage and Greece by 146 BC, Egypt by 30 BC. Christ 0

1250 Frederick II dies. Middle Ages end. Free Italian towns start Renaissance

1419 Madeira island Discovered. African coastline explored by 1498

Ancient Egypt

Ancient Greece

Ancient Rome

Renaissance Italy

Portuguese Empire

b = Birth time

1716 Black slave trade to Brazil at its peak. Millions of blacks enslaved or killed

1497 Renaissance art and architecture. Birth of Science. Copernican revolution (1543)

117 AD Rome at peak: Trajan in Mesopotamia. Christianity preached in Rome by Saints Peter, Paul against slavery by 69 AD

434 BC Pericles’ Age. Democracy peak. Arts and Science peak. Aristotle

1154 BC Luxor and Karnak temples edified by Ramses II by 1260 BC

p = Peak time

d = Death time

P = Peak ordinate

1822 Brazil independent, other colonies retained

1564 Council of Trent ends Catholic and Spanish rule

273 AD Aurelian builds new walls around Rome after Military Anarchy, 235–270 AD

323 BC Alexander the Great dies. Hellenism starts in Near East

5:749 × 10−3

2:193 × 10−3

2:488 × 10−3

(continued)

1999 3:431 × 10−3 Last colony, Macau, lost to Republic of China

1660 Cimento shut. Bruno burned 1600. Galileo died 1642

476 AD Western Roman Empire ends. Dark Ages start in West. Not in East

30 BC Cleopatra’s death: last Hellenistic queen

689 BC Assyrians invade 30 BC Cleopatra’s death: 8:313 × 10−4 Egypt in 671 BC, leave 669 last Hellenistic queen BC

s = Decline = senility time

Table 1 Birth, peak, decline and death times of nine Historic Western Civilizations (3100 BC–2035 AD), plus the relevant peak heights. They are shown in Fig. 2 as nine b-lognormal probability density functions (pdfs)

572 Evo-SETI Theory and Information Gap Among Civilizations

1524 Verrazano first in New York bay. Cartier in Canada, 1534

1588 Spanish Armada Defeated British Empire’s expansion starts

1898 Philippines, Cuba, Puerto Rico seized from Spain

French Empire

British Empire

USA Empire

b = Birth time

1402 Canary islands are conquered by 1496. In 1492 Columbus discovers America

Spanish Empire

Table 1 (continued)

1972 Moon Landings, 1969–72: America leads the world

1904 British Empire peak. Top British Science: Faraday, Maxwell, Darwin, Rutherford

1812 Napoleon I dominates continental Europe and reaches Moscow

1798 Largest extent of Spanish colonies in America: California settled since 1769

p = Peak time 1898 Last colonies lost to the USA

d = Death time

2001 9/11 terrorist attacks: decline. Obama 2009

2035 Singularity? Will the USA yield to China?

1947 1974 After World Wars One and Britain joins the EEC Two, India gets independent and loses most of her colonies

1870 Napoleon III defeated. 1962 Third Republic starts Algeria lost as most World Wars colonies. Fifth Republic starts in 1958

1805 Spanish fleet lost at Trafalgar

s = Decline = senility time

0:013

8:447 × 10−3

4:279 × 10−3

5:938 × 10−3

P = Peak ordinate

2 A Simple Proof of the B-Lognormal’s Pdf 573

574

Evo-SETI Theory and Information Gap Among Civilizations

This pdf has two parameters: 1. μ turns out to be the mean value of the Gaussian and the abscissa of its peak. Since the independent variable x may take up any value between −∞and +∞, i.e. it is a real variable, so μ must be real too. 2. σ turns out to be the standard deviation of the Gaussian and so it must be a positive variable. 3. Since the Gaussian is a pdf, it must fulfill the normalization condition ∞ −∞

(x−μ)2

e− 2 σ 2 dx = 1 √ 2π σ

(2)

and this is the equation we need in order to “discover” the b-lognormal. Just perform in the integral (2) the substitution x = ln t (where ln is the natural log). Then (2) is turned into the new integral ∞ 0

(ln t−μ)2

e_ 2σ 2 dt = 1. √ 2π σ t

(3)

But this (3) may be regarded as the normalization condition of another random variable, ranging “just” between zero and +∞, and this new random variable we call “lognormal” since it “looks like” a normal one except that x is now replaced by ln t and t now also appears at the denominator of the fraction. In other words, the lognormal pdf is ⎧ ⎨

_

(ln(t)−μ)2 2

lognormal(t; μ, σ ) = e √2π2 σσ ·t ⎩ holding for 0 ≤ t ≤ ∞, −∞ ≤ μ ≤ +∞, σ ≥ 0.

(4)

Just one more step is required to jump from the “ordinary lognormal” (4) (i.e. the lognormal starting at t = 0) to the b-lognormal, that is the lognormal starting at any instant b (“b” stands for “birth”). Since this simply is a shifting along the time axis from 0 to the new time origin b, in mathematical terms it means that we have to replace t by (t − b) everywhere in the pdf (4). Thus the b-lognormal pdf must have the equation ⎧ ⎨ ⎩

b_lognormal(t; μ, σ, b) =

_

(ln(t−b)−μ)2

e√ 2 σ 2 2π σ ·(t−b)

holding for t ≥ b, and up to t = ∞.

(5)

The b-lognormal (5) is sometimes called “three-parameter lognormal” by statisticians. But we prefer to call it b-lognormal in order to stress its biological meaning as the key probability density representing the LIFETIME of all living beings.

3 History Formulae

575

3 History Formulae Having so defined “life” as a finite b-lognormal, this author was able to show that, given one’s birth b, death d and (somewhere in between) one’s senility s, then the two parameters μ (a real number) and σ (a positive number) of the b-lognormal (5) are given by the two equations 

d−s √ σ = √d−b s−b μ = ln(s − b) +

(d−s)(b+d−2s) . (d−b)(s−b)

(6)

These were called “History Formulae” by this author for their use in Mathematical History. The mathematical proof of (6) is found in Ref. [6], pages 227–231 and follows directly from the definition of s (as descending inflexion point) and d (as interception between the descending tangent straight line at s and the time axis). In previous versions of his Evo-SETI Theory, the author gave an apparently different version of the History Formulae (6) reading 

σ =

√ d−s √ d−b s−b

μ = ln(s − b) +

2s 2 −(3d+b)s+d 2 +bd . (d−b)s−bd+b2

(7)

This simply was because he had not yet factorized the fraction of the second equation (with apologies).

4 Death Formula One more interesting result discovered by this author, and firstly published by him in 2012 (Ref. [6] Chap. 6, Eq. (6.30), page 163) is the following “Death Formula” (its proof is obtained by inserting the History Formulae (6) into the equation for the b-lognormal peak abscissa, p = b + eμ−σ :   p−b s + b · ln s−b   . d= p−b +1 ln s−b 2

(8)

This formula allows one to compute the death time d if the birth time b, the peak time p and the senility time s are known. The difficulty is that, while b and p are usually well known, s is not so, thus jeopardizing the practical usefulness of the Death Formula (8).

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Evo-SETI Theory and Information Gap Among Civilizations

5 Birth-Peak-Death (BPD) Theorem This difficulty in estimating s for any b-lognormal led the author to discover the Birth-Peak-Death Theorem described in this section, that he only obtained on April 4, 2015. Ask the question: can a given b-lognormal be entirely characterized by the knowledge of its birth, peak and death only? Yes is the answer, but, unfortunately, no exact formula exists yielding s in terms of (b; p; d): only numerically approximated formulae exist, and they are given in this section. Proof Start from the exact Death Formula (8) and expand it into a Taylor series with respect to s around p say to order 2. The result given by the Maxima symbolic manipulator is d = p + 2(s − p) −

3(s − p)2 + ··· 2(b − p)

(9)

This (9) is quadratic equation in s that, once solved for s, yields the second-order approximation for s in terms of (b; p; d) √ 2 − p 2 + (3d − b) p − 3bd + 2b2 + p + 2b s= 3

(10)

In the practice, (10) is a “reasonable” numeric approximation yielding s as a function of (b; p; d), and is certainly much better that the corresponding first-order approximation given by the linear equation d = p + 2(s − p) + · · ·

(11)

whose solution simply is s=

p+d 2

(12)

i.e. s (to first approximation) simply is the middle point between p and d, as geometrically obvious. However, if one really wants a better approximation than the quadratic one (10), it is possible to expand the Death Formula (8) into a Taylor series with respect to s around p to third order, finding d = p + 2(s − p) −

5(s − p)3 3(s − p)2 + 2 + ··· 2(b − p) 6b − 12 pb + 6 p 2

(13)

This (13) is a cubic (i.e. third-degree polynomial) in s that may be solved for s by virtue of the well-known Cardan [Girolamo Cardano (1501–1576)] formulae that

5 Birth-Peak-Death (BPD) Theorem

577

we will not repeat here since they are exact but too lengthy to be reproduced in this paper. As a matter of fact, it might even be possible to expand the Death Formula (8) to fourth order in s around p, that would lead to the fourth-degree algebraic equation (a quartic) in s 5(s − p)3 3(s − p)2 + 2 2(b − p) 6b − 12 pb + 6 p 2 4 (s − p) − 3 + ··· 2b − 6 pb2 + 6 p 2 b − 2 p 3

d = p + 2(s − p) −

(14)

and then solve (14) for s by virtue of the exact four formulae of Lodovico Ferrari (1522–1565) (he was Cardan’s pupil) that are huge and occupy a whole page each one. However, this game may not go on forever: the fifth-degree algebraic equation is not solvable by virtue of radicals and so we must stop with degree 4. Also, the reader should notice that the inputs to Table 1 of this paper were (b; p; d) and not (b; s; d), as the author had always done previously, for instance in deriving the whole of Chap. 7 of Ref. [6] back in 2012. This improvement is remarkable since it allowed a fine-tuning of Table 1 with respect to all similar previous material. In other words still, it is easier to assign birth, peak and death rather than birth, senility and death. That’s why the Theorem described in this section was called Birth-Peak-Death Theorem.

6 Information Entropy (SHANNON ENTROPY) as the Measure of a Civilization’s Advancement Please notice that: 1. The data in Table 1 and the resulting b-lognormals in Fig. 2 are experimental results, meaning that we just took what described in History textbooks (with a lot of words) and translated that into the simple b-lognormals shown in Fig. 2. In other words, a new branch of knowledge was forged: we love to call it “Mathematical History”. More about this in future papers. 2. The envelope of all the above b-lognormals “looks like” a simple exponential curve. In Fig. 2 two such exponential envelopes were drawn: the one going from the peak of Ancient Greece (the Pericles age in Athens, cradle of Democracy) to the peak of the British Empire (Victorian age, the age of Darwin and Maxwell) and to the peak of the USA Empire (Moon landings in 1969–72). This notion of b-lognormal envelope will later be precisely quantified in our “Peak-Locus Theorem”. In addition, we also drew the exponential envelope passing through the peaks of Renaissance Italy and today’s USA. 3. It’s now high time to introduce a “measure of evolution” namely a function of the three parameters μ, σ and b accounting for the fact that “the experimental

578

Evo-SETI Theory and Information Gap Among Civilizations

Fig. 2 clearly shows that, the more the time elapses, the more highly peaked, and narrower and narrower, the b-lognormals are”. In Ref. [6], pages 685–686, this author showed that the requested measure of evolution is the (Shannon) entropy, namely the information entropy of each infinite b-lognormal that fortunately has the simple equation Hinfinite_b−lognormal (μ, σ ) = ln

√

 1 2π σ + μ + . 2

(15)

If measured in bits, as customary in Shannon’s Information Theory, (15) becomes

Hinfinite_b−lognormal_in_bits (μ, σ ) =

ln

√  2π σ + μ + ln 2

1 2

.

(16)

This is the b-lognormal entropy definition that was used in Refs. [1–9] and we are going to use in this paper also. In reality, Shannon’s entropy is a measure of the disorganization of an assigned pdf f X (x), rather than a measure of its organization. To change it into a measure of organization, we should just drop the minus sign appearing in front of the Shannon definition of entropy for any assigned pdf f X (x): ∞ H =−

f X (x) · ln f X (x)d x.

(17)

−∞

The final goal of all these mathematical studies is of course to “prepare” the future of Humankind in SETI, when we will have to face other Alien Civilizations whose past may be the future for us (Table 2).

7 Information Gaps, Namely Entropy Differences Among the Nine Historic Western Civlizations The key formula (for instance applied here to the Egypt vs. Greece case) H = H _of_Egypt − H _of_Greece = 1.467 bits/individual

(18)

σ = 0:181

σ = 0:277

σ = 0:366

μ = 6:018

μ = 6:801

μ = 5:583

776 BC = −776 First Olympic Games, from which Greeks compute years

753 BC = −753 Rome founded Italy seized by Romans by 270 BC, Carthage and Greece by 146 BC, Egypt by 30 BC. Christ 0

1250 Frederick II dies Middle Ages end. Free Italian towns start Renaissance

1419 μ = 5:828 Madeira island Discovered. African coastline explored by 1498

Ancient Greece

Ancient Rome

Renaissance Italy

Portuguese Empire

σ = 0:440

σ = 0:242

μ = 7:632

3100 BC = −3100 Lower and Upper Egypt unified. First Dynasty

Ancient Egypt

σ Computed by the History Formulae

μ Computed by the History Formulae

b = Birth time

H_Portugal = 9:004 bits/individual (continued)

H_Italy = 8:217 bits/individual

H_Rome = 9:390 bits/individual

H_Greece = 9:548 bits/individual

H_Egypt = 11:011 bits/individual

H = Shannon ENTROPY of the b-lognormal pdf

Table 2 Birth = b, μ and σ of the nine Historic Western Civilizations (3100 BC–2035 AD) shown in Fig. 2 and Table 1, plus the Shannon ENTROPY of the relevant b-lognormal probability density function as the Civilization Level MEASURE

7 Information Gaps, Namely Entropy Differences Among the Nine … 579

σ = 0:051

σ = 0:073

σ = 0:396

μ = 5:981

μ = 5:831

μ = 4:462

1524 Verrazano first in New York bay. Cartier in Canada, 1534

1588 Spanish Armada Defeated. British Empire’s expansion starts

1898 Philippines, Cuba, Puerto Rico seized from Spain

French Empire

British Empire

USA Empire

σ Computed by the History Formulae σ = 0:116

μ Computed by the History Formulae

1402 μ = 5:994 Canary islands are conquered by 1496. In 1492 Columbus discovers America

b = Birth time

Spanish Empire

Table 2 (continued)

H_USA = 7:148 bits/individual

H_Britain = 6:677 bits/individual

H_France = 6:388 bits/individual

H_Spain = 7:586 bits/individual

H = Shannon ENTROPY of the b-lognormal pdf

580 Evo-SETI Theory and Information Gap Among Civilizations

7 Information Gaps, Namely Entropy Differences Among the Nine …

581

shows us how much Greece was more advanced than Egypt, since the Greek blognormal in Fig. 2 is clearly “more peaked” than the Egyptian b-lognormal. In other words, since the AREA under each b-lognormal always equals 1 (normalization condition), as long as the time elapses the b-lognormals become “more and more peaked” and they also last “less and less” in time. Not by chance the Egyptian Civilization lasted millennia, while all modern Civilizations lasted some centuries at best. Table 3 shows all possible ENTROPY DIFFERENCES = INFORMATIONS GAPS among our nine Civilizations and is presented here for the first time. This table is what mathematicians call an “antisymmetric, or antimetric, or skew-symmetric matrix”.

8 Conclusion This author thinks that the best conclusion to this paper is the popular description of his Evo-SETI Theory given in 2015 by the American author Michael Shermer in his recent book “The Moral Arc”, Ref. [10] and https://www.amazon.it/The-MoralArc-Science-Humanity/dp/0805096914.

−1.958 −3.157 −2.868 −2.397

−3.425 −4.335 −3.864

Britain

USA

−0.541

−2.008

Portugal −4.624

−1.328

−2.795

Italy

France

−0.154

−1.622

Rome

Spain

1.467 0

0 −1.467

Greece

Greece

Egypt

Egypt

INFORMATION GAP in bits/individual

−2.242

−2.713

−3.002

−1.804

−0.386

−1.173

0

0.154

1.622

Rome

−1.069

−1.540

−1.829

−0.630

0.7872

0

1.173

1.328

2.795

Italy

−1.856

−2.327

−2.616

−1.418

0

−0.7872

0.386

0.541

2.008

Portugal

−0.438

−0.909

−1.198

0

1.418

0.630

1.804

1.958

3.425

Spain

0.759

0.289

0

1.198

2.616

1.829

3.002

3.157

4.624

France

0.471

0

−0.289

0.909

2.327

1.540

2.713

2.868

4.335

Britain

0

−0.471

−0.759

0.438

1.856

1.069

2.242

2.397

3.864

USA

Table 3 INFORMATION GAPS = (Shannon) ENTROPY DIFFERENCES in bits/individual among the nine Historic Western Civilizations (3100 BC–2035 AD) shown in Fig. 2 and Table 1

582 Evo-SETI Theory and Information Gap Among Civilizations

References

583

References 1. C. Maccone, The Statistical Drake Equation, Paper IAC-08-A4.1.4 presented on October 1st, 2008, at the 59th International Astronautical Congress (IAC) held in Glasgow, Scotland, UK, 29 Sept––29–3 Oct 2008 2. C. Maccone, The statistical Drake equation. Acta Astronaut. 67, 1366–1383 (2010) 3. C. Maccone, The statistical Fermi paradox. J. Br. Interplanet. Soc. 63, 222–239 (2010) 4. C. Maccone, SETI and SEH (statistical equation for habitables). Acta Astronaut. 68, 63–75 (2011) 5. C. Maccone, A mathematical model for evolution and SETI. Orig. Life Evol. Biosph. (OLEB) 41, 609–619 (2011) 6. C. Maccone, Mathematical SETI, A 724-Pages Book Published by Praxis-Springer in the Fall of 2012, edition 2012, 2012, ISBN, ISBN-10: 3642274366|ISBN-13: 978-3642274367 7. C. Maccone, SETI, evolution and human history merged into a mathematical model. Int. J. Astrobiol. 12(3), 218–245 (2013) 8. C. Maccone, Evolution and mass extinctions as lognormal stochastic processes. Int. J. Astrobiol. 13(4), 290–309 (2014) 9. C. Maccone, Evo-SETI entropy identifies with molecular clock. Acta Astronaut. 115, 286–290 (2015) 10. M. Shermer, The Moral Arc, Henry Holt and Company, New York, 2015 (See in particular page 436 for the reference to this author’s Evo-SETI Theory)

Kurzweil’s Singularity as a Part of Evo-SETI Theory

Abstract Ray Kurzweil’s famous 2006 book “The Singularity Is Near” predicted that the Singularity (i.e. computers taking over humans) would occur around the year 2045. In this chapter we prove that Kurzweil’s prediction is in agreement with the “Evo-SETI (Evolution and SETI)” mathematical model that this author has developed over the last ten years in a series of mathematical papers published in both Acta Astronautica and the International Journal of Astrobiology. The key ideas of Evo-SETI are: 1. Evolution of life on Earth over the last 3.5 billion years is a stochastic process in the number of living Species called Geometric Brownian Motion (GBM). It increases exponentially in time and is in agreement with the Statistical Drake Equation of SETI (see ref. [5]). 2. The level of advancement of each living Species is the (Shannon) ENTROPY of the b-lognormal probability density (i.e. a lognormal starting at the positive time b (birth)) corresponding to that Species. (Peak-Locus Theorem of Evo-SETI Theory). 3. Humanity is now very close to the point of minimum radius of curvature of the GBM exponential, called “GBM knee”. We claim that this knee is precisely Kurzweil’s Singularity, in that before the Singularity the exponential growth was very slow (these are animal and human Species made of meat and reproducing sexually over millions of years), whereas, after the Singularity, the exponential growth will be extremely rapid (computers reproducing technologically faster and faster in time). But how is this paper structured in detail? Well, first of all (Part 1) we describe what the GBM is, and why it reflects the stochastic exponential increase that occurred in Darwinian Evolution for over 3.5

© Springer Nature Switzerland AG 2020 C. Maccone, Evo-SETI, https://doi.org/10.1007/978-3-030-51931-5_18

585

586

Kurzweil’s Singularity as a Part of Evo-SETI Theory

billion years. Please notice that the denomination “Geometric Brownian Motion” (taken from Financial Mathematics) is incorrect since the GBM is NOT a Brownian motion as understood by physicists (i.e. a stochastic process whose probability density function (pdf) is a Gaussian). On the contrary, the GBM is a lognormal process, i.e. a process whose pdf is a lognormal pdf. Next (Part 2) we compute the time when the GBM knee occurs (i.e. the time of minimum radius of curvature) and find what we call the knee equation, i.e. the relationship between t_knee, ts (the time of the origin of life on Earth) and B (the rate of growth of the GBM exponential). This equation holds good for any time assumed to be the Singularity time, either in the past, or now, or in the future. Then (Part 3) Ray Kurzweil’s claim that the Singularity is near becomes part of our Evo-SETI Theory in that t_knee is set to zero (i.e. approximately nowadays, when compared to the 3.5 billions of years of past Darwinian Evolution of life on Earth). This leads to a very easy form of the GBM exponential as well as to the discovery of a pair of important new equations: 1. The inverse proportionality between the average number of Species living NOW on Earth and B, the pace of evolution. In other words, it would be possible to find B were the biologists able to tell us “fairly precisely” how many Species live on Earth nowadays. Unfortunately, this is not the case since, when it comes to insects and so on, the number of Species is so huge that it is not even known if it ranges in the millions or even in the billions. 2. More promising appears to be another new equation, that we discovered, relating the time of the origin of life on Earth, ts (that is known fairly precisely to range between 3.5 and 3.8 billion years ago) and the average number of living Species NOW. Finally, the mathematical machinery typical of the Evo-SETI Theory is called into action: 1. The Peak-Locus Theorem stating that the GBM exponential is where ALL PEAKS of the b-lognormals running left-to-right are located, so that the blognormals become higher and higher and narrower and narrower (with area = 1 as the normalization). 2. The Shannon ENTROPY as EVOLUTION MEASURE of the b-lognormals, more correctly with the sign reversed and starting at the time of the origin of life on Earth, that is rather called EvoEntropy. 3. After this point, one more paper should be written to describe… how the blognormal’s “width” would correctly describe the “average duration in time” of each Species (before the Singularity) and of each COMPUTER Species (after the Singularity)… 4. …but this is “too much to be done now”, and so we leave it to a new, forthcoming paper. Keywords Ray Kurzweil · Singularity · Darwinian evolution · Molecular clock · Entropy · SETI

1 Geometric Brownian Motion (GBM) Is Key to Exponential …

587

1 Geometric Brownian Motion (GBM) Is Key to Exponential Stochastic Evolution 1.1 Darwinian Evolution as the Exponential Increase of the Number of Living Species Consider Darwinian Evolution. To assume that the number of Species increased exponentially over the 3.5 billion years of evolutionary time span is certainly a gross oversimplification of the real situation. However, we will assume this exponential increase of the number of living Species in time just temporarily in this section while in Sect. 1.2 we will do much better by virtue of Geometric Brownian Motions (GBMs). In other words, we assume that 3.5 billion years ago there was on Earth only one living Species, whereas now there may be (say) 50 million living Species or more. Note that the actual number of Species currently living on Earth does not really matter as a number for us: we just want to stress the exponential character of the growth of Species. Thus, we shall assume that the number of living Species on Earth increases in time as E(t) (standing for “exponential in time”): E(t) = Ae Bt

(1)

where A and B are two positive constants that we will determine numerically. This assumption is obviously in agreement with the classical Malthusian theory of population growth. But it also is in line with the more recent “Big History” viewpoint about the whole evolution of the Universe, from the Big Bang up to now, requesting that progress in evolution occurs faster and faster, so that only an exponential growth is compatible with the requirements that (1) approaches infinity for t → ∞, and all its time derivatives are exponentials too, apart from constant multiplicative factors. Let us now adopt the convention that the current epoch corresponds to the origin of the time axis, i.e. to the instant t = 0. This means that all the past epochs of Darwinian Evolution correspond to negative times, whereas the future ahead of us (including finding ETs) corresponds to positive times. Setting t = 0 in (1), we immediately find E(0) = A

(2)

proving that the constant A equals the number of living Species on Earth right now. We shall assume A = 50 million species = 5.107 species.

(3)

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Kurzweil’s Singularity as a Part of Evo-SETI Theory

To also determine the constant B numerically, consider the two values of the exponential (1) at two different instants t 1 and t 2 , with t 1 < t 2 , that is 

E(t1 ) = Ae Bt1 E(t2 ) = Ae Bt2 .

(4)

Dividing the lower equation by the upper one, A disappears and we are left with an equation in B only: E(t2 ) = e B(t2 −t1 ) . E(t1 )

(5)

ln(E(t2 )) − ln(E(t1 )) . t2 − t1

(6)

Solving this for B yields B=

We may now impose the initial condition stating that 3.5 billion year ago there was just one Species on Earth, the first one (whether this was RNA is unimportant in the present simple mathematical formulation): 

t1 = −3.5 × 109 years E(t1 ) = 1 when ln(E(t1 )) = ln(1) = 0.

(7)

The final condition is of course that today (t 2 = 0) the number of Species equals A given by (3). Upon replacing both (7) and (3) into (6), the latter becomes:   ln 5.107 1.605 × 10−16 ln(E(t2 )) = . =− B=− t1 −3.5 × 109 years sec

(8)

Having thus determined the numerical values of both A and B, the exponential in (1) is thus fully specified. This curve is plot in Fig. 1 just over the last billion years, rather than over the full range between −3.5 billion years and now.

1 Geometric Brownian Motion (GBM) Is Key to Exponential …

589

Fig. 1 Exponential curve representing the growing number of Species on Earth up to now, without taking the mass extinctions into any consideration at all

1.2 Darwinian Evolution Was Just a Particular Realization of Geometric Brownian Motion in the Number of Living Species Consider again the exponential curve described in the previous section. The most frequent question that non-mathematically minded persons ask this author is: “then you do not take the mass extinctions into account”. Our answer to this objection is that our exponential curve is just THE MEAN VALUE of a certain stochastic process that may run above and below that exponential in an unpredictable way. Such a stochastic process is called Geometric Brownian Motion (abbreviated GBM) and is described, for instance, at the web site: http://en.wikipedia.org/wiki/Geomet ric_Brownian_motion, from which Fig. 2 is taken.

Fig. 2 Geometric Brownian Motion (GBM)

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Kurzweil’s Singularity as a Part of Evo-SETI Theory

It shows two particular realizations of the stochastic process. Their mean values are two exponentials with different values of A and B, that is of μ and σ (i.e. of μ = 1, σ = 0.2 and μ = 0.5, σ = 0.5) in the relevant lognormal probability density functions that we will not write here since irrelevant to the goals of this chapter. In other words, mass extinctions that occurred in the past are indeed taken into account as unpredictable fluctuations in the number of living Species that occurred in the PARTICULAR REALIZATION of the GBM between −3.5 billion years and now. So, extinctions are “unpredictable vertical downfalls” in that GBM plot that may indeed happen from time to time, but we don’t know when. Also notice that: 1. Any particular realization of the GBM occurred over the last 3.5 billion years is very much unknown to us in its numeric details, but… 2. We won’t care either, inasmuch as the theory of stochastic processes only cares about such statistical quantities like the mean value and the standard deviation curves that are deterministic curves in time with well-known equations.

2 Knee of Any Exponential 2.1 Every Exponential in Time Has Just a Single Knee: The Instant at Which Its Curvature Is Highest Consider the easiest possible exponential curve as a function of the time t having the equation y(t) = et .

(9)

Now ask the question: what is the expression of the curvature κ for such a simple exponential in time? The answer may be found in textbooks about the Calculus, as well as at the Wikipedia site. https://en.wikipedia.org/wiki/Curvature and reads

κ(t) = 

 2  d y  dt 2  1+

If we insert (9) into (10), we get



dy dt

2 23

.

(10)

2 Knee of Any Exponential

591

ksimple exponential (t) =

 t e 

et =

3 .  1 + e2t 2 1 + (et )2 3 2

(11)

Now, the radius of curvature R(t) is defined as the reciprocal of the curvature (10), that is   2 23 1 + dy dt  2  . R(t) = d y   dt 2 

(12)

Then, in the particular case of the simple exponential (9), its radius of curvature provided by (12) reads 3

1 + e2t 2 Rsimple exponential (t) = . et

(13)

where the absolute value at the denominator of (12) disappeared since the exponential is a positive curve. Let us now ask the fundamental question: for the simple exponential (9), what is the minimum radius of curvature? To answer this question we must compute the derivative of (13) with respect to t, set it equal to zero, and then solve the resulting equation with respect to t, thus finding the time t of the minimum curvature. Let us do so: after a few steps, one gets d Rsimple exponential (t) = 0= dt

3 2

1 3

1 + e2t 2 2e2t − 1 + e2t 2 et ... e2t

(14)

leading to

1 + e2 t

 21

 1 + e2 t = 0 that is 2e2t − 1 = 0 · 3et − et

(15)

and finally tsimple exponential knee

1 ln(2) = −0.346. = ln √ =− 2 2

(16)

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Kurzweil’s Singularity as a Part of Evo-SETI Theory

We call (16) the abscissa of the knee of the simple exponential (9) because for t < t simple exponential knee the growth of the exponential (9) is very slow, while for t > t simple exponential knee the growth of the exponential (9) is very fast. Thus, the simple exponential’s knee time is the “dividing time” in between very slow and very fast growth. Better still, the knee time is the tangent point between the simple exponential (9) and its osculating circle https://en.wikipedia.org/wiki/Osculating_circle having the minimum radius of curvature, or, if you so prefer, the maximum curvature. And the calculations we have made so far show that every exponential only has a single knee, and not many. Actually we may now complete the calculation of the only exponential knee point by inserting (16) into (9) and so obtaining   ysimple exponential tsimple exponential knee = etsimple exponential knee ln(2) 1 = e− 2 = √ = 0.707. 2

(17)

As for the numeric value of the radius of curvature of the simple exponential at its knee, upon inserting (16) into (13), after a few reductions one finds    3 2 2 ln √12 3 

1+e 1 + e2t 2   = Rsimple exponential (t) = et ln √12 e 3 [3] 2 = 2.60. = 2

(18)

This value of 2.60 for the radius of curvature is evidently correct if one just has a look to Fig. 3.

Fig. 3 The simple exponential y = et (left to right: the raising blue curve) and the osculating red circle having the minimum radius of curvature, i.e. the maximum ln(2) curvature, just for the value tsimple exponential knee = − ln(2) = −0.346. (For interpretation 2 of the references to color in this 2 figure legend, the reader is referred to the web version of this article.)

2 Knee of Any Exponential

593

We will skip here the calculations of the coordinates of the centre of the osculating circle.

2.2 (GBM) Exponential as Mean Value of the Increasing Number of Species Since the Origin of Life As we pointed out already, the key idea of Evo-SETI Theory is that the increasing number of Species living on Earth since the time of the Origin of Life (about 3.5 billion years ago) is a lognormal stochastic process L(t) with an ARBITRARY (i.e. to be defined experimentally) MEAN VALUE denoted mL (t) (Please see Refs [8, 9] for many more mathematical details). In the present paper, however, we shall only assume the much easier model having the EXPONENTIAL (rather than the ARBITRARY ONE) MEAN VALUE given by

m GBM (t) = N s · e B(t−ts)

⎧ ⎨ t ≥ ts with Ns > 0 ⎩ B > 0.

(19)

This is the so-called Geometric Brownian Motion (GBM) subcase of the general lognormal process L(t), i.e. of the general theory developed in Refs. [8, 9]. We assume the time t to start at ts (“time of start”) and we assume the mean value at this starting time ts to be equal to the known positive number Ns. In fact (19) yields m GBM (ts) = N s.

(20)

If, as in all papers previously published by this author, ts means the time of the Origin of Life on Earth, then (20) has Ns = 1, meaning that there was only ONE Species on Earth when life started, and this only Species is today assumed by Evolution specialists to be RNA. The first question posed by the GBM exponential mean value (19) is: what is the practical meaning of the positive constant B? The answer is easily found by considering (19) nowadays, namely by inserting t = 0 into (19). One then gets m GBM (0) = Average Number of Species NOW = N s · e−Bts .

(21)

Solving (21) for B yields the meaning of B:  B=

ln

Average Number of Species NOW Ns

−ts

 .

(22)

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Kurzweil’s Singularity as a Part of Evo-SETI Theory

In all his papers published prior the 2017, this author assumed the following two CONVENTIONAL values for ts and B  ts = −3.5 × 109 years = (23) Number of Species NOW = 50 millions. In terms of B, inserting (23) into (22) one gets  =

ts = −3.5 × 109 years −9 B = 5.0650095895406915×10 . years

(24)

In the sequel of this paper we shall discuss the validity of the two numbers appearing in (23) and (24).

2.3 Deriving the Knee Time for GBMs The next question we face is: at what time does the KNEE occur for the GBM exponential (19)? To answer this question we must insert (19) instead of y(t) into the expression (12) for the radius of curvature. The result is   2  23 1 + dtd N s · e B(t−ts)   . RGBM (t) =  d2   dt 2 N s · e B(t−ts) 

(25)

Noticing that the derivative of any increasing exponential certainly is a positive quantity, we may get rid of the absolute value at the denominator of (25), getting

RGBM (t) =

  2  23 1 + dtd N s · e B(t−ts) d2 dt 2

N s · e B(t−ts)

.

(26)

Since the knee is the value of t for which (26) is minimum, we must then differentiate (26) with respect to t and set the relevant derivative equal to zero. Upon differentiating the fraction in (26) just with respect to, Ns·eB(t − ts) initially, and then multiplying this times the derivative of Ns·eB(t − ts) with respect to t, one gets

2 Knee of Any Exponential

595

  d N s · e B(t−ts) d RGBM (t) d RGBM (t) . 0= =  dt dt d N s · e B(t−ts) d RGBM (t)  N s Be B(t−ts) =  d N s · e B(t−ts) ⎧   2  21 d 2 2 ⎪ ⎨ 23 1 + dtd N s · e B(t−ts) 2 dt N s · e B(t−ts) dtd 2 N s · e B(t−ts) . dtd 2 N s · e B(t−ts) =  2 2 ⎪ d ⎩ B(t−ts) N s · e dt 2   d B(t−ts) 2  23 d 3 B(t−ts) ⎫ ⎪ ⎬ 1 + dt e · dt 3 e N s.Be B(t−ts) . − (27)  2 2 ⎪ d ⎭ B(t−ts) e 2 dt

Since (27) must equal zero, only the quantity within braces must equal zero, yielding 

2  21 3 d d 0= N s · e B(t−ts) 2 N s · e B(t−ts) 1+ 2 dt dt 3    2 2 d 3 1 + dtd N s · e B(t−ts) · dt 3 N s · e B(t−ts) − .  2 2 d B(t−ts) N s · e dt 2

(28)

Upon performing all derivatives shown in (28) one gets

1 0 = 3 1 + N s 2 B 2 e2B(t−ts) 2 N s B 3

1 + N s B 2 e2B(t−ts) 2 .(N s)3 B 3 e B(t−ts) B(t−ts) e − . (N s)4 B 4 e2B(t−ts)

(29)

1 Again, since (29) must equal zero, the whole factor 1 + N s 2 B 2 e2B(t−ts) 2 · N s · e B(t−ts) may be taken out, leaving just 0 = 3N s B −

1 + N s 2 B 2 e2B(t−ts) . N s Be2B(t−ts)

(30)

596

Kurzweil’s Singularity as a Part of Evo-SETI Theory

The last Eq. (30) must be solved for t, thus yielding the knee abscissa, t knee, of the exponential (19). Rearranging (30) one gets 2N s 2 B 2 e2B(t−ts) = 1

(31)

1 . 2N s 2 B 2

(32)

yielding e2B(t−ts) =

Now (32) is a quadratic equation in eB(t −ts) yielding the two roots

1 B(t − ts) = ln ± √ 2N s B

(33)

that is tGBM Knee = ts +

  ln ± √2N1 s B B

.

(34)

Clearly, the minus sign inside the log in (34) must be discarded (since any log argument must always be positive) and so (34) becomes

tGBM Knee = ts −

ln

√

 2N s B B

(35)

This is the fundamental GBM knee equation, at the root of all further discussions in the sequel of this paper. Please notice that there are four parameters in this Eq. (35): (ts, Ns, B, t GBM knee ). The sequel of this paper is a discussion of the “four-parameter” Eq. (35) in relationship to what really has happened to the number of living Species on Earth in the past, and perhaps in the future….

2.4 Knee-Centered Form of the GBM Exponential It is interesting to re-cast the GBM exponential (19) in a new form where the time t is centered around the GBM knee time (35), rather than around the time-of-start ts. To do so, just solve (35) for ts

2 Knee of Any Exponential

597

ts = tGBM Knee +

ln

√

 2N s B B

(36)

and insert (36) into (19). The result is m GBM (t) = N se B(t−ts) = N se Bt · e−B·ts

√ ln( 2N s B ) −B tGBM Knee + e B(t−tGBM Knee ) B Bt = N se · e = . √ 2B

(37)

That is, the knee-centered GBM exponential reads m GBM (t) =

e B(t−tGBM Knee ) . √ 2B

(38)

This form (38) of the GBM exponential shows very neatly that: 1. For all times before the knee time t GBM knee the exponential growth is very slow, i.e. like the negative exponential e−t for t → ∞. 2. For all times after the knee time t GBMknee the exponential growth is very fast, i.e. like the positive exponential et for t → ∞. 3. The constant √12B is just a scale factor that does not change the essence of remarks (1) and (2) at all.

2.5 Finding When the GBM Knee Will Occur According to the Author’s Conventional Values for ts and B As pointed out in (23), prior to 2017 this author always assumed that the time of the origin of life on Earth was −3.5 billion years and that the current number of Species living on Earth was 50 millions. Was this actually the case, when will the GBM knee occur? In the past? Or nowadays? Or in the future? To find out, let us insert the assumed numeric values (23) into the GBM knee Eq. (35) with the B given by the lower Eq. (24). The surprising result is tGBM Knee for this author’ s assumed inputs = 2.0272472028487825 · 108 years ≈ 200 million years into the future!

(39)

Clearly, the author’s assumed values (23) MUST BE WRONG if the Singularity is near, as Kurzweil suggested. So, how may we get out of this paradox?

598

Kurzweil’s Singularity as a Part of Evo-SETI Theory

3 Kurzweil’s “the Singularity Is Near” (Is Nowadays) 3.1 Ray Kurzweil’s 2006 Book “the Singularity Is Near” In 2006 the seminal book “The Singularity Is Near” (Ref. [4]) was published by the American inventor and futurist Ray Kurzweil about artificial intelligence and the future of humanity (see, for instance the site. https://en.wikipedia.org/wiki/The_Singularity_Is_Near). Figure 4 shows the book cover and a picture of Raymond Kurzweil (born on February 12, 1948). Summarizing (too much!) this book’s content, the SINGULARITY is the time when computers will take over humans by virtue of their own superior intellectual capabilities (artificial intelligence = AI) and so, loosely said, poor humans “made of meat” will stop existing, automatic machines will dominate the Earth, and finally expand into space. This author’s admiration for Kurzweil’s book led him to create the mathematical model presented in this paper within the framework of this author’s Evo-SETI (Evolution and SETI) mathematical theory (see Refs. [6–9]).

Fig. 4 Front cover of the 2006 book by Raymond “Ray” Kurzweil “The Singularity Is Near” taken from the Wikipedia site https://en.wikipedia.org/wiki/The_Singularity_Is_Near plus a picture of Ray Kurzweil (born February 12, 1948) shot on or prior to July 5, 2005, taken from the Wikipedia site https://en.wikipedia.org/wiki/Ray_Kurzweil

3 Kurzweil’s “the Singularity Is Near” (Is Nowadays)

599

Fig. 5 Evolution of the universe, from the 10 billion years ago to nowadays regarded as the increasing GBM stochastic process in the number of Civilizations populating the Universe

3.2 Kurzweil’s Singularity Is the GBM’s Knee in Our Evo-SETI Theory Now the reader is ready to understand why we developed the mathematical theory of the GBM knee in Parts 1 and 2 of the present paper. We claim that: Kurzweil’s Singularity is the GBM’s knee in our Evo-SETI Theory. This claim is rather obvious if you look at the exponential shown in Fig. 5. This also is Fig. 5 Ref. [11]. It depicts the GBM exponential assuming that, since about 10 billion years ago, the number of Civilizations in the Universe (i.e. not just in the Milky Way Galaxy, as assumed by the famous Drake equation of traditional SETI) “exploded”. These are just guesses of course, but the behaviour of the exponential before and after its own knee is obvious and immediately reveals that the knee is one only. Please see Fig. 5, i.e. Fig. 5 of the author’s 2014 paper “SETI as a Part of Big History”, Acta Astronautica, 101 (2014), 67–80.

3.3 Measuring the Pace of Evolution B by Measuring the Average Number m0 of Species Living on Earth Right Now In Evo-SETI Theory the present time of Darwinian Evolution is the zero instant t = 0. So, if we wish to “merge” the Evo-SETI Theory with Kurzweil’s claim that “The Singularity Is Near”, we simply have to let

600

Kurzweil’s Singularity as a Part of Evo-SETI Theory

tGMB Knee = 0

(40)

into the GBM knee Eq. (35), that now becomes

ts =

ln

√  2N s B B

.

(41)

Better still, since the number of living Species at the time of the origin of life ts was just one (RNA), letting Ns = 1 into (41), the latter becomes simply ts =

ln

√

 2B

B

.

(42)

This we like to call the SINGULARITY-EVO-SETI (SES) equation. Contrary to the four-parameter Eq. (35), the SES Eq. (41) only has two parameters (ts and B) and we are now going to “play” with them to see what the conclusions are. In addition, the Singularity-centered GBM exponential is now even easier than the knee-centered exponential (38). In fact, by virtue of (40), (38) now becomes e Bt m SINGULARITY - CENTERED EXPONENTIAL (t) = √ . 2B

(43)

This Eq. (43) we will call the “SINGULARITY-CENTEREDEXPONENTIAL” (SCE) equation. In the sequel of this chapter we will use it to find out the b-lognormals that it envelops, i.e. the so-called family of Running-b-Lognormals (RbLs, in the language of Evo-SETI Theory). Finally, consider the very important numeric value of the Singularity-CenteredExponential (43) at the present time t = 0. This numeric value is very important because, happening now, we may hope to be able to measure it experimentally as a part of current Biology. Upon defining the new experimental quantity m0 as the current value of the average number of Species living on Earth, i.e. upon defining m0 ≡ m SINGULARITY - CENTERED EXPONENTIAL (0) = m L (0)

(44)

then (43) yields immediately 1 m0 = √ . 2B

(45)

Alternatively, solving (45) for B, one gets 1 . B=√ 2m0

(46)

3 Kurzweil’s “the Singularity Is Near” (Is Nowadays)

601

This is an important new result of our Singularity-Evo-SETI Theory inasmuch as it relates to each other two numerical quantities like m0 and B that were regarded as completely unrelated prior to this paper. In other words still, we have discovered how to numerically measure the pace of Evolution in the Singularity-Evo-SETI Theory if we just can measure, or at least estimate, the average number m0 = m L (0) of Species living on Earth right now. This is why we call (46) “the pace equation” with m0 = m L (0) .

3.4 An Unexpected Discovery: The “Origin-to-Now” (“OTN”) Equation Relating the Time of the Origin of Life on Earth ts to m0 the Average Number of Species Living on Earth Right Now Let us stop a moment to ponder over what we did so far in this paper: 1. We assumed that Darwinian Evolution was just a realization of the stochastic process called Geometric Brownian Motion (GBM) in the number of Species living on Earth since the origin of life. This assumption is the same as assuming that the mean value of the number of living Species is an exponential increasing in time from the initial value of 1 when RNA appeared on Earth to the (unknown but measurable in principle) current value m0. 2. We then merged our Evo-SETI GBM Theory with Kurzweil’s claim that the Singularity is near, practically nowadays (Eq. 40). 3. As a consequence, we discovered the simple pace Eq. (46) relating the pace B to the current average value m0 of Species living on Earth. 4. But… we did not relate yet the time of the origin of life on Earth, ts, to m0. Well, we will do so right now, and solve the resulting numerical equation in m0 that proves to be the “final, resolvent equation” of our Singularity-Evo-SETI Theory. 5. To do so, just re-write the SES Eq. (41) in terms of the pace Eq. (46). One then gets the “Origin-to-Now” equation

ts =

ln

√

 2B

B

=

√

2 m0 ln

1 m0

√ = − 2 m0 ln(m0).

(47)

Since ts is a negative number, one might prefer to take the minus sign in front of it, thus rewriting (47) as follows −ts √ = m0 ln(m0) 2

(48)

thus making it plainly clear that, if we want to solve (48) for m0 in terms of the better-known ts, we can solve (48) numerically only, and not analytically.

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Kurzweil’s Singularity as a Part of Evo-SETI Theory

3.5 Solving the “Origin-to-Now” Equation Numerically for the Two Cases of −3.5 and −3.8 Billion Years of Life Development We now proceed to solve (48) graphically for the two different cases −3.5 and −3.8 billion years of life development on Earth. Let us rewrite (48) with x instead of m0 and in the form: √ 2 x ln(x). (49) 1= −ts One might say that, solving (49) numerically is the same as finding the numerical value of the “independent variable” x for which the function of m0 on the righthand-side of (49) equals just 1. This function of m0 is shown in Fig. 6 hereafter. One immediately notices that the function of x given by (49) is increasing and reaches the value of 1 just for x = 1.333 × 108 . In other words, the Singularity-Evo-SETI Theory predicts that, if life originated 3.5 billion years ago, NOWADAYS there should be 133 million living Species on Earth, on the average. And a little more refined numerical solution to (49), that we omit for the sake of brevity, replaced this 133 million by 132.4 billion Species living on Earth on the average right now. In this case, the pace of evolution, B, derived from (46), would be Bfor ts = −3.5 billion years =

5.341 × 10−9 . year

(50)

0.8

0.6 2 9

x ln ( x)

3.5 10

0.4

0.2

0 0

7

3.333 10

7

6.667 10

8

1 10

8

1.333 10

8

1.667 10

8

2 10

x

Fig. 6 Solving (49) numerically to find out that, if life started 3.5 billion years ago, then the average number m0 of living species today should be 133 million, or, better, 132.4 million

3 Kurzweil’s “the Singularity Is Near” (Is Nowadays)

603

0.8

0.6 2 3.8 10

9

x ln ( x) 0.4

0.2

0 0

7

7

7

8

8

2.857 10 5.714 10 8.571 10 1.143 10 1.429 10 1.714 10

8

2 10

8

x

Fig. 7 Solving (49) numerically to find out that, if life started 3.8 billion years ago, then the average number m0 of living Species today should be 142.9 million, or, better, 143.1 million

Let us now suppose that life originated on Earth 3.8 billion years ago. Then Fig. 7 (a graph absolutely similar to Fig. 4 with just 3.8 billion years ago replacing the previous 3.5 billion years ago) shows that the average number m0 of Species living on Earth today should be around 143.1 million. Clearly, life, having had 0.3 billion years (300 million years) more time to evolve, would have increased the average number of Species living nowadays from 132.4 million to 143.1 million. In this case, the pace of evolution, B, derived from (46), would be Bfor ts=−3.8 billion years =

4.941 × 10−9 . year

(51)

i.e. slightly less than (50).

3.6 No Way for the Biologists to Measure m0 Experimentally! But… unfortunately the biologists are in wild disagreement among themselves when it comes to estimate m0 numerically: it could range between 8.7 million ± 1.3 million according to the United Nations Environmental Program (web site: http:// www.unep.org/newscentre/default.aspx?DocumentID=2649&ArticleID=8838) to 1 trillion according to the site http://www.livescience.com/54660-1-trillion-specieson-earth.html.

604

Kurzweil’s Singularity as a Part of Evo-SETI Theory

The whole discussion seems to be well summarized by this sentence, that we reproduce here from the last site we mentioned: “Calculating how many Species exist on Earth is a tough challenge. Researchers aren’t even sure how many land animals are out there, much less the numbers for plants, fungi or the most uncountable group of all: microbes.”

4 Upper and Lower Standard Deviation Curves 4.1 Lognormal PDF of the GBM Up to now, we only dealt with the mean value of the number of living Species and imposed that it must be an exponential, i.e. the relevant stochastic process must be a GBM. But we did not consider both the upper and lower standard deviation curves above and below the exponential, respectively, that are crucial in order to estimate how much each different realization of the GBM “spreads” around its own mean value. Now we do so. The relevant mathematics is a little more complicated than those of the previous Sections, and so the reader might find it helpful to take advantage of a symbolic manipulator to calculate especially the relevant integrals: Maxima was used by this author, but Mathematica and Maple would be helpful too. We must first of all recall from, say, Refs. [8–11], that the probability density function (pdf) of the GBM is the lognormal in the number n ≥ 0 of living Species, as given by the equation

GBM lognormal pdf(n; t, ts, N s, B, σ ) =

e

−

2 2 ln(n)−ln(N s)−B(t−ts)+ σ (t−ts) 2 2 σ 2 (t−ts)

√ √ 2π nσ t − ts

.

(52)

This pdf has one more new positive parameter σ > 0 whose numerical value we will finally determine at the end of this Section in terms of the previously determined numerical value of ts, Ns = 1, B and m0. Then, the mean value of the pdf (52) is given by the integral

∞ n· 0

e

−

ln(n)−ln(N s)−B(t−ts)+

√

2σ 2 (t−ts)

σ 2 (t−ts) 2

√ 2π nσ t − ts

2

e Bt dn = N se B(t−ts) = √ . 2B

(53)

that is just the same as (19) and (37). It is actually possible to compute all the moments of (i.e. k = 1, 2, …) of the lognormal pdf (52), getting

4 Upper and Lower Standard Deviation Curves

∞ nk · 0

e

−

ln(n)−ln(N s)−B(t−ts)+

σ 2 (t−ts) 2

2 σ 2 (t−ts)

√ √ 2π nσ t − ts

605

2

dn = N s k e−

2Bk(t−ts)+k(k−1)σ 2 (t−ts) 2

.

(54)

For k = 0, (54) yields the normalization condition of the pdf (52). For k = 1, the mean value (53) is found again. From (54) one might derive of the descriptive statistics of the lognormal pdf (52), that we omit here for the sake of brevity. We are just interested in the k = 2 case yielding the mean value of the square, that is

∞ n2 · 0

e

−

2 2 ln(n)−ln(N s)−B(t−ts)+ σ (t−ts) 2 2σ 2 (t−ts)

√ √ 2π nσ t − ts

dn = N s 2 e2B(t−ts)+σ

2

(t−ts)

  = GBM2 .

(55)

Now the GBM variance comes, that, by virtue of (53) and (55) turns out to be     2 2 σGBM = GBM2 − GBM2 = N s 2 e2B(t−ts)+σ (t−ts) − N s 2 e2B(t−ts)  2  = N s 2 e2B(t−ts) eσ (t−ts) − 1 .

(56)

The square root of (56) is the GBM standard deviation   σGBM = N se B(t−ts) eσ 2 (t−ts) − 1 = GBM . eσ 2 (t−ts) − 1.

(57)

This is a new function of the time, that, upon replacing the mean value (43), explicitly reads e B t  σ 2 (t−ts) e − 1. σGBM (t) = √ 2B

(58)

The square root factor in (57) 

eσ 2 (t−ts) − 1

(59)

is known to statisticians as the “variation coefficient” (since it gives the pace of how the ratio σ GBM /GBM changes in time). Then, (59) shows that the variation coefficient vanishes at the initial GBM instant ts, which is the same as saying that all three curves of the mean value, upper standard deviation and lower standard deviation have the same initial point at ts, or, in other words still, the GBM starts at ts with probability 1. And all the above simply means that the two standard deviation curves have the two equations

606

Kurzweil’s Singularity as a Part of Evo-SETI Theory

⎧ ⎨ upper stdev(t) = ⎩ lower stdev(t) =





eB t √ 1 + eσ 2 (t−ts) 2B   eB t √ 1 − eσ 2 (t−ts) 2B

 −1  −1 .

(60)

4.2 Finding the GBM Parameter σ The standard deviation (58) still contains the unknown parameter σ that we are now going to find. To this end, consider the current (i.e. t = 0) value of the standard deviation curve (58). Letting t = 0 into (58) one gets  σGBM (0) =

e−ts·σ 2 − 1 √ 2B

(61)

This equation may then be solved for the unknown parameter σ, yielding, after a few steps, 

 ln 1 + 2B 2 (σGBM (0))2 σ = −ts

(62)

Let us now stop a moment and ponder over what we are doing: 1. σ GBM (0) standard deviation nowadays, that is the (unknown) value above and below the (supposed to be known to biologists, but not really so in the practice) m0 of the number of Species living on Earth nowadays. So, why don’t we re-write δm0 instead of σ GBM (0) in (62)? 2. And why don’t we replace B by m0 in (62) by virtue of (46)? 3. Well, if we take both suggestions into account, (62) is turned into its final form

σ =

   δm0 2  ! ln 1 + m0 −ts

(63)

Yet, we don’t know the numerical value of δm0 in (63): how can we express it in terms of the numerical values of other known quantities? The answer lies in the OTN (Origin-To-Now) Eq. (47) that we rewrite here for convenience √ ts = − 2m0 ln(m0).

(64)

What we need is the error analysis of this Eq. (64). Differentiating it, one gets √ δts = − 2[1 + ln(m0)]δm0.

(65)

4 Upper and Lower Standard Deviation Curves

607

Then, dividing (65) by (64) one gets the relationship among the relative errors on ts and m0 δts [1 + ln(m0)] δm0 = · ts ln(m0) m0

(66)

Solving (66) for the relative error on m0 yields δm0 ln(m0) δts = · . m0 1 + ln(m0) ts

(67)

Multiplying this by m0 yields the δm0, i.e. the standard deviation of the number of Species living on Earth nowadays δm0 =

m0 ln(m0) δts · . 1 + ln(m0) ts

(68)

As for σ, (67) is the expression to be inserted into (63) in order to get the numeric value of σ, since all other quantities are now numerically known. By doing so, we get for σ the final expression

σ =

   ln(m0) ln 1 + 1+ln(m0) · ! −ts

δts ts

2 .

(69)

4.3 Numerical Standard Deviation Nowadays for the Two Cases of Life Starting 3.5 and 3.8 Billion Years Ago Now the time is ripe to get down to the numbers. The standard deviation on ts, δts, in between the two Cases of life starting −3.5 and −3.8 billion years ago, respectively, is clearly the average between them δts =

−(3.5 + 3.8) × 109 = −3.65 × 109 . 2

(70)

As for Case 1 (ts = − 3.5·109 years ago), with m0 given by the results of Fig. 4, namely m0 = 132.4 million = 1.324 × 108 then (68) yields a standard deviation of

(71)

608

Kurzweil’s Singularity as a Part of Evo-SETI Theory

δm0 = 1.3106591479540047 × 108 .

(72)

Jee… but this standard deviation is nearly as much big as the mean value (71) itself! Does this make sense? We think it does. In fact, it simply stresses the huge uncertainly on the number of living Species nowadays… just to give a hand to the “poor” Biologists, who don’t even know if m0 is in the order of the hundreds of millions (as we claim) or (crazy!) even up to trillions!. As for Case 2 ts = −3.8 × 109 years ago, with m0 given by the results of Fig. 4, namely m0 = 143.1 million = 1.341 × 108

(73)

then (68) yields a standard deviation of δm0 = 1.4168786255929899 × 108 .

(74)

Same comments as for Case 1!.

4.4 Numerical σ for the Two Cases of Life Starting 3.5 and 3.8 Billion Years Ago Finally σ. We know from the theory that σ is positive, and we know from the practice (see Ref. [8], in particular pages 231–233) that it is “usually smaller than 1”. So, what do we expect from (69) in the two respective Cases 1 and 2? As for Case 1, (69) yields σ = 1.3970082115699428 × 10−5

(75)

while, for Case 2, (69) yields σ = 1.3972208698944956 × 10−5 .

(76)

Jee… again! These two numbers are very small, and they differ from each other only on the fifth significant figure: “surprises” of the Singularity-EvoSETI Theory!.

5 Evo-entropy of the b-Lognormals Having Their Peaks …

609

5 Evo-entropy of the b-Lognormals Having Their Peaks on the GBM Exponential 5.1 Peak-Locus Theorem The Peak-Locus Theorem is a new mathematical discovery of ours playing a central role in Evo-SETI Theory. In its most general formulation, it holds good for any lognormal process L(t) and any arbitrary mean value mL (t). In the GBM case, when mL (t) is an exponential, it is shown “in principle” in Fig. 8. Consider first the b-lognormal equation, i.e. the equation of a lognormal probability density function (pdf) starting at any positive instant b > 0 (b stands for “birth” since b is the birth time of a living being (e.g. RNA, an animal, a human, or a civilization) [ln(t−b)−μ]2

e− 2σ 2 . b lognormal pdf(t; μ, σ, b) = √ 2π σ (t − b)

(77)

Then, the Peak-Locus theorem states that the family of b-lognormals each having their peaks exactly located upon the mean value curve mL (t), is given by the following three equations, specifying the three parameters μ(p), σ (p) and b(p) in (77), as three functions of the independent variable p that is the b-lognormal’s peak:

Fig. 8 Darwinian Exponential as the geometric LOCUS OF THE PEAKS of b-lognormals for the GBM case. Each b-lognormal is a lognormal starting at a time (t = b = birth time) and represents a different SPECIES that originated at time b of the Darwinian Evolution. This is CLADISTICS, as seen through the glasses of our Evo-SETI model. It is evident that, when the generic “Running blognormal” moves to the right, its peak becomes higher and higher and narrower and narrower, since the area under the b-lognormal always equals 1. Then, the (Shannon) ENTROPY of the running b-lognormal is the DEGREE OF EVOLUTION (apart from a minus sign in front!) reached by the corresponding SPECIES (or living being, or a civilization, or an ET civilization) in the course of Evolution

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Kurzweil’s Singularity as a Part of Evo-SETI Theory

⎧ ⎪ ⎪ μ( p) = ⎪ ⎨

2

eσ L p 4π[m L ( p)]2 σ L2

−p

σ L2 2

p

σ ( p) = √2πe m2 ( p) ⎪ ⎪ L ⎪ 2 ⎩ b( p) = p − eμ( p)−[σ ( p)] .

(78)

The proof of (78) is quite lengthy and was given as a special pdf file (written in the language of the Maxima symbolic manipulator) that the reader may freely download in the web site of Ref. [11]. In the particular case of √the Singularity-centered Exponential (43), the peak-locus theorem (78) with σ L = 2B (see Ref. [11]) yields L ⎧ B2 ⎪ ⎨ μ( p) = 2π − Bp σ ( p) = √Bπ ⎪ B2 2 ⎩ b( p) = p − eμ( p)−[σ ( p)] = p − e−Bp− 2π .

(79)

These Eq. (79) will be used in a moment to find the EvoEntropy(p) of the “Running b-lognormal” “enveloped” by the GBM exponential.

5.2 Entropy as Measure of Evolution The Shannon Entropy H of any pdf f X (x), typical of Information Theory, is measured in bits and defined as follows 1 H =− ln 2

∞



 f X (x) · log( f X (x)) d x.

(80)

−∞

If the pdf is the b-lognormal in time (80), then the integral (80) may be computed exactly (see Chapter [30] of Ref. [7]), and, in terms of the μ(p) and σ (p) of the Peak-Locus Theorem (78), turns out to be    √ 1 1 ln 2π σ ( p) + μ( p) + . H ( p) = − ln(2) 2

(81)

5 Evo-entropy of the b-Lognormals Having Their Peaks …

611

This is a function of the peak abscissa p of the so-called “Running b-lognormal”, that is the b-lognormal pdf constrained above the time axis and having its peak on the mean value mL (t). Its three parameters (μ, σ, b) are given by the general peak-locus theorem (78). Inserting (79) into (81) the latter becomes   " √  B2 1 1 ln 2B + . H ( p) = − Bp + ln(2) 2π 2

(82)

Thus, (82) yields the Shannon Entropy of each member of the family of ∞1 blognormals (the family’s parameter is p) peaked upon the mean value curve (43). The b-lognormal Entropy (82) is thus the Measure of the Amount of Evolution of that b-lognormal: it measures “the decreasing disorganization in time of the particular Species represented by that b-lognormal”, let it be a cell, a plant, a human or even a civilization. Entropy is thus disorganization decreasing in time. However, one would prefer to use a measure of the “increasing organization” of a given Species in time. The EvoEntropy of p EvoEntropy( p) = −[H ( p) − H (ts)]

(83)

(EvoEntropy means “Entropy of Evolution”) is a function of p that has a minus sign in front, thus changing the decreasing trend of the Shannon Entropy (80) into the increasing trend of our EvoEntropy (83). In addition, our EvoEntropy starts at zero at the initial time ts of the origin of life on Earth, as expected: EvoEntropy(ts) = 0.

(84)

Inserting (82) into the EvoEntropy definition (83) we find that all the non-pdependent terms cancel against each other and so only the two terms are left yielding the time difference between p and ts In other words, in the case of our SingularityEvo_SETI-Theory, the EvoEntropy (83) becomes just an exact linear function of the time p Singularity GBM EvoEntropy( p) =

B · ( p − ts). ln(2)

(85)

This is, of course, a straight line in the time p starting at the time ts of the Origin of Life on Earth and increasing linearly thereafter. It is measured in bits/individual and is shown in Fig. 9.

612

Kurzweil’s Singularity as a Part of Evo-SETI Theory EvoEntropy of the LATEST SPECIES in bits/individual 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0

3.5

3

2.5

2

1.5

1

0.5

0

Time in billions of years before present (t=0)

Fig. 9 EvoEntropy (in bits per individual) of the latest Species appeared on Earth during the last 3.5 billion years. This shows that a Man (nowadays) is 25.575 bits more evolved than the first form of life (RNA?) 3.5 billion years ago. As you see, the EvoEntropy’s STRAIGHT LINE does not change before (depicted here) and after (not depicted here) the SINGULARITY occurring nowadays

But… THIS IS THE SAME LINEAR BEHAVIOUR IN TIME AS THE MOLECULAR CLOCK (see Refs. [16, 17]), that is the technique in molecular evolution that uses fossil constraints and rates of molecular change to deduce the time in geologic history when two Species or other taxa diverged. The molecular data used for such calculations are usually nucleotide sequences for DNA or amino acid sequences for proteins. So, we have discovered that the Entropy in our Evo-SETI model and the Molecular Clock are the same linear time function, apart for multiplicative constants (depending on the adopted units, like bits, seconds, etc.). This conclusion appears to be of key importance to understand “where a newly discovered exoplanet stands on its way to develop LIFE” (see also Ref. [1]).

6 The Korotayev-Markov Alternative Evolution Theory …

613

6 The Korotayev-Markov Alternative Evolution Theory with a Cubic-like Mean Value in Time 6.1 Peak-Locus Theorem When the Mean Value Is a Polynomial in the Time So far in this chapter we only considered the case when the mean value mL (t) is the exponential (19), and this is of course the case when the stochastic process L(t) is the GBM. However, our Peak-Locus Theorem (78) holds good for any continuous mean value mL (t), as shown by the detailed proof in Maxima code making up Appendix 3 to Ref. [14]. Particularly important is the case when the mean value function mL (t) equals a generic polynomial in t, namely #

polynomial degree

m polynomial (t) =

ck t k

(86)

k=0

ck being the coefficient of the k-th power of the time t in the polynomial (86). We just confine ourselves to mention here that the case where (86) is a seconddegree polynomial (i.e. a parabola in t) may be used to describe the Mass Extinctions (as described in Refs. [1, 2]) that plagued life on Earth over the last 3.5 billion years, as proven in Sect. 9 of Ref. [15]. A summary of the statistical properties of the L(t) process when its mean value is the polynomial (86) is given in Table 1.

6.2 Finding the Cubic When Its Maximum and Minimum Times Are Given, in Addition to the Five Conditions to Find the Parabola Having completely solved in Ref. [15]. the problem of deriving the equations of the lognormal process L(t) for which the mean value is an assigned parabola, the next step is to derive the Cubic (i.e. the third-degree polynomial in t) now assumed to be the mean value of the lognormal process L(t). The relevant calculations are longer than for the parabola case, but still manageable. Unfortunately, similar calculations turn out to be too long and complicated for even higher-order polynomials like a Quartic

Upper standard deviation curve

Standard deviation

Variance

Mean value curve

Particular M polynomial (t) function

Probability distribution Probability density function 2

σ P(t) = e Mpolynomial (t) e m P(t) + σ P(t)

k=0

$polynomial degree

ck t k

σ2 2

t

 $  polynomial degree 2 2 eσ t − 1 = ck t k e σ t − 1 k=0 # 

% polynomial degree σ2 2 = e Mpolynomial (t) e 2 t 1 + eσ t − 1 = ck t k k=0  % 2 · 1 + eσ t − 1

P(t) ≡ m P(t) = e Mpolynomial (t) e 2 t =   2 2 2 σ P(t) = e2Mpolynomial (t) eσ t eσ t − 1

σ2

ln(n)−Mpolynomial (t)   − 2σ 2 t p(t) pd f n; Mpolynomial (t), σ, t = √ 1 √ e for n ≥ 0 2πσ tn  $ 2 polynomial degree Mpolynomial (t) = ln ck t k − σ2 t k=0



Lognormal distribution of stochastic processes with polynomial mean

(continued)

Table 1 Summary of the properties of the polynomial lognormal distribution that applies to the stochastic process L(t) = lognormally changing number of ET communicating civilizations in the Galaxy, as well as the number of living Species on Earth over the last 3.5 billion years. Clearly, everywhere in this table the time variable t should be replaced by (t − ts) was our lognormal stochastic process L(t) to start with a positive value Ns at the initial time ts  Stochastic process 1) N umber o f E T Civili zations(in S E T I ) p(t) = 2) N umber o f Living Species(in Evolution)

614 Kurzweil’s Singularity as a Part of Evo-SETI Theory

Kurtosis

Skewness

Median (=fifty-fifty probability value for P(t))

Value of the mode Peak

Mode (=abscissa of the lognormal peak)

All the moments, i.e. k-th moment

Lower standard deviation curve

Table 1 (continued)

 2 σ2 P k (t) = ek Mpolynomial (t) ek 2 t

√ 1 √ 2πσ t

K4 (K 2 )2

(K 2 ) 2

= e4 σ 2t

+ 2 e3 σ

2t

2t

+ 3 e2 σ 2t

−6

· e−Mpolynomial (t) · e

median = m = e Mpolynomial (t)  2  2 K3 σ t +2 eσ t − 1 3 = e

f P(t) (n mode ) =

n mode ≡ n peak = e Mpolynomial (t) e−σ



σ2 2

t

# 

% polynomial degree σ2 2 m P(t) + σ P(t) = e Mpolynomial (t) e 2 t 1 − eσ t − 1 = ck t k k=0  % 2 · 1 − eσ t − 1

6 The Korotayev-Markov Alternative Evolution Theory … 615

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Kurzweil’s Singularity as a Part of Evo-SETI Theory

or a Quintic: namely, the analytic solutions appear to be prohibitive for polynomials higher than the Cubic considered in this section, and we shall not describe here our failed attempts in this regard. Let us start by writing down the Cubic starting at the starting-time ts: Cubic(t) = a(t − ts)3 + b(t − ts)2 + c(t − ts) + d.

(87)

We must determine the cubic’s four coefficients (a, b, c, d) in terms of the following seven inputs: 1. The initial time (starting time) ts. 2. The initial numeric value Ns of the stochastic process L(t) at ts, namely L cubic (ts) = Ns. To be precise, we assume that it is certain (i.e. with probability 1) that the process L(t) will take up the value Ns at the initial time t = ts, and so will its mean value, with a zero standard deviation there. 3. The final time (ending time) te of our lognormal L(t) stochastic process. 4. The final numeric value Ne of the mean value of the stochastic process L(t) at te, namely we define L cubic (te) = N e.

(88)

5. In addition to the assumption (88), we also must assume that L(t) will have a certain standard deviation δNe above and below the mean value (88) at the endtime t = te. These first five inputs are just the same as the five inputs described in Sect. 3.1 of Ref. [15] for the GBMs, and in Sect. 9.2 of the same Ref. [15] for the Parabola Model, that in both cases we then used to describe the Mass Extinctions as stochastic lognormal processes. But now, for the Cubic case we are forced to introduce two more inputs: 6. The time of the Cubic’s maximum, t Max, and 7. The time of the Cubic’s minimum, t min. It is intuitively clear that, in order to handle the four-coefficient Cubic (87), more conditions are necessary than just the previous five conditions, necessary to handle both the GBM (52) and the three-coefficient parabola. However, it was not initially obvious to this author how many more conditions would have been necessary and especially which ones. The answers to these two questions came out only by doing the actual calculations, as we now describe for the particular case when the two conditions (6) and (7) reveal themselves sufficient to determine the Cubic (87) completely. This particular way of determining the Cubic is important in the study of Darwinian Evolution as described by the contemporary Russian scientist Andrey Korotayev and his colleague Alexander V. Markov, as we shall study in the next section.

6 The Korotayev-Markov Alternative Evolution Theory …

617

Going now over to the actual calculations, we start by noticing that, because of the two initial conditions (1) and (2), the Cubic (87) yields, respectively 

Cubic(ts) = d Cubic(ts) = N s.

(89)

These, inserted into the Cubic (87), change that into Cubic(t) = a(t − ts)3 + b(t − ts)2 + c(t − ts) + N s.

(90)

We then invoke the two final conditions (3) and (4) that translate into the single equation Cubic(te) = N e.

(91)

In other words, (91) changes the Cubic (87) into N e − N s = a(te − ts)3 + b(te − ts)2 + c(te − ts).

(92)

The only three unknowns in (92) are the three still unknown Cubic coefficients (a, b, c). But actually (92) is a relationship among these three coefficients (a, b, c). Thus, in reality, we only need two more conditions yielding, for instance, both b and c as functions of a, respectively, and we would then insert them both into (92) that would then become an equation in the only unknown a. Solving that equation for a would then solve the problem completely, yielding then both b and c as functions of all known quantities. So, let us now look for these two still missing conditions on (a, b, c). To this end, the key idea is that every Cubic has both a Maximum and a minimum. To find them, the Cubic’s (87) first derivative with respect to t must be set equal to zero: dCubic(t) = 3a(t − ts)2 + 2b(t − ts) + c = 0. dt

(93)

Solving this quadratic for t yields the two roots: 

having set

t1 = ts + t2 = ts +

√ −b− b2 −3ac √3a −b+ b2 −3ac 3a

= ts + X 1 = ts + X 2

(94)

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Kurzweil’s Singularity as a Part of Evo-SETI Theory



X1 = X2 =

√ −b− b2 −3ac √3a −b+ b2 −3ac . 3a

(95)

Now notice that the two Eq. (94) yield the abscissas (i.e. the instants) of the two stationary points of the quadratic (93), but we don’t know which ones, namely we don’t know which one is the Maximum and which one is the minimum. Then, if we suppose that the abscissas (i.e. the instants) of the Maximum and the minimum of the Cubic (87) are assigned, i.e. they are known, then X 1 and X 2 are known also, since they are the same thing as the Maximum and the minimum except for the additive time ts, the starting time of the Cubic (87). By doing so, we have indeed taken the two conditions (6) and (7) into account. Adding the Eq. (95) and then solving for b yields b=−

3a(X 1 + X 2 ) 2

(96)

that is the expression of b as a function of a that we were seeking. Similarly, multiplying the Eq. (95) and then solving for c yields the required expression of c as a function of a c = 3a X 1 X 2 .

(97)

So, we just need inserting the two Eqs. (96) and (97) into (92) to get an equation in the only unknown a that is  3a(X 1 + X 2 ) N e − N s = (te − ts) · a(te − ts)2 − (te − ts) + 3a X 1 X 2 . (98) 2 Solving (98) for a yields a=

2(N e − N s) .

2 (te − ts) 2(te − ts) − 3(X 1 + X 2 )(te − ts) + 6X 1 X 2

(99)

Next we find b inserting (99) into (96) b=

−3(X 1 + X 2 )(N e − N s) 

(te − ts) 2(te − ts)2 − 3(X 1 + X 2 )(te − ts) + 6X 1 X 2

and we also find c inserting (99) into (97)

(100)

6 The Korotayev-Markov Alternative Evolution Theory …

c=

6X 1 X 2 (N e − N s) .

(te − ts) 2(te − ts)2 − 3(X 1 + X 2 )(te − ts) + 6X 1 X 2

619

(101)

Thus, the Cubic (90) is now obtained by inserting (99)–(101) into (90), with the result Cubic(t) = (N e − N s) ·



(t − ts) 2(t − ts)2 − 3(X 1 + X 2 )(t − ts) + 6X 1 X 2  + N s.

(te − ts) 2(te − ts)2 − 3(X 1 + X 2 )(te − ts) + 6X 1 X 2

(102)

A glance to (102) immediately reveals that both the boundary conditions given by the lower Eq. (89) and by Eq. (91), respectively, are indeed fulfilled. But the important point is to notice that the Cubic (102) is symmetric in X 1 and X 2 , namely that the Cubic (102) does not change at all if X 1 and X 2 are interchanged. Again, this is another way to say that “we don’t know which one, out of X 1 and X 2 , corresponds to the abscissa of the Maximum and the abscissa of the minimum”. The answer to this apparent “surprise” is that it all depends on the factor (Ne − Ns) in front of the fraction in (102): 1. If Ne > Ns then the Cubic’s coefficient of t 3 in (102) is positive. Then, the cubic “starts” at −∞, grows up to its Maximum, then goes down to its minimum, and finally “climbs up again on the right”. In other words, the Maximum is reached before the minimum. And this will be the case of the Markov-Korotayev’s Cubic of Evolution that we shall study in the next section. 2. If Ne < Ns, it’s the other way round. That is, the Cubic “starts” at +∞, gets down to its minimum first, then it climbs up to its Maximum, and finally gets down to −∞ on the right. In other words, its minimum comes before its Maximum. But there is still a better form of (102) that we wish to point out. This comes from the replacement of X 1 and X 2 in (102) by virtue of the explicit abscissa of the Maximum, t Max, and of the minimum, t min, related to X 1 and X 2 via (94), that is (assuming for simplicity that Ne > Ns, as in the Markov-Korotayev case): 

√

b −3ac tMax = ts + −b− √ = ts + X 1 3a −b+ b2 −3ac tmin = ts + = ts + X 2 3a 2

(103)

from which one gets 

X 1 = tMax − ts X 2 = tmin − ts.

(104)

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Kurzweil’s Singularity as a Part of Evo-SETI Theory

Thus, inserting (104) into (102), we reach our final version of the Cubic mean value of the L(t) lognormal stochastic process Cubic(t) = (N e − N s)    

(t − ts) 2(t − ts)2 − 3 tMax + tmin − 2ts (t − ts) + 6(tMax − ts) tmin − ts    

· (te − ts) 2(te − ts)2 − 3 tMax + tmin − 2ts (te − ts) + 6(tMax − ts) tmin − ts + Ns

(105)

Our careful reader might have noticed that the condition (5) was not used to derive the Cubic (105). Well, this is because the condition (5) does not affect the Cubic (105): it only affects the standard deviation σ Cubic (t) and the two corresponding upper and lower standard deviation curves above and below the mean value Cubic (105). This fact is evident from the fourth equation in Table 1, clearly showing that the time function M L (t) and the positive parameter σ have nothing to do with each other, namely they are independent of each other, just as the mean value and the variance of the Gaussian (normal) distribution are totally independent of each other. In conclusion, the positive parameter σ is determined by (63) just rewritten in this section’s notation:    δ N e 2  ln 1 + ! Ne σ = . (106) te − ts We are now ready to write down the two equations of the upper and lower standard deviation curves. They are actually the same as the two equations at the 7th and 8th lines in Table 1 that we re-write here in the current “Cubic” notation.

6.3 Markov-Korotayev Biodiversity Regarded as a Lognormal Stochastic Process Having a Cubic Mean Value Let us now refer to the important mathematical paper [3] in the new research field nowadays called Big History. Also interesting is the Wikipedia site http://en.wikipe dia.org/wiki/Andrey_Korotayev, whose words we now report almost just the same. According to this site, in 2007–2008 the Russian scientist Andrey Korotayev, in collaboration with Alexander V. Markov, showed that a “hyperbolic” mathematical model can be developed to describe the macrotrends of biological evolution. These authors demonstrated that changes in biodiversity through the Phanerozoic correlate much better with the hyperbolic model (widely used in demography and

6 The Korotayev-Markov Alternative Evolution Theory …

621

macrosociology) than with the exponential and logistic models (traditionally used in population biology and extensively applied to fossil biodiversity as well). The latter models imply that changes in diversity are guided by a first-order positive feedback (more ancestors, more descendants) and/or a negative feedback arising from resource limitation. Hyperbolic model implies a second-order positive feedback. The hyperbolic pattern of the world population growth has been demonstrated by Korotayev to arise from a second-order positive feedback between the population size and the rate of technological growth. According to Korotayev and Markov, the hyperbolic character of biodiversity growth can be similarly accounted for by a feedback between the diversity and community structure complexity. They suggest that the similarity between the curves of biodiversity and human population probably comes from the fact that both are derived from the interference of the hyperbolic trend with cyclical and stochastic dynamics (Refs. [12, 13]). This author was struck by Fig. 10 (taken from the above-mentioned Wikipedia site) showing the increase, but not monotonic increase, of the number of Genera (in thousands) during the 541 million years making up for the Phanerozoic. Thus, this author came to wonder whether the red curve in Fig. 10 could be regarded as the Cubic mean value curve of a lognormal stochastic process, just as the exponential mean value curve is typical of Geometric Brownian Motions. ⎧    ⎨ upper_ standard_ deviation_ curve(t) = m Cubic(t) + σCubic(t) = Cubic(t) · 1 + eσ 2 (t−ts) − 1    ⎩ lower_ standard_ deviation_ curve(t) = m Cubic(t) − σCubic(t) = Cubic(t) · 1 − eσ 2 (t−ts) − 1 .

(107)

Fig. 10 During the Phanerozoic the biodiversity shows a steady but not monotonic increase from near zero to several thousands of genera

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Kurzweil’s Singularity as a Part of Evo-SETI Theory

This author’s answer to the above question is “yes”: we may indeed use our Cubic (105) to represent the red line in Fig. 10, thus reconciling the Markov-Korotayev theory with our theory requiring that the profile curve of Evolution must be the Cubic mean value curve of a certain lognormal stochastic process (and certainly not a GBM in this case). Let us thus consider the following numerical inputs to the Cubic (105) that we derive “by a glance to Fig. 10” (the precision of these numerical inputs is really unimportant at this early stage of “matching” the two theories (ours and the Markov-Korotayev’s) since we are just looking for the “proof of concept”, and better numeric approximations might follow in the future): ⎧ ts = −530 ⎪ ⎪ ⎨ Ns = 1 ⎪ te = 0 ⎪ ⎩ N e = 4000.

(108)

In words, the first two Eq. (108) mean that the first of the Genera appeared on Earth about 530 million years ago, i.e. before that time the number of Genera on Earth was zero. Also in words, the last two Eq. (108) mean that, at the present time t = 0, the number of Genera on Earth is 4000 on the average. Now, “on the average” means that, nowadays, a standard deviation of about 1000 (plus or minus) affects the average value of 4000. This is shown in Fig. 10 by the grey stochastic process called “all genera”. And this is re-phrased mathematically by invoking the (5) condition of Sect. 6.2, and assigning the fifth numeric input δ N e = 1000.

(109)

Then, as a consequence of the four numeric boundary inputs (108) plus the standard deviation on the current value of Genera (109), Eq. (106) yields the numeric value of the positive parameter σ

σ =

   δ N e 2  ! ln 1 + N e te − ts

= 0.011.

(110)

Having thus assigned numeric values to the first five conditions of Sect. 6.2, only conditions (6) and (7) remain to be assigned. These are the two abscissae of the Maximum and minimum, respectively, that a glance to Fig. 7 makes us establish as (of course in millions of years ago):

6 The Korotayev-Markov Alternative Evolution Theory …

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Biodiversity during the Phanerozoic 5000

Thousands of Genera

4000

3000

2000

1000

0

600

550

500

450

400

350

300

250

200

150

100

50

0

Millions of Years Ago

Fig. 11 Our cubic mean value curve (thick red solid curve) plus and minus the two standard deviation curves (thin solid blue curves) give more mathematical information than just the previous Fig. 10. In fact, we now have the two standard deviation curves of the lognormal stochastic process L(t) that are completely missing in the Markov-Korotayev theory and in their plot shown in Fig. 8. We thus claim that our Cubic mathematical theory of the lognormal stochastic process L(t) is a “more profound mathematization” than the Markov-Korotayev theory of Evolution since it is stochastic, rather than just deterministic. This completes our “stochastic extension” of the Markov-Korotayev Evolution model. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article)



tMax = −400 tmin = −220.

(111)

Inserting then these seven numeric inputs into the Cubic (105) as well as into both the Eq. (107) of the upper and lower standard deviation curves, the final plot shown in Fig. 11 is produced.

7 Conclusions At this point we have to face the crucial question: “What is the meaning of the b-lognormals before and after the SINGULARITY?”. The answer to this question comes along these lines: 1. As for the time BEFORE the Singularity, the vertical axis is the overall number of SPECIES living on Earth at a certain time of Darwinian Evolution. This interpretation seems to be correct for the whole of the 3.5 (or 3.8) billion years of Darwinian Evolution before present. And that implied a numerical value for B in the order of 10−9 years (see Ref. [8]).

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2. However, when this author described the b-lognormals of the most important Historic Western Civilizations (see Ref. [8]), the value of B was about 10−5 , i.e. four orders of magnitude higher. In other words, the two B of Darwinian Evolution and of the History of human civilizations differ by about four orders of magnitude, the latter being about 10,000 times faster than the former. 3. Not to mention the value of B after the Singularity, that is of course “experimentally” unknown to us. In addition, after the Singularity, the living Species before must be replaced by generations of computers, meaning that each b-lognormal represents a more and more advanced family of computers ruling “life” on Earth after the Singularity. 4. And… Aliens (i.e. Alien MACHINES) visiting the Earth would be astonished to find out that, before the Singularity, i.e. until 2016 or so, the inhabitants of Earth were still made out of MEAT, as in the Science Fiction story by Terry Bissonhttp://www.eastoftheweb.com/short-stories/UBooks/TheyMade.shtml. 5. After this point, one more paper should be written to describe… how the blognormal’s “width” would correctly describe the “average duration in time” of each Species (before the Singularity) and of each COMPUTER Species (after the Singularity)… 6. …but this is “too much to be done now”, and so we have leave it to a new, forthcoming paper.

APPENDIX, i.e. our Evo-SETI SINGULARITY THEOREM. A NEW …

APPENDIX, i.e. our: Evo-SETI SINGULARITY THEOREM. A NEW RESULT connecting three quite different facts like: (1) The time ts of the origin of life on Earth (RNA); (2) The number m0 of Species living NOW ~ 140 million; (3) The SINGULARITY is just (a few years from) NOW !

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References

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References 1. C. Maccone, The Statistical Drake Equation, paper #IAC-08-A4.1.4 presented on October 1st, 2008, at the 59th International Astronautical Congress (IAC) held in Glasgow, Scotland, UK, 29 Sept–3 Oct 2008 2. C. Maccone, SETI and SEH (statistical equation for habitables). Acta Astronaut. 68, 63–75 (2011) 3. C. Maccone, A mathematical model for evolution and SETI. Orig. Life Evolut. Biospheres 41, 609–619 (2011) 4. R. Kurzweil, The Singularity is Near, first published in 2005 by Viking Press, a member of Penguin Group (USA) Inc., original copyright © Ray Kurzweil, 2005 All rights reserved, Italian edition 2008 by the title of La singolarità è vicina, copyright © Apogeo s.r.l., translation by Virginio B. Sala. © copyright 2014 by Maggioli Editore, Santarcangelo di Romagna (Ravenna, Italy) 5. C. Maccone, The statistical Drake equation. Acta Astronaut. 67, 1366–1383 (2010) 6. C. Maccone, The statistical Fermi paradox. J. Br. Interplanet. Soc. 63, 222–239 (2010) 7. C. Maccone, SETI, Evolution and human history merged into a mathematical model. Int. J. Astrobiol. 12(3), 218–245 (2013) 8. C. Maccone, Mathematical SETI, a 724-pages book published by Praxis-Springer in the fall of 2012, ISBN, ISBN-10: 3642274366. ISBN-13: 978-3642274367. Edition: 2012, 2012 9. M. Nei, K. Sudhir, Molecular Evolution and Phylogenetics (Oxford University Press, 2000) 10. M. Nei, Mutation-Driven Evolution (Oxford University Press, 2013) 11. L.W. Alvarez, W. Alvarez, F. Asaro, H.V. Michel, Extraterrestrial cause for the cretaceous– tertiary extinction. Science 208, 1095–1108 (1980) 12. C. Maccone, New Evo-SETI results about civilizations and molecular clock. Int. J. Astrobiol. (2016). http://journals.cambridge.org/action/displayAbstract?fromPage=online& aid=10256336&fileId=S1473550415000506 13. W. Alvarez, In the Mountains of Saint Francis: Discovering the Geologic Events that Shaped Our Earth, Kindle, 2008 14. C. Maccone, Evolution and mass extinctions as lognormal stochastic processes. Int. J. Astrobiol. 13(4), 290–309 (2014) 15. A.V. Korotayev, A.V. Markov, Mathematical modeling of biological and social phases of big history, in Teaching and Researching Big History—Exploring a New Scholarly Field, ed. by L. Grinin, D. Baker, E. Quaedackers, A. Korotayev (Uchitel Publishing House, Volgograd, Russia, 2014), pp. 188–219 16. A. Markov, A. Korotayev, Phanerozoic marine biodiversity follows a hyperbolic trend. Palaeoworld 16(4), 311–318 (2007) 17. A. Markov, A. Korotayev, Hyperbolic growth of marine and continental biodiversity through the Phanerozoic and community evolution. J. Gen. Biol. 69(3), 175–194 (2008)

Energy of Extra-Terrestrial Civilizations According to Evo-SETI Theory

Abstract Consider two great scientists of the past: Kepler (1571–1630) and Newton (1642–1727). Kepler discovered his three laws of planetary motion by observing Mars: he knew experimentally that his three laws were correct, but he didn’t even suspect that all three mathematical laws could be derived as purely mathematical consequences of a “superior” mathematical law. The latter was the Law of Gravitation that Newton gave the world together with his supreme mathematical discovery of the Calculus, necessary for that mathematical derivation. We think we did the same for the “molecular clock”, the experimental law of genetics discovered in 1962 by Emile Zuckerkandl (1922–2013) and Linus Pauling (1901–1994) and derived by us as a purely mathematical consequence of our mathematical Evo-SETI Theory. Let us now summarize how this mathematical derivation was achieved. Darwinian evolution over the last 3.5 billion years was an increase in the number of living species from one (RNA?) to the current (say) 50 million. This increasing trend in time looks like being exponential, but one may not assume an exact exponential curve since many species went extinct in the past, especially in the five, big mass extinctions. Thus, the simple exponential curve must be replaced by a stochastic process having an exponential mean value. Borrowing from financial mathematics (the “Black-Sholes models”), this “exponential” stochastic process is called Geometric Brownian Motion (GBM). Its probability density function (pdf) is a lognormal (and not a Gaussian) (Proof: see Ref. Maccone in Mathematical SETI. Praxis-Springer, Zürich, 2012 [5], Chap. 30, and Ref. Maccone in Int J Astrobiol 12(issue 3):218–245, 2013 [6], and, more in general, Refs. Maccone in Orig Life Evol Biospheres (OLEB) 41:609–619, 2011 [4] and Maccone in Int J Astrobiol 13(issue 4):290–309, 2014 [7]). Lognormal also is the pdf of the statistical number of communicating ExtraTerrestrial (ET) civilizations in the Galaxy at a certain fixed time, like a snapshot: this result was obtained in 2008 by this author as his solution to the Statistical Drake Equation of SETI (Proof: see Ref. Maccone in Acta Astronaut 67:1366–1383, 2010 [3]). Thus, the GBM of Darwinian evolution may also be regarded as the extension in time of the Statistical Drake equation (Proof: see Ref. Maccone in Int J Astrobiol 12(issue 3):218–245, 2013 [6]). But the key step ahead made by this author in his Evo-SETI (Evolution and SETI) mathematical theory was to realize that life also is just a b-lognormal in time: every living organism (a cell, a human, a civilization, even an ET civilization) is born at a certain time b (“birth”), grows up to a peak p (with an ascending inflexion point in between, © Springer Nature Switzerland AG 2020 C. Maccone, Evo-SETI, https://doi.org/10.1007/978-3-030-51931-5_19

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a for adolescence), then declines from p to s (senility, i.e. descending inflexion point) and finally declines linearly and dies at a final instant d (death). In other words, the infinite tail of the b-lognormal was cut away and replaced by just a straight line between s and d, leading to simple mathematical formulae (“History Formulae”) allowing one to find this “finite b-lognormal” when the three instants b, s, and d are assigned. Next our crucial Peak-Locus Theorem comes. It means that the GBM exponential may be regarded as the geometric locus of all the peaks of a one-parameter (i.e. the peak time p) family of b-lognormals. Since b-lognormals are pdf-s, the area under each of them always equals 1 (normalization condition) and so, going from left to right on the time axis, the b-lognormals become more and more “peaky”, and so they last less and less in time. This is precisely what happened in Human History: civilizations that lasted a millennium each (like Ancient Greece and Rome) lasted just a few centuries in later times (like the Italian Renaissance and Portuguese, Spanish, French, British and USA Empires) but they were more and more advanced in the “level of civilization”. This “level of civilization” is what physicists call entropy. Also, in Refs. Maccone (Mathematical SETI. Praxis-Springer, Zürich, 2012 [5]), Maccone (Int J Astrobiol 12(issue 3):218–245, 2013 [6]), this author proved that, for all GBMs, the (Shannon) Entropy of the b-lognormals in his Peak-Locus Theorem grows linearly in time. The Molecular Clock (Refs. Felsenstein in Inferring phylogenies. Sinauer Associates Inc., Sunderland, Massachusetts, 2004 [1], https://en.wikipedia.org/wiki/ Molecular_clock, Maccone in Acta Astronaut 115:286–290, 2015 [8] , Maruyama in Stochastic problems in population genetics. Springer, Berlin, 1977 [9], Nei and Sudhir, Molecular evolution and phylogenetics. Oxford University Press, Oxford, 2000 [10], Nei in Mutation-driven evolution. Oxford University Press, Oxford, 2013 [11]), well known to geneticists since 1962, shows that the DNA base-substitutions occur linearly in time since they are neutral with respect to Darwinian selection. This is Kimura’s neutral theory of molecular evolution. The conclusion is that the Molecular Clock and the linear increase of EvoEntropy in time are just the same thing! In other words, we derived the Molecular Clock mathematically as a part of our Evo-SETI Theory. In addition, our EvoEntropy, i.e. the Shannon Entropy of the b-lognormal (with the minus sign reversed and starting at zero at the time of the origin of Life on Earth) is just the new Evo-SETI Scale to measure the evolution of life on Exoplanets (measured in bits). That was the situation prior to the present paper, firstly presented at the SETI II Session of the Adelaide IAC in October 2017. In fact, just as classical thermodynamics entails both energy and entropy, so our Evo-SETI Theory needs entailing the energy used by a living Species or Civilization along its whole lifetime in addition to its entropy (i.e. Molecular Clock). In other words still, while the Molecular Clock is a measure of the advancement in evolution, the energy required to get that advancement is another topic not faced by this author prior to 2017. However, in the present paper we were able to add the consideration of energy in addition to entropy by replacing the b-lognormal probability densities previously used by a new curve, finite in the time, that we call “logpar”. This logpar is made up by an ascending b-lognormal in the time between the birth and the peak of the living organism, followed by a descending parabola in the time between its peak and death. The logpar curve is not normalized to one: the area under the

Energy of Extra-Terrestrial Civilizations According to Evo-SETI Theory

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logpar curve may be any positive number since it represents the energy requested by the organism to live over its entire lifetime “birth-to-death”. In other words still, we mathematically demonstrate in this paper that just three instants (birth b, peak p and death d) must be assigned in order to make the mathematical logpar curve perfectly described. The history of the Roman Civilization fits to this description in that we not only know when Rome was funded (753 bc) and reached its peak (117 ad) but also when it collapsed (in the West), i.e. 476 ad. Its energy is then estimated in terms of money (Sestertii) and so we dare to say that our Evo-SETI Theory so extended adequately describes not only the Entropy but also the Energy of Rome. In conclusion, we think that our invention of the logpar power curve provides a new, formidable mathematical tool for our Evo-SETI mathematical description of Life, History and SETI. Keywords Entropy · Energy · Logpar · Roman Civilization

1 Part 1 1.1 Logpar Curves and Their History Equations 1.1.1

Introduction to Logpar “Finite Lifetime” Curves

In this paper we introduce the new notion of a finite lifetime as the new logpar curve. The idea is easy: we seek to represent any lifetime by virtue of just three points in time: birth, peak, death (b, p, d). No other point in between. That is, no other “senility point” s as those appearing in all blognormals that this author had published in his Evo-SETI Theory prior to 2017. In fact, it is easier and more natural to describe someone’s lifetime just in terms of birth, peak and death, than in terms of birth, senility and death. In other words, it is so uncertain to define in the practice when the senility time arrives. Let us look at Fig. 1. The first part, the one on the left, i.e. prior to the peak time p, is just a b-lognormal: it starts at birth time b, climbs up to the adolescence time a (ascending inflexion point of the b-lognormal) (in reality the adolescence time should more properly be called “puberty time” since it marks the beginning of the reproduction capacity for that individual) and finally reaches the peak time at p (maximum, i.e. the point of zero first derivative of the b-lognormal). All this is just ordinary b-lognormal stuff, as we have been “preaching” since about 2012. But now the novelty comes, i.e. the second part, the one on the right: that is just a parabola having its vertex exactly at the peak time p. Notice that this definition automatically implies that the tangent line at the peak is horizontal, i.e. the same for both the b-lognormal and the parabola. Notice also that, after the peak, the parabola plunges down until it reaches the time axis at the death time d. Therefore this new definition of death time d is different from the old definition of d applying to blognormals alone, as we did prior to 2017.

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Fig. 1 Representation of the history of the Roman civilization as a LOGPAR finite curve. Rome was funded in 753 bc, i.e. in the year −753 in our notation, or b = 753. Then the Roman republic and empire (the latter since the first emperor, Augustus, roughly after 27 bc) kept growing in conquered territory until it reached its peak (maximum extension, up to Susa in current Iran) in the year 117 ad, i.e. p = 117, under emperor Trajan. Afterwards it started to decline and loose territory until the final collapse in 476 ad (d = 476, Romulus Augustulus, last emperor). Thus, just three points in time are necessary to summarize the history of Rome: b = 753; p = 117; d = 476: No other intermediate point, like senility in between peak and death, is necessary at all since we now used a logpar rather than a b-lognormal, as this author had done prior to 2017. The numbers along the vertical axis will be explained in Sects. 1.1.10 and 2.1.4

And this is the LOGPAR (b-LOGnormal plus PARabola) new CURVE FINITE IN TIME (namely ranging in time just between birth and death). We introduce the logpar for the first time in the present paper and we study it with surprising results.

1.1.2

Finding the Parabola Equation of the Right Part of the Logpar

We shall now cast into appropriate mathematics the above popular description of what a logpar curve is. Consider the equation of a parabola in the time t having vertical axis along the t = p vertical line: y = α(t − p)2 + β(t − p) + γ

(1)

where α, β and γ are the three coefficients of the time that we must determine according to the assumptions shown in Fig. 1. To find them, we must invoke the three conditions that we know to hold by virtue of a glance to Fig. 1: 1. #1 CONDITION: the height of the peak is P, just the same as the height of the peak of the b-lognormal on the left in Fig. 1. Thus, inserting the two equations of the peak, namely

1 Part 1

647



t=p y=p

(2)

into (1), the latter yields immediately P=γ

(3)

that, when inserted back into (1), changes it into y = α(t − p)2 + β(t − p) + P.

(4)

2. #2 CONDITION: the tangent straight line at both the b-lognormal and the parabola at the peak abscissa p is horizontal. In other words, the first derivative of (4) at t = p must equal zero. Differentiating (4) with respect to t, equalling that to zero and then solving for β yields β = −2α(t − p).

(5)

Inserting (5) into (4), the latter is turned into y = −α(t − p)2 + P.

(6)

3. #3 CONDITION: at the death time d, one must have y = 0, yielding from (6) the equation 0 = −α(d − p)2 + P.

(7)

Solving (7) for α one gets α=

P . (d − p)2

(8)

Finally, inserting (8) into (6) the desired equation of the parabola is found  (t − p)2 . y(t) = P 1 − (d − p)2 

(9)

As confirmation, one may check that (9) immediately yields the two conditions 

y( p) = P y(d) = 0.

(10)

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1.1.3

Finding the b-Lognormal Equation of the Left Part of the Logpar

As for the b-lognormal between birth and peak, making up the left part of the logpar curve, we already know all its mathematical details from the previous many papers published by this author on these topics, but we shall summarize here the main equations for the sake of completeness. The equation of the b-lognormal starting at b reads (log(t−b)−μ)2

e− 2σ 2 . b_ lognormal(t; μ, σ, b) = √ 2π σ (t − b)

(11)

Tables listing the main equations that can be derived from (11) were given by this author in Refs. [5, 6] and we shall not re-derive them here again. We just confine ourselves to reminding that: 1. The abscissa p of the peak of (11) is given by p = b + eμ−σ . 2

(12)

Proof. Take the derivative of (11) with respect to t and set it equal to zero. Then solve the resulting equation for t, that now becomes p, and (12) is found. 2. The ordinate P of the peak of (11) is given by σ2

e 2 −μ P=√ . 2π σ

(13)

Proof. Rewrite p instead of t in (11) and then insert (12) instead of p. Then simplify to get (13). 3. The abscissa of the adolescence point (that should actually be better named “puberty point”) is the abscissa of the ascending inflexion point of (11). It is given by a = b + e−

σ

√

2 σ 2 +4 − −3σ 2 2

+μ

(14)

Proof. Take the second derivative of (11) with respect to t and set it equal to zero. Then solve the resulting equation for t, that now becomes a, and (14) is found. 4. The ordinate of the adolescence point is given by e−

σ

√

2 σ 2 +4 + σ4 4

√

2π σ

−μ− 21

(15)

Proof. Just rewrite a instead of t in (11) and then insert (14) and simplify the result.

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Let us now notice that, within the framework of the logpar theory described in this paper, we may not say that (11) fulfills the normalization condition ∞ b_ log normal(t;μ,σ ,b)dt = 1

(16)

b

since (11) here is only allowed to range between b and p. Rather than adopting (16), we must thus replace (16) by the integral of (11) between b and p only. Fortunately, it is possible to evaluate this integral in terms of the error function defined by 2 er f (x) = √ π

x

e−t dt. 2

(17)

0

In fact, the integral of the b-lognormal (11) between b and p turns out to be given by p b

p

e−(log(t−b)−μ) b_ log normal(t; μ, σ, b)dt = dt √ 2πσ (t − b) b √ √  2μ 1 + er f 2 ln( p−b)− 2σ = 2 2

Now, inserting (12) instead of p into the last erf argument, a remarkable simplification occurs: μ and b both disappear and only σ is left. In addition, the erf property erf(−x) = −erf(x) allows us to rewrite =

  1 + er f − √σ2 2

=

1 − er f



σ √ 2



2

.

(18)

In conclusion, the area under the b-lognormal between birth and peak is given by p b_ log normal(t; μ, σ, b)dt =

  1 − er f − √σ2 2

(19)

b

This result will prove to be of key importance for the further developments described in the present paper.

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1.1.4

Energy of Extra-Terrestrial Civilizations According to Evo-SETI Theory

Area Under the Parabola on the Right Part of the Logpar Between Peak and Death

We already proved that the parabola on the right part of the logpar curve has the equation [10]. Now we want to find the area under this parabola between peak and death, that is  d   d (t − p)2 P P 1− dt = P(d − p) − (t − p)2 dt (d − p)2 (d − p)2 p

p

(d − p)3 P = P(d − p) − 2 (d − p) 3 2P(d − p) . = 3

(20)

In conclusion, the area under our parabola between peak and death is given by (20), that we now rewrite as   d   (t − p)2 2P(d − p) P 1− dt = . (d − p)2 3

(21)

p

1.1.5

Area Under the Full Logpar Curve Between Birth and Death

We are now in a position to compute the full area A under the logpar curve, that is given by the sum of Eqs. (19) and (21), that is 1 − er f 2



σ √ 2

 +

2P(d − p) =A 3

(22)

This is the most important equation in this paper. In fact, if we want the logpar be a truly probability density function (pdf), we must assume in (22) A=1

(23)

But, surprisingly, we shall not do so! Let us rather ponder over what we are doing: 1. We are creating a “Mathematical History” model where the “unfolding History” of each Civilization in the time is represented by a logpar curve. 2. The knowledge of only three points in time is requested in this model: b, p and d.

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3. But the area under the whole curve depends on σ as well as on μ, as we see upon inserting (13) instead of P into (22), that is 1 − er f



σ √ 2

2



σ2

e 2 −μ 2(d − p) +√ = A(μ, σ ). · 3 2π σ

(24)

4. Also p is to be replaced by its expression (12) in terms of σ and μ, yielding the new equation 1 − er f 2



σ √ 2



σ2

e 2 −μ 2(d − b − eμ−σ ) +√ = A(μ, σ ). · 3 2π σ 2

(25)

5. The meaning of (25) is that birth and death are fixed, but the position of the peak may move according to the different numeric values of σ and μ. 6. In addition to that, we “dislike” the presence of the error function erf in (25) since this is not an “ordinary” function, i.e. it is one of the functions that mathematicians call “higher transcendental functions”, having complicated formulae describing them. Thus, we would rather get rid of erf. How may we do so? 1.1.6

The Area Under the Logpar Curve Depends on Sigma Only, and Here Is the Area Derivative with Respect to Sigma

The simple answer to the last question (6) is “by differentiating both sides of (25) with respect to σ”. In fact, the derivative of the erf function (17) is just the “Gaussian” exponential der f (x) 2 2 = √ · e−x · dx π

(26)

and so the erf function itself will disappear by differentiating (25) with respect to σ. In fact, the derivative of the first term on the left hand side of (25) simply is, according to (26), ⎡ d ⎣ dσ

1 − er f 2



σ √ 2

⎤

σ2

e− 2 ⎦ = −√ · 2π

(27)

As for the derivative with respect to σ of the second term on the left hand side of (25) we firstly notice that σ appears three times within that term. Thus, the relevant derivative is the sum of three terms, each of which includes the derivative of one of the three terms multiplied by the other two terms unchanged. In equations, one has:

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  ⎡ ⎤ σ σ2 2 e 2 −μ 2(d − b − eμ−σ ) ⎦ d ⎣ 1 − er f √2 +√ · dσ 2 3 2π σ σ2

2 2 e− 2 = √ 3 π 3

√ σ2 σ2 2 e− 2 2(−eμ−σ + d − b)e 2 −μ −√ − √ 3 πσ2 2π √ σ2 2 2(−eμ−σ + d − b)e 2 −μ + . √ 3 π

(28)

Several alternative forms of this Eq. (28) are possible, and that is rather confusing. However, using a symbolic manipulator (this author did so by virtue of Maxima), a few steps lead to the following form of (28): d A(μ(σ ), σ ) d A(σ ) ≡ dσ dσ √ √ σ2 σ2 3 σ2 σ2 2(d − p)e− 2 2(d − p)e− 2 2 2 e− 2 e− 2 + √ =− √ + √ −√ 3 π ( p − b)σ 2 3 π ( p − b) 3 π 2π ( pσ 2 − 2dσ 2 + bσ 2 − 2 p + 2d)e− = √ 3 2π( p − b)σ 2

σ2 2

.

(29)

This (29) is the derivative of the area with respect to sigma.

1.1.7

Exact “History Equations” for Each Logpar Curve

We now take a further, crucial step in our analysis of the logpar curve: we impose that the derivative of the area with respect to sigma, i.e. (29), is zero d A(σ ) = 0. dσ

(30)

This means that we are imposing the logpar energy to be minimal. That is somehow reminiscent of the Least Action principle of physics, though it is not exactly so. Thus, let us rewrite the imposed condition (30) by virtue of the last expression in (29) that, getting rid of both the exponential and the denominator, immediately boils down to pσ 2 − 2dσ 2 + bσ 2 − 2 p + 2d = 0.

(31)

1 Part 1

653

This is just the quadratic equation in σ σ 2 ( p − 2d + b) = 2( p − d)

(32)

and so we finally get σ2 =

2(d − p) . 2d − (b + p)

(33)

This is the most important new result discovered in the present paper. It is the logpar History Equation for σ: √ √ 2 d−p σ =√ . 2d − (b + p)

(34)

In other words, given the input triplet (b, p, d) then (33) immediately yields the exact σ2 of the b-lognormal left part of the logpar curve. It was discovered by this author on November 22, 2015, and led not only to this paper, but to the introduction of the Energy spent in a lifetime by a living creature, or by a whole civilization whose “power-vs-time” behaviour is given by the logpar curve, as we will understand better in the coming sections of this paper. At the moment we confine ourselves to taking the limit of both sides of (34) for d → ∞, with the result √ √ √ √ 2 d−p 2 d = lim √ = 1. lim σ = lim √ d→∞ d→∞ 2d(b + p) d→∞ 2d

(35)

Since we already know σ to be positive, (35) shows that σ may range in between zero and one only 0 0 rather than at time zero. The lifetime of any living form may then be expressed as a b-lognormal starting at b, reaching puberty at the ascending inflexion point a (“adolescence (end)”), raising up to the peak time p, then starting to decline at the descending inflexion point s (“senility”) and finally going down along a straight line up to the intercept d with the time axis, that is the “death” of the individual. Based on all this, the author was able to derive several mathematical consequences like the Central Limit Theorem of Statistics re-cast in the language of Evo-SETI theory: from the lifetime of each individual to the lifetime of the “big b-lognormal” of the whole population itself to which the individual belongs (“E-Pluribus-Unum Theorem”). In addition, this author discovered the “Peak-Locus Theorem” translating Cladistics in term of Evo-SETI: each SPECIES created by Evolution over 3.5 billion years is a b-lognormal whose peak lies on the exponential in the number of alive Species. More correctly still, this exponential is not the exact curve telling us exactly how many Species were on Earth at a given time in the past: on the contrary the exponential is the mean value of a stochastic process called “Geometric Brownian Motion” (GBM) in the mathematics of finances, so that also the Mass Extinctions of the past are incorporated in Evo-SETI Theory as all-lows of the GBM. But then: what is the Shannon ENTROPY of each b-lognormal representing a Species? Answer: the Shannon ENTROPY (with a reversed sign) is the MEASURE OF HOW EVOLVED THAT SPECIES WAS, or is now, compared to other Species of the past and of the future. That means MEASURING EVOLUTION, at long last: i.e. just a number in bits, typical of Shannon’s Information Theory, rather than a mountain of words! One more key point: what is the equivalent of the MOLECULAR CLOCK in Evo-SETI Theory? Answer: it is the STRAIGHT LINE behavior in time of the Shannon Entropy if the exponential is the Peak-Locus curve of all the b-lognormals representing the various Species (called “Evo-Entropy” in our papers). Concluding © Springer Nature Switzerland AG 2020 C. Maccone, Evo-SETI, https://doi.org/10.1007/978-3-030-51931-5_20

675

676

The Evo-SETI Unit of Evolution is EE = Earth Evolution = 25.575 bit

top remark: this author was able to GENERALIZE his Peak-Locus Theorem from the simple exponential case to the GENERAL CASE when the mean-value PeakLocus is not just an exponential, but rather an ARBITRARY CURVE that you may chose at will: for instance it as a polynomial of the third degree in the time in the Markov-Korotayev (2007) model of evolution, leading then to a non-linear EvoEntropy. A neat mathematical tool for future biologists willing to understand Evolution by the statistically simple Evo-SETI Theory! And the Evo-SETI UNIT of evolution is 25.575 bits if life on Earth started 3.5 billion years ago. It should be given a name. We propose EE (Earth Evolution). Keywords Cladistics · Biological evolution · Molecular clock · Entropy · SETI

1 Purpose of This Chapter This Chapter describes recent developments in a new statistical theory describing Evolution and SETI by mathematical equations. This we call the Evo-SETI mathematical model of Evolution and SETI. Now the question is: whenever a new exoplanet is discovered, where does that exoplanet stand in its evolution towards life as we have it on Earth nowadays, or beyond? This is the central question of Evo-SETI. In this paper we show that the (Shannon) Entropy of b-lognormals answers such a question, thus allowing the creation of an Evo-SETI SCALE. And we propose that the Evo-SETI UNIT of evolution is about 25.575 bits if life on Earth started 3.5 billion years ago. This unit we propose to be called EE (Earth Evolution). So, a planet like Mars will have EE  l, while an exoplanet hosting ETs more advanced than us will have EE > 1.

2 During the Last 3.5 Billion Years Life Forms Increased Like a (Lognormal) Stochastic Process Let us look at Fig. 1: on the horizontal axis is the time t, with the convention that negative values of t are past times, zero is now, and positive times are future times. The starting point on the time axis is ts = −3.5 × 109 years i.e. 3.5 billion years ago, the time of the origin of life on Earth that we assume to be correct. If the origin of life started earlier than that, say 3.8 billion years ago, the coming equations would still be the same and their numerical values will only be slightly changed. On the vertical axis is the number of Species living on Earth at time t, denoted L(t). This “function of the time” we don’t know in detail, and so it must be regarded as a random function, or stochastic process, with the notation L(t) standing for “life at time t”. In this paper we adopt the convention that capital letters represent random variables,

2 During the Last 3.5 Billion Years Life Forms Increased Like a (Lognormal) …

677

Fig. 1 Biological evolution as the increasing number of living Species on Earth between 3.5 billion years ago and now. The red solid curve is the mean value of the GBM stochastic process given by (22), while the blue dot-dot curves above and below the mean value are the two standard deviation curves given by (11) and (12), respectively. The “Cambrian Explosion” of life, that on Earth started about 542 million years ago, is evident in the above plot just before the value of −0.5 billion years in time, where all three curves “start leaving the time axis and climbing up”. Notice also that the starting value of Species living 3.5 billion years ago is ONE by definition, but it “looks like” zero in this plot since the vertical scale is too small to show the difference between one and zero. Notice finally that nowadays (i.e. at time t = 0) the two standard deviation curves have exactly the same distance from the middle mean value curve, i.e. 30 million living Species more or less the mean value of 50 million Species. These are assumed values that we used just to exemplify the GBM mathematics: biologists might wish to assume other numeric values. (For interpretation of the references to colour inthis figure legend, the reader is referred to the web version of this article.)

i.e. stochastic processes if they depend on the time, while lower-case letters mean ordinary variables or functions.

3 Mean Value of the Lognormal Process L(t) The most important ordinary, continuous function of the time associated with a stochastic process like L(t) is its mean value, denoted by m L (t) ≡ L(t).

(1)

The probability density function (pdf) of a stochastic process like L(t) is assumed in Evo-SETI theory to be a b-lognormal, that is, its equation reads

678

The Evo-SETI Unit of Evolution is EE = Earth Evolution = 25.575 bit

 n ≥ 0, L(t) pd f (n; M L (t), σ, t) = √ with √ t ≥ ts, 2πσ L t − ts n  σ ≥ 0, and L M L (t) = arbitrary function of t. e

−

[ln(n)−M L (t)]2 2σ L2 (t−ts)

(2)

This assumption is in line with the extension in time of the statistical Drake equation, namely foundational and statistical equation of SETI, as shown in Ref. [3]. The real variable ts in (2) is the “time of start” of life on Earth. In numbers, we assume it to be ts = −3.5 billion years, but scientists like biologists and paleontologists say that higher values in the past might be correct, like ts = −3.8 billion years if not higher. Finally about the notation: writing t s would have been neater than ts but we could not do so since the Maxima symbolic manipulator, that we used to do all calculations, would interpret t s as “the component s of the vector t” that is of course not what we want. The mean value (1) is of course related to the pdf (2) by the relevant integral in the number n of living Species on Earth at time t, that is ∞ m L (t) ≡

n·√ 0

e

−

[ln(n)−M L (t)]2 2σ L2 (t−ts)

dn. √ 2π σ L t − tsn

(3)

The “surprise” is that this integral (3) may be computed exactly with the key result that the mean value mL (t) is given by m L (t) = e M L (t) e

σ L2 2

(t−ts)

.

(4)

In turn, the last equation has the “surprising” property that it may be inverted exactly, i.e. solved for M L (t): M L (t) = ln(m L (t)) −

σ L2 (t − ts). 2

(5)

4 L(t) Initial Conditions at ts Now about the initial conditions of the stochastic process L(t), namely about the value L(ts). We shall assume that the positive number L(ts) = N s is always exactly known, i.e. with probability one:

(6)

4 L(t) Initial Conditions at ts

679

Pr{L(ts) = N s} = 1.

(7)

In the practice, Ns will be equal to 1 in the theories of evolution of life on Earth or on an exoplanet (i.e., there must have been a time ts in the past when the number of living species was just one, let it be RNA or something else), and it will be equal to the number of living species just before the asteroid/comet impact in the theories of mass extinction of life on a planet. The mean value mL (t) of L(t) also must equal the initial number Ns at the initial time ts, that is m L (ts) = N s.

(8)

Replacing t by ts in (4), one then finds m L (ts) = e M L (ts)

(9)

that, checked against (8), immediately yields N s = e M L (ts) that is M L (ts) = ln(N s).

(10)

These are the initial conditions for the mean value. After the initial instant ts, the stochastic process L(t) unfolds oscillating above or below the mean value in an unpredictable way. Statistically speaking, however, we expect L(t) “not to depart too much” from mL (t) and this fact is graphically shown in Fig. 1 by the two dot-dot blue curves above and below the mean value solid red curve mL (t). These two curves are the upper standard deviation curve upper_ st_ dev_ curve(t) = m L (t)[1 +

 2 eσL (t−ts) − 1]

(11)

 2 eσL (t−ts) − 1]

(12)

and the lower standard deviation curve lower_ st_ dev_ curve(t) = m L (t)[1 −

respectively (Proof: see Table 2 of Ref. [6]). Notice that both (11) and (12), at the initial time t = ts, equal the mean value m(ts) = Ns, that is, with probability one again, the initial value Ns is the same for all the three curves shown in Fig. 1. The function of the time  2 (13) coefficient_ of_ variation(t) = eσL (t−ts) − 1 is called “coefficient of variation” by statisticians since the standard deviation of L(t) (be careful: this is just the standard deviation ΔL (t) of L(t) and not either of the above two “upper” and “lower” standard deviation curves given by (11) and (12),

680

The Evo-SETI Unit of Evolution is EE = Earth Evolution = 25.575 bit

respectively) is  2 st_ dev_ curve(t) ≡ Δ L (t) = m L (t) eσL (t−ts) − 1

(14)

Indeed, (14) shows that the coefficient of variation (13) is the ratio of Δ(t) to mL (t), i.e. it expresses how much the standard deviation “varies” with respect to the mean value. Having understood this fact, it is then obvious that the two curves (11) and (12) are obtained as  2 m L (t) ± Δ L (t) = m L (t) ± m L (t) eσL (t−ts) − 1

(15)

respectively.

5 L(t) Final Conditions at te > ts Now about the final conditions for the mean value curve as well as for the two standard deviation curves. Let us call te the ending time of our mathematical analysis, namely the time beyond which we don’t care any more about the values assumed by the stochastic process L(t). In the practice, this te is zero (i.e. now) in the theories of evolution of life on Earth or on an exoplanet, or the time when the mass extinction ends (and life starts growing up again) in the theories of mass extinction of life on a planet. First of all, it is clear that, in full analogy to the initial condition (8) for the mean value, also the final condition has the form m L (te) = N e

(16)

where Ne is a positive number denoting the number of species alive at the end time te. But we don’t know what random value will L(te) take. We only know that its standard deviation curve (14) will take at time te a certain positive value that will differ by a certain amount δNe from the mean value (16). In other words, we only know from (14) that one has  2 δ N e = Δ L (te) = m L (te) eσL (te−ts) − 1

(17)

Dividing (17) by (16) the common factor mL (te) disappears, and one is left with δNe = Ne

 eσL (te−ts) − 1. 2

(18)

5 L(t) Final Conditions at te > ts

681

Solving this for σ L finally yields    2 ln 1 + δNNee σL = . √ te − ts

(19)

This equation expresses the so far unknown numerical parameter σ L in terms of the initial time ts plus the three final-time parameters (te, Ne, δNe). Thus, in conclusion, we have shown that, once the five parameters (ts, Ns, te, Ne, δNe) are assigned numerically, the lognormal stochastic process L(t) is determined completely. Finally notice that the square of (19) may be rewritten in the following different form:  ⎧ ⎫  2 1  2  te−ts ⎨ ⎬ ln 1 + δNNee δ N e = ln 1 + (20) σ L2 = ⎩ ⎭ te − ts Ne from which we infer the formula 

e

σ L2

=e

ln

 1 2 te−ts 1+( δNNee )



= 1+



δNe Ne

1 2  te−ts

.

(21)

This Eq. (21) enables us to get rid of eσL replacing it by virtue of the four boundary parameters supposed to be known: (ts, te, Ne, δNe). It will be later used in Sect. 8 in order to rewrite the Peak-Locus Theorem in terms of the boundary conditions, rather 2 than in terms of eσL . 2

6 Important Special Cases of m(t) 1. The particular case of (1) when the mean value m(t) is given by the generic exponential m GBM (t) = N0 eμG B M t = or, alternatively, = A e B t

(22)

is called Geometric Brownian Motion (GBM), and is widely used in financial mathematics, where it represents the “underlying process” of the stock values (Black-Sholes models). This author used the GBM in his previous models of Evolution and SETI (Refs. [3–6]), since it was assumed that the growth of the number of ET civilizations in the Galaxy, or, alternatively, the number of living Species on Earth over the last 3.5 billion years, grew exponentially (Malthusian growth). Notice that, upon equating the two right-hand-sides of (4) and (22), we

The Evo-SETI Unit of Evolution is EE = Earth Evolution = 25.575 bit

682

find e MGBM (t) e

2 σG BM 2

(t−ts)

= N0 eμG B M (t−ts) .

(23)

Solving this equation for M GBM (t) yields   σG2 B M (t − ts). MGBM (t) = ln N0 + μG B M − 2

(24)

This is (with ts = 0) just the “mean value showing at the exponent” of the well-known GBM pdf, i.e. 

GBM(t) pd f (n; N0 , μ, σ, t) =

e

−

   2 2 ln(n)− lnN0 + μ− σ2 t 2σ 2 t

√ √ 2π σ tn

, (n ≥ 0).

(25)

We conclude this short description of the GBM as the exponential sub-case of the general lognormal process (2) by warning that “GBM” is a misleading name, since GBM is a lognormal process and not a Gaussian one, as the Brownian Motion is indeed. 2. As we mentioned already, another interesting particular case of the mean value function mL (t) in (3) is when it equals a generic polynomial in t starting at ts, namely 

polynomial_degree

m polynomial (t) =

ck (t − ts)k .

(26)

k=0

with ck being the coefficient of the k-th power of the time t in the polynomial (26). We just confine ourselves to mention that the case where (26) is a second-degree polynomial (i.e. a parabola in t) may be used to describe the Mass Extinctions on Earth over the last 3.5 billion years (see Ref. [7]). 3. We must also introduce the notion of b-lognormal [ln(t−b)−μ]2

e− 2σ 2 b-lognormal_pdf(t; μ, σ, b) = √ 2π σ (t − b)

(27)

holding for t ≥ b = birth, and meaning the lifetime of a living being, let it be a cell, a plant, a human, a civilization of humans, or even an ET civilization (Ref. [6], in particular pages 227–245).

7 Boundary Conditions When m(t) Is a First, Second or Third Degree …

683

7 Boundary Conditions When m(t) Is a First, Second or Third Degree Polynomial in the Time (t − ts) In Ref. [7] the reader may find a mathematical model of Biological Evolution different from the GBM model described in terms of GBMs. That is the Markov-Korotayev model, for which this author proved the mean value (1) to be a Cubic(t) i.e. a third degree polynomial in t. We summarize hereafter the key formulae proven in Ref. [7] about the case when the assigned mean value mL (t) is a polynomial in t starting at ts, that is: 

polynomial_degr ee

m L (t) =

ck (t − ts)k .

(28)

k=0

1. The mean value is a straight line. Then this straight line simply is the line through the two points (ts, Ns) and (te, Ne), that, after a few rearrangements, turns out to be: m straight_ line (t) = (N e − N s)

t − ts + N s. te − ts

(29)

2. The mean value is a parabola, i.e. a quadratic polynomial in t. Then, the equation of such a parabola reads:   t − ts t − ts m parabola (t) = (N e − N s) 2− + N s. te − ts te − ts

(30)

Equation (30) was actually firstly derived by this author in Ref. [7], pages 299– 301, in relationship to Mass Extinctions (i.e. it is a decreasing function of the time). 3. The mean value is a cubic. Then, in Ref. [7], pages 304–307 this author proved, in relation to the Markov-Korotayev model of Evolution, that the cubic mean value of the L(t) lognormal stochastic process is given by the cubic equation in t m cubic (t) = (N e − N s) ·

(t − ts)[2(t − ts)2 − 3(tMax + tmin − 2ts)(t − ts) + 6(tMax − ts)(tmin − ts)] (te − ts)[2(te − ts)2 − 3(tMax + tmin − 2ts)(te − ts) + 6(tMax − ts)(tmin − ts)]

+ N s.

(31)

Notice that, in (31) one has, in addition to the usual initial and final conditions Ns = mL (ts) and Ne = mL (te), two more “middle conditions” referring to the two instants (t M , t m ) at which the Maximum and the minimum of the cubic Cubic(t) occur, respectively:

The Evo-SETI Unit of Evolution is EE = Earth Evolution = 25.575 bit

684



tmin = time_of_the_Cubic_Minimum tMax = time_of_the_Cubic_Maximum.

(32)

8 Peak-Locus Theorem The Peak-Locus theorem is a new mathematical discovery of ours playing a central role in Evo-SETI. In its most general formulation, it holds good for any lognormal process L(t) and any arbitrary mean value mL (t). In the GBM case, it is shown in Fig. 2. The Peak-Locus theorem states that the family of b-lognormals each having its peak exactly located upon the mean value curve (1), is given by the following three equations, specifying the parameters μ(p), σ (p) and b(p), appearing in (27) as three functions of the independent variable p, the b-lognormal’s peak: that is, if rewritten directly in terms of mL (p): ⎧ ⎪ ⎪ μ( p) = ⎪ ⎨

2

eσ L p 4π[m L ( p)]2 σ L2

p

−p

σ L2 2

σ ( p) = √2πe m2 ( p) ⎪ ⎪ L ⎪ 2 ⎩ b( p) = p − eμ( p)−[σ ( p)]

.

(33)

Fig. 2 Biological exponential as the geometric LOCUS OF THE PEAKS of b-lognormals for the GBM case. Each b-lognormal is a lognormal starting at a time (t = b = birth time) and represents a different SPECIES that originated at time b of the Biological Evolution. This is CLADISTICS, as seen through the glasses of our Evo-SETI model. It is evident that, when the generic “Running blognormal” moves to the right, its peak becomes higher and higher and narrower and narrower, since the area under the b-lognormal always equals 1. Then, the (Shannon) ENTROPY of the running b-lognormal is the DEGREE OF EVOLUTION reached by the corresponding SPEQES (or living being, or a civilization, or an ET civilization) in the course of Evolution (see, for instance, Refs. [1, 9, 11, 12])

8 Peak-Locus Theorem

685

The Proof of (33) is lengthy and was given as a special pdf file (written in the language of the Maxima symbolic manipulator) that the reader may freely download in the web site of Ref. [7]. But we now present an important new result: the Peak-Locus Theorem (33) rewritten not in terms of σ L anymore, but rather in terms of the four boundary parameters supposed to be known: (ts, te, Ne, δNe). To this end, we must insert (21) and (20) into (33), with the result ⎧ ⎪ ⎪ ⎪ ⎪ μ( p) = ⎪ ⎨

 p 2 te−ts 1+( δNNee )

− ln

4π[m L ( p)]2  p 2 2(t−ts) 1+( δNNee ) √ 2π m L ( p) μ( p)−[σ ( p)]2

  p  2 2(t−ts) 1 + δNNee

⎪ ⎪ σ ( p) = ⎪ ⎪ ⎪ ⎩ b( p) = p − e

.

In the particular GBM case, the mean value is (22) with μGBM = B, σ L = and N 0 = Ns = A. Then, the Peak-Locus theorem (33) with ts = 0 yields: ⎧ 1 ⎪ ⎨ μ( p) = 4π A2 − Bp, 1 σ = √2π A , . ⎪ ⎩ b( p) = p − eμ( p)−σ 2

(34)

√

2B

(35)

In this simpler form, the Peak-Locus theorem was already published by the author in Refs. [4–6], while its most general form is (33) and (34).

9 EvoEntropy(p) as Measure of Evolution The (Shannon) Entropy of the b-lognormal (27) is H ( p) =

  √ 1 1 ln( 2π σ ( p)) + μ( p) + . ln(2) 2

(36)

This is a function of the peak abscissa p and is measured in bits, as in Shannon’s Information Theory. By virtue of the Peak-Locus Theorem (33), it becomes   2 eσ L p 1 1 H ( p) = − ln(m L ( p)) + . ln(2) 4π [m L ( p)]2 2

(37)

One may also rewrite (37) directly in terms of the four boundary parameters (ts, te, Ne, δNe) upon inserting (21) into (37), with the result:

686

The Evo-SETI Unit of Evolution is EE = Earth Evolution = 25.575 bit

⎧ p  δ N e 2 te−ts ⎪ ⎨ 1 + Ne 1 H ( p) = − ln(m L ( p)) + ⎪ 4π [m L ( p)]2 ln(2) ⎩

⎫ ⎪ 1⎬ . 2⎪ ⎭

(38)

Thus, (37) or (38) yield the Entropy of each member of the family of ∞1 blognormals (the family’s parameter is p) peaked upon the mean value curve (1). The b-lognormal Entropy (37) is thus the Measure of the Amount of Evolution of that b-lognormal: it measures “the decreasing disorganization in time of what that b-lognormal represents”, let it be a cell, a plant, a human or even a civilization. Entropy is thus disorganization decreasing in time. However, that one would prefer to use a measure of the “increasing organization” in time. The EvoEntropy of p EvoEntropy( p) = −[H ( p) − H (ts)]

(39)

(Entropy of Evolution) is a function that has a minus sign in front, thus changing the decreasing trend of the (Shannon) Entropy (36) into the increasing trend of our EvoEntropy (39). This “need to change the sign” in front of (38) is actually an “old story” started in 1944 by the famous physicist Erwin Schrodinger and continued in 1953 by the French-American physicist Leon Brillouin when he coined the word Negentropy to mean entropy with the reversed sign. More at the site https://en.wik ipedia.org/wiki/Negentropy. In addition, our EvoEntropy starts at zero at the initial time ts, as expected: EvoEntropy(ts) = 0.

(40)

Summarizing, the EvoEntropy of the PEAK TIME p of the “running b-lognormal” (i.e. the PEAK TIME p of the generic b-lognormal defining the generic Species in time) is defined by: EvoEntropy( p) = −[H ( p) − H (ts)]

(41)

and this EvoEntropy has the property of starting at the time of the origin of life on Earth and then increasing thereafter: EvoEntropy(ts) = 0.

(42)

By virtue of (37), and keeping (8) in mind, the EvoEntropy (41) becomes EvoEntropy( p)_of_ the_ Lognormal_ Process_ L(t)    2 2 eσL ts eσ L p m L ( p) 1 − + ln . = ln(2) 4π N s 2 Ns 4π [m L ( p)]2

(43)

9 EvoEntropy(p) as Measure of Evolution

687

Alternatively, we may rewrite (43) directly in terms of the five boundary parameters (ts, Ns, te, Ne, δNe) upon inserting (21) into (43), thus finding: EvoEntropy( p)_of_ the_ Lognormal_ Process_ L(t) ⎧ ⎫ p ts   δ N e 2 te−ts  δ N e 2 te−ts ⎪ ⎪   ⎨ 1 + Ne 1 + Ne m L ( p) ⎬ 1 − + ln = . ⎪ ln(2) ⎪ 4π N s 2 Ns 4π [m L ( p)]2 ⎩ ⎭

(44)

Let us now remark that the standard deviation at the end time, δNe, really is irrelevant to compute the EvoEntropy (44). In fact, the EvoEntropy (44) is just a continuous curve, and not a stochastic process. Keeping this in mind, we see that the “true” EvoEntropy curve (44) is obtained by “squashing down” (44) into the mean value curve mL (t), and that only happens if we let δ N e = 0.

(45)

Inserting (45) into (44), the latter simplifies dramatically into EvoEntropy( p)_ of_ the_ Lognormal_ Process_ L(t)    1 1 m L ( p) 1 − + ln = ln(2) 4π N s 2 4π [m L ( p)]2 Ns

(46)

which is the final form of the EvoEntropy (43) and (44) that we will use in the sequel. We may now see very neatly that the final EvoEntropy (46) is made up by three terms: 1. The constant term 1 4π N s 2

(47)

whose numeric value in the particularly important case Ns = 1 boils down to 1 = 0.07958 4π

(48)

1 4π [m L ( p)]2

(49)

that is “almost zero”. 2. The “inverse square” term −

that rapidly falls down to zero for mL (t) approaching infinity. In other words, this inverse-square term may become “almost negligible” for large values of the time p.

688

The Evo-SETI Unit of Evolution is EE = Earth Evolution = 25.575 bit

3. And finally the “dominant term”, i.e. the logarithmic term 

m L ( p) ln Ns

 (50)

that actually is the leading term (“dominant term”) in the EvoEntropy (46) for large values of the time p. In conclusion, the EvoEntropy (46) in essence depends basically upon its logarithmic term (50) and so its shape in time must be similar to the shape of a logarithm i.e. nearly vertical at the beginning of the curve and then progressively approaching the horizontal shape. This curve has no maxima nor minima, nor inflexions.

10 Perfectly Linear EvoEntropy When the Mean Value Is Perfectly Exponential (GBM): This Is Just the Molecular Clock! In the GBM case (22), that is when the mean value is given by the exponential m G B M (t) = N se

σ L2 (t − ts) = N se B(t−ts) 2

(51)

the EvoEntropy (43) becomes just an exact linear function of the time p since the first two terms inside the braces in (43) just cancel against each other. Proof: just insert (51) into (43) and then simplify: EvoEntropy( p)_of_ GBM ⎧ ⎫ ⎪ ⎪ ⎛ ⎞ ⎪ ⎪ 2 σL ⎪ ⎪ 2 2 ⎬ 2 ( p−ts) N se eσ L p 1 ⎨ eσL ts ⎝ ⎠ − + ln =   2 ⎪ ln(2) ⎪ 4π N s 2 Ns σ L2 ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ 4π N se 2 ( p−ts)    2 2 2 σL 1 eσL ts eσ L p ( p−ts) = − + ln e 2 2 ln(2) 4π N s 2 4π N s 2 eσL ( p−ts)   2 eσL ts 1 σ L2 1 ( p − ts) − + = 2 ln(2) 4π N s 2 2 4π N s 2 eσL (−ts)   2 2 1 eσL ts eσL ·(ts) σ L2 = ( p − ts) − + ln(2) 4π N s 2 4π N s 2 2  2  σL 1 1 {B · ( p − ts)}. = ( p − ts) = ln(2) 2 ln(2)

(52)

10 Perfectly Linear EvoEntropy When the Mean Value Is Perfectly Exponential …

689

In other words, the GBM EvoEntropy is given by GBM_ EvoEntropy( p) =

B · ( p − ts). ln(2)

(53)

This is, of course, a straight line in the time p starting at the time ts of the Origin of Life on Earth and increasing linearly thereafter. It is measured in bits/individual and is shown in Fig. 3. But … THIS IS THE SAME LINEAR BEHAVIOUR IN TIME AS THE MOLECULAR CLOCK! And the Molecular Clock is the technique in molecular evolution that uses fossil constraints and rates of molecular change to deduce the time in geologic history when two species or other taxa diverged. The molecular data used for such calculations are usually nucleotide sequences for DNA or amino acid sequences for proteins (see Refs. [2, 10]). So, we have discovered that the EvoEntropy of the GBM exponential (i.e. of the mean Biological Evolution increasing exponentially in the last 3.5 billion years) and the Molecular Clock are the same linear function of the time (here denoted by p),

Fig. 3 EvoEntropy (in bits per individual) of the latest Species appeared on Earth during the last 3.5 billion years if the mean value is an increasing exponential, i.e. if our lognormal stochastic process L(t) is a GBM. This straight line shows that a Man (nowadays) is 25.575 bits more evolved than the first form of life (RNA) that started 3.5 billion years ago

The Evo-SETI Unit of Evolution is EE = Earth Evolution = 25.575 bit

690

apart for multiplicative and additive constants depending on the adopted units, like bits, seconds, etc. This conclusion appears to be of key importance to understand “where a newly discovered exoplanet stands on its way to develop LIFE”.

11 Introducing the EE Evo-SETI Unit: Information Equal to the EvoEntropy Reached by the Evolution of Life on Earth Nowadays The purpose of this paper really is to propose our new Evo-SETI UNIT of evolution. From the above discussion it follows that the numeric value of the Evo-SETI unit is about 25.575 bits if life on Earth started 3.5 billion years ago. This unit we propose to call EE (EvoEntropy), so that we define EE as E E = Evo_ SETI_ unit_ of_ Evolution_ as_ the_ difference_ in_I n f or mation_Content_between_R N A_and_H umans = 25.575 bits.

(54)

Thus, a planet like Mars will have an Evo-SETI evolution much less than 1 EE in case it hosted any primitive form of life, even in the past. And just 0 EE in case it did not host any form of life at all. On the contrary, an exoplanet hosting an ExtraTerrestrial Civilization much more advanced than the Human one will have a EE larger or much larger than 1 according to the higher degree of Evolution reached by that Civilization when compared to Humans. This is our Evo-SETI SCALE quantifying the Evolution of Life in the Universe and, at the same time, proving once again that the Molecular Clock discovered in 1962 by Emil Zuckerkandl (1922–2013) and Linus Pauling (1901–1994) is indeed a correct, fundamental law of nature.

12 Conclusions The Biological Evolution of life on Earth over the last 3.5 to roughly 4 billion years has hardly been cast into any “profound” mathematical form. The molecular clock is an exception in that Zuckerkandl and Pauling cast it in a “straight line” form, i.e. in the easiest possible geometrical form. Since 2012 this author has tried to do profound mathematics about the evolution of life on earth by resorting to lognormal probability distributions in the time, starting each at a different time instant b (birth) and called b-lognormals (Refs. [3–8, 10]). His discovery (in the years 2010–2015) of the Peak-Locus Theorem valid for any enveloping mean value (and not just the exponential one (GBM), see the Appendix 1

12 Conclusions

691

to Ref. [8]) has made it possible the use of the Shannon Entropy of Information Theory as the correct mathematical tool measuring the Evolution of life in bits/individual. In conclusion, what happened on Earth over the last 4 billion years is now summarized by a few simple statistical equations, but is just about the evolution of life on Earth, and not on other Exoplanets. The extension of our Evo-SETI Theory to life on other Exoplanets will be possible only when SETI, the current scientific Search for ExtraTerrestrial Intelligence, will achieve the first Contact between Humans and an Alien Civilization.

13 Appendix A: Supplementary Data Supplementary data related to this article can be found at https://dx.doi.org/10.1016/ j.actaastro.2018.05.010.

References 1. J. Felsenstein, Inferring Phylogenies (Sinauer Associates Inc., Sunderland, Massachusetts, 2004) 2. https://en.wikipedia.org/wiki/Molecular_clock 3. C. Maccone, The statistical Drake equation. Acta Astronaut. 67, 1366–1383 (2010) 4. C. Maccone, A mathematical model for evolution and SETI. Orig. Life Evol. Biospheres 41, 609–619 (2011) 5. C. Maccone, Mathematical SETI (Praxis-Springer, Zürich, Fall of 2012), 724 p. ISBN, ISBN10: 3642274366. ISBN-13: 978-3642274367 6. C. Maccone, SETI, evolution and human history merged into a mathematical model. Int. J. Astrobiol. 12(3), 218–245 (2013) 7. C. Maccone, Evolution and mass exctinctions as lognormal stochastic processes. Int. J. Astrobiol. 13(4), 290–309 (2014) 8. C. Maccone, New evo-seti results about civilizations and molecular clock. Int. J. Astrobiol. 16, 40–59 (2017) 9. T. Maruyama, Stochastic Problems in Population Genetics. Lecture Notes in Biomathematics #17 (Springer, Berlin, 1977) 10. C. Maccone, Kurzweil’s singularity as a part of evo-seti theory. Acta Astronaut. 132, 312–325 (2017) 11. M. Nei, K. Sudhir, Molecular Evolution and Phylogenetics (Oxford University Press, Oxford, 2000) 12. M. Nei, Mutation-Driven Evolution (Oxford University Press, Oxford, 2013)

Evo-SETI Quartics Yielding ET Civilizations’ Energy

Abstract Evo-SETI Theory is a mathematical model to compute both the energy and the information amount (=Shannon entropy) available to advanced ET Civilizations. This mathematical model was gradually developed by this author over the last 10 years in a dozen mathematical papers published in Acta Astronautica, International Journal of Astrobiology and Life. In the present chapter we explore for the first time the Evo-SETI “Quartics”, i.e. polynomials of the fourth degree in the time yielding an advanced ET Civilization’s power curve in between the Civilization’s birth and death. Alternatively, this Quartic curve might represent the power needed by a Living Being to survive for the whole of its lifetime. Then the integral of such a curve in the time is of course a polynomial of the fifth degree representing the energy absorbed by that Civilization or Living Being. For instance, we know that the Sun was born about 4.5 billion years ago and will presumably reach its death as a red giant in, say, four to five billion more years. Thus, one might firstly compute the Quartic power curve of the Sun over about ten billion years of time. Then the integral (total area) under such a curve is the Sun’s total energy output over 10 billion years. Computing how much of that energy has reached and will reach the Earth over the Sun’s lifetime will give us an upper bound for the energy available to humans now and in the future. In other words, we may try to quantify the advancement of the Human Civilization in terms of energy and, consequently, in terms of Shannon Entropy (= Information), assuming that this EvoEntropy is a stochastic process with exponential mean value (Geometric Brownian Motion = GBM), as we always did in Evo-SETI Theory. The same mathematical model applies to all stars, and so to all ET civilizations as well. Keywords SETI · Quartics · Energy · Civilizations · Life

1 Introduction to Evo-SETI Theory Evo-SETI (Evolution and SETI) is a mathematical model developed by this author in the last ten years in a series of mathematical papers published in journals like “Life”, the “International Journal of Astrobiology” and “Acta Astronautica”. The list of these papers is found in Ref. [1] at the end of this paper and references therein. © Springer Nature Switzerland AG 2020 C. Maccone, Evo-SETI, https://doi.org/10.1007/978-3-030-51931-5_21

693

694

Evo-SETI Quartics Yielding ET Civilizations’ Energy

The goal of Evo-SETI theory is to mathematically describe the evolution of the universe since the Big Bang and up to humans as we know it scientifically. This is obtained by assuming that the evolution of, say, Living Species in the sense of Biology is represented mathematically by b-lognormals, i.e. probability densities in the time running in between the birth time b and the death time d. This author discovered a theorem, that he calls “Peak-Locus Theorem”, showing that the geometric locus of all b-lognormals in time corresponds to the exponential increase in time of the number of Living Species on Earth. However, understanding that mathematics is not necessary in order to understand this paper, that is thus self-contained herewith.

2 Quartic Curve in the Time t Representing the Power (Measured in Watts) of a Civilization Between Its Inception (b = Birth) and Its End (d = Death) A Quartic (or “Quartic function”) is a polynomial of the fourth degree in its independent variable (website https://en.wikipedia.org/wiki/Quartic_function). As such, every Quartic has five coefficients of the different five power terms of its own independent variable. That is, denoting by t the independent variable (i.e. time, in this book), one has by definition poly4(t) ≡ at 4 + bt 3 + ct 2 + dt + e.

(1)

In (1) a, b, c, d, e are the five real coefficients. In order to determine these five coefficients, we need five equations. This paper is devoted to finding these five equations and solving them for the five unknown coefficients a, b, c, d, e, a rather uneasy task since the resulting equations are usually awfully long.

3 Conditions Imposed on Our Quartic Power Curve We start by considering a polynomial of the fourth degree in the time t with t constrained to range only in between the “birth-time” b and the “death time” d, that is b ≤ t ≤ d.

(2)

In Evo-SETI Theory, this polynomial represents the power (measured in Watts) of a Civilization or of a Living Being as long as the time elapses. It is thus natural to assume that t starts at the value zero at the initial time b (standing for “birth time”) and ends at zero at the death time d. That is, we assume that this polynomial (denoted by poly4(t) in the sequel) has zero value at each of the two boundary conditions (2),

3 Conditions Imposed on Our Quartic Power Curve

695

that is 

poly4(b) = 0 bir th = initial condition . poly4(d) = 0 death = final condition

(3)

4 Separating Our Quartic Curve into the Product of the Two Above Boundary Conditions Times a Quadratic Polynomial in the Time Because of the two boundary conditions (2) and (3) we may now separate the fourthdegree polynomial into the product of the following three parts: 1. The monomial where t ≥ b multiplied by… 2. The monomial where t ≤ d multiplied by… 3. Multiplied by a second-degree polynomial with three unknown coefficients A, B, C. This separated polynomial (4) is the first step to avoid writing every time the long sequence of five summed monomials, each having a different degree in t. In other words, we may rewrite our poly4(t) as   poly4(t) = (t − b)(d − t) At 2 + Bt + C .

(4)

This Eq. (4) we like to call the “Separated Quartic”. Here A, B, C are the three unknown coefficients of the quadratic part 

At 2 + Bt + C



(5)

of the Quartic (4) the we must now determine. How? The answer is that we need three more equations in order to determine A, B, C. In the next section we prove that two more equations are provided by the derivatives of the Quartic at b and d, respectively. One more explanation about the work we are doing is now necessary. This is about the NASA computer algebra code “Maxima” that we used to do all the calculations described in this paper. Maxima is downloadable for free at the site https://max ima.sourceforge.net/ and here https://en.wikipedia.org/wiki/Maxima_(software) are a few descriptions of its strong mathematical capabilities. But Maxima also has its own way of writing equations “alphabetically”: for instance Eq. (4) is rewritten by Maxima in the form   poly4(t) = (t − b)(t − d) C + Bt + At 2

(6)

696

Evo-SETI Quartics Yielding ET Civilizations’ Energy

that differs from (4) just because of a minus sign. This sign difference does not matter at all since, at the moment, we did not yet work out the coefficients of the Quartic (1). Having so said, we will use (6) rather than (4) for all subsequent calculations in this paper. The Appendix to this paper contains the full Maxima code written by this author just for this chapter.

5 Expressing the Separated Quartic in Terms of b, d and the Derivatives Db and Dd of this Quartic at the Initial and End Times, Respectively The derivative of (6) with respect to the time t is given by the sum of three terms:     dpoly4(t) = (t − b) At 2 + Bt + C + (t − d) At 2 + Bt + C dt + (t − b)(t − d)(2 At + B).

(7)

Therefore, the first derivative of poly4(t) at t = b is found by letting t = b into (7) and reads   dpoly4(b) ≡ Db = (b − d) Ab2 + Bb + C . dt

(8)

Similarly, the first derivative of poly4(t) at t = d is found by letting t = d into (7) and reads   dpoly4(d) ≡ Dd = (d − b) Ad 2 + Bb + C . dt

(9)

Dividing (8) by (9) we get Db Ab2 + Bb + C =− 2 . Dd Ad + Bb + C

(10)

We now solve the last Eq. (10) for C thus obtaining an expression yielding C in terms of the four known quantities b, d, Db and Dd, plus the unknown A. We shall not write down all the relevant, lengthy steps, and just report here the final result, reading:   (Ddb + Dbd)B + Ddb2 + Dbd 2 A C =− Db + Dd

(11)

5 Expressing the Separated Quartic in Terms of b, d and the Derivatives …

697

On the other hand, another [and unrelated to (11)] expression of C is found upon solving (8) for C, and reads, after a few steps C =−

Bb(b − d) + Ab2 (b − d) − Db b−d

(12)

If one now divides (11) by (12), C disappears from the resulting equation, and this resulting equation contains A and B only, all expressed in terms of the known four quantities b, d, Db and Dd plus A. We don’t write this resulting equation since it’s quite lengthy. However, if this lengthy equation is solved for B, then one gets: B=−

  A d 3 − bd 2 − d 2 tb + d 3 − Db − Dd (b − d)2

.

(13)

At last, we succeeded in finding the two Eqs. (13) and (12) yielding the two unknowns B and C in terms of the four known quantities ts, te, Dts and Dte plus the unknown A. The next and “final” step is obviously to insert both (12) and (13) into the separated Quartic (4) and rearrange the so-obtained Quartic in decreasing powers of t from t 4 down to t 0 , that is the known term of the Quartic. These calculations are awfully lengthy again and we won’t show them, but the results are the five coefficients of the full Quartic (1) that we describe in the next section.

6 The General Quartic and Its Five Coefficients Expressed in Terms of b, d and the Derivatives Db and Dd of the Quartic at the Initial and End Times Hereafter are all the five Quartic coefficients expressed in terms of the four known quantities b, d, Db and Dd plus the coefficient A of the highest-power term, t 4 : clearly, if A > 0 then the Quartic approaches +∞ for t → ±∞, and viceversa −∞ if A < 0. The latter case won’t be of interest to us since the power of a Civilization is obviously a positive curve in the time. So we assume A > 0 from now on, and, in conclusion (and without any lengthy proof) we get the five coefficients: ⎧ General Quartic coefficient of t 4 ⎪ ⎪ ⎪ ⎪ General Quartic coefficient of t 3 ⎪ ⎪ ⎨ General Quartic coefficient of t 2 ⎪ ⎪ General Quartic coefficient of t 1 ⎪ ⎪ ⎪ ⎪ ⎩ General Quartic coefficient of t 0

= A>0 = Db+Dd 2 − 2 A(b + d) (d−b)   2 = A b + 4bd + d 2 − =

d Dd+2bDd+2d Db+bDb (d−b)2

2bd Dd+b2 Dd+d 2 Db+2bd Db (d−b)2

− 2 Abd(b + d)

Db = known term = bd bd A − bDd+d 2 (d−b)

.

(14)

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Evo-SETI Quartics Yielding ET Civilizations’ Energy

7 Energy of the General Quartic, i.e. Area Under the General Quartic and the Time Axis, i.e. Integral of the Quartic Between b and d Since the Quartic is the power curve of a given Civilization between birth and death, or the total energy used by a Living Being between birth and death, the area under that power curve is the Energy on that Civilization or Living Being. In other words, the Energy needed by that Civilization or Being for the whole of its own lifetime between birth and death is the definite integral of the Quartic between birth and death: d

d Quartic(t)dt =

Energy = b

poly4(t)dt.

(15)

b

We thus find and compute the following definite integral: d



(d − b)2 2 A(d − b)3 − 5(Dd − Db) poly4(t)dt = 60

(16)

b

yielding the total energy needed (i.e. produced) by that Civilization in order to survive throughout its whole lifetime, or the same for a Being during its lifetime. Equation (16) is the most important new result of this paper (Fig. 1).

8 Defining the Smooth-Start Quartic, That Is the Quartic with Zero Derivative at b In order to apply Evo-SETI Theory to Biology and History, it is important to consider the case of a “Smooth-Start Quartic”, that is the special case of (14) when the first derivative of the Quartic at birth time b vanishes. This means a “smooth transition from no-life into life”. Let us thus insert Db = 0 into (14). The result is (Fig. 2)

(17)

8 Defining the Smooth-Start Quartic, That Is the Quartic with Zero Derivative …

699

Fig. 1 Plot of a general quartic. Suppose b = 1 and d = 3 as an example of general (= arbitrary) quartic in t. The above plot shows that the general quartic in t is TOO GENERAL for our Evo-SETI models of a living being or a civilization, because the quartic power curve may become NEGATIVE and so the total energy is FALSE, being the ALGEBRAIC SUM of a positive (= above time axis) plus negative (= below time axis) parts

Fig. 2 Smooth-start quartic. Suppose b = 1 and d = 5, Dd = −100 and A = 10. The above graph shows that at t = b = 1 the tangent to the Quartic is indeed horizontal, but the tangent is not horizontal at t = d = 5. In loose terms, this means “smooth birth but not-so-smooth death”: a useful feature to represent a host of applications of our quartic power curves to biology and history

700

Evo-SETI Quartics Yielding ET Civilizations’ Energy

⎧ Smooth-Start Quartic coefficient of t 4 ⎪ ⎪ ⎪ ⎪ Smooth-Start Quartic coefficient of t 3 ⎪ ⎪ ⎨ Smooth-Start Quartic coefficient of t 2 ⎪ ⎪ Smooth-Start Quartic coefficient of t 1 ⎪ ⎪ ⎪ ⎪ ⎩ Smooth-Start Quartic coefficient of t 0

=A Dd = (d−b) 2 − 2 A(b + d)   2 = A b + 4bd + d 2 − =

2

d Dd+2bDd (d−b)2

− 2 Abd(b + d)

bDd = known_ term = bd bd A − (d−b) 2 2bd Dd+b Dd (d−b)2

.

(18)

9 Energy of the Smooth-Start Quartic The Energy of the Smooth-Start Quartic is immediately found by inserting (17) into (16) and reads d

(d − b)2 2 A(d − b)3 − 5Dd Smooth − Start_ Quartic(t)dt = 60

(19)

b

This also is a very important new result, next to (16).

10 Defining the Symmetric Quartic, i.e. When the Derivatives Db and Dd of the Quartic at the Initial and End Times are Equal to Each Other We call Symmetric Quartic the particular case of the General Quartic described by (14) when the derivatives Db and Dd of the General Quartic at the initial and end times are equal to each other: Dd = Db

(20)

Then, Eq. (14) get easier and read, respectively: ⎧ ⎪ Symmetric Quartic coefficient of t 4 ⎪ ⎪ ⎪ ⎪ Symmetric Quartic coefficient of t 3 ⎪ ⎪ ⎨ Symmetric Quartic coefficient of t 2 ⎪ ⎪ ⎪ Symmetric Quartic coefficient of t 1 ⎪ ⎪ ⎪ ⎪ ⎩ Symmetric Quartic coefficient of t 0

=A 2Db = (d−b) 2 − 2 A(b + d)   2 = A b + 4bd + d 2 − 3Db(b+d) . (d−b)2 Db(b2 +4bd+d 2 ) = − 2 Abd(b + d) 2 (d−b)

= known term = bd bd A − Db(b+d) (d−b)2 (21)

11 Energy of the Symmetric Quartic, i.e. Area Under the Symmetric Quartic …

701

11 Energy of the Symmetric Quartic, i.e. Area Under the Symmetric Quartic, i.e. Integral of the Symmetric Quartic Between b and d The Energy of the Symmetric Quartic is obviously found upon inserting Db instead of Dd into (16). It thus happens that the second term inside the braces cancels out, and one is just left with: d Energy of the Symmetric Quartic =

Symmetric_ Quartic(t)dt b

=

A(d − b)5 . 30

(22)

In view of the applications, (22) again is one of the most important new discoveries published in this paper.

12 The Special Case of the Symmetric Quartic with Zero Derivatives at Both b and d For applications to Living Beings, it is important to assume that both derivative of the Quartic at birth and death are zero: 

Db = 0 Dd = 0

(23)

Inserting (23) into (21), one then finds that the coefficients of the Quartic with zero derivative at both b and d get easier once more, and are given by ⎧ Symmetric Quartic with zero derivatives at birth and death, coefficient of t 4 ⎪ ⎪ ⎪ ⎪ 3 ⎪ ⎨ Symmetric Quartic with zero derivatives at birth and death, coefficient of t Symmetric Quartic with zero derivatives at birth and death, coefficient of t 2 ⎪ ⎪ ⎪ Symmetric Quartic with zero derivatives at birth and death, coefficient of t 1 ⎪ ⎪ ⎩ Symmetric Quartic with zero derivatives at birth and death, coefficient of t 0

=A = −2 A(b + d)   = A b2 + 4bd + d 2 . = −2 Abd(b + d) = Ab2 d 2

(24)

In addition to (24), we just state that, for the easy case (24), the Quartic peak is in the middle point between b and d Symmetric Quadric peak time =

b+d . 2

(25)

702

Evo-SETI Quartics Yielding ET Civilizations’ Energy

Fig. 3 An example of a symmetric quartic with b = 1, d = 3, Db = 0, Dd = 0, A = 10

Also, the two inflexion points for the Symmetric Quartic with zero derivatives at b and d occur at the times √ √  3−3 d+ − 3−3 b left inflection = − ( √ ) 6( √ ) . (26) 3+3)d+(3− 3)b right inflection = ( 6

Figure 3 shows an example of a Symmetric Quartic with b = 1, d = 3, Db = 0, Dd = 0, A = 10.

(27)

13 Energy of the Symmetric Quartic with Zero Derivatives at Both b and d The Energy of the Symmetric Quartic with zero derivatives at both b and d is derived from (22). However, since Db does not appear in (22), the result is just the same, that is Energy of the Symmetric Quadric with zero derivatives at both b and d d =

Symmetric_ Quadric(t)dt b

13 Energy of the Symmetric Quartic with Zero Derivatives …

=

703

A(d − b)5 30

(28)

14 Determining the Last Unknown, A, If One Knows the Energy that a Civilization or a Living Being Used (or Produced) During Its Own Lifetime In the practical applications of our Quartics, other scientists will have estimated the total Energy that a Civilization or a Living Being used (or produced) during its own Lifetime. Thus, solving (28) for A shows how A is actually expressed in terms of that known Energy: A=

30 (d − b)5

 ·

Energy of the Symmetric Quadric with zero derivatives at both b and d

 (29)

15 Conclusions We have done new research work by using Quartics to represent the power curve (measured in Watts) of a Civilization or of a Living Being. The integral of such a power curve in the time is obviously the Energy that this Civilization or Living Being consumed in its whole lifetime. In SETI, one might estimate the energy that the central star pours upon its planets and so this paper might become a good mathematical model for the Lifetime of a Civilization in SETI. Alternatively, this paper might help to estimate how much energy a Living Being needs to live its whole life, thus providing a valuable contribution to Astrobiology.

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Evo-SETI Quartics Yielding ET Civilizations’ Energy

Appendix

QUARTICS as Power Curves modelling Civilizations and Living Beings. ENERGY as their integral over their lifetime. This Maxima file was completed by Claudio Maccone (e-mail: [email protected]) on September 12, 2019, for presentation ar the International Astronautical Congress (IAC) in Washington DC, SETI 2 Session, on Tuesday, October 22, 2019, in the afternoon. Clearing Maxima's memory from all previous calculations. (%i90) kill(all); (%o0) done Maxima is aware that b t ] CRUCIAL STEP: SEPARATING THE QUADRIC into the product of three time factors. (%i2) Separated_Quadric:(t-b)*(t-d)*(A*t^2+B*t+C); (%o2) ( t - b ) ( t - d ) ( C + t B + t 2 A ) Derivative of the Separated Quadric with respect to the time. (%i3) Separated_Quadric_derivative:diff(Separated_Quadric,t); (%o3) ( t - d ) ( C + t B + t 2 A ) + ( t - b ) ( C + t B + t 2 A ) + ( t - b ) ( t - d ) ( B + 2 t A ) Defining Db = derivative of the Separated Quadric at birth t=b. (%i4) Separated_Quadric_derivative_in_b:Db=subst(b,t,Separated_Quadric_derivative); (%o4) Db = ( b - d ) ( C + b B + b 2 A ) Equation yielding C in terms of b, d, Db, Dd. (%i5) C_vs_A_and_B:first(solve(Separated_Quadric_derivative_in_b,C)); (%o5) C = -

( b d - b 2 ) B + ( b 2 d - b 3 ) A + Db d -b

Eliminating C from the Quadric equation.

Appendix

705

(%i6) Separated_Quadric_with_A_and_B:Separated_Quadric,C_vs_A_and_B; (%o6) ( t - b ) ( t - d ) -

( b d - b 2 ) B + ( b 2 d - b 3 ) A + Db d -b

+ t B + t2 A

Defining Dd = derivative of the Separated Quadric at death t=d. (%i7) Separated_Quadric_derivative_in_d:Dd=subst(d,t,Separated_Quadric_derivative); (%o7) Dd = ( d - b ) ( C + d B + d 2 A ) (%i8) Db_over_Dd:radcan(Separated_Quadric_derivative_in_b/Separated_Quadric_derivative_in_d); (%o8)

Db Dd

=-

C + b B + b2 A C+d B+d2 A

One more expression of C as a function of A and B, different from the previous one. (%i9) new_C_vs_A_and_B:first(solve(Db_over_Dd,C)); (%o9) C = -

( b Dd + d Db ) B + ( b 2 Dd + d 2 Db ) A Dd + Db

Eliminating C among its two and different previous expressions. (%i10) only_A_and_B:radcan(C_vs_A_and_B/new_C_vs_A_and_B); (%o10) 1 =

( ( b d - b 2 ) Dd + ( b d - b 2 ) Db ) B + (( b 2 d - b 3 ) Dd + ( b 2 d - b 3 ) Db ) A + Db Dd + Db 2 ( ( b d - b 2 ) Dd + ( d 2 - b d ) Db ) B + (( b 2 d - b 3 ) Dd + ( d 3 - b d 2 ) Db ) A

Resulting B as a function of b, d, Db and Dd only (plus A, that we did not touch). (%i11) B_vs_A:factor(first(solve(only_A_and_B,B))); (%o11) B = -

d 3 A - b d 2 A - b 2 d A + b 3 A - Dd - Db (d - b)2

Inserting the expressions of both B and C into the Separated Quadric, only containing A. (%i12) Separated_Quadric_with_A_only:Separated_Quadric_with_A_and_B,B_vs_A; -

( b d - b 2 ) ( d 3 A - b d 2 A - b 2 d A + b 3 A - Dd - Db ) (d - b) 2

(%o12) ( t - b ) ( t - d ) ( t

(d 3

A -b

d2

A - b2

d

A + b3

(d - b)2

d -b A - Dd - Db )

+ ( b 2 d - b 3 ) A + Db

-

+ t2 A )

Re-Writing the last equation so that all the five coefficients of the Quadric become evident.

706

Evo-SETI Quartics Yielding ET Civilizations’ Energy (%i13) quartic_in_t:distrib(rat(Separated_Quadric_with_A_only,t)); (%o13)

t ( (- 2 b d 4 + 2 b 2 d 3 + 2 b 3 d 2 - 2 b 4 d ) A + ( 2 b d + b 2 ) Dd + ( d 2 + 2 b d ) Db ) d 2 - 2 b d + b2

t 2 ( (d 4 + 2 b d 3 - 6 b 2 d 2 + 2 b 3 d + b 4 ) A + ( - d - 2 b ) Dd + ( - 2 d - b ) Db ) d 2 - 2 b d + b2 t 3 ( (- 2 d 3 + 2 b d 2 + 2 b 2 d - 2 b 3 ) A + Dd + Db ) d2-2

b

d + b2

+ t4 A +

+

(b2 d 4 - 2 b3 d 3 + b4 d 2) A d2-2

b

d + b2

-

b 2 d Dd d2-2

b d 2 Db d 2 - 2 b d + b2

t4 coeff (%i14) t4_coeff:coeff(quartic_in_t,t,4); (%o14) A t3 coeff (%i15) t3_coeff:factor(coeff(quartic_in_t,t,3)); (%o15) -

2 d 3 A - 2 b d 2 A - 2 b 2 d A + 2 b 3 A - Dd - Db (d - b)2

(%i16) factor(-(2*b^3*A-2*d*b^2*A-2*d^2*b*A+2*d^3*A)); (%o16) - 2 ( d - b ) 2 ( d + b ) A (%i17) easy_t3_coeff:-2*(d+b)*A+(Dd+Db)/(d-b)^2; (%o17)

Dd + Db (d - b)2

- 2 (d + b) A

(%i18) radcan(t3_coeff-easy_t3_coeff); (%o18) 0 t2 coeff (%i19) t2_coeff:factor(coeff(quartic_in_t,t,2)); (%o19)

d 4 A + 2 b d 3 A - 6 b 2 d 2 A + 2 b 3 d A + b 4 A - d Dd - 2 b Dd - 2 d Db - b Db (d - b)2

(%i20) factor(b^4*A+2*d*b^3*A-6*d^2*b^2*A+2*d^3*b*A+d^4*A); (%o20) ( d - b ) 2 ( d 2 + 4 b d + b 2 ) A (%i21) factor(-Db*b-2*Dd*b-2*Db*d-Dd*d); (%o21) - ( d Dd + 2 b Dd + 2 d Db + b Db )

+

b d + b2

-

Appendix

707

(%i22) easy_t2_coeff:(d^2+4*b*d+b^2)*A-(d*Dd+2*b*Dd+2*d*Db+b*Db)/((d-b)^2); (%o22) ( d 2 + 4 b d + b 2 ) A -

d Dd + 2 b Dd + 2 d Db + b Db (d - b)2

(%i23) radcan(t2_coeff-easy_t2_coeff); (%o23) 0 t1 coeff (%i24) t1_coeff:factor(coeff(quartic_in_t,t,1)); (%o24) -

2 b d 4 A - 2 b 2 d 3 A - 2 b 3 d 2 A + 2 b 4 d A - 2 b d Dd - b 2 Dd - d 2 Db - 2 b d Db (d - b)2

(%i25) factor(-(2*b*d^4*A-2*b^2*d^3*A-2*b^3*d^2*A+2*b^4*d*A)); (%o25) - 2 b d ( d - b ) 2 ( d + b ) A (%i26) factor(-(-2*b*d*Dd-b^2*Dd-d^2*Db-2*b*d*Db)); (%o26) 2 b d Dd + b 2 Dd + d 2 Db + 2 b d Db (%i27) easy_t1_coeff:-2*b*d*(d+b)*A+(2*b*d*Dd+b^2*Dd+d^2*Db+2*b*d*Db)/((d-b)^2); (%o27)

2 b d Dd + b 2 Dd + d 2 Db + 2 b d Db (d - b)2

- 2 b d (d + b) A

(%i28) radcan(t1_coeff-easy_t1_coeff); (%o28) 0 t0 coeff = known term (%i29) t0_coeff:factor(coeff(quartic_in_t,t,0)); (%o29)

b d ( b d 3 A - 2 b 2 d 2 A + b 3 d A - b Dd - d Db ) (d - b)2

(%i30) factor(d*b^3*A-2*d^2*b^2*A+d^3*b*A); (%o30) b d ( d - b ) 2 A (%i31) easy_t0_coeff:b*d*(b*d*A+((-b*Dd-d*Db)/(d-b)^2)); (%o31) b d b d A +

- b Dd - d Db (d - b)2

(%i32) radcan(t0_coeff-easy_t0_coeff); (%o32) 0 Quartic in t.

708

Evo-SETI Quartics Yielding ET Civilizations’ Energy (%i33) quartic_in_t:A*t^4+easy_t3_coeff*t^3+easy_t2_coeff*t^2+easy_t1_coeff*t+easy_t0_coeff; (%o33) t 2 ( d 2 + 4 b d + b 2 ) A -

d Dd + 2 b Dd + 2 d Db + b Db

2 b d Dd + b 2 Dd + d 2 Db + 2 b d Db

b d A+

(d - b)2 - b Dd - d Db (d - b)2

(d - b)2

- 2 b d (d + b) A + t 3

+t

Dd + Db (d - b)2

- 2 (d + b) A + b d

+ t4 A

Verifying that one has quartic_in_t(b)=0 and quartic_in_t(d)=0 (%i34) radcan(subst(b,t,quartic_in_t)); (%o34) 0 (%i35) radcan(subst(d,t,quartic_in_t)); (%o35) 0 Verifying that the derivative is Db and Dd at b and d, respectively. (%i36) radcan(subst(b,t,diff(quartic_in_t,t))); (%o36) Db (%i37) radcan(subst(d,t,diff(quartic_in_t,t))); (%o37) Dd Example of GENERAL = ARBITRARY quartic_in_t. This shows that the GENERAL quartic in t is TOO GENERAL for our Evo-SETI models of Living Being and Civilization, because the quartic may become NEGATIVE and so the Energy is FALSE, being the ALGEBRAIC SUM of a positive (= above time axis) plus negative (=below time axis) parts. (%i38) q_numer:[b=1,d=3,Db=%pi,Dd=%pi,A=1]; (%o38) [ b = 1 , d = 3 , Db = π , Dd = π , A = 1 ] (%i39) q_ex:ev(quartic_in_t,q_numer,numer); (%o39) t 4 + ( 0.5 π - 8 ) t 3 + ( 22 - 3.0 π ) t 2 + ( 5.5 π - 24 ) t + 3 ( 3 - 1.0 π ) (%i40) plot2d(q_ex, [t,-0.2,3.5]); (%o40) Times of the two maxima and of the minimum in between: TOO LENGHTY !!! We won't write it!!! This is the type of LENGHTY results that must be AVOIDED when handling Quartics. And we did so :-). Times of the two inflection points.

Appendix

709

(%i41) two_inflections:(solve(diff(quartic_in_t,t,2)=0,t)); (%o41) [ t = - ( 3 sqrt( ( 4 d 6 - 24 b d 5 + 60 b 2 d 4 - 80 b 3 d 3 + 60 b 4 d 2 - 24 b 5 d + 4 b 6 ) A 2 + ( (- 4 d 3 + 12 b d 2 - 12 b 2 d + 4 b 3 ) Dd + (4 d 3 - 12 b d 2 + 12 b 2 d - 4 b 3 ) Db ) A + 3 Dd 2 + 6 Db Dd + 3 Db 2 ) + (- 6 d 3 + 6 b d 2 + 6 b 2 d - 6 b 3 ) A + 3 Dd + 3 Db ) / ( ( 12 d 2 - 24 b d + 12 b 2 ) A ) , t = ( 3 sqrt( ( 4 d 6 - 24 b d 5 + 60 b 2 d 4 - 80 b 3 d 3 + 60 b 4 d 2 - 24 b 5 d + 4 b 6 ) A 2 + ( (- 4 d 3 + 12 b d 2 - 12 b 2 d + 4 b 3 ) Dd + (4 d 3 - 12 b d 2 + 12 b 2 d - 4 b 3 ) Db ) A + 3 Dd 2 + 6 Db Dd + 3 Db 2 ) + (6 d 3 - 6 b d 2 - 6 b 2 d + 6 b 3 ) A - 3 Dd - 3 Db ) / ( ( 12 d 2 - 24 b d + 12 b 2 ) A ) ] (%i42) factor(4*d^6-24*b*d^5+60*b^2*d^4-80*b^3*d^3+60*b^4*d^2-24*b^5*d+4*b^6); (%o42) 4 ( d - b ) 6 (%i43) factor((-4*d^3+12*b*d^2-12*b^2*d+4*b^3)); (%o43) - 4 ( d - b ) 3 ENERGY. ENERGY of a Civilization or a Living Being over its whole LIFETIME, birth-to-death. This is the definite integral of the Quartic between b and d. We must just factorize the lengthy result in a couple of equivalent EASIER FORMS. (%i44) energy:radcan(integrate(quartic_in_t,t,b,d)); (%o44) ( 2 d 5 - 10 b d 4 + 20 b 2 d 3 - 20 b 3 d 2 + 10 b 4 d - 2 b 5 ) A + ( - 5 d 2 + 10 b d - 5 b 2 ) Dd + ( 5 d 2 - 10 b d + 5 b 2 ) Db 60

(%i45) factor(A*(2*d^5-10*b*d^4+20*b^2*d^3-20*b^3*d^2+10*b^4*d-2*b^5)); (%o45) 2 ( d - b ) 5 A (%i46) factor((-5*d^2+10*b*d-5*b^2)*Dd+(5*d^2-10*b*d+5*b^2)*Db); (%o46) - 5 ( d - b ) 2 ( Dd - Db ) (%i47) easy_energy:(2*A*(d-b)^5-5*(d-b)^2*(Dd-Db))/60; (%o47)

2 ( d - b ) 5 A - 5 ( d - b ) 2 ( Dd - Db ) 60

(%i48) bis_easy_energy:((d-b)^2*(2*A*(d-b)^3-5*(Dd-Db)))/60; (%o48)

( d - b ) 2 ( 2 ( d - b ) 3 A - 5 ( Dd - Db ) ) 60

(%i49) ratsimp(energy-easy_energy); (%o49) 0 (%i50) ratsimp(energy-bis_easy_energy); (%o50) 0

710

Evo-SETI Quartics Yielding ET Civilizations’ Energy

So far we just worked with the full, general Quartic, with no extra-constraint at all. But now we impose upon the Quartic the new constraint the its derivative at birth equals zero. Quartic in t with Db=0, smooth start. (%i51) Smooth_Start_Quartic:subst(0,Db,quartic_in_t); (%o51) t 2 ( d 2 + 4 b d + b 2 ) A Dd (d - b)2

d Dd + 2 b Dd

- 2 (d + b) A + b d b d A -

(d - b) b Dd (d - b)2

2

+t

2 b d Dd + b 2 Dd (d - b)2

- 2 b d (d + b) A + t 3

+ t4 A

Smooth start Quartics EXAMPLE (%i52) Smooth_Start_example:[b=1,d=5,Dd=-100,A=10]; (%o52) [ b = 1 , d = 5 , Dd = - 100 , A = 10 ] (%i53) Smooth_poly4:Smooth_Start_Quartic,Smooth_Start_example; (%o53) 10 t 4 -

505 t 3 2015 t 2 2675 t 4

+

4

-

4

+

1125 4

(%i54) plot2d(Smooth_poly4,[t,0,6]); (%o54) Finding the minimun and Maximum (= Peak) of every Smooth_Start Quadric. (%i55) min_and_Max_for_smooth_quartic:solve(diff(Smooth_Start_Quartic,t)=0,t); (%o55) [ t = - ( sqrt( ( 4 d 6 - 24 b d 5 + 60 b 2 d 4 - 80 b 3 d 3 + 60 b 4 d 2 - 24 b 5 d + 4 b 6 ) A 2 + (- 4 d 3 + 12 b d 2 - 12 b 2 d + 4 b 3 ) Dd A + 9 Dd 2 ) + (- 6 d 3 + 10 b d 2 - 2 b 2 d - 2 b 3 ) A + 3 Dd ) / ( ( 8 d 2 - 16 b d + 8 b 2 ) A ) , t = ( sqrt( ( 4 d 6 - 24 b d 5 + 60 b 2 d 4 - 80 b 3 d 3 + 60 b 4 d 2 - 24 b 5 d + 4 b 6 ) A 2 + (- 4 d 3 + 12 b d 2 - 12 b 2 d + 4 b 3 ) Dd A + 9 Dd 2 ) + (6 d 3 - 10 b d 2 + 2 b 2 d + 2 b 3 ) A - 3 Dd ) / ( ( 8 d 2 - 16 b d + 8 b 2 ) A ) , t = b ] (%i56) factor(4*d^6-24*b*d^5+60*b^2*d^4-80*b^3*d^3+60*b^4*d^2-24*b^5*d+4*b^6); (%o56) 4 ( d - b ) 6 (%i57) factor((-4*d^3+12*b*d^2-12*b^2*d+4*b^3)); (%o57) - 4 ( d - b ) 3 (%i58) radicand:A^2*4*(d-b)^6-(-4*(d-b)^3)*Dd*A-9*Dd^2; (%o58) 4 ( d - b ) 6 A 2 + 4 ( d - b ) 3 Dd A - 9 Dd 2 (%i59) radcan(radicand-(4*d^6-24*b*d^5+60*b^2*d^4-80*b^3*d^3+60*b^4*d^2-24*b^5*d+4*b^6)*A^2+(-4 (%o59) 0

Appendix

711

(%i60) factor((-6*d^3+10*b*d^2-2*b^2*d-2*b^3)); (%o60) - 2 ( d - b ) 2 ( 3 d + b ) (%i61) inflections_for_smooth_quartic:solve(diff(Smooth_Start_Quartic,t,2)=0,t); (%o61) [ t = - ( 3 sqrt( ( 4 d 6 - 24 b d 5 + 60 b 2 d 4 - 80 b 3 d 3 + 60 b 4 d 2 - 24 b 5 d + 4 b 6 ) A 2 + (- 4 d 3 + 12 b d 2 - 12 b 2 d + 4 b 3 ) Dd A + 3 Dd 2 ) + (- 6 d 3 + 6 b d 2 + 6 b 2 d - 6 b 3 ) A + 3 Dd ) / ( ( 12 d 2 - 24 b d + 12 b 2 ) A ) , t = ( 3 sqrt( ( 4 d 6 - 24 b d 5 + 60 b 2 d 4 - 80 b 3 d 3 + 60 b 4 d 2 - 24 b 5 d + 4 b 6 ) A 2 + (- 4 d 3 + 12 b d 2 - 12 b 2 d + 4 b 3 ) Dd A + 3 Dd 2 ) + (6 d 3 - 6 b d 2 - 6 b 2 d + 6 b 3 ) A - 3 Dd ) / ( ( 12 d 2 - 24 b d + 12 b 2 ) A ) ] Now we impose a DIFFERENT RESTRICTION on the Quadric: we no longer request a smooth-birth, but rather request that BOTH derivatives at birth ad death ARE EQUAL to each other. The result is the PERFECTLY SYMMETRIC Quartic before and after its own MID-POINT (b+d)/2. SYMMETRIC Quartic => Dd=Db. (%i62) sym_t4_coeff:subst(Db,Dd,t4_coeff); (%o62) A (%i63) sym_t3_coeff:subst(Db,Dd,easy_t3_coeff); (%o63)

2 Db (d - b)2

- 2 (d + b) A

(%i64) sym_t2_coeff:subst(Db,Dd,easy_t2_coeff); (%o64) ( d 2 + 4 b d + b 2 ) A -

3 d Db + 3 b Db (d - b)2

(%i65) sym_t1_coeff:subst(Db,Dd,easy_t1_coeff); (%o65)

d 2 Db + 4 b d Db + b 2 Db (d - b)2

- 2 b d (d + b) A

(%i66) sym_t0_coeff:subst(Db,Dd,easy_t0_coeff); (%o66) b d b d A +

- d Db - b Db (d - b)2

(%i67) sym_quartic:sym_t4_coeff*t^4+sym_t3_coeff*t^3+sym_t2_coeff*t^2+sym_t1_coeff*t+sym_t0_coeff; (%o67) t 2 ( d 2 + 4 b d + b 2 ) A 2 Db (d - b)2

3 d Db + 3 b Db

- 2 (d + b) A + b d b d A +

ENERGY of Symmetric Quartic.

2

(d - b) - d Db - b Db (d - b)2

+t

d 2 Db + 4 b d Db + b 2 Db

+ t4 A

(d - b)2

- 2 b d (d + b) A + t 3

712

Evo-SETI Quartics Yielding ET Civilizations’ Energy (%i68) def_sym_quartic_energy:sym_quartic_energy=radcan(integrate(sym_quartic,t,b,d)); (%o68) sym_quartic_energy =

( d 5 - 5 b d 4 + 10 b 2 d 3 - 10 b 3 d 2 + 5 b 4 d - b 5 ) A 30

(%i69) bis_easy_energy; (%o69)

( d - b ) 2 ( 2 ( d - b ) 3 A - 5 ( Dd - Db ) ) 60

(%i70) sym_quartic_energy:subst(Dd,Db,bis_easy_energy); (%o70)

(d - b)5 A 30

Special case of the Symmetric Quadric: it is SMOOTH at boh birth and death. That is, ZERO first derivative at both b and d. (%i71) subst(0,Db,sym_t4_coeff); (%o71) A (%i72) factor(subst(0,Db,sym_t3_coeff)); (%o72) - 2 ( d + b ) A (%i73) factor(subst(0,Db,sym_t2_coeff)); (%o73) ( d 2 + 4 b d + b 2 ) A (%i74) factor(subst(0,Db,sym_t1_coeff)); (%o74) - 2 b d ( d + b ) A (%i75) factor(subst(0,Db,sym_t0_coeff)); (%o75) b 2 d 2 A (%i76) sym_quartic_with_zero_derivatives_at_b_and_d:subst(0,Db,subst(0,De,sym_quartic)); (%o76) t 4 A - 2 ( d + b ) t 3 A + ( d 2 + 4 b d + b 2 ) t 2 A - 2 b d ( d + b ) t A + b 2 d 2 A (%i77) easy_sym_quartic_with_zero_derivatives_at_b_and_d:factor(sym_quartic_with_zero_derivatives_at_b_ (%o77) ( t - b ) 2 ( t - d ) 2 A (%i78) def_energy_easy_sym_quartic_with_zero_derivatives_at_b_and_d:energy_easy_sym_quartic_with_zero (%o78) energy_easy_sym_quartic_with_zero_derivatives_at_b_and_d = ( d 5 - 5 b d 4 + 10 b 2 d 3 - 10 b 3 d 2 + 5 b 4 d - b 5 ) A 30

(%i79) ratsimp(((d^5-5*b*d^4+10*b^2*d^3-10*b^3*d^2+5*b^4*d-b^5)*A)/30); (%o79)

( d 5 - 5 b d 4 + 10 b 2 d 3 - 10 b 3 d 2 + 5 b 4 d - b 5 ) A 30

Appendix

713

(%i80) factor(d^5-5*b*d^4+10*b^2*d^3-10*b^3*d^2+5*b^4*d-b^5); (%o80) ( d - b ) 5 (%i81) def_Energy_of_the_Symmetric_Quadric:Energy_of_the_Symmetric_Quadric=(A*((d-b)^5))/30; (%o81) Energy_of_the_Symmetric_Quadric =

(d - b)5 A 30

Two minima and the Maximum = PEAK in exactly in between for the Symmetric Quadric. (%i82) min_and_Max_for_easy_symmetric_Quadric:solve(diff(sym_quartic_with_zero_derivatives_at_b_and_ (%o82) [ t = d , t =

d +b 2

,t=b]

The two inflection points of the Symmetric Quadric. (%i83) inflections_for_easy_symmetric_Quadric:(solve(diff(sym_quartic_with_zero_derivatives_at_b_and_d,t (%o83) [ t = -

( 3 - 3) d + (- 3 - 3) b 6

,t=

( 3 + 3) d + (3 - 3 ) b 6

]

Numerical example of the Symmetric Quadric. (%i84) numbers_1:[A=2,b=1,d=3]; (%o84) [ A = 2 , b = 1 , d = 3 ] (%i85) plot2d(2*(3-t)^2*(1-t)^2,[t,0,4]); (%o85) (%i86) peak_height_in_the_middle_point:peak_height=factor(subst((d+b)/2,t,easy_sym_quartic_with_zero_der (%o86) peak_height =

(d - b)4 A 16

(%i87) peak_height_in_the_middle_point,numbers_1; (%o87) peak_height = 2 Given the middle-point peak height, find A. (%i88) factor(first(solve(peak_height_in_the_middle_point,A))); (%o88) A =

16 peak_height (d - b)4

(%i89) ev(subst(100,peak_height,%),numer); (%o89) A =

1600 (d - b)4

Reference 1. C. Maccone, Energy of extraterrestrial civilizations according to Evo-SETI theory. Acta Astronaut. 144, 202–213 (2018). See therein also the references to prior related papers published by this author

KLT for an Expanding Universe with SETI Applications

Abstract The relativistic KLT was first considered by one of the authors as early as the 1990s, but that was confined to special relativity only. In this paper, we attempt, for the first time, to extend the KLT calculations to the Friedman-Lemaitre-RobertsonWalker (FLRW) metric of the expanding universe. It is claimed that our new results could be of interest for SETI applications about incoming signals from receding galaxies. Keywords KLT · Friedman-Lemaitre-Robertson-Walker (FLRW) metric · SETI

1 Introduction SETI, the Search for ExtraTerrestrial Intelligence, is today mainly the search for ET radio signals. Two types of such ET signals possibly exist: 1. Stationary, intentional signals suitable to be analyzed by the Fast Fourier Transform (FFT) (and that is “Classical SETI”), and 2. Transients, i.e. non-stationary leakage signals that might be emitted by moving ET sources like spaceships possibly flying around even at relativistic speeds. For such signals the FFT won’t work, and it will be necessary to resort to the KLT instead. The KLT (acronym for Karhunen-Loève Transform, for various equivalent definitions see the website https://en.wikipedia.org/wiki/Principal_component_analysis) is a mathematical algorithm superior to the classical FFT in many regards: 1. The KLT can filter signals out of the background noise over both wide and narrow bands. That is in sharp contrast to the FFT that rigorously applies to narrow-band signals only. 2. The KLT can be applied to random functions that are non-stationary in time, i.e. whose autocorrelation is a function of the two independent variables t 1 and t 2 separately. 3. The KLT can detect signals embedded in noise to unbelievably small values of the Signal-to-Noise Ratio (SNR), like 10−3 or so. © Springer Nature Switzerland AG 2020 C. Maccone, Evo-SETI, https://doi.org/10.1007/978-3-030-51931-5_22

715

716

KLT for an Expanding Universe with SETI Applications

An excellent filtering algorithm such as the KLT, however, comes with a cost that one must be ready to pay for especially in SETI: its computational burden is much higher than for the FFT. In fact, for an autocorrelation matrix of size N, the calculations must be of the order of N2 or even higher, rather than N * log(N). Nevertheless, the unbelievable recent progress in programmable cards, capable of doing Teraflop calculations at moderate prices, makes us believe that the KLT no longer is a dream, but rather a technological reality already at work. In this Chapter “KLT for an Expanding Universe with SETI Applications”, we study KLT filtering even for signals emitted by relativistic sources, as described by this author in Ref. [26]. We also provide a full list of References about the research work done over 40 years by author Claudio Maccone and his Co-Workers and/or Students: these are References [2-8, 10–25, and 27]. Thanks for your interest into the mathematically difficult research topics of the KLT.

2 Claudio Maccone’s 1981 Exact Analytical Solution Yielding the KLT for All Time-Rescaled Gaussian Stochastic Processes (that is Gaussian Noises) In 1980 author Claudio Maccone obtained his Ph.D. at the Department of Mathematics of the University of London (UK) King’s College by solving analytically the integral equation of the KLT for all Gaussian stochastic processes where the time does not elapse uniformly, but rather is dilated or shrunk like a power law of the time, that is τ (t) = constant · t 2H

(1)

Special Relativity tells us that (1) is the time-rescaling law of a moving spacecraft or a moving galaxy, or a moving Fast Radio Burst) where the proper time τ is increasing or decreasing in terms of the coordinate time t like the power 2H if 2H is higher or smaller than 1, respectively. The parameter H is actually the Hurst exponent so named for historic reasons (see the website https://en.wikipedia.org/wiki/Hurst_ exponenthttps://en.wikipedia.org/wiki/Hurst_exponent).

3 The Machinery of Maccone’s 1981 Exact KLT for Relativistic Frames in Arbitrary Motion It is well known that in special relativity two time variables exist: the coordinate time t, which is the time measured in the fixed reference frame, and the proper time τ , which is the time shown by a clock rigidly connected with the moving body. They are related by

3 The Machinery of Maccone’s 1981 Exact KLT for Relativistic …

t  τ (t) =

1−

v 2 (s) ds c2

717

(2)

0

where v(t) is the body velocity and c is the speed of light. We are devoted to the relativistic interpretation of the Brownian motion whose time variable is the proper time, B(τ ), rather than the coordinate time, B(t). The bulk of the results was given by the author in a purely mathematical form, with no reference to relativity, in Ref. [9]. However, a summary of that work is now given in a form suitable for the physical developments that will follow. Consider standard Brownian motion (Wiener-Lévy process) B(t), with mean zero, variance t, and initial condition B(0) = 0. A white noise integral is the process X(t) defined by t X (t) =

f (s)d B(s)

(3)

0

where f (t) is assumed to be continuous and non-negative. Evidently, X(0) = 0, and it can be proved that ⎛ X (t) = B ⎝

t

⎞ f (s)ds ⎠ 2

(4)

0

Thus X(t) is a time-rescaled Gaussian process, with mean zero and t 1 ∧t2

E{X (t1 )X (t2 )} =

f 2 (s)ds

(5)

0

as autocorrelation (covariance); t1 ∧ t2 denotes the minimum (smallest) of t 1 and t 2 . Now the KL theorem states that X (t) =

∞ 

Z n ϕn (t) (0 ≤ t ≤ T )

(6)

n=1

where: (1) the functions φn (t) are the autocorrelation eigenfunctions to be found from T E{X (t1 )X (t2 )}ϕn (t2 )dt2 = λn ϕn (t1 ) 0

(7)

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KLT for an Expanding Universe with SETI Applications

where the constants λn are the corresponding eigenvalues; and (2) the Z n are orthogonal random variables, with mean zero and variance Z n , that is: E{Z m Z n } = λn δmn .

(8)

This theorem is valid for any continuous-parameter second-order process with mean zero and known autocorrelation. The series (6) converges in mean square, and uniformly in t. Finally, if X(t) is Gaussian, as in Eqs. (3), (4), and (5), the random variables Z n are Gaussian also, and, since orthogonal, they are independent. After these preliminaries, we can state the main result of Ref. [9]. The white noise integral (3), or the equivalent time-rescaled Gaussian process (4), has the KL expansion X (t) =

∞ 

Z n Nn





t

f (s)ds · Jν(t) γn  0T

f (t) 0

n=1

t 0

f (s)ds



f (s)ds

.

(9)

Here: 1. The order of the Bessel functions ν(t) is not a constant, but is rather the time function

 χ 3 (t) d χ  (t) · (10) ν(t) = − 2 f (t) dt f 2 (t) with χ (t) =

 f (t)

t

f (s)ds

(11)

0

2. The constants γn are the (increasing) positive zeros of the equation 

 f (T ) · γn  ∂ Jν(T ) (γn )  ν (T ) = 0 χ (T ) · Jν(T ) (γn ) + χ (T ) ·  T Jν(T ) (γn ) + ∂ν 0 f (s)ds (12) 

In general, (12) can only be solved numerically. 3. The normalization constants Nn follow from the normalization condition ⎡ Nn2 ⎣

T 0

⎤2 f (s)ds ⎦ ·

1

 2 x Jν((x)) (γn x) d x = 1

(13)

0

in which the new Bessel functions order ν((x)) is (10) changed by aid of the transformation

3 The Machinery of Maccone’s 1981 Exact KLT for Relativistic …

t

719

T f (s)ds = x

0

f (s)ds

(14)

0

4. The eigenvalues are determined by ⎡ λn = ⎣

⎤2

T

f (s)ds ⎦

0

1 (γn )2

(15)

5. The Gaussian random variables Z n are independent and orthogonal, and have zero mean and variance λn . The proof of this theorem may be sketched as follows: firstly the KLT integral equation for the autocorrelation (5) is an integral equation of the Volterra type and so it may be transformed into a differential equation with two boundary conditions; and secondly, the resulting differential equation is reduced to the standard Bessel differential equation by changing the independent variable (the time) and separating the unknown function into functions each of one variable only. Let us now go back to relativity. Since from (4) it plainly appears that the rescaled time of the new Brownian motion is given by t f 2 (s)ds

(16)

0

we merely have to equate (16) and (2) to get the relationship among the arbitrary time-rescaling function f (t) and the arbitrary body velocity v(t): t 

t

1−

f (s)ds = 2

0

v 2 (s) ds c2

(17)

0

By differentiating and taking the positive square root, it follows that: 1

v 2 (t) 4 f (t) = 1 − 2 c

(18)

This formula is the starting point to study the KL expansion (6) for a relativistic body. Inversion of (18) leads at once to:  (19) v(t) = c 1 − f 4 (t)

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KLT for an Expanding Universe with SETI Applications

Now, the reality of the motion requires the radicand to be non-negative, whence, taking the positive sign in front of all square roots, we find f (t) ≤ 1

(20)

This is the fundamental upper bound imposed on the “arbitrary” function f (t) by special relativity. In other words, since the speed of light can in no case be exceeded, so f (t) must not exceed one. As already pointed out, the lower bound on f (t), required by the presence of the radicals in (17) and (19), is zero. Therefore 0 ≤ f (t) ≤ 1 (0 ≤ t ≤ T )

(21)

is the physical range of the (otherwise arbitrary) function f (t). We also want to point out the Newtonian limit of the results. By this we mean the limit as c → ∞. Then, as we see from (18), lim f (t) = 1

c→∞

(22)

and the time-rescaled process under consideration reduces to standard Brownian motion, B(t). This agrees, of course, with (2), stating that the proper time τ becomes the same as the coordinate time t.

4 The 1981 Exact KLT Machinery in Case of Relativistic Frames in Uniform Motion An easy and instructive example of how Maccone’s exact KLT works is the simplest possible case of (2) is when the velocity ν(t) is a constant in time, i.e. when the body’s motion is uniform. Then the time-rescaling function f (t) is a constant K as well 1

v 2 (t) 4 f (t) = 1 − 2 =K c

(23)

Let us now recall the property of the Brownian motion √ called self-similarity to the order 1/2 and expressed by the formula B(c t) = cB(t) where c is any real positive constant (see (1.20) of Ref. [5] for the relevant proof). From this and from (23), one gets at once ⎛ t ⎞    X (t) = B ⎝ K 2 ds ⎠ = B K 2 t = K B(t) 0

(24)

4 The 1981 Exact KLT Machinery in Case of Relativistic Frames …

721

Thus the uniform proper-time Brownian motion B(τ ) = X (t) equals the uniform coordinate-time Brownian motion B(t) multiplied by the constant K, that is 1

v2 4 B(τ ) = 1 − 2 B(t) c

(25)

The KL expansion of B(τ ) is, of course, the same as that of B(t) apart from the multiplicative factor K . And the relevant eigenfunctions are just sines. To provide an example of how the machinery outlined in Sect. 1 actually works, we shall now prove this result. From (11): χ (t) =



t

K

√ K ds = K t

(26)

0

And K χ  (t) = √ 2 t

(27)

The order ν(t) of the Bessel functions is then found from (10): 

 3 1 χ 3 (t) d χ  (t) K 3t 2 d = − 2 ν(t) = − 2 √ f (t) dt f 2 (t) K dt 2K t 

  3  1 1 t− 2 3  = = −K t 2 − = 4K 4 2

(28)

in which both the time t and the constant K have disappeared from the result. Simplifications of this kind are vital to make the mathematical investigations feasible. Since ν = 21 , the relevant Bessel function is (Ref. [1], p. 54)  J 21 (x) =

2 sin x πx

(29)

Thus, from (26), (28) and (29), the KL expansion follows: X (t) =

 K

t

K ds 0

=K

∞  n=1

∞  n=1

Z n Nn

Z n Nn J 21 

t

γn  0T 0

2T t sin γn π γn T

K ds



K ds



(30)

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KLT for an Expanding Universe with SETI Applications

In this expression the normalization constants Nn are yet to be found. To this end, we must know the γn given by (12). That is,   √ K γn K  J 1 (γn ) = 0 √ J 1 (γn ) + K T T 2 T 2 K 0 ds 2

(31)

1 J 1 (γn ) + γn J 1 (γn ) = 0. 2 2 2

(32)

or, simplifying,

But this is a special case of the more general Bessel functions formula ν Jν (z) + z Jν (z) = z Jν−1 (z)

(33)

so that (32) actually amounts to J− 21 (γn ) = 0

(34)

since γn = 0. One has  J− 21 (x) =

2 cos x πx

(35)

so that (34) finally becomes the boundary condition: cos γn = 0

(36)

In this case we find the exact γn expression to be γn = nπ −

π with (n = 1, 2, . . .) 2

(37)

Reverting now to the normalization constants Nn , (13) yields

 1 = Nn2

2 

T

K ds 0



0 1

1

 2 x J 21 (γn x) d x

2 sin2 (γn x)d x π γn 0  1  Nn2 K 2 T 2 = Nn2 K 2 T 2 − sin γ cos γ γ = n n n π γn2 π γn

= Nn2 K 2 T 2

from which

(38)

4 The 1981 Exact KLT Machinery in Case of Relativistic Frames …

√ √ γn π Nn = KT

723

(39)

As for the eigenvalues λn , from (15) they are given by λn =

K 2T 2 γn2

(40)

and these also are the variances of the independent Gaussian random variables Z n . It is interesting to point out that the variance property 2 = c2 σ Z2 σcZ

(41)

and (40) yield the following proportionality among the proper-time random variables Z n and the coordinate-time random variables Z n0 (corresponding to the case ν(t) ≡ 21 , or, from (18), f (t) ≡ 1): Z n = K Z n0

(42)

Thus the KL expansion of the proper-time Brownian motion is 

  2 t sin γn T T n=1    ∞  2 t 0 sin γn = K B(t) =K Zn T T n=1

B(τ ) =

∞ 

Zn

(43)

and (25) is found once again. In other words, passing from an inertial reference frame to another one, the random variables Z n just change their variance according to (41), whereas the time eigenfunctions are the same.

5 Friedman-Lemaître-Robertson-Walker (FLRW) Metric and the Friedman Equations with  Now about General Relativity and Cosmology. The Friedman-Lemaître-Robertson-Walker (FLRW) metric is the mathematical foundation of Cosmology: https://en.wikipedia.org/wiki/Friedmann%E2%80%93L ema%C3%AEtre%E2%80%93Robertson%E2%80%93Walker_metric. We did all calculations by virtue of Maxima, the symbolic manipulator freely downloadable from the website http://maxima.sourceforge.net/. We assume the reader to be familiar the basics of General Relativity and so our FLRW metric is given by the following covariant metric tensor (“lg” means “lower g”, i.e. the covariant metric tensor)

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KLT for an Expanding Universe with SETI Applications

⎡

a2 1−kr 2

⎢ ⎢ 0 lg = ⎢ ⎣ 0 0

0 0 2 2 0 a r 0 sin(θ )a 2 r 2 0 0

⎤ 0 ⎥ 0 ⎥ ⎥ 0 ⎦ −c2

(44)

In (44) the only time function is a(t) i.e. the scale factor (related to the radius of the universe supposed to be a three-sphere), and the constant k equals +1 for the closed universe (elliptical geometry), −1 for the open universe (hyperbolic geometry) and 0 if the spacetime is Euclidean (Einstein-DeSitter universe). Then Maxima works out all the relevant Riemann, Ricci and Einstein tensors with the result that the Einstein equations are just the two Friedman equations with the cosmological term in  −c2 k + a 2 c2 − d2 a= 2 dt 2a

 d 2 a dt

(45)

and #  2 $ 3 c2 k + dtd a a2

= 8πρG

(46)

Alternatively, one of the two Friedman equations may be replaced by a third (independent) equation that follows from the Bianchi identities fulfilled by the stressenergy tensor T μν and expressed by the equation T μν;ν = 0

(47)

in which “;” denotes covariant differentiation. But we will not dig further into General Relativity issues here, except by pointing the following neat diagram taken from the Wikipedia website https://en.wikipedia. org/wiki/Age_of_the_universe that appears below. It shows the behaviour in time of the various solutions to the Friedman equations, the one currently accepted (2019) by the Cosmology community being the “Lambda Cold Dark Matter” (CDM) (Fig. 1).

6 KLT Time-Rescaling Function f (t) for Motion of Spaceships, Particles and Light Inside Any Expanding Universe Given Its a(t) Function This section is the most important section of the present research paper. It computes the KLT time-rescaling function f (t) for the motion of a spaceship, or an elementary particle, or even a beam of light or neutrinos or gravitational waves inside any expanding universe given its a(t).

6 KLT Time-Rescaling Function f (t) for Motion of Spaceships, …

725

Fig. 1 Different models of the expansion of the universe in time produced by General Relativity. On the horizontal axis, time t = 0 is now, past times are negative times and future times are positive times, all measured in billions of years (Gyr = 109 years). On the vertical axis are the ratios between the radius on the universe at any time a(t) and the radius of the universe now, i.e. the constant a0 (to be later specified more precisely). The most accepted model as of 2019 seems to be the “Lambda Cold Dark Matter” (CDM model). All other models only have a historic value, though much easier mathematically. Of course, the creation of a comprehensive numeric code, especially for CDM, might lead to new insights in Cosmology

The key idea is to rewrite the FLRW metric in such a way that “it finally comes to coincide with the proper-time Formula (2) of Special Relativity when the a(t) is practically a constant with respect to the “little motion” from one star to another one inside the huge expanding universe. In other words still: consider a Star Trek spaceship travelling “just within the Milky Way”: then the proper time aboard will be much less than the 13.772 billion years elapsed since the Big Bang (=age of the universe) and so the radius of the universe a(t) didn’t almost change at all, and (2) is practically still valid apart from (slight) factors in front of v(t). Let us now cast all this into equations. Of course we start from the covariant metric tensor (44) of the FLRW metric. By rewriting the metric instead of the tensor, one gets c2 dτ 2 = c2 dt 2 − a 2 (t) · dΣ 2

(48)

in which dΣ 2 is the metric of the spatial part of the spacetime 3-sphere making up the universe. This dΣ 2 is expressed in 3-dimensional spherical coordinates in (44) but it could be expressed in other coordinate system that we are not interested in, and, in addition, while (44) is a space-like metric (as requested by Maxima) (48) is a time-like metric, but that does only change the signs, not the physics. So, a few steps

726

KLT for an Expanding Universe with SETI Applications

bring (48) into the time-like form t



dτ =

1 − a 2 (s) ·

  1 dΣ 2 ds c2 ds

1 − a 2 (s) ·

v 2 (s) ds c2

0

t  =

(49)

0

The next crucial step is to check (49) against the KLT time-rescaling expressed by (16), that is t 

t

1 − a 2 (s) ·

f (s)ds = 2

0

v 2 (s) ds c2

(50)

0

Differentiating both sides of (50) with respect to t one gets  f (t) = 2

1 − a 2 (t) ·

v 2 (t) c2

(51)

and finally

v 2 (t) f (t) = 1 − a (t) · 2 c 2

 41 (52)

This is the desired KLT time-rescaling function for all motions taking place with a certain speed-law v(t) inside the expanding universe with scale factor a(t). This is also the most important new equation in this paper. We now wish to point out a few cases of possible applications of (52): 1. Motion of a “Star Trek spaceship” with “flight law” equal to ν(t) inside an expanding universe having the expansion law a(t). 2. Motion of a constant-speed νconst beam of particles inside an expanding universe having the expansion law a(t). 3. Motion of electromagnetic waves, neutrinos and gravitational waves (all having speed equal to c), inside an expanding universe having the expansion law a(t).

7 Motion of a “Star Trek Spaceship” with “Flight Law” Equal to ν(t) …

727

7 Motion of a “Star Trek Spaceship” with “Flight Law” Equal to ν(t) Inside an Expanding Universe Having the Expansion Law a(t) In general, no analytic development related to (52) is known to these authors at present time. However, the following particular case of ν(t) is called “hyperbolic motion of special relativity”, and was studied in mathematical detail since 1909 because the relevant differential equation may be easily solved exactly, see https://en.wikipedia. org/wiki/Hyperbolic_motion_(relativity). It represents the motion of a relativistic spaceship having a constant proper acceleration g in the spaceship’s reference frame, that is, the spaceship departs from Earth at t = 0 and keeps accelerating at a constant acceleration, say g = 9.8

m s2

(53)

so that humans aboard would feel the same weight they have on Earth! An excellent idea for Science Fiction (SF) writers as well: for instance, Poul Anderson https://en. wikipedia.org/wiki/Poul_Anderson exploited this idea in his SF novel “Tau Zero”, website https://en.wikipedia.org/wiki/Tau_Zero, after which the Tau Zero Foundation is named https://tauzero.aero/. So, the speed-law for this hyperbolic motion reads gt ν(t) = %  2 1 + gct

(54)

And the time-rescaling function (52) for the KLT telecommunications between the receding spaceship and the Earth is found by inserting (54) into (52) 

 g t 2

f (t) = 1 − a (t) ·

c

2

1+

 g t 2

 14 (55)

c

We stop at this point, but it’s clear that much more work, both analytical and numeric, should be done in order to detail the KLT-optimized telecommunications between the relativistic spaceship and the Earth.

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KLT for an Expanding Universe with SETI Applications

8 Motion of a Constant-Speed νconst Beam of Particles Inside an Expanding Universe Having the Expansion Law a(t) This topic should be of interest for particle physicists and cosmologists. Upon inserting νconst instead of the arbitrary time function ν(t) into (52) one gets  vconst f (t) = 1 − a 2 (t) · 2 c

1 4

(56)

But this is the same as (25) (apart from constant numeric factors) and so this is the case of a uniform motion, the KLT of which was described already in Sect. 4 of this paper. Nothing more to add, then.

9 Motion of c-Speed Electromagnetic Waves, Neutrinos and Gravitational Waves in an Expanding Universe Having the Expansion Law a(t) Same as (56) with c replacing νconst so that (56) boils down to  1 f (t) = 1 − a 2 (t) 4

(57)

10 Applications of the KLT to SETI SETI, the Search for ExtraTerrestrial Intelligence, is an area of astronomy that has been growing considerably since its inception in 1960 by Frank Drake with Project Ozma. A good summary of 60 years of SETI searches may be found at the website https://en.wikipedia.org/wiki/Search_for_extraterrestrial_intelligence. The advent of the Breakthrough Listen Project (https://en.wikipedia.org/wiki/Bre akthrough_Listen) after 2015 has already produced a huge set of SETI observations amounting to 1015 bytes (i.e. 1 TeraByte = 1 TB) of SETI data open to analysis by anybody. In other words, anyone is free to search the SETI data by virtue of the algorithm they wish. Clearly, the KLT is one such algorithm. In Italy we have been developing algorithms to compute the KLT at least since the publication of Claudio Maccone’s 2009 “Deep Space Flight and Communications” book (Ref. [26]). In particular, a whole school of young SETI radio astronomers developing KLT algorithms for radio astronomy and SETI already produced valuable scientific papers like Ref. [27].

11 Conclusions

729

11 Conclusions This paper has extended the KLT studies to General Relativity for the first time, to the best of these two authors’ knowledge. But this is just a beginning. Innumerable applications to SETI and to Cosmology are to be envisaged by young researches in these fields.

References 1. F. Biraud, SETI at the Nançay radio-telescope. Acta Astronaut. 10, 759–760 (1983) 2. K. Karhunen, Uber lineare methoden in der wahrscheinlichkeits-rechnung. Ann. Acad. Sci. Fenn., Ser. A 1, Math. Phys. 37, 3–79 (1946) 3. M. Loève, Fonctions aleatoires de second ordre. Rev. Sci. 84(4), 195–206 (1946) 4. M. Loève, Probability Theory: Foundations, Random Sequencies (Van Nostrand, Princeton, NJ, 1955) 5. C. Maccone, Telecommunications, KLT and Relativity, vol. 1, IPI Press, Colorado Springs, Colorado, USA, 1994, ISBN: 1-880930-04-8. This book embodies the results of some thirty research papers published by the author about the KLT in the fifteen years span 1980–1994 in peer-reviewed journals 6. S. Montebugnoli, C. Bortolotti, D. Caliendo, A. Cattani, N. D’Amico, A. Maccaferri, C. Maccone, J. Monari, A. Orlati, P.P. Pari, M. Poloni, S. Poppi, S. Righini, M. Roma, M. Teodorani, SETI-Italia 2003 status report and first results of a KL Transform algorithm for ETI signal detection, paper IAC-03-IAA.9.1.02 presented at the 2003 International Astronautical Congress held in Bremen, Germany, 29 Sept–3 Oct 2003 7. C. Maccone, Advantages of the Karhunen-Loève transform over fast Fourier transform for planetary radar and space debris detection. Acta Astronaut. 60, 775–779 (2007)

Annotated Bibliography1 : Il Nuovo Cimento: 8. C. Maccone, Special relativity and the Karhunen-Loe’ve expansion of Brownian motion, Nuovo Cimento. Ser. B 100, 329–342 (1987)

Bollettino dell’Unione Matematica Italiana: 9. C. Maccone, Eigenfunctions and energy for time-rescaled Gaussian processes. Boll. Unione Mat. Ital. Ser. 6(3-A), 213–219 (1984) 10. C. Maccone, The time-rescaled Brownian motion B(t2H), Boll. Unione Mat. Ital. Ser. 6(4–C), 363–378 (1985) 11. C. Maccone, The Karhunen–Loe’ve expansion of the zero-mean square process of a timerescaled Gaussian process. Boll. Unione Mat. Ital. Ser. 7(2–A), 221–229 (1988)

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Journal of the British Interplanetary Society: 12. C. Maccone, Relativistic interstellar flight and genetics. J. Br. Interplanet. Soc. 43, 569–572 (1990)

Acta Astronautica: 13. C. Maccone, Relativistic interstellar flight and Gaussian noise. Acta Astronaut. 17(9), 1019– 1027 (1988) 14. C. Maccone, Relativistic interstellar flight and instantaneous noise energy. Acta Astronaut. 21(3), 155–159 (1990)

KLT for Data Compression: 15. C. Maccone, The data compression problem for the “GAIA” astrometric satellite of ESA. Acta Astronaut. 44(7–12), 375–384 (1999) 16. R.S. Dixon, M. Klein, On the detection of unknown signals, in Proceedings of the Third decennial US-USSR Conference on SETI held at the University of California at Santa Cruz, 5–9 Aug 1991. Later published in the Astronomical Society of the Pacific (ASP) Conference Series (Seth Shostak, editor), vol. 47, 1993, pp. 128–140 17. C. Maccone, Karhunen–Loève Versus Fourier Transform for SETI, in J. Heidmann, M. Klein, ed. by Proceedings of the Third Bioastronomy Conference held in Val Cenis, Savoie, France, 18–23 June 1990, Lecture Notes in Physics, vol. 390, Springer, Berlin, 1990, pp. 247–253

After these seminal works were published, the importance of the KLT for SETI was finally acknowledged by the SETI Institute experts in: 18. R. Eckers, K. Cullers, J. Billingham, L. Scheffer (ed.), SETI 2020, SETI Institute, 2002, pp. 234, note 13. The authors say: “Currently (2002) only the Karhunen Loeve (KL) transform [Mac94] shows potential for recognizing the difference between the incidental radiation technology and white noise. The KL transform is too computationally intensive for present generation of systems. The capability for using the KL transform should be added to future systems when the computational requirements become affordable”

The paper [Mac94] referred to in the SETI 2020 statement mentioned above is: 19. C. Maccone, The Karhunen–Loe’ve transform: a better tool than the Fourier transform for SETI and relativity. J. Br. Interplanet. Soc. 47, 1 (1994)

References

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An early paper about the KLT for SETI-Italia: 20. S. Montebugnoli, C. Maccone, SETI-Italia Status Report 2001, a paper presented at the 2001 IAF Conference held in Toulouse, France, 1–5 Oct 2001

An early paper about the possibility of a “Fast” KLT: 21. A.K. Jain, A fast Karhunen–Loe’ve Transform for a class of random processes, IEEE Trans. Commun. COM 24, 1023–1029 (1976)

Papers about the KLT and BAM-KLT: 22. F. Schilliro’, S. Pluchino, C. Maccone, S. Montebugnoli, Istituto Nazionale di Astrofisica (INAF)—Istituto di Radioastronomia (IRA), Rapporto Tecnico, La KL Transform: considerazioni generali sulle metodologie di analisi ed impiego nel campo della Radioastronomia, Technical Report (in Italian only), Jan 2007 23. C. Maccone, Innovative SETI by the KLT, in Proceedings of the “Bursts, Pulses and Flickering” Conference held at Kerastari, Greece, 13–18 June 2007 at POS (Proceedings of Science) website, http://pos.sissa.it//archive/conferences/056/034/Dynamic2007_034.pdfS 24. S. Yatawatta, Personal Communication, 17 June 2008

A paper about the KLT for Relativistic Interstellar Flight: 25. C. Maccone, Relativistic optimized link by KLT. J. Br. Interplanet. Soc. 59, 94–98 (2006)

The “final” 2009 book by the author about the Sun as a Gravitational Lens, the relevant FOCAL space missions, and the KLT, including the relativistic KLT: 26. C. Maccone, Deep Space Flight and Communications, a 400-pages technical treatise published by Praxis-Springer in 2009. ISBN 978-3-540-72942-6

A recent paper about the KLT implementation on the Sardinia Radio Telescope (SRT): 27. A. Melis et al., A real-time KLT implementation for radio-SETI applications. Proc. SPIE 9914, 99143D-1 (2016)

SETI Space Missions

Abstract This is a review Chapter about over thirty years of this author’s activity to promote space missions enabling SETI searches from space, rather than from the Earth only. It all started on January 14, 1987, when this author visited for the first time the SETI community at NASA Ames Research Center in Mountain View, California, meeting for the first time persons like Frank Drake, John Billingham and Jill Tarter, the leaders of American SETI. Later in June of the same year this author also attended the Bioastronomy conference at Balatonfüred, Hungary, where he met the Soviet SETI counterparts, notably Nicolay Kardashev and his school, plus Ivan Almar from Hungary, and he was caught forever in the SETI business. On that occasion he also presented his first SETI space mission, Quasat, that never became a reality. For many years afterward, this author attended all the important SETI meetings worldwide, now as a Member of the SETI Committee of the International Academy of Astronautics (IAA) based in Paris. Until June 10, 2010, when he firstly presented his plan to legally protect the central part of the Moon Farside at the United Nations COPUOS in Vienna on behalf of the IAA. The Moon Farside is the only place in space, and not too far from the Earth, where radio transmissions and noises produced by Humanity on Earth may not reach since the spherical body of the Moon blocks them, acting like a shield. Thus, protecting the Moon Farside from all kinds of non-scientific future exploitations (e.g. real estate, tourists, industry and military) has long been a concern for many far-sighted space scientists as well as for several IAA Academicians. We started facing this problem in the 1990s, when the French radio astronomer Jean Heidmann of the Paris Meudon Observatory firstly promoted an IAA Cosmic Study about which areas of the Moon Farside should be reserved for scientific uses only. But Heidmann passed away on July 2, 2000, and this author took over his IAA Cosmic study, so that a paper describing both the scientific and legal aspects of the problem was published in 2008. For ten more years “nothing happened”, but the years 2018–2019 saw China setting up a relay satellite (Queqiao) at the Lagrangian point L2 hovering about 60,000 km above the Moon Farside. Unfortunately, the undeclared but quite real “current, new race to the Moon” complicates matters terribly. All the space-faring nations now keep their eyes on the Moon, and only the United Nations might have a sufficient authority to Protect the Farside and keep safe its unique “radio-noise free” environment. But time is money, and the “Moon Settlers” may well reach the Moon before the United © Springer Nature Switzerland AG 2020 C. Maccone, Evo-SETI, https://doi.org/10.1007/978-3-030-51931-5_23

733

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Nations come to agree about any official decision concerning the Farside Protection. Quite an URGENT ISSUE. In this review paper, we propose that the new “Moon Village” supported by the vision of the ESA Director General, Jan Woerner, be located OUTSIDE the PAC (Protected Antipode Circle) obviously not to interfere with the detection of radiation coming from space, but also SOUTH OF THE PAC, to be “close” to the South Pole as much as needed to benefit of water there. It thus appears the best venue for the “Moon Village” would be on or around the 180° meridian and south to the −30° in latitude of the PAC, possibly much more south of that, almost at the South Pole, thus resolving Moon Village VENUE ISSUE. Keywords Chang’e 4 · Queqiao · Moon farside · Crater daedalus · Moon village · Legal issues

1 Introduction This review Chapter “SETI Space Missions” is a summary of the scientific research work and international activity done by this author for over thirty years about space missions related to SETI. Though no intentional SETI space mission did ever fly up to now, papers and proposals were written by this and other authors to explore the possibility of adapting parts of other planned space missions for the benefit of SETI. In particular, as of 2019 the most important SETI space missions to come are probably to the Moon Farside Missions. These near-future missions make up for the vast majority of the scientific and political arguments discussed in the present paper, as we shall see in Sect. 4 and following Sections.

2 A SETI Space Mission that Never Was: Quasat This author’s interest into SETI space missions began with the Biostronomy Conference held in Balatonfured, Hungary, June 22–27, 1987. There he presented his paper entitled “The Quasat Satellite and its SETI Applications”, now available at the website https://link.springer.com/chapter/10.1007/978-94-009-2959-350. Quasat (standing for “Quasar satellite”) was intended to be the first space VLBI ever radio astronomy satellite, and was a joint NASA-ESA project on which this author became involved while employed at Alenia Spazio SpA in Turin in 1987. This author goal’s was to make changes to the Quasat spacecraft turning it into first SETI spacecraft ever. Some Quasat technical information is found at website https://ntrs.nasa. gov/archive/nasa/casi.ntrs.nasa.gov/19850015511.pdf. Unfortunately, Quasat never became a reality. In fact, the Space Shuttle Challenger disaster, https://en.wikipe dia.org/wiki/Space_Shuttle_Challenger_disaster that occurred on January 28, 1986, forced NASA to divert most available funds towards the construction of a new Shuttle,

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and so NASA withdrew from the Quasat project. ESA alone could not make for the necessary funds to continue the Quasat Project, and so Quasat was cancelled by both NASA and ESA. The task of pursuing the first space VLBI mission was then undertaken by Japan, and the first ever space VLBI satellite successfully launched and operated was the VSOP-HALKA Japanese space mission in the years 1997– 2005, as per the website https://en.wikipedia.org/wiki/HALCA. No SETI usage of VSOP-HALCA was ever planned or attempted.

3 This Author’s Activity About Exploiting for SETI the Moon Farside Radio Quietness in the Fifteen Years 1995–2010 This author kept working on scientific space missions at Alenia Spazio SpA in Turin until December 30, 2004, after which he enjoyed an “early retirement” at age 56, resulting in 24 h a day to study and work as he pleased. So, in 2009 he could publish his book on the FOCAL space mission to 550 AU to exploit the Sun as a gravitational lens, as per the websites https://en.wikipedia.org/wiki/FOCAL_(spa cecraft) and https://www.springer.com/gp/book/9783540729426. And in the fifteen years 1995–2010 he also concentrated on the issue of exploiting the radio-noise free environment then existing on the Moon Farside in order to do there excellent radio astronomy in general and SETI in particular.

4 This Author’s First Presentation Ever at the United Nations COPUOS About Legally Protecting the Central Part of the Moon Farside Against Man-Made Radio Pollution On June 10th, 2010, this author made a presentation in front of the United Nations Committee on the Peaceful Uses of Outer Space (COPUOS) in Vienna. His presentation was of course archived by the United Nations Office of Outer Space Affairs (OOSA) and is now freely downloadable at the UNOOSA web site http://www.uno osa.org/pdf/pres/copuos2010/tech-06E.pdf. Figure 1 shows him while making his presentation on June 10, 2010, while Fig. 2 shows his related United Nations badge. That presentation’s goal was to ask the United Nations to legally protect a circular piece of land on the Moon Farside centered around the Antipode (the point opposite to the Earth at the Farside center). The word “protect” means here to reserve that “Protected Antipode Circle (PAC)” for use by scientists only, especially radio astronomers, forbidding any other non-scientific activity by realtors, industry, tourists and the military. A neat description of this project was recently re-published in Maccone, 2019, Ref. [5] and will not be repeated here. However, our readers are

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Fig. 1 Claudio Maccone while presenting the Protected Antipode Circle (PAC) at the United Nations COPUOS (Committee on the Peaceful Uses of Outer Space) at Vienna, Austria, on June 10, 2010. As the picture shows, he was sitting at the IAA Permanent Seat at COPUOS

Fig. 2 Claudio Maccone’s badge used for his speech described above

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strongly advised to read that paper carefully, just to know the reasons why they and the International Astronomical Union (IAU) in particular, should support this author’s project before it gets too late. Thanks. One might naively expect that, after such an official presentation at the United nations, something should have happened towards the Moon Farside’s protection, either at the political level or the scientific level, or at both. But you are to be disappointed: nothing happened at all for the next five years after 2010. Why? Because no national “big” space agency, nor any private entrepreneur, had the technological capability of sending a spacecraft to the Moon Farside before until about 2015, actually until 2018, when it did happen. The International Astronomical Union should have been the #1 promoter of the Farside Protection but that large international institution didn’t care either, or was just unaware even of the urgency of this issue: they love too much topics like “dark matter”, or what will happen when the Andromeda Galaxy collides with the Milky Way in just about 3 billion years, and similar “learned topics”. So, the IAU astronomers didn’t “get down” to legal-scientific topics like the Moon Farside Protection. The situation started to change only in the summer of 2015, when the new Director General of ESA, Prof. Jan Woerner, began his term. Woerner declares himself to be a civil engineer, very much interested in building a Moon Village possibly on the Farside, though the Moon Village location was not specified yet. Next, a host of eager “Moon bases constructors” established the Moon Village Association (November 2017). But nobody cared to ask to reserve the central part of the Farside, specifically crater Daedalus, for usage by radio astronomers only. This author’s voice was a voice in the desert, once again.

5 Defining PAC, the “Protected Antipode Circle” The need to keep the Farside of the Moon free from human-made RFI (Radio Frequency Interference) has long been discussed by the international scientific community. In particular, in 2005 this author reported to the IAA (International Academy of Astronautics) the results of an IAA “Cosmic Study” that had been started back in 1994 by the late French radio astronomer Jean Heidmann (1923–2000) and had been completed by this author after Heidmann’s death (see, for instance, Maccone [2, 3]). The center of the Farside, specifically crater Daedalus, is ideal to set up a future radio telescope (or a phased array) to detect radio waves of all kinds that it is impossible to detect on Earth because of the ever-growing RFI. Nobody, however, seems to have established a precise border for the circular region around the Antipode of the Earth (i.e. zero latitude and 180° longitude both East and West) that should be protected from wild human exploitation when several nations will have reached the capability of easy travel to the Moon.

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We now describe the PAC, the Protected Antipode Circle. This is a large circular piece of land about 1820 km in diameter, centered around the Antipode of the Earth on the Farside. The same Circle is also defined by spanning an angle of 30° at the Moon center along the Earth-Moon axis in all directions reaching the Farside, and so also in longitude and in latitude. In other words still, the PAC spans a total angle of 60° at the cone vertex right at the center of the Moon. There are three sound scientific reasons for defining PAC this way: (1) PAC is the only area of the Farside that will never be reached by the radiation emitted by future human space bases located at both the L4 and L5 Lagrangian points of the Earth-Moon system (the geometric proof of this fact is trivial); (2) PAC is the most shielded area of the Farside, with an expected attenuation of man-made RFI ranging from 15 to 100 dB or higher; (3) PAC does not overlap with other areas of interest to human activity except for a minor common area with the Aitken Basin, the southern depression supposed to have been created 3.8 billion years ago during the “big wham” between the Earth and the Moon and supposed to possibly contain some frozen water. Figure 3 shows a photo of the Farside of the Moon, the two parallels at plus and minus 30° drawn by solid red lines, and PAC, the Protected Antipode Circle, shown as the red, solid circle centered at the Antipode and tangent to the above two parallels at plus and minus 30°. In view of these unique features, we propose PAC to be officially recognized by the United Nations as an INTERNATIONALLY and LEGALLY PROTECTED AREA, where no radio contamination by humans will possibly take place now and in the future, for the benefit of scientists of all Humankind. Two related References are Ref. [1] and, most recently, Ref. [4].

6 Need for RFI-Free Radio Astronomy, as Pointed Since 1974 by Both ITU and Jean Heidmann (1923–2000) In order to detect radio signals of all kinds, as radio astronomers do, it is mandatory to firstly reject all man-made RFI (Radio Frequency Interference). But RFI is produced in ever increasing amounts by the technological growth of civilization on Earth, and has now reached the point where large bands of the spectrum are blinded by legal or illegal transmitters of all kinds. Since the 1980s, the late French radio astronomer Jean Heidmann (1923–2000) pointed out that Radio astronomy from the surface of the Earth is doomed to die in a few decades if uncontrolled growth of RFI continues. Heidmann also made it clear, however, that advances in modern space technology could bring Radio astronomy to a new life, was Radio astronomy done from the Farside of the Moon, obviously shielded by the Moon spherical body from all RFI produced on Earth. Since the mid 1980s Heidmann was already referring to the ITU (International Telecommunications Union) documents in this regard, for example, the document ITU-R RA.479-5 of 1974 and updated in 1979. Article S22 if this ITU document is shown hereafter in Fig. 4 (Fig. 5).

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Fig. 3 PAC, the Protected Antipode Circle, is the circular piece of land (1820 km in diameter along the Moon surface) that we propose to be reserved for scientific purposes only on the Farside of the Moon. At the center of PAC is the Antipode of the Earth (on the equator and at 180° in longitude) and, near to the Antipode, is crater Daedalus, an 80 km crater proposed by the author in 2005 as the best location for the future Lunar Farside Radio Telescope. Inside Daedalus, the expected attenuation of the man-made RFI (Radio Frequency Interference) coming from the Earth is in the order of 100 dB or higher

7 A Short Review About the Five Lagrangian Points of the Earth-Moon System In view of the following developments in this paper, we present now a short review about the five Lagrangian points of the Earth-Moon system, shown in Fig. 6. Their existence was discovered mathematically by Leonhard Euler (1707–1783) and Joseph-Louis Lagrange (1735–1813) more than 150 years before any idea about spaceflight was understood starting about 1950. For more details derivation, the readers about their mathematical might wish to consult the Wikipedia site https://en.wikipedia.org/wiki/Lagrangianpoint.

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Fig. 4 As early as 1974 had the International Telecommunications Union (ITU) defined the socalled “ITU Shielded Zone of the Moon”, covering almost the whole of the Farside. But nobody cared in the practice up to now

8 The Quiet Cone Above the Moon Farside Depends on the Orbits of Telecommunication Satellites Orbiting the Earth In this section we briefly describe the notion of Quiet Cone above the Moon Farside. That is important in order to calculate the orbits of present and future artificial satellites around the Moon. Consider Fig. 7. A telecommunication satellite in a circular orbit around the Earth emits radio waves from, say, the point G of its circular orbit. The waves propagate along a straight line until they graze the surface of the Moon at point L. Then, if we make the whole Fig. 7 rotate around the Earth-Moon axis, a CONE is departing from the grazing circle on the surface of the Moon up to its own Apex A. This cone is the portion of space above the Farside of the Moon that will never be reached by these specific electromagnetic waves. Thus, it is appropriately called the QUIET CONE, where “quiet” means “radio-silent”. Figure 8 was taken, just as the satellite (called “RadioMoon” here) in a circular orbit previous Fig. 5, from the author’s two papers around the Moon spend inside the

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Fig. 5 The area of the Farside covered by the ITU 1974 Article S22 is nearly the whole of the Farside, and so it is a much larger than the PAC proposed by this author at the United Nations COPUOS on June 10, 2010, for legal protection against electromagnetic emissions of all kinds. Yet nobody cared in the years before 2015 because nobody cared about sending spacecrafts to the Moon Farside. This picture changed suddenly in 2015 when the new Director General of ESA, Jan Woerner, started advocating the creation of a Moon Village, though where it should be built is unclear up to now (September 2019). Even more, the picture changed in 2018, when China sent a few spacecraft to explore the Farside and a Relay satellite (Queqiao) at the Lagrangian point L2, about 65,000 km above the Moon and along on the Earth-Moon axis, necessary to keep the radio link between the Earth and the spacecrafts exploring the Farside

Quiet Cone. Maccone [2, 3] and shows the basic geometry needed to compute how much time would a satellite (called “RadioMoon” here) in a circular orbit around the Moon spend inside the Quiet Cone.

9 Selecting Crater Daedalus Near the Farside Center, i.e. the Near the Earth Antipode It is easy to imagine that the time will come when commercial wars among the big industrial trusts running the telecommunications business by satellites will lead them to grab more and more space around the Earth, pushing their satellites into orbits

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Fig. 6 The five Earth-Moon Lagrangian Points (i.e. the points where the Earth and Moon gravitational pulls on a spacecraft cancel out, keeping into account the rotation of the Moon around the Earth!): (1) Let R denote the Earth-Moon distance that is 384,400 km. Then, the distance between the Moon and the Lagrangian point L1 equals 0.1596003 * R, that is 61,350 km. Consequently the Earth-to-L1 distance equals 0.8403997 * R, that is 323,050 km. (2) The distance between the Moon and the Lagrangian point L2 equals 0.1595926 * R, that is 61,347 km. (3) The distance between the Earth and the Lagrangian point L3 equals 1.007114 * R, that is 387135 km. (4) The two “triangular” Lagrangian Points L4 and L5 are just at same distance R from Earth and Moon

Fig. 7 A telecommunication satellite in a circular orbit around the Earth emits radio waves from, say, the point G of its circular orbit. The waves propagate along a straight line until they graze the surface of the Moon at point L. Then, if we make the whole Fig. 3 rotate around the Earth-Moon axis, a CONE is departing from the grazing circle on the surface of the Moon up to its own Apex A. This cone is the portion of space above the Farside of the Moon that will never be reached by these specific electromagnetic waves. Thus, it is appropriately called QUIET CONE, where “quiet” means “radio silent”

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Fig. 8 The basic geometry needed to compute how much time would a satellite (called “RadioMoon” here) in a circular orbit around the Moon spend inside the Quiet Cone

with apogee much higher than the geostationary one. Then, a “safe” crater must be selected on the Farside along the Moon equator. Where exactly on the Farside? The answer is found by considering the two Lagrangian points L4 and L5 of the Earth-Moon system, as described in Fig. 6. They both fulfill the “lucky condition” of being gravitationally stable, i.e. a future man-made satellite or even a Space Station placed in a (rather small) region of space surrounding either L4 or L5 will not move away from its position. On the other hand, it will never be possible to put a satellite into a circular orbit around the Earth at a distance higher than the distance of the Lagrangian point L1 nearest to the Earth, that is located at 323,050 km (Lagrangian points are, by definition, the points of zero orbital velocity in the two-body problem!). In words, this means the following: the Moon Farside Sector in between the two longitudes 150 E and 150 W will never be blinded by RFI coming from satellites orbiting the Earth alone, nor from radiation coming from future artificial satellites or space stations located at the gravitationally stable Lagrangian points L4 and L5. This RADIO QUIET SECTOR of the Moon Farside we call the PRISTINE SECTOR of the Moon Farside, as shown in Fig. 6. At its center is the Antipode to Earth on the Moon surface, that is the point exactly opposite to the Earth direction on the other side of the Moon. And our theorem simply proves that the Antipode is the most shielded point on the Moon surface from radio waves coming from the Earth. An intuitive and obvious result, really. So, where are we going to locate our SETI Farside Moon base? Just take a map of the Moon Farside and look. One notices that the Antipode’s region (at the crossing of the central meridian and of the top parallel in the figure) is too a rugged region to establish a Moon base. Just about 5° South along the 180° meridian, however, one finds a large crater about 80 km in diameter. This crater is called Daedalus. So, this author proposes to establish the first RFI-free base on the Moon just inside crater Daedalus, the most shielded crater of all on the Moon Farside from Earth-made radio pollution! (Fig. 9)

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Fig. 9 AS11-44-6609 (July 1969)—An oblique of the Crater Daedalus on the Lunar Farside as seen from the Apollo 11 spacecraft in lunar orbit. The view looks southwest. Daedalus (formerly referred to as I.A.U. Crater No. 308) is located at 179° east longitude and 5.5° south latitude. Daedalus has a diameter of about 50 statute miles (~80 km). This is a typical scene showing the rugged terrain on the Farside of the Moon, downloaded from the web site: http://spaceflight.nasa.gov/gallery/ima ges/apollo/apollo11/html/as11_44_6609.html

10 Our 2010 Vision of the Moon Farside for RFI-free Science, Likely not Valid After 2018 As we said, the Pristine Sector, being included between the longitudes 150 E and W, makes angles orthogonal to the directions of L4 and L5. The result is this author’s 2010 vision of the Farside of the Moon shown Fig. 10, a vision probably not “defendable” any more after China’s 2018 Moon Farside Missions. Figure 10 shows a diagram of the Moon as seen from above its North Pole with the different “exploitation regimes” proposed by this author. One sees that: (1) The near side of the Moon is left totally free to activities of all kinds: scientific, commercial and industrial. (2) The Farside of the Moon is divided into three thirds, namely three sectors covering 60° in longitude each, out of which:

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Fig. 10 Our 2010 vision of the Moon Farside with the Daedalus Crater Base for RFI-free Radio astronomy, Bioastronomy and SETI science. Future International Space Stations (ISS) might be located at both the L4 and L5 Earth-Moon Points in the decades to come. Only Point L2 should have been kept free at all times. But this 2010 vision is probably not “defendable” any more after China’s 2018 Moon Farside Missions. In 2019 the Farside is radio polluted already in the S and X bands, tough probably not so in other bands. Only negotiations among all space-faring countries might be able to still preserve some kind of RFI-free environment in the Moon Farside

(a) The Eastern Sector, in between 90° E and 150° E, can be used for installation of radio devices, but only under the control of the International Telecommunication Union (ITU-regime). (b) The Central Sector, in between 150° E and 150° W, must be kept totally free from human exploitation, namely it is kept in its “pristine” radio environment totally free from man-made RFI. This Sector is where crater Daedalus is, a ~100 km crater located in between 177° E and 179° W and around 5° of latitude South. At the moment, this author is not aware of how high is the circular rim surrounding Daedalus. (c) The Western Sector, in between 90° W and 150° W, can be used for installation of radio devices, but only under the control of the International Telecommunication Union (ITU-regime). Also: (3) The Eastern Sector is exactly opposite to the direction of the Lagrangian point L4, and so the body of the Moon completely shields the Eastern Sector from RFI produced at L4. Thus, L4 may be fully exploited. (4) The Western Sector is exactly opposite to the direction of the Lagrangian point L5, and so the body of the Moon completely shields the Western Sector from RFI produced at L5. Thus, L5 may be fully exploited in this author’s vision. In other words, this author’s vision achieves the full bilateral symmetry around

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the plane passing through the Earth-Moon axis and orthogonal to the Moon’s orbital plane. (5) Of course, L2 should not be utilized at all, since it faces crater Daedalus just at the latter’s zenith. Any RFI-producing device located at L2 would flood the whole of the Farside, and should be ruled out. However, since 2018 China already positioned the Queqiao communication relay satellite emitting on both X and S bands right there. It is now (2019) obvious that NEGOTIATIONS ABOUT THE FREQUECY SPECTRUM above the Farside ARE NECESSARY AND URGENT!!!

11 The Further Two Lagrangian Points L1 and L2 of the Sun-Earth System: Their “Polluting” Action on the Farside of the Moon There still is an unavoidable drawback, though. This is coming from the further two Lagrangian points L1 and L2 of the SunEarth system, located along the Sun-Earth axis and outside the gravitational sphere of influence of the Earth that has a radius of about 924,646 km around the Earth. Precisely, the Sun-Earth L1 point is located at a distance of 1,496,557.035 km from the Earth towards the Sun, and the L2 point at the (virtually identical) distance of 1,496,557.034 km from the Earth in the direction away from the Sun, that is toward the outer solar system. These two points have the “nice” property of moving around the Sun just with the same angular velocity as the Earth does, while keeping also at the same distance from the Earth at all times. Thus, they are ideal places for scientific satellites. Actually, the Sun-Earth L1 Point has already been in use for scientific satellite location since the NASA ISEE III spacecraft was launched on 12 August 1978 and reached the Sun-Earth L1 region in about a month. On December 2, 1995, the ESANASA “Soho” spacecraft for the exploration of the Solar Corona was launched. On February 14, 1996, Soho was inserted into a halo orbit around the Sun-Earth L1 point, where it is still librating now (2007). As for the Sun-Earth L2 point, there are plans to let the NASA’s SIM (Space Interferometry Mission) satellite be placed there, as will be ESA’s GAIA astrometric satellite as well. So, all these satellites do “POLLUTE” the otherwise RFI-free Farside oof the Moon when the Farside is facing them. Unfortunately, the Moon Farside is facing the Sun-Earth L1 point for half of the Moon’s synodic period, about 14.75 days, and it is facing the Sun-Earth L2 point for the next 14.75 days. Really all the time! This radio pullution of the Moon Farside by scientific satellites located at the Lagrangian Points L1 and L2 of the Sun-Earth system is, unfortunately, UNAVOIDABLE. We can only hope that telecom satellites will never be put there. As for the scientific satellites already there or on the way, the radio frequencies they use are well known and usually narrow-band. This should help the Fourier transform of

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Fig. 11 The next two closest Lagrangian Points to the Earth are the Lagrangian Points L1 and L2 of the Sun-Earth system. These are located along the Sun-Earth axis at the distances of about 1.5 million kilometers from the Earth toward the Sun (L1) and outward (L2). Unfortunately, spacecrafts located in the neighborhood of these L1 and L2 Sun-Earth Points do send electromagnetic waves to the Farside of the Moon. Examples are the ISEE-III and Soho spacecrafts, already orbiting around L1, and more spacecrafts will do so in the future around both L1 and L2 (Courtesy of the late Dr. Robert “Bob” Farquar, Johns Hopkins University Applied Physics Laboratory, Laurel, MD, USA). In addition to the five Lagrangian Points of the Earth-Moon system (already described in Fig. 4)

the future spectrum analyzers to be located on the Moon Farside to get rid of these transmissions completely (Fig. 11).

12 Attenuation of Man-Made RFI on the Moon Farside In a paper presented by this author at the International Astronautical Congress held in Valencia in October 2006, Ref. [6] his co-worker Salvatore “Salvo” Pluchino succeeded in computing the RFI attenuation on the Farside [6]. A basic result proven there are the RFI attenuation values shown in Table 1. The precise line frequencies of high scientific importance are taken from the 2006 paper by Pluchino, Antonietti and Maccone. In practice, these are the attenuations of man-made RFI to be expected at crater Daedalus and within the PAC. It should also be stated that these are the attenuation values assuming that the Moon is not surrounded by a very thin ionosphere. Since a very tiny Lunar Ionosphere might possibly exist, however, the values below might be slightly incorrect.

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Table 1 Attenuation in the lunar equatorial plane and at lunar longitude at λ = 180° (near the Daedalus crater) for radio waves having some of the most important frequencies used by radio astronomers to explore the universe Origin of radio waves

Radio frequency f

Source in GEO (dB)

Source in orbit at L1 distance (dB)

Source still at L4 or L5 (dB)

ELF

0.003 MHz

−27.39

−15.57

−14.61

VLF

0.030 MHz

−37.39

−25.10

−23.94

Jupiter’s storm

20 MHz

−65.63

−53.33

−52.16

Deuterium

327.384 MHz

−77.77

−65.48

−64.30

Hydrogen

1420.406 MHz

−84.14

−71.85

−70.68

Hydroxyl radical

1612.231 MHz

−84.69

−72.40

−71.23

Formaldehyde

4829.660 MHz

−89.46

−77.17

−75.99

Methanol

6668.518 MHz

−90.86

−78.56

−77.39

Water vapor

22.235 GHz

−96.09

−83.79

−82.62

Silicon monoxide

42.519 GHz

−98.90

−86.61

−85.44

Carbon onoxide 109.782 GHz

−103.02

−90.73

−89.56

Water vapor

−105.25

−92.95

−91.78

183.310 GHz

13 A New Mathematical Contribution of Ours: The Blocking Equation for Electromagnetic (em) Waves Emitted at Height H Above the Earth and Reaching the Earth-Moon Axis at a Distance X Above Moon Farside This section is devoted to the mathematical proof of a new key equation that we discovered in August 2019 and call the “Blocking equation”. This equation shows that the vast majority of electromagnetic (em) waves coming from the surface of the Earth are indeed blocked by the Moon’s spherical body if the radio receiver of these waves is located at the Lagrangian point L2 of the Earth-Moon System, as indeed is the case for China’s Queqiao relay satellite. In our calculations all distances are measured in km, and we conventionally assume that the distance of the Earth-Moon Lagrangian point L2 above the Moon’s surface and along the EarthMoon axis is 64.900 km. For an easy description of this L2 point, see the Wikipedia site https://en.wikipedia.org/wiki/Lagrangianpoint. With reference to Fig. 7, we use the notations: (1) EM is the Earth-Moon distance, assumed to be equal to 384,402 km, with the Moon assumed for simplicity to be in a circular orbit around the Earth. (2) rE is the Earth’s radius, assumed for simplicity to be equal to 6378.1 km, with the Earth being assumed to be spherical.

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(3) rM is the Moon’s radius equal to 1738.1 km, with the Moon assumed to be spherical for simplicity. Proof of the Blocking Equation Just consider the two similar and rectangular triangles having centers at the center of the Earth and the Moon, respectively. The two hypotenuses are given by (EM + x) and x, respectively, and so we may immediately write down the proportion x EM + x = rE + h rM

(1)

This (1) is a linear algebraic equation in x, that, solved for x, yields the x(h) function, i.e. the distance x from the Moon center, along the Earth-Moon axis, at which em waves, emitted by an Earth satellite in a circular orbit around the Earth at height h above the Earth surface, intercept the Earth-Moon axis: x(h) =

EM ·rM (r E + h) − r M

(2)

A glance to (2) reveals that (2) is an equilateral hyperbola conveniently shifted form the origin in the (h, x) plane. The plot of this equilateral hyperbola (in blue) is given in Fig. 2. On the vertical axis one has the height h (in km) above the Earth’s surface from which em waves are emitted towards the Moon. On the horizontal axis one has the distance x beyond the Moon’s center at which these em waves intercept the Earth-Moon axis. After inserting the above numerical values (1), (2) and (3), the Eq. (2) is converted into the numeric equilateral hyperbola equation x = (6.6774471419999993 ∗ 108 )/(h + 4641.0)

(3)

(This author did all calculations and plots by virtue of NASA’s symbolic manipulator “Maxima”, and so two minus signs are unfortunately retained in the equilateral hyperbola equation appearing in Fig. 12, but that’s just formal). In Fig. 7 the horizontal red line shows the distance of 64,900 km along the EarthMoon axis where the Earth-Moon Lagrangian point L2 is located (approximatively). One can see that all em waves emitted in between zero height (h = 0), i.e. just from the surface of the Earth, where Humans live, and about h ≈ 5000 km are indeed blocked by the Moon’s sphere and cannot reach the Lagrangian point L2. On the contrary, em waves emitted from satellites orbiting the Earth at heights higher than about 5000 km pass beyond the Moon’s spherical body and hit the Lagrangian point L2, where the Queqiao relay satellite now stands. In other words still, Queqiao and all future similar satellites located at L2 will be affected by em radiation coming from satellites orbiting the Earth at heights higher than about 5000 km (Fig. 12).

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Fig. 12 Blocking Equation plot. Showing how the Moon’s spherical body blocks em waves produced by a satellite in circular orbit around the Earth at height h above the Earth’s surface. The height h above the Earth’s surface is shown in km on the horizontal axis, ranging between zero (i.e. em waves produced by Humanity at the surface of the Earth, namely the vast majority of em waves in the vicinity of the Earth) and 35,000 km, i.e. the height of geostationary satellites, like the telecommunication relay satellites in the Earth’s equatorial plane

It is possible to make the above estimated blocking distance of 5000 km more precise by inverting (2), then yielding the h(x) function (the inversion is immediate) that is about 2.216 times the distance of L2 from the Moon’s center. h(x) =

EM ·rM +rM −rE x

(4)

Upon replacing the L2 distance of 64,900 km and all other numeric data (1), (2), (3) into (4), the latter yields h_ blocking = 5637.164 km

(5)

Finally, setting h(x) = 0 into (4) and then solving the resulting equation for x, one gets the distance above the Moon and along the Earth-Moon axis at which em waves emitted by Humanity on Earth are no longer blocked by the Moon spherical body: distance_ above_ Moon_ of_ no_ loger_ blocked_ em_ waves_ emitted_ by_ Humanity_ at_ Earth_ surface EM ·rM ≈ 143732.120 km =x= rE −rM

(6)

14 The Years 2018–2019

751

14 The Years 2018–2019 The year 2018 has seen a couple of innovative and politically quite important Chinese space missions (for more details, please see the Wikipedia site https://en.wikipedia. org/wiki/Chang%27e4#Queqiaosatellite, from which we now freely quote): (1) Direct communications with Earth are impossible on the Farside of the Moon, since transmissions are blocked by the Moon itself. Communications must go through a communications relay satellite, which is placed at a location that has a clear view of both the Farside landing side and the Earth. On May 20, 2018, the Chinese National Space Agency (CNSA) launched the Queqiao (“Magpie Bridge”) relay satellite to a halo orbit around the Earth-Moon L2 Lagrangian point. The relay satellite has a mass of 425 kg, and it uses a 4.2-m antenna to receive X band signals from the lander and rover of the Chang’e 4 space mission to the Farside and relay them to Earth control on the S band. This spacecraft took 24 days to reach L2, using a lunar swing-by to save fuel. On June 14, 2018, Queqiao finished its final adjustment burn and entered the L2 halo mission orbit, which is about 65,000 km above the Moon Farside. This is the first lunar relay satellite ever at this location, but there might be more in the future, when other space-faring nations decide to explore the Farside. (2) The Chang’e 4 spacecraft, including a robotic lander and a rover, was launched on December 7, 2018, and landed on the Farside on January 3, 2019, in the Von Karman crater (180 km in diameter), that is in the South Pole-Aitken Basin on the Moon Farside. These two space missions are marking an historic change in the Moon exploration. In fact, not only this is the first time that the exploration of the Moon Farside is attempted, but the Moon Farside “radio silence” that existed prior the Queqiao arrival at L2 is now partially polluted for the first time by the radio emissions of Queqiao in both X and S bands. (3) As we mentioned already, the International Academy of Astronautics (IAA, based in Paris) had been studying since the 1990s the possibility of setting up a future radio telescope inside crater Daedalus, an 80 km crater practically located at the center of the Farside. One might also say that Daedalus is very close to the Antipode, the point along the Earth-Moon axis exactly opposite to the Earth on the Farside. On 10 June 2010, this author presented at the United Nations Committee on the Peaceful Uses of Outer Space (COPUOS, based in Vienna) his proposal to create a “Protected Antipode Circle” (PAC) area around the Antipode to let a future radio telescope landed inside Daedalus to observe the radio sky as “purely” as it was before Humans discovered radio waves. In other words, this author’s proposal to the UN was to ask the space-faring nations to come to an agreement reserving the PAC for astronomical exploration only, since crater Daedalus is the closest location to Earth where the radio pollution produced by Humankind does not still practically reach. This activity would be especially important for SETI, the Search for ExtraTerrestrial Intelligence that NASA now calls “Searching for Technosignatures” just to avoid the “toopolitical” word SETI. Actually, setting up more than one radio telescope inside

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SETI Space Missions

Daedalus would enable interferometry, the best technique for radio astronomical searches ever. Even more precisely, these radio telescopes would not be dishes at all: they would be phased arrays, i.e. planar deployable surfaces with dipoles commanded by computers, of course much easier to be landed inside Daedalus than big dishes. (4) However, the communications relay satellite Queqiao now at L2 changes this idyllic picture. In fact, it is obvious that the future astronomical observatory Daedalus will have to abandon radio exploration on the X and S bands because blinded by Queqiao, unless negotiations between China and the other space-faring nations will reach a general agreement on which bands are reserved for what. This is a task for international organizations like ITU (International Telecommunications Union), the American CORF (Committee on Radio Frequencies) and the European CRAF (Committee of Radio Astronomy Frequencies), plus other related organizations worldwide, like the Moon Village Association (MVA). (5) On March 27, 2019 this author, on behalf of the IAA, intends to organize in Paris an IAA International Conference by the title of “Moon Farside 2019”: the first attempt ever to reach an international agreement about the frequencies allowed or not allowed over the PAC in the future.

15 The Farside Spectrum Still Is not Polluted (in August 2019) Except in the S, X and UHF Bands Let us now take a closer look at the electromagnetic radio spectrum as given, for instance at the Wikipedia site https://en.wikipedia.org/wiki/Radiospectrum from which the two Fig. 13 are taken.

Fig. 13 IEEE radar bands. Frequency bands in the microwave range are designated by letters. This convention began around World War 2 with military designations for frequencies used in radar, which was the first application of microwaves. Unfortunately there are several incompatible naming systems for microwave bands, and even within a given system the exact frequency range designated by a letter varies somewhat between different application areas. One widely used standard is the IEEE radar bands established by the US Institute of Electrical and Electronic Engineers

16 Queqiao and Chang’e 4 Communications Bands Above the Moon Farside

753

Fig. 14 It plainly appears that only the S, X and UHF bands are involved in the whole mission. No other bands are involved at all. In particular, the L band, crucial to SETI searches, is basically not involved, except for the UHF band used by the rover communicate with the lander, that should have a modest power (unknown to this author at the moment). Thus, had we radio telescope nowadays working inside crater Daedalus for SETI, the Chinese Farside spacecrafts would NOT hamper its searches for ETs at all: good news for SETI scientists

16 Queqiao and Chang’e 4 Communications Bands Above the Moon Farside Please look at Fig. 14, summarizing well the telecommunications involved with the Chang’4 and Queqiao spacecrafts, as well as with the rover exploring the Farside (taken from the Planetary Society website http://www.planetary.org/multimedia/ space-images/spacecraft/change-4-mission.html).

17 COSMOLOGY: Need for Ultra-Low Frequency Radio Astronomy in Space Within the Quiet Cone Above the Moon Farside, I.E. just at the Lagrangian Point L2, Where Queqiao Is Let us now consider ultra-low frequency radio astronomy (frequencies below ~20 MHz). At these frequencies, “single dish” (or “single phased array”, if you wish) radio astronomy is quite helpless due to very low angular resolution: it will be

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SETI Space Missions

Fig. 15 COSMOLOGY. The Netherlands-China Low-frequency Explorer (NCLE) instrument with three 5-meter antennas for Queqiao, now (January 2019) in place in a halo orbit around the Lagrangian point L2 about 65,000 km right above the Farside, and so within the Quiet Cone shielded from man-made radio pollution coming from Earth. This NCLE is essentially made by three 5-mlong ribs (very low frequency antennas) here shown under construction in the Netherlands before they were sent to China for the Queqiao launch, in the spring of 2018. Principal Investigator of NCLE is renowned astronomer Heino Falcke, sponsored by ESA to build NCLE in the Netherlands

totally knocked-out by the confusion effect. The only way out is to employ interferometry. However, employing radio interferometry on the Farside will bring its own issues in the technology. The case in question is well exemplified by Cosmology, as we now briefly describe. NCLE (Netherlands-China Low Frequency Explorer) is considered a pathfinder mission for a future low-frequency space-based or moon-based radio interferometer which has the detection and tomography of the 21-cm Hydrogen line emission from the Dark Ages period as the principal science objective. In simple words, the hydrogen line that nowadays is at 1420 MHz, i.e. 21 cm, was at a much lower frequency about 300,000 years after the Big Bang, and cosmologists hope to detect it at L2 by virtue of NCLE (Figs. 15 and 16).

18 SETI (or “Technosignatures”, According to NASA’s 2018 “New Jargon”) NASA did about a year of SETI searches in between October 12, 1992 and October 3, 1993, but was forbidden by Congress to do any SETI activity in the next 25 years (though Astrobiology was supported by NASA in the meantime). Surprisingly, NASA started again to take an interest in SETI in 2018. On September 26–28,

18 SETI (or “Technosignatures”, According to NASA’s …

755

Fig. 16 Title of the article published by renowned Cosmologist Joseph Silk in Nature, January 4, 2018

2018, the first NASA Technosignatures Workshop was held in Houston at the Lunar and Planetary Science (LPI) Institute, website https://www.hou.usra.edu/mee tings/technosignatures2018/agenda/. Maccone was invited to make a presentation about “SETI in Europe” and he did so, adding the Moon Farside Protection also, see the website https://www.hou.usra.edu/meetings/technosignatures2018/presentat ion/?video=maccone.mp4.

19 Summary About This Author’s Work to Legally Protecting Radio Astronomy on the Farside and Within the Quiet Cone in the Space Above the Farside The goal of this paper was to make the readers sensitive to the importance of protecting the Central Farside of the Moon from any future anti-scientific exploitation. The Farside of the Moon is a unique place for us in the whole universe: it is close to the Earth, but protected from the radio emissions that we ourselves are creating in an ever increasing amount, making our radio telescopes on Earth blinder and blinder. In particular, we think we gave sound scientific reasons why the PAC, Protected Antipode Circle, should be declared an internationally protected area under the Protection of the United Nations, or, in absence of that institution, by a direct agreement among the space-faring nations. To achieve this goal, in the mentioned IAA Cosmic Study we analysed the given legal situation and presented a procedure for establishing the PAC. This would encompass the declaration of the PAC as ‘international scientific preserves’ or ‘scientific reserves’, although there is a possibility of ramification on the level of domestic law and regulation. This effort would require a broad constituency of nations and could be pursued based on Art. 7(3) of the Moon Agreement. Furthermore, we showed that the existing Radio Regulations could guarantee the necessary communication frequencies for a future base on the lunar Farside, as well as the necessary radio silence for successful astronomic endeavours within the PAC.

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SETI Space Missions

We also summed up several ideas for the scientific use of the Farside to gain further knowledge of the universe (and its inhabitants, that is SETI). The Farside cannot be left to the realtor’s speculations! And this is an urgent matter! Some international agreement must be taken for the benefit of all Humankind.

20 CONCLUSIONS: The Moon Village Should Be Located Outside the PAC and Along the 180 Degrees Meridian, Possibly Close to the South Pole We conclude this review paper by proposing (again, see also Maccone [5]) that the new “Moon Village” supported by the vision of the ESA Director General, Jan Woerner, be located OUTSIDE the PAC (obviously not to interfere with the detection of radiation coming from space) but also SOUTH OF THE PAC, to be “close” to the South Pole as much as needed to benefit of water there. It thus appears the best venue for the “Moon Village” would be on or around the 180 degree meridian and south to the -30 degree in latitude of the PAC, possibly much more south of that, almost at the South Pole, thus resolving Moon Village VENUE ISSUE..

References 1. L. David, February 10, 2019, http://www.leonarddavid.com/protect-the-moons-farside-a-scient ists-plea-for-quiet-zone/. 2. C. Maccone, The quiet cone above the farside of the moon. Acta Astronaut. 53(2003), 65–70 (2003) 3. C. Maccone, Moon farside radio lab. Acta Astronaut. 56(2005), 629–639 (2005) 4. C. Maccone, Protected antipode circle on the farside of the moon. Acta Astronaut. 63(2008), 110–118 (2008) 5. C. Maccone, Moon farside protection, moon village and PAC (Protected Antipode Circle). Acta Astronaut. 154(2019), 233–237 (2019) 6. S. Pluchino, N. Antonietti, C. Maccone, Protecting the moon farside Radiotelescopes from RFI produced at future Lagrangian-Points Space Stations, paper IAC-06-D4.1.01 presented at the International Astronautical Congress held in Valencia (Spain), October 2–6, 2006

Power and Energy of Civilizations by Logpar and Logell Power Curves (with Ancient Rome Example)

Abstract This chapter presents a new mathematical model for the ENERGY that a living being needs in order to live its whole life between birth and death. This also applies to a civilization made up by many living beings. The model is based on a LOGELL POWER CURVE that is a curve in the time made up by a lognormal probability density in between birth and peak, followed by the descending quarter of an ellipse between peak and death (LOGELL means LOGnormal plus ELLipse). We derive analytic equations yielding the ENERGY in terms of three free parameters only: the time of birth b, the time of the power peak p, and the time of death occurs, d. The author’s previously published papers about his so-called Evo-SETI Theory (Evo-SETI stands for Evolution and SETI) cover the biological evolution over the last 3.5 billion years described as an increase in the number of living Species from one (RNA) to the current (say) 50 million. Past mass extinctions make this evolution become a stochastic process having an exponential mean value, called Geometric Brownian Motion (GBM). In those papers, a lifetime, rather than a logell, was a b-lognormal, i.e. a lognormal starting at instant b (birth) and descending straight to death at its descending inflexion point. Our mathematical discovery of the PeakLocus Theorem showed that the GBM exponential is the geometric locus of all the peaks of the b-lognormals. Since b-lognormals are probability densities, the area under each of them always equals 1 (normalization condition) and so, going from left to right on the time axis, the b-lognormals become more and more “peaky”, and so they last less and less in time. This “level of civilization” is what physicists call (Shannon) ENTROPY of information, meaning that the higher Species have higher information content than the lower Species. This author also proved mathematically that, for all GBMs, the (Shannon) Entropy of the b-lognormals grows LINEARLY in time. The Molecular Clock, well known to geneticists since 1962, shows that the DNA base-substitutions occur LINEARLY in time since they are neutral with respect to Darwinian selection. The conclusion is that the Molecular Clock and the LINEAR increase of EvoEntropy in time are just the same thing! In other words, we derived the existence of the Molecular Clock mathematically as a part of our Evo-SETI Theory. Finally, this linearly growing entropy is just the new EvoSETI SCALE to measure the evolution of life on Exoplanets (measured in bits). In conclusion, our invention of the logell power curve, described in this paper, provides a new

© Springer Nature Switzerland AG 2020 C. Maccone, Evo-SETI, https://doi.org/10.1007/978-3-030-51931-5_24

757

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Power and Energy of Civilizations by Logpar and Logell …

mathematical tool for our Evo-SETI mathematical description of Life, History and SETI.

1 Part 1: Logell Curves and Their History Equations 1.1 Introduction to Logell “Finite Lifetime” Curves The starting idea is easy: we seek to represent and summarize the lifetime of any living being by virtue of just three points in the time: birth, peak, death (BPD). No other point in between them is needed. That is, no other “senility point” s is needed such as the one appearing in all b-lognormals that this author had published in his Evo-SETI Theory prior to 2017. In fact, it is easier and more natural to describe someone’s lifetime just in terms of birth, peak and death, than in terms of birth, senility and death, since it is uncertain to define when senility arrives, and actually hard to define in the practice for any individual or for any civilization. Please look at Fig. 1. The first part, the one on the left, i.e. prior to the peak time p, is just a b-lognormal: it starts at birth time b, climbs up to the adolescence time a (ascending inflexion point of the b-lognormal) (in reality the adolescence time should more properly be called “puberty time” since it marks the beginning of the reproduction capacity for that individual) and finally reaches the peak time at p (maximum, i.e. the point of zero

Fig. 1 Representation of the History of the Ancient Roman civilization as a LOGELL curve, finite in the time. Rome was funded in 753 B.C., i.e. in the year −753 in our notation, or b = −753. Then the Roman republic and empire (the latter since the first emperor, Augustus, roughly after 27 B.C.) kept growing in conquered territory until it reached its peak (maximum extension, up to Susa in current Iran) in the year 117 A.D., i.e. p = 117, under emperor Trajan. Afterwards it started to decline and loose territory until the final collapse in 476 A.D. (d = 476, Romulus Augustulus, last emperor). Thus, just three points in time are necessary to summarize the History of Rome: b = −753, p = 117, d = 476. The numbers along the vertical axis will be explained later

1 Part 1: Logell Curves and Their History Equations

759

Fig. 2 Representation of the History of the Roman civilization as a LOGPAR finite curve. This logpar curve in time is made up by a b-lognormal in between birth and peak, and a parabola in between peak and death. It was used by this author in the years 2017 and 2018 in Refs. [1, 2]. Please have a look at these references for the full logpar mathematical description, culminating in the two Logpar History Formulae expressing the b-lognormal’s two parameters μ (a real number) and σ (a positive number) in terms of the three assigned real numbers b, p and d, with the condition b< p 0 and − ∞ < μ < ∞. Tables listing the main equations that can be derived from (7) were given by this author in Refs. [1, 2] and we shall not re-derive them here again. We just confine ourselves to reminding that: 1. The abscissa p of the peak of (7) is given by p = b + eμ−σ . 2

(8)

Proof. Take the derivative of (7) with respect to t and set it equal to zero. Then solve the resulting equation for t, that now becomes p, and (8) is found. 2. The ordinate P of the peak of (7) is given by σ2

e 2 −μ . P=√ 2π σ

(9)

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Power and Energy of Civilizations by Logpar and Logell …

Proof. Rewrite p instead of t in (7) and then insert (8) instead of p. Then simplify to get (9). 3. The abscissa of the adolescence point (that should actually be better named “puberty point”) is the abscissa of the ascending inflexion point of (7). It is given by a = b + e−

σ

√

2 σ 2 +4 − 3σ2 2

+μ

(10)

Proof. Take the second derivative of (7) with respect to t and set it equal to zero. Then solve the resulting equation for t, that now becomes a, and (10) is found. 4. The ordinate of the adolescence point is given by e−

σ

√

2 σ 2 +4 + σ4 4

√

−μ−

1 2

(11)

2π σ

Proof. Just rewrite a instead of t in (7) and then insert (10) and simplify the result. Let us now notice that, within the framework of the logell theory described in this paper, we may NOT say that (7) fulfills the b-lognormal’s normalization condition ∞ b_lognormal(t; μ, σ, b)dt = 1.

(12)

b

In fact, (7) here is only allowed to range in between b and p. Thus, rather than adopting (12), we must thus replace (12) by the integral of (7) between b and p only. Fortunately, it is possible to evaluate this integral in terms of the error function defined by 2 er f (x) = √ π

x

e−t dt. 2

(13)

0

In fact, the integral of the b-lognormal (7) between b and p turns out to be given by p

p b_lognormal(t; μ, σ, b)dt =

b

b

−

(log(t−b)−μ)2 2σ 2

e dt = √ 2π σ (t − b)

1 + er f

√

√

2 ln( p−b)− 2μ 2σ

2

Now, inserting (8) instead of p into the last erf argument, a remarkable simplification occurs: μ and b both disappear and only σ is left. In addition, the erf property er f (−x) = − er f (x) allows us to rewrite

1 Part 1: Logell Curves and Their History Equations

=



1 + er f − √σ2 2

=

763

1 − er f



σ √ 2

.

2

(14)

In conclusion, the area under the b-lognormal between birth and peak is given by p b_lognormal(t; μ, σ, b)dt =

1 − er f



σ √ 2

2

.

(15)

b

This result will prove to be of key importance for the further developments described in the present paper.

1.4 Area Under the Ellipse on the Right Part of the Logell Between Peak and Death We already proved that the ellipse on the right part of the logell curve has Eq. (5). Now we want to find the area under this ellipse between peak and death. If one remembers that the area of the whole ellipse with semi-axes a and b equals π a b, it is obvious that our area is just a quarter of that. The same results is of course found by evaluating the definite integral (we leave this calculation to the reader as an exercise) 

d P

1−

p

π (t − p)2 dt = P(d − p). 2 4 (d − p)

(16)

1.5 Area Under the Full Logell Curve Between Birth and Death We are now in a position to compute the full area A under the logell curve, that is given by the sum of Eqs. (15) and (16), that is 1 − er f 2



σ √ 2

+

π P(d − p) = A 4

(17)

This is one of the most important equations in this paper. In fact, if we want the logell be a truly probability density function (pdf), we must assume in (17) A=1

(18)

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Power and Energy of Civilizations by Logpar and Logell …

But, surprisingly, we shall NOT do so! Let us rather ponder over what we are doing: 1. We are creating a “Mathematical History” model where the “unfolding History” of each Civilization in the time is represented by a logell curve. 2. The knowledge of only three points in time is requested in this model: b, p and d. 3. But the area under the whole curve depends on σ as well as on μ, as we see upon inserting (9) instead of P into (17), that is 1 − er f



2

σ √ 2



σ2

e 2 −μ π · (d − p) = A(μ, σ ). +√ 2π σ 4

(19)

4. Also p is to be replaced by its expression (8) in terms of σ and μ, yielding the new equation 1 − er f 2



σ √ 2



σ2 2

e + √

−μ(σ )

2π σ

·

2 π d − b − eμ(σ )−σ 4

= A(σ ).

(20)

5. The meaning of (20) is that if birth and death are fixed, but the position of the peak may freely move in between them according to the different living beings or civilizations that we are going to consider. In other words, (20) yields different numeric values of σ and μ(σ ) according to where the peak is in between birth and death.. 6. In addition, we would like to get rid of the error function er f in (20). How may we do so?

1.6 The Area Under the Logell Curve Depends on Sigma Only, and Here Is the Area Derivative w.r.t. Sigma The simple answer to the last question (6) is “by differentiating both sides of (20) with respect to σ ”. In fact, the derivative of the er f function (13) is just the “Gaussian” exponential 2 d er f (x) 2 = √ · e−x . dx π

(21)

and so the er f function itself will disappear by differentiating (20) with respect to σ . Actually, the derivative of the first term on the left hand side of (20) simply is, according to (21),

1 Part 1: Logell Curves and Their History Equations

⎤ ⎡ σ σ2 d ⎣ 1 − er f √2 ⎦ e− 2 =−√ . dσ 2 2π

765

(22)

As for the derivative with respect to σ of the second term on the left hand side of (20) we firstly notice that σ appears three times within that term. Thus, the relevant derivative is the sum of three terms, each of which includes the derivative of one of the three terms multiplied by the other two terms unchanged. In equations, one has:

⎤

⎡ σ μ−σ 2 σ2 e 2 −μ π d − b − e d ⎣ 1 − er f √2 ⎦ +√ · dσ 2 4 2π σ

σ2 √ μ−σ 2 √ − σ2 σ2 − π − e + d − b e 2 −μ πe 2 e 2 = −√ + − 3 5 2π 22 22 σ2

√ σ2 2 π eμ−σ − d + b e 2 −μ . − 5 22

(23)

Several alternative forms of this Eq. (23) are possible, and that is rather confusing. However, using a symbolic manipulator (this author did do so by virtue of NASA’s Maxima), a few more steps lead to the following form of (23): e− d A(μ(σ ), σ ) d A(σ ) ≡ = dσ dσ

σ2 2

+μ

√     2  2 π (d − b) σ 2 − 1 eσ + eμ σ 2 (π − 4) + π . √ 8 π σ2 (24)

We may further simplify (24) by inserting (8) and so letting μ disappear. The result is √    2   σ2 2 e− 2 +μ 2 π (d − p) σ 2 − 1 eσ + eσ ( p − b) σ 2 (π − 4) + π d A(σ ) = √ dσ 8 πσ2 √ 2     σ2 e− 2 +μ 2eσ π (d − p) σ 2 − 1 + ( p − b) σ 2 (π − 4) + π = √ 8 π σ2 √   σ2 e 2 +μ 2 σ 2 [π (d − b) + (π − 4)( p − b)] − (π (d − p)) = . (25) √ 8 π σ2 This (25) is the derivative of the logell area with respect to sigma.

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Power and Energy of Civilizations by Logpar and Logell …

1.7 Exact “History Equations” for Each Logell Curve We now take a further, crucial step in our analysis of the logell curve: we IMPOSE that the derivative of the area with respect to sigma, i.e. (25), is zero d A(σ ) = 0. dσ

(26)

What does that mean? Well, (26) is the Evo-SETI equivalent of the LEAST ACTION PRINCIPLE in physics! This conclusion does not show up at the moment, but it will at the end of this paper. For the time being rewrite the imposed condition (26) by virtue of the last expression in (25) that, getting rid of both the exponential and the denominator, boils down to σ 2 [π (d − b) + (π − 4)( p − b)] − (π (d − p)) = 0.

(27)

The last equation is a quadratic equation in σ σ 2 [π (d − b) + (π − 4)( p − b)] = π (d − p)

(28)

That, solved for σ 2 , immediately yields σ2 =

π (d − p) π (d − b) + (π − 4)( p − b)

(29)

This is the most important new result discovered in the present paper: the LOGELL HISTORY EQUATION FOR σ √ √ π d−p σ =√ . π (d − b) + (π − 4)( p − b)

(30)

In words, given the input triplet (b, p, d), then (30) immediately yields the exact σ of the elliptical left part of the logell curve. It was discovered by this author on September 4, 2018, and led not only to this paper, but to the introduction of the ENERGY spent in a lifetime by a living creature, or by a whole civilization whose “power-vs-time” behaviour is given by the logell curve, as we will understand better in the coming sections of this paper. At the moment we confine ourselves to taking the limit of both sides of (30) for d → ∞, with the result √ √ √ √ π d−p π d = lim √ = 1. lim σ = lim √ d→∞ d→∞ d→∞ π (d − b) + (π − 4)( p − b) πd

(31)

1 Part 1: Logell Curves and Their History Equations

767

Since we already know that σ must be positive, (31) really shows that σ may range between zero and one only 0 < σ < 1.

(32)

With the remark that the two limiting values σ → 0 and σ → 1 are “unphysical”, since one may not die at birth nor die at an infinite age, respectively. Next to (30) one of course has a similar LOGELL EQUATION FOR μ, that is immediately derived from (8) and (30). To this end, just take the log of (8) to get μ = ln( p − b) + σ 2

(33)

that, invoking (29), one finds the desired logell equation for μ μ = ln( p − b) +

π (d − p) . π (d − b) + (π − 4)( p − b)

(34)

In conclusion, our key two LOGELL HISTORY EQUATIONS are 

√ √

π d−b σ = √π(d−b)+(π−4)( p−b) π(d− p) μ = ln( p − b) + π(d−b)+(π−4)( . p−b)

(35)

1.8 Considerations on the Logell History Equations Some considerations on the logell History Formulae (35) are now of order: 1. All these formulae are exact, i.e. no Taylor series expansion was used to derive them. 2. But they were obtained by equalling to zero the derivative with respect to σ of the total area under the logell curve given by (20). 3. Therefore the logell History Formulae (35) are the equations of a minimum (we shall later show that this is indeed a minimum and not a maximum) of the A(σ ) function expressing the total area (20) as a function of σ . This minimum is the MINIMUM ENERGY PRINCIPLE of our Evo-SETI Theory, i.e. “in our lifetime, we always act in such a way as to minimize the energy that we are using”.

768

Power and Energy of Civilizations by Logpar and Logell …

1.9 Logell Peak Coordinates Expressed in Terms of (b, p, d) Only Of particular importance for all future logell applications is the expression of the peak coordinates ( p, P) expressed in terms of the input triplet (b, p, d) only. Since the peak abscissa p is assumed to be known, we only have to derive the formula for the peak ordinate P. That is readily obtained by inserting the logell History Formulae (35) into the peak height expression (9). After a few rearrangements, it is found to be given by √ P=

(π − 4)( p − b) + π (d − b) − e √ √ 2 π d − p( p − b)

π(d− p) 2 [(π−4)( p−b)+π(d−b)]

(36)

1.10 History of Rome as an Example of How to Use the Logell History Formulae Let us go back to the History of Rome as summarized in the caption to Fig. 1. First of all, let us write down neatly the key three numeric input values in the History of Rome that were already mentioned in the caption to Fig. 1: ⎧ ⎨ b = −753 Rome_input_triplet = p = 117 ⎩ d = 476.

(37)

Then the logell History Eqs. (35) immediately yield numerical values of the logell σ and μ for Rome, that we shall hereafter denote by σ R and μR, respectively  Rome_logell_doublet

σ R = 0.602 μR = 7.131.

(38)

Next the study of the logell peak comes. We already know from (37) that the abscissa of the peak of the Roman civilization was in 117 A.D. under Trajan p Rome = 117.

(39)

But the logell peak ordinate for Rome must be found by virtue of (36). One thus gets Plog ell_Rome = 6.358 × 10−4 = 0.0006358.

(40)

2 Part 2: Energy as the Area Under Logell Power Curves

769

2 Part 2: Energy as the Area Under Logell Power Curves 2.1 Area Under Any Logell Power Curve and Its Meaning as “Lifetime Energy” of that Living Being The logell history formulae (35) 

√ √

π d− p σ = √π(d−b)+(π−4)( p−b) π(d− p) μ = ln( p − b) + π(d−b)+(π−4)( . p−b)

(41)

are “much similar” to the logpar history formulae (39) of Ref. [1], that is 

√ √

2 d− p σ = √2d−(b+ p) μ = ln( p − b) +

2(d− p) . 2d−(b+ p)

(42)

From this point in the paper onward, we may thus proceed along the lines outlined in Ref. [1]. What is the physical meaning of the area (20)? If we consider the logell curve as the curve of the power (measured in watts) of the Roman civilization along the whole of its history course, then the area under this curve, i.e. the integral of the logell between birth and death is the total energy (measured in joules) spent by the civilization in its whole lifetime: ENERGY_spent_in_the_Civilization_LIFETIME d =

POWER_of_that_Civilization(t)dt b

d =

logell_curve_of_that_Civilization(t)dt.

(43)

b

In other words, if we know the power curve of any living being that lived in the past, like a cell, or an animal, or a human, or a Civilization of humans or of any other living forms (including ExtraTerrestrials), the integral of that power curve, i.e. logell curve, between birth and death, is the TOTAL ENERGY spent by that living form during the whole of its lifetime. One more point regarding the last statement: if we assume that all Humans have potentially the same amount of energy to spend during their whole lifetime, then the logell of great men who “died young” (like Mozart, for instance) must have the same area below their logell and so a much higher peak since they lived shorter than others.

770

Power and Energy of Civilizations by Logpar and Logell …

Let us next consider the condition that usually one’s death comes after one’s peak, but, in exceptional cases, one’s death may also come just at one’s peak (as, for instance, for death in a car accident), that is d≥p

(44)

We now wish to find the value of the ENERGY AT THE PEAK of one’s lifetime. Then, taking the limit of (19) for d → p from above and noticing from (13) that er f (0) = 0 we find Energy( p) = lim

1 − er f



σ √ 2



2 1 1 − er f (0) = . = 2 2 d→ p

= lim

1 − er f



√ √ π d− p √ √ 2 π(d−b)+(π−4)( p−b) 2

d→ p

(45)

This (45) may look as a surprising result: why should the Energy of someone at its peak be equal to just 1/2, and not to any other positive value (in joules)? Well, we will soon solve this matter of extending the Energy at peak from 1/2 to any other positive value as we did already in Ref. [1]. But, for the time being, please just content yourself of using the conventional value 1/2 to simplify the calculations. On the contrary, if we take the limit of (19) for d → ∞ (this limit is called “immortality limit” since the living being or civilization is now supposed to live for an infinite amount of time) we immediately see that

lim A(μ, σ ) = lim

d→∞



⎧ ⎨ 1 − er f √σ

d→∞⎩

2

2

⎫ σ2 ⎬ e 2 −μ π +√ · (d − p) = ∞. ⎭ 2π σ 4

(46)

In other words: “if you want to live for an infinite amount of time, you need and infinite amount of energy to do so” (as it is obvious). Next question: what is the time of the energy minimum? We skip all the lengthy calculations made by Maxima and just write here the result: √ dabscissa_ of_ logell_ minimum_ Energy = b +

−π 2 + 8π − 12 + 2 ( p − b) π

.

(47)

This is the abscissa of the minimum of the Energy: we could prove that it is really a minimum, rather than a maximum, by computing the second derivative, but we shall not do so here for the sake of brevity. Let us rather remark that, for the case (37) of Ancient Rome, (47) yields the year of Rome’s minimum energy as 301 A.D. This was the time of emperor Diocletian, certainly a very troubled time, since it was the time of the largest persecution ever against the Christians, site: https://en.wik ipedia.org/wiki/Diocletian. Tourists in today’s Rome, visiting the Basilica of “Santa Maria degli Angeli e dei Martiri”, might just think of the 40,000 Christian slaves

2 Part 2: Energy as the Area Under Logell Power Curves

771

that died to build the Diocletian’s Baths, i.e. just at the minimum of Pagan Rome’s Civilization.

2.2 Discovering an Oblique Asymptote of the Energy Function Energy (D) While the Death Instant D Is Increasing Indefinitely This author discovered (on September 4, 2018) that the following logell total energy (48) has an oblique asymptote for d → ∞. Before we derive the equation of this oblique asymptote, however, some careful understanding of what d means is in order. We always said that the logell theory described in this paper necessitates the three inputs (b, p, d). However, in this section, we are going to consider higher and higher values of the death instant d so that the area under the logell, i.e. the energy of the phenomenon of which the logell is the power, may assume any assigned value. Thus, in this section, the death time d becomes a sort of new independent variable D rather than just one of the three fixed inputs (b, p, d). In other words still, we will be careful to make the distinction between 1. The fixed, i.e. known, death instant d and 2. The movable, i.e. independent variable D allowing us to extrapolate into the future the logell having the three fixed input values (b, p, d). Having so said, the logell total ENERGY provided by Maxima upon inserting (41) into (20) and then rearranging, must more correctly be rewritten as a function of D rather than d and reads

√ √ π D− p 1 − er f √2√π(D−b)+(π−4)( p−b) logell_total_energy(D) = 2 π(D− p) √ √ D − p π (D − b) + (π − 4)( p − b)e− 2π D+(2π−8) p+(8−4π)b + · 5 2 2 ( p − b) (48) Figure 4 shows Rome’s both LOGELL total energy in blue and the LOGPAR total energy in red, the latter as mathematically described in Ref. [1]. 1. The solid curve in blue is the LOGELL ENERGY curve as given by Eq. (48) of this paper. The dot-dot straight line in blue is its oblique asymptote. Let now consider the definition of oblique asymptote given in elementary Calculus textbooks: if the limit lim [Energy(D) − (m D + q)]

D→∞

(49)

772

Power and Energy of Civilizations by Logpar and Logell …

Fig. 4 Rome’s total energy as a function of D (i.e. d, the death time regarded now as the independent variable): The solid curve in red is the LOGPAR ENERGY curve as described in Ref. [1]. The dot-dot straight line in red is its oblique asymptote

exists, then the Energy curve Energy(D) approaches more and more the straight line yoblique_ asymptote (D) = m D + q.

(50)

Differentiating (50) with respect to D we immediately see that the angular coefficient m of the oblique asymptote is given by the limit for D → ∞ of the first derivative of the energy (48). In fact, the first derivative of (48) is given by the following lengthy expression provided by Maxima d logell_energy = ((π 2 D 2 − 4π p D − 2π 2 bD + 4π bD + π 2 p 2 dD − 8π p 2 + 16 p 2 − 2π 2 bp + 20π bp − 32bp + 2π 2 b2 − 12π b2 − 12π b2 πp

πD

+ 16b2 )%e 2π D+2π p−8 p−4πb+8b − 2π D+2π p−8 p−4πb+8b )/(25/2 ( p  − b) D − p(π D + (π − 4) p + (4 − 2π )b)3/2 )

(51)

Taking the limit of (51) for D → ∞ we get the energy asymptote’s angular coefficient m:

2 Part 2: Energy as the Area Under Logell Power Curves

773

√ 0.1900 d Energy(D) π = 5√ . = D→∞ dD p−b 2 2 e( p − b)

m = lim

(52)

The same would of course been found had we considered the limit lim

D→∞

Energy(D) m D+q = lim = m. D→∞ D D

(53)

As for the asymptote’s intercept with the vertical axis, q, (50) shows that it is given by the limit q = lim [Energy(D) − m D].

(54)

D→∞

Thus, (50), (52) and Maxima yielded the result

q=

1 − er f 2



√1 2



√ √ √ 3 e πp 2π − 2 2 − 5√ + √ √ . 4 e π 2 2 e( p − b)

(55)

In conclusion, the oblique asymptote to the Logell Energy (48) is given by

y(D) =

π D + π ( p − 2b) − 4( p − b) + 5√ √ 2 2 e π ( p − b)

1 − er f 2



√1 2

.

(56)

3 Part 3: Mean Power in a Logell Lifetime 3.1 Mean Power in a Logell Lifetime In this section we are going to consider the notion of mean value of a logell power curve. Having abandoned the normalization condition for our logell curves, clearly we may not use the same mean value definition of a random variable typical of probability theory. However it’s easy to use the Mean Value Theorem for Integrals instead. This is a variation of the mean value theorem which guarantees that a continuous function has at least one point where the function equals the average value of the function. To translate the Mean Value Theorem for Integrals into a mathematical equation holding for logell curves, we have to start from the Area equation, that is the logell energy Eq. (48) with d replaced by D, and divide that area by the length of the (D − b) segment in order to get the point along the vertical axis such that the area of the rectangle equals the Area. This is the required Mean Power Value over a lifetime and is given by

774

Power and Energy of Civilizations by Logpar and Logell …

A(b, p, D) logell_total_energy(D) = D−b D−b

⎤ √ √ π D− p 1 − er f √2√π(D−b)+(π−4)( p−b) ⎥ ⎥ 2 ⎥· π(D− p) √ √ ⎥ D − p π (D − b) + (π − 4)( p − b)e− 2 π D+(2π−8) p+(8−4π)b ⎦

Mean_POWER_over_a_lifetime = ⎡ 1 = D−b

⎢ ⎢ ·⎢ ⎢ ⎣

+

5

2 2 ( p − b) (57)

Just to check the correctness of (57), consider the limit of the Mean POWER over a lifetime (57) for D → ∞. The calculation implies the use of L’Hospital’s rule, and the result is Asymptotic_Mean_Power_over_a_lifetime = lim (Mean_Power_over_a_lifetime) D→∞ ! logell_total_energy(D) = lim D→∞ D−b √ π = 5√ . 2 2 e( p − b)

(58)

But, this is, of course, the same as angular coefficient (52) of the logell energy asymptote, since this angular coefficient is just the limit of the derivative of the logell total energy for D → ∞. So, everything makes sense. By this we have completed the study of the mean along the vertical axis, i.e. the power axis. However, one might still wish to find, in some sense, “the mean value of what lies on the horizontal axis”, i.e. the lifetime mean value. That is done in the next section.

4 Part 4: Logell Lifetime Mean Value 4.1 Lifetime Mean Value It is natural to seek for some mathematical expression yielding the mean value of a lifetime, meaning the mean value along the time axis of the (D − b) time segment representing the lifetime of a living organism, or a civilization or even an ET civilization. We propose the following definition of such a lifetime mean value:

4 Part 4: Logell Lifetime Mean Value

775

p lifetime_mean_value =

D t · b_lognormal(t; μ, σ, b) dt +

t · ellipse(t) dt = p

b

(59) inserting the b-lognormal (7) and the ellipse (5) into (59), the latter is turned into p = b

(log(t−b)−μ)2

e− 2σ 2 t· √ dt + 2π σ (t − b)

D

 t · P 1−

p

(t − p)2 dt. (D − p)2

(60)

The first integral may be computed in terms of the error function er f (x) given by (13), and the result is p b

2

(log(t−b)−μ) e e− 2σ 2 t· √ dt = 2π σ (t − b)

+

σ2 2

+μ

 2

 p−b)+μ √ 1 − er f σ −log( 2σ

2 

 b 1 − er f log(√p−b)−μ 2σ

(61)

2

that may be further simplified by resorting to (8), with the result p b

−

(log(t−b)−μ)2 2σ 2

e t· √ dt = 2πσ (t − b)

e

σ2 2

+μ

 √  1 − er f 2σ 2

+

  b 1 − er f √σ2 2

.

(62)

Re-expressing now (62) in terms of the Logell History Formulae (35), it finally takes the lengthy but exact form given by Maxima as follows: p b

−

(log(t−b)−μ)2 2σ 2

√ √ √ 3π(d− p) 2 π d− p ( p − b)%e 2((π−4)( p−b)+π(d−b)) erf √(π−4)( p−b)+π(d−b)

e t· √ dt = − 2 2π σ (t − b)

√ √ π d− p 3π(d− p) b erf √2√(π−4)( (b − p)%e 2((π−4)( p−b)+π(d−b)) b p−b)+π(d−b) − + − 2 2 2

(63)

As for the second integral in (60), i.e. the ellipse integral, it is promptly computed as follows D



 3  D−p (t − p)2 t · P · 1− dt = 3 p − 6dp 2 + 3d 2 p a sin 2 p−d (D − p) p  √   −6dp 2 + 6d 2 p − 2d 3 P + −D + 2 p − d D − d 2D 2

! + 2 p3

776

Power and Energy of Civilizations by Logpar and Logell …

  − p D − 3 p 2 + 4dp − 2d 2 P /(6 p − 6d)

(64)

Inserting for P its expression (36), after some rearranging we conclude that the ellipse integral is given by 

D t·P· p

1−

(t − p)2 dt (D − p)2



 π(D. p) π( p.D) π(D − b) + (π − 4)( p − b) %e 2(π(D−b)+(π−4)( p−b)) + πD+π p−4 p−2πb+4b (3 p 3 !! ! D−p 2 2 3 2 2 3 + 2 p − 6dp + 6d p − 2d −6dp + 3d p)asin p−d



√   √ −D + 2 p − d D − d / 2π( p − b) D − p +  π(D − b) + (π − 4)( p − d)(2D 2 − p D − 3 p 2

π(D. p) π( p.D) + 4dp − 2d 2 )% e 2(π(D−b)+(π−4)( p−b)) + πD+π p−4 p−2πb+4b √

 / 2π( p − d) D − p /(6 p − 6d) (65)

=

In conclusion, the mean lifetime is found by summing (63) and (65) and reads

√ √ √ 3π(d− p) 2 π d− p ( p − b)%e 2((π−4)( p−b)+π(d−b)) erf √(π−4)( p−b)+π(d−b)

lifetime_mean_value = − 2

√ √ π d− p 3π(d− p) √ b erf 2√(π−4)( p−b)+π(d−b) (b − p)%e 2((π−4)( p−b)+π(d−b)) b − + − 2 2 2  π(D− p) p−D) + πD+π π( 2(π(D−b)+(π−4)( p−b)) p−4 p−2πb+4b π(D − b) + (π − 4)( p − b)%e ! !!  3  D−p + 2 p 3 − 6dp 2 + 6d 2 p − 2d 3 3 p − 6dp 2 + 3d 2 p asin p−d

√   √ / 2π( p − b) D − p + ( −D + 2 p − d D − d  π(D − b) + (π − 4)( p − b)(2D 2 − p D − 3 p 2 + 4dp π(D− p)

π( p−D)

− 2d 2 )%e 2(π(D−b)+(π−4)( p−b)) + πD+π p−4 p−2πb+4b √

 / 2π( p − b) D − p /(6 p − 6d)

(66)

Just to give a numerical example, let us find the mean lifetime of the Civilization of Rome. The first integral (63), by virtue of the Rome input triplet (37), yields the numeric value of the mean b-lognormal, i.e. mean_value_of_Rome_b-lognormal = −35.6

(67)

4 Part 4: Logell Lifetime Mean Value

777

This means four years before the battle of Actium https://en.wikipedia.org/ wiki/Battle_of_Actium, fought on 2 September 31 B.C.: a crucial event that saw Cleopatra’s Egypt being absorbed as just one more Roman province. On the other hand, the quarter-of-ellipse integral (16) and the Rome input triplet (37) yield for the quarter-of-ellipse mean value the year Rome_quarter - of - ellipse_mean_value = 49.1.

(68)

This year 49 A.D. was a year falling during the empire of Claudius (41–54 A.D.), and, most importantly, was just about 16 years after Jesus Christ had been crucified in Jerusalem. So, by summing up the two Eqs. (67) and (68), we reach the important conclusion that the mean value of the overall Rome’s LOGELL History power curve falls just around the year 13.5 A.D., i.e. within the time of Augustus, first Roman emperor (he died August 19th, 14 A.D.): mean_value_of_Rome_LOGELL_History_Power_Curve = 13.5 A.D. = Time_of_Augustus. (69) This is a noteworthy result. Our Evo-SETI Theory, in the LOGELL form described in this paper, predicts that the “most important year in the History of Rome happened … just in the time of Augustus, and just when Jesus Christ was living his transition from a boy into an adult man”. By far the most important formation time in every man’s life, and the more so in Jesus’s own life!

5 Part 5: Conclusions: Which One Is Better? Logell or LOGPAR? 5.1 Conclusions About Rome’s Civilization In this paper we have discussed for the first time two alternative mathematical models for our Evo-SETI Theory: 1. The LOGELL model, where one’s life is described by a power curve made up by a b-lognormal in between birth and peak plus a quarter of an ellipse in between peak and death. 2. The LOGPAR model, where one’s life is described by a power curve made up by a b-lognormal in between birth and peak plus descending parabola in between peak and death. Which one better describes the true history of a civilization? As for the case of Rome, the two models provide the following results:

778

Power and Energy of Civilizations by Logpar and Logell …

1. LOGELL: minimum energy at the time of Diocletian, i.e. at the crucial transition between pagan and Christian Rome (313 A.D. Edict of Milan by Constantine and Licinius). Then, full recovery (at the old Trajan level of 117 A.D.) around the year 800 A.D. by Charlemagne. 2. LOGPAR: minimum energy around the year 378 A.D. (first serious defeat inflicted by the Barbarians to the Romans at the battle of Hadrianople) and full recovery only at the end of the Middle Ages about 1400 A.D. (Italian Renaissance). The personal opinion of this author is that the LOGPAR model is more appropriate than the LOGELL one, at least in the case of Rome.

5.2 Conclusions About Evo-Seti Theory as of 2018 More and more exoplanets are now being discovered by astronomers either by observations from the ground or by virtue of space missions, like “Kepler”, “Gaia”, and other future space missions. As a consequence, a recent estimate sets at 40 billion the number of Earth-sized planets orbiting in the habitable zones of sun-like stars and red dwarf stars within the Milky Way galaxy. With such huge numbers of “possible Earths” in sight, Astrobiology and SETI are becoming research fields more and more attractive to a number of young scientists. Mathematically innovative papers like the Evo-SETI ones, revealing unsuspected relationships like the one between the Molecular Clock and the Entropy of b-lognormals in Evo-SETI Theory, should thus be welcome. But in Refs. [1, 2] and in this paper we did more than just in all previous Evo-SETI papers. While just preserving all the advantages of the b-lognormal probability density functions, we kept these b-lognormals good only for the first part of the curve: the one between birth and peak. The second part, between peak and death, was replaced in the present paper for the first time by just a simple descending quarter-of-ellipse, thus avoiding any inflexion point like the “senility” point typical of b-lognormals that was so difficult to estimate numerically in most cases. Thus LOGELL curves, just as LOGPAR curves, have greatly simplified the description of any finite phenomenon in time like the lifetime of a cell, or a human, or a civilization (like the Rome one used in this paper as an example) or even like an ET civilization. In other words, we abandoned the normalization condition of b-lognormals retaining just their shape, but not the unit area underneath. This transformed both logpars and logells into power curves, both in the popular sense where “power” means “political and military power” and in the strictly physical sense, where “power” means a curve measured in Watts. And the area under such a logpar or logell is indeed the ENERGY associated to the whole lifetime between birth and death. So, for the first time in the creation of our Evo-SETI Theory, we were able to add ENERGY to the ENTROPY previously considered already. And energy and entropy are the two

5 Part 5: Conclusions: Which One Is Better? Logell or LOGPAR?

779

pillars of classical Thermodynamics, making Evo-SETI even more neatly applicable to a host of biological as well as physical phenomena. Finally, there is one more crucial step ahead that we made by introducing both logpars and logells. Without mentioning it much so far, we actually “stumbled” into the PRINCIPLE OF LEAST ACTION. This is of course the #1 mathematical tool of all theoretical physicists: just think of all the unified theories of gravitation, where a certain action function is postulated, then the Least Action Principle (or Hamilton’s Principle) and the relevant Euler-Lagrange differential equations are derived, and finally (hopefully) solved, yielding the trajectory of particles. Well, the ACTION has the dimension of an ENERGY MULTIPLED BY THE TIME, and this is precisely what we did when finding the area under the logpars and the logells and considering their integrals in between birth and death. So we claim that… the logpars and logells are the optimal trajectory of our Evo-SETI Theory, also in regard to the Least Action Principle. The future will reveal if our conjectures are right and largely applicable to Astrobiology.

References 1. C. Maccone, Energy of extraterrestrial civilizations according to Evo-SETI theory. Acta Astronaut. 144, 202–213 (2018) 2. C. Maccone, Life expectancy and life energy according to Evo-SETI theory. Int. J. Astrobiol. (2018)

Logpars and Energy of Nine Historically Important Civilizations

Abstract Before you read this chapter, please read or re-read Chapter entitled “Energy of ExtraTerrestrial Civilizations according to Evo-SETI Theory”. Thanks. There you will find the mathematical proofs of all the results that are just used in this chapter without any repetition of the relevant mathematical proofs. Now about Evo-SETI. Since we did not discover any ExtraTerrestrial Civilization yet, we must resort to Terrestrial Historic Civilizations in order to find out mathematically “how things unfolded in time” for the Energy of each Human Historic Civilization. In this chapter we do so for nine important historic civilizations, hereafter given with the relevant Input Triplet (b, p, d) in years (negative = B.C. i.e. before Christ’s birth assumed to have occurred at zero time, or positive = A.D. i.e. after Christ’s birth): (1) Ancient Egypt (b = −3100 First Dynasty, p = −1154 last years before crisis, d = −30 Cleopatra’s death). (2) Ancient Greece (b = −776 First Olympic Games, p = −438 Democracy & Parthenon in Athens, d = −30 Cleopatra’s death). (3) Ancient Rome (b = −753 Romulus Foundation of Rome, p = 117 Trajan’s death, d = 476 last western emperor deposed). (4) Renaissance Italy (abbreviated in the sequel as just “Italy”) (b = 1250 Frederick II’s death, p = 1527 Rome sacked, d = 1660 Accademia del Cimento ended). (5) Portugal (b = 1419 Madeira discovered, p = 1716 top of black slave trade across Atlantic, d = 1999 Macau lost to China). (6) Spain (b = 1402 Canarias conquered by Castilians, p = 1798 maximum extent of Spanish Empire, d = 1898 last colonies Cuba, Puerto Rico and the Philippines handed over to the United States). (7) France (b = 1525 Verrazano discovers New York Bay, p = 1812 Napoleon reaches Moskow, d = 1962 Algeria independent). (8) Britain (b = 1588 Victory over Spanish Armada, p = 1904 Maximum Extent of British Empire, d = 1974 Britain joins EEC and loses most important colonies). (9) USA (b = 1898 Colonial Victory over Spain, p = 1972 Americans first land on the Moon, d = 2065 Singularity?). Many other Civilizations were important too for Human History, but we apologize we could hardly take them into account because the relevant Energy calculations were quite long, and our Maxima computer code, given as the last part of this chapter was becoming unmanageable for a modest PC as ours. © Springer Nature Switzerland AG 2020 C. Maccone, Evo-SETI, https://doi.org/10.1007/978-3-030-51931-5_25

781

782

Logpars and Energy of Nine Historically Important Civilizations

Keywords Biological evolution · Molecular clock · Entropy · SETI

1 Summary of Chapter “Energy of Extra-Terrestrial Civilizations According to Evo-SETI Theory” Results that Will Be Used in This Chapter It will be remembered from Chapter “Energy of Extra-Terrestrial Civilizations According to Evo-SETI Theory” that the LOGPAR (LOGnormal +PARabola) is our preferred representation of a POWER CURVE (measured in Watts) for any FINITETIME PHOENOMENON. Or instance, the lifetime of the Sun may be represented as the Logpar (Fig. 1). In this chapter, however, we wish to apply the notion of a Logpar to Human History. We thus hope to pave the way to SETI scientists so as to enable them to estimate the lifetime of Alien Civilizations when they will find the first one or the first few of them by virtue of SETI observations. So, let us remind from Chapter “Energy of Extra-Terrestrial Civilizations According to Evo-SETI Theory” that the Logpar equation is

Logpar_ pdf =

⎧ ⎪ ⎨

−

(ln(t−b)−μ)2

σ2 = e√2π σ2(t−b)   2 ⎪ ⎩ = P 1 − (t− p) 2 (d− p)

for b ≤ t ≤ p for p ≤ t ≤ ∞

(1)

for which the peak coordinates are given by

Fig. 1 History of the Sun as a LOGPAR power curve over about 10 billion years of stellar evolution: the blue part is the ascending b-lognormal, while the red part is the descending parabola. Million of years on the horizontal axis, with zero now, and arbitrary units (to be transformed into Watts) on the vertical axis

1 Summary of Chapter 19 Results that Will Be Used in This Chapter

 Peak_ coordinates =

p = b + eμ−σ P=

783 2

(2)

σ2 e 2 −μ

√

. 2π σ

and the range of the two Logpar numeric parameters are

−∞ ≤ μ ≤ ∞ σ > 0.

(3)

Then, the consequences of the above Logpar assumptions are the following. (1) Only three numerical inputs (called input triplet) (b, p, d) (4) are necessary and sufficient to specify the logpar curve completely. These three inputs are the instants of the birth b, of the peak p and of the death d of the civilization. Just the same applies to all living forms of any kind, like cells, animals, plants, humans, and even extraterrestrial beings, though as of 2020 we know nothing about them yet. (2) If we know the input triplet, then the two Logpar parameters (σ and μ) are immediately computed from the two Logpar History Formulae that we discovered on November 22, 2015: 

√ √

2 d− p σ = √2d−(b+ p) 2(d− p) μ = ln( p − b) + 2d−(b+ . p)

(5)

(3) But, then, the integral (with respect to the time) of the Logpar power curve is the ENERGY (measured in Joules) that the civilization needed to exist for the whole of its lifetime, or, if you so prefer, you may say that this ENERGY was SPENT by the civilization—it all depends on how you like to describe things: words are often ambiguous but equations are not so for scientists. In equations: ENERGY_ spent_ in_ the_ Civilization_ LIFTIME

d =

POWER_ of_ that_ Civilization(t)dt b

d =

Logpar_ of_ that_ Civilization(t)dt =

1 − er f 2



σ √ 2

+

2(d − p)P . 3

b

(6) (4) The last line in (6) was obtained by this author upon imposing that the total lifetime energy given by (6) is a MINIMUM function of σ . This MINIMUM TOTAL ENERGY PRINCIPLE is for Evo-SETI Theory something of capital importance, like the LEAST ACTION PRINCIPLE in physics (please see the website https://en.wikipedia.org/wiki/Principle_of_least_

784

Logpars and Energy of Nine Historically Important Civilizations

Fig. 2 Representation of the History of the Ancient Roman civilization as a LOGPAR curve, finite in the time. Rome was funded in 753 B.C., i.e. in the year −753 in our notation, or b = −753. Then the Roman republic and empire (the latter since the first emperor, Augustus, roughly after 27 B.C.) kept growing in conquered territory until it reached its peak (maximum extension) in the year 117 A.D., i.e. p = 117, under emperor Trajan. Afterwards it started to decline and loose territory until the final collapse in 476 A.D. (d = 476, Romulus Augustulus, last emperor). The numbers along the vertical axis should be normalized in Watts (or… Sextertii, i.e. the “money equivalent of power”)

action). However, this author honestly admits that he could not dig into this crucial matter as he would have liked, and he just passes that on to future generations for further exploration. To discover that not only physics but also biology and astrobiology obey THE SAME MINIMUM ENERGY PRINCIPLE would be just… wonderful!!! (5) As a practical example of how to apply Logpars to a historic human civilization, consider the history of Ancient Rome from its foundation in 753 bc to it peak in 117 ad to its fall in 476 ad. Just look at Fig. 2 hereafter, please. Thus, just three points in time are necessary to summarize the History of Ancient Rome: ⎧ ⎨ b = −753 Ancient_ Rome_ input_ triplet = (7) p = 117 ⎩ d = 476. From (7) and the Logpar History equations (5), one immediately derives the numerical values of the two parameters σ and μ for Ancient Rome: Ancient_ Rome_ Logpar_ Doublet =

σ R = 0.672 μR = 7.221.

(8)

2 Finding the Logpars of Nine Civilizations that Made …

785

Fig. 3 LOGPARS of Ancient Egypt, Ancient Greece and Ancient Rome. On the horizontal axis: time in years between 3100 bc (First Dynasty in Egypt) and 476 ad (Last Western Roman Emperor: Romulus Augustulus). The logpar of Ancient Greece clearly stands out much higher than the logpars of Ancient Aegypt and Ancient Rome. This means that Ancient Greece was the “most culturally advanced civilization of the Ancient World, as indeed even the Roman poet Horace so admitted when he wrote “Graecia capta ferum victorem coepit” (“Greece, subdued by the Romans, actually captured them by her culturally superior civilization”. Vertical scale in arbitrary units to be normalized later: we would suggest “Overall Civilization Index”, but don’t have time to explore that more in depth now. Apologies

2 Finding the Logpars of Nine Civilizations that Made the History of the World We are now ready to understand both the (long but detailed with comments) Maxima Appendix to this chapter. So, we’ll just look at the plots produced by that Maxima code and comment these plots, since they visually describe how good the Logpar version of Evo-SETI Theory is. We start with the three “main” Ancient Civilizations: Egypt, Greece, Rome. Figure 3 shows the three resulting Logpars power curves. Clearly, the Greece Logpar stands out much higher than the preceding Egypt’s and even of the following Rome’s. Please read the caption (Figs. 4 and 5).

3 Peak Height of Each of the Nine Civilizations The peak height of each Logpar is given by the lower Eq. (2). Upon inserting the Logpar History Formulae (5) the lower Eq. (2) is changed into

786

Logpars and Energy of Nine Historically Important Civilizations

Fig. 4 LOGPARS of six modern (and all Western) civilizations: Renaissance Italy (1250–1660), Portugal (1419–1999), Spain (1402–1898), France (1525–1962), Britain (1588–1974), USA (1898– 2065 (Singularity?)). While the first five civilizations are “approximately similar in lasting for 4–5 centuries each and equally profound in roughly contributing each the same amount of progress”, the last civilization (USA) clearly stands out for both its BREVITY IN TIME and MUCH HIGHER CONTRIBUTION to Science and Progress: will the Singularity come in 2065 ad?

Fig. 5 NINE LOGPARS of the three Ancient Civilizations shown in Fig. 3 plus the six Modern (and all Western) civilizations shown in Fig. 4. The thousands-year-gap in between the two groups are the Dark Ages (Middle Ages)

3 Peak Height of Each of the Nine Civilizations

787

√ 2d − ( p + b) e− P= √ √ 2 π d − p( p − b)

d− p 2d−( p+b)

.

(9)

A glance at Fig. 9 shows that, out of the nine civilizations, the seven middle civilizations (Greece to Britain) were “very roughly equally advanced” for the about 500 years of each civilization’s duration, while the USA clearly stand out much higher and shorter nowadays: are we approaching the Singularity? The nine numeric values of the Peak Heights are found in the Maxima code in the Appendix.

4 Total Energy of Each Civilization The total energy needed (and so produced) by each civilization over its whole lifetime is the integral of the Logpar between b and d, and is given by (6). When re-cast in terms of each input triplet (b, p, d) (by using the Logpar History Formulae) the TOTAL ENERGY OF EACH CIVILIZATION is thus given by

TOTAL_ ENERGY_ of_ each_ civilization =

1 − er f



√ d− p √ 2d−(b+ p)

2 √ √ d − p 2d − (b + p)e− + √ 3 π ( p − b)

d− p 2d−(b+ p)

.

(10) The nine numeric values of the Total Energy are found in the Maxima code in the Appendix.

5 After-Peak Energy and Longterm History, that Is, Letting D Approach Infinity The longterm history of the nine civilizations is the previous Logpar mathematical theory extrapolated to the limit when the death instant d becomes larger and larger, i.e. d is let to approach infinity. Why do we study Longterm History of any Civilization? Because, when the first ET civilization will be detected, we want to know where that ET civilization stands in its whole history. That will allow us to make a comparison with longterm history of human civilizations. And this is the best we can do as of 2020, when NO ET civilization has been discovered yet by SETI researches.

788

Logpars and Energy of Nine Historically Important Civilizations

Let us start now by recalling from Fig. 2 of Chapter “Energy of Extra-Terrestrial Civilizations According to Evo-SETI Theory” that: (1) The time range of longterm history (denoted by d, since we are pushing the death time d further and further out in time) starts at that civilization’s power peak p and then increases up to infinity, that is p ≤ d ≤ ∞.

(11)

(2) Inserting the initial value d = p into the Logpar History Formulae (5), one immediately gets σ = 0. Since one has er f (0) = 0

(12)

then (6) yields INITIAL_AFTER - PEAK_ENERGY_of_the_Civilization  1 − er f √02 1 − er f (0) 1 = = . = 2 2 2 In other words, ALL AFTER-PEAK ENERGY CURVES start at the value 0.5 as for the Ancient Rome case

(13) 1 2

=

6 SURPRISE: The MINIMUM AFTER-ENERGY Value Is THE SAME for All Nine Civilizations! Figure 6 shows that, after the value d = p, the red AFTER-PEAK ENERGY curve has a minimum at √  √ 5+3 p+ 1− 5 b (14) d= 4 as was proven in Chapter “Energy of Extra-Terrestrial Civilizations According to Evo-SETI Theory”. For the Ancient Rome case (i.e. the triplet (7)) the minimum (14) falls at the year 386 ad, i.e. not much later after the battle of Adrianople of 378 ad, when the Visigoths defeated the Romans seriously for the first time https://en. wikipedia.org/wiki/Battle_of_Adrianople. Equation (14) is equation (‰117) if the Maxima code in the Appendix. What is the numeric value of the AFTER-ENERGY MINIMUM for each civilization? To find it, we just insert (14) into the total energy of the civilization (10) and rearrange, simplifying the resulting equation. The result is equation (‰119) of the

6 SURPRISE: The MINIMUM AFTER-ENERGY Value …

789

Fig. 6 AFTER-PEAK ENERGY, i.e. LONGTERM ENERGY of Ancient Rome extrapolated to Renaissance time (about 1500 ad) and beyond up to the year 2000 ad

Maxima code, that is 

√√ 5−1 √ √ er f √ √√ − 5−1· 5+1·e 2 5+1 + A=− √ 3 2 3 · 22 · π

√1 5+3

+

1 = 0.378. 2

(15)

This author is an upright person, and so he now openly “confesses” that he is unable to understand why the SAME numerical value (15) applies to ALL civilizations. In other words, the Maxima code equation (‰118) contains a number of various roots with b and p, but all that stuff CANCELS OUT and both b and p DISAPPEAR leaving the purely numeric result (15). Why is it so? Mistery!!!

7 AFTER-PEAK-ENERGY Curves for All Nine Civilizations In conclusion, all the nine civilizations AFTER-PEAK ENERGY CURVES are plot in Fig. 7. Figure 8 is the same as Fig. 7, but it is extrapolated into the future up to 3000 ad.

790

Logpars and Energy of Nine Historically Important Civilizations

Fig. 7 AFTER-PEAK ENERGY CURVES, i.e. LONGTERM ENRGY curves for nine civilizations up to 2000 ad

Fig. 8 AFTER-PEAK ENERGY CURVES, i.e. LONGTERM ENRGY curves for nine civilizations up to 3000 ad

7 AFTER-PEAK-ENERGY Curves for All Nine Civilizations

791

Figure 8 neatly shows that each of the nine curves approaches its own ASYMPTOTE for d → ∞ given by asymptote(d) = LONGTERM_ENERGY(d) 2d − ( p + b) + = √ √ 3 2π e( p − b)

1 − er f



√1 2



2

(16)

as proven in Chapter “Energy of Extra-Terrestrial Civilizations According to Evo-SETI Theory”, and (16) is equation (‰175) of the Maxima code in the Appendix. The derivative of (16) with respect to the longterm time d is the LONGTERM POWER of that civilization 2 . LONGTERM_ POWER(d) = √ √ 3 2π e( p − b)

(17)

Clearly, both the Longterm Energy (16) and Longterm Power (17) are higher and higher according to the smaller and smaller values of ( p − b), that means: the faster was a civilization to develop from scratch up to its peak, the more energy and power it will keep in the long run.

8 Central Star, Longterm Energy and the Evolution of a Civilization There Figure 9 plots all nine LONGTERM ENERGY ASYMPTOTES. Why is this useful for SETI? Well, we certainly know the energy emitted by the central star around which a hypotetical civilization revolves. We also know how old that star is from the known star type. Thus, we may estimate how much energy will bless that civilization over long times. Checking that against the asymptotes in Fig. 9, even for million of years, we have some hope to find out what the probability is for that stellar system “to be inhabited by a human-type civilization”, or, at least this is what the present author hopes and why he wrote this new (unpublish) chapter in his Evo-SETI book.

9 LONGTERM POWER for All Nine Civilizations Finally, the LONGTERM POWER for all nine historic human civilizations is given by Fig. 10. Clearly, the USA stand out as today’s leading Power.

792

Logpars and Energy of Nine Historically Important Civilizations

Fig. 9 LONGTERM ENERGY ASYMPTOTES for each civilization up to 3000 ad

Fig. 10 LONGTERM POWER, given by (17), for the nine historic human civilizations

10 MAXIMA CODE for All Calculations Described in This Chapter The Maxima code for all calculations described in this chapter now follows. It had been completed by this author on February 3rd, 2018, but never published.

Appendix

Appendix LOGPAR (=LOGnormal + PARabola) Power Curves of Nine Western Civilizations (3100 BC–2100 AD)

793

794

Logpars and Energy of Nine Historically Important Civilizations

Finding the Expression of the Analytic LOGPAR

Finding the Expression of the Analytic LOGPAR

Numeric TRIPLET Input for Each Civilization

795

796

Logpars and Energy of Nine Historically Important Civilizations

Numeric LOGPARs of Each Civilization

Numeric LOGPARs of Each Civilization

797

798

Logpars and Energy of Nine Historically Important Civilizations

Numeric LOGPARs of Each Civilization

799

800

Logpars and Energy of Nine Historically Important Civilizations

Plotting the LOGPARs

Plotting the LOGPARs

801

802

Logpars and Energy of Nine Historically Important Civilizations

P = Power Peak of Every Civilization

P = Power Peak of Every Civilization

803

804

Logpars and Energy of Nine Historically Important Civilizations

Total Energy of Every Civilization

Total Energy of Every Civilization

805

806

Logpars and Energy of Nine Historically Important Civilizations

Total Energy of Every Civilization

807

808

Logpars and Energy of Nine Historically Important Civilizations

Total Energy of Every Civilization

809

810

Logpars and Energy of Nine Historically Important Civilizations

Longterm History (that is letting d -> infinity). Longterm Energy and Power for all Civilizations

Longterm History (that is letting d -> infinity

811

812

Logpars and Energy of Nine Historically Important Civilizations

Longterm History (that is letting d -> infinity

813

814

Logpars and Energy of Nine Historically Important Civilizations

Longterm History (that is letting d -> infinity

815

816

Logpars and Energy of Nine Historically Important Civilizations

Longterm History (that is letting d -> infinity

817

818

Logpars and Energy of Nine Historically Important Civilizations

Longterm History (that is letting d -> infinity

819

820

Logpars and Energy of Nine Historically Important Civilizations

Longterm History (that is letting d -> infinity

821

822

Logpars and Energy of Nine Historically Important Civilizations

MOLECULAR CLOCK as a Stochastic Process: Evo-Entropy (Shannon Entropy of Evolution) of a Geometric Brownian Motion (GBM) with a LINEAR MEAN VALUE

Abstract In May 2020 this author made what is likely to become one of most important mathematical discoveries in Evo-SETI Theory: how to trasform the Molecular Clock from a simple straight line in the time into a STOCHASTIC PROCESS in the time. Why is that so important? Because that is the central problem affecting the Molecular Clock in all its astrobiological applications. Keywords Molecular Clock · Lognormal Process with ARBITRARY time mean · Molecular Evolution of Species

1 Introduction Which is the most promising application of our Evo-SETI theory to biology and astrobiology? Our answer is “the Molecular Clock”. In fact, the Molecular Clock has been known to biologists since 1962, and so they have no doubt about its validity. However, their Molecular Clock basically is just a STRAIGHT LINE in the time, rather than a stochastic process whose mean value is a straight line. Well, precisely that stochastic process, rather than just the pure straight line, is what we are going to discuss in this Chapter “MOLECULAR CLOCK as a Stochastic Process: Evo-Entropy (Shannon Entropy of Evolution) of a Geometric Brownian Motion (GBM)”. In other words, we are going to transform the notion of Molecular Clock from just a pure straight line in the time into that line plus an upper standard deviation curve above the straight line plus a lower standard deviation curve below the straight line. The present Chapter “MOLECULAR CLOCK as a Stochastic Process: Evo-Entropy (Shannon Entropy of Evolution) of a Geometric Brownian Motion (GBM)” is the fitting climax to this book. It deals with the crucial applications of our Geometric Brownian Motion (GBM) stochastic process the Molecular Clock Data, with enormous applications to Biology now in sight.

© Springer Nature Switzerland AG 2020 C. Maccone, Evo-SETI, https://doi.org/10.1007/978-3-030-51931-5_26

823

824

MOLECULAR CLOCK as a Stochastic …

2 Summary of the Mathematical Appendix Found in Chapter “OVERCOME Theorem, that is PEAK-LOCUS Theorem, Proven by Maxima” of This Book Let us firsty review the mathematical Appendix to Chapter “OVERCOME Theorem, that is PEAK-LOCUS Theorem, Proven by Maxima” of this book. That Appendix is written in the language of NASA’s Maxima symbolic manipulator, but other authors might wish to re-do-it by Mathematica or Maple: up to them. For the reader’s benefit, we now present a Summary of the mathematical Appendix to Chapter “OVERCOME Theorem, that is PEAK-LOCUS Theorem, Proven by Maxima”. If one ASSUMES the equation (‰84) i.e. (‰94), that is B=

√ √ σ2 s L2 ≡ L that is s L = 2 B 2 2

(1)

then the mean value of the Evo-Entropy of the GBM is a straight line, which is what in Biology is called the Molecular Clock. This straight-line equation is the last formula in the Appendix, i.e. straight line in the time t =

B (t − ts) with (t ≥ ts). ln(2)

(2)

In (2) one has: 1. ts = time-of-start of Life on Earth = 3.5 billion years ago = −3.5 × 109 years (assumed value in this book). 2. We assume that the first living form on Earth was RNA, the first molecule “capable of reproducing itself”, so originating, in turn, the DNA molecule, i.e. the basis for life on Earth. 3. We denote by N s the number of living forms at time of the origin of life on Earth. Since we have just assumed that RNA was the first living form, then we must assume N s = 1, but other authors might prefer other small positive values, referring to the various models about the origin of life. 4. We denote by N e (number of living Species at the END time) the number of living Species nowadays on Earth, since, for us, nowadays is the zero time. In our convention, past times are negative times and future times are positives times. 5. We assume that the current numeric value of N e is 50 million = 5 × 107 , that seems “reasonable” to many biologists, though not to all biologists. We keep this number just as a practical example only. 6. As per equation ‰88 of the Appendix to Chapter “OVERCOME Theorem, that is PEAK-LOCUS Theorem, Proven by Maxima”, the constant B has the dimensions

2 Summary of the Mathematical Appendix Found …

825

of one-over-time, and is given by (ln is the natural log, i.e. the log to base e = 2.71828…)   6   ln 50×10 ln NN es 1 5.06 × 10−9 1.605 × 10−16 B= = ≈ ≈ . 9 te − ts 0 − (−3.5 × 10 year) year sec (3) Please realize that this is the pace of evolution of life on Earth. In other words, we know pretty well that life started 3.5 billion years ago, or so, and (3) expresses just that. On other exoplanets, (3) might have different numerical values that we don’t know yet, i.e. the pace of evolution of life on a planet depends on the physical and chemical conditions of that exoplanet, unknown to us. 7. As per equation ‰31 of the Appendix to Chapter “OVERCOME Theorem, that is PEAK-LOCUS Theorem, Proven by Maxima”, the positive real number deltaNe = δ N e > 0

(4)

is the value of the standard deviation above and below the mean value of the number of Living Species nowadays. This value is unknown in general, but is extremely important for all applications of statistics to the evolution of life. We now explain its meaning by virtue of a diagram. Figure 1 shows the Markov-Korotayev model (see Chapter “Evo-SETI: A Mathematical Tool for Cladistics, Evolution, and SETI” of this book) of the evolution of life on Earth over the last 542 million years, that is since the Cambrian explosion of life.

Fig. 1 Biodiversity since 542 million years ago: https://en.wikipedia.org/wiki/Andrey_Korotayev

826

MOLECULAR CLOCK as a Stochastic …

Fig. 2 Markov-Korotayev red-curve of Fig. 1 as this mean-value blue curve of Evo-SETI Theory

As we saw in Chapter “Evo-SETI: A Mathematical Tool for Cladistics, Evolution, and SETI” of this book, this author proved in 2017 that the Markov-Korotayev model is just the particular case of his Evo-SETI Theory when the mean value is a CUBIC, as shown in Fig. 2. In other words, we have extended the Markov-Korotayev model by passing from the simple Markov-Korotayev curve to our GBM lognormal stochastic process. The GBM mean value is a cubic curve, but now also the two standard-deviation curves are given ABOVE and BELOW the mean-value cubic, as shown again in Fig. 4, that is actually Fig. 5 of the “Life” paper in Chapter “Evo-SETI: A Mathematical Tool for Cladistics, Evolution, and SETI”. Now the mathematics. Please note that the CUBIC mean value in the time is given by the truly m(t) cubic equation in t (that is a polynomial of the third degree in t) that we call m(t) and is given by (N e − N s)(t − ts)(6(tm − ts)(t M − ts) − 3(t − ts)(t M − 2 ts + tm) + 2(t − ts)2 )   + Ns (te − ts) 6(tm − ts)(t M − ts) − 3(te − ts)(t M − 2 ts + tm) + 2(te − ts)2

(5)

On the contrary, the two standard deviation curves in Fig. 3 are NOT cubic curves (though they “look so” in all Figs. 1, 2, 3 and 4). Actually, the upper standard equation curve has the equation given by ‰32 in the Appendix to Chapter “OVERCOME Theorem, that is PEAK-LOCUS Theorem, Proven by Maxima”, that is    2 u_st_dev_curve(t) := m(t) 1 + %es L (t−ts) − 1

(6)

2 Summary of the Mathematical Appendix Found …

827

Fig. 3 Not just a mean-value cubic, but upper- and lower standard-deviation curves above and below

Fig. 4 The Cubic mean value curve (thick red solid curve) ± the two standard deviation curves (thin solid blue and green curve, respectively) provide more mathematical information than Fig. 2. One is now able to view the two standard deviation curves of the lognormal stochastic process, Eqs. (11) and (12), that are completely missing in the Markov-Korotayev theory and in their plot shown in Fig. 4. This author claims that his Cubic mathematical theory of the Lognormal stochastic process L(t) is a more profound mathematization than the Markov-Korotayev theory of Evolution, since it is stochastic, rather than simply deterministic

828

MOLECULAR CLOCK as a Stochastic …

in which s L is given by the very important formula (that we will soon use again in this chapter) (here s L is the same as σ L ):    δ2 log NNee2 + 1 sL = . √ te − ts

(7)

Alternatively, the same upper standard deviation curve (6) may be rewritten directly as a function of the four assigned input parameters (ts, te, N e, δ N e) as given by equation ‰35 of the Appendix to Chapter “OVERCOME Theorem, that is PEAK-LOCUS Theorem, Proven by Maxima”: ⎛

m(t)⎝



δ 2N e N e2

+1

t−ts te−ts

⎞ − 1 + 1⎠

(8)

Similarly, the lower standard deviation curve has the equation ‰42 of the Appendix to Chapter “OVERCOME Theorem, that is PEAK-LOCUS Theorem, Proven by Maxima”:    2 1_st_dev_curve(t) := m(t) 1 − %esL (t−ts) − 1

(9)

that may me re-expressed in terms of the four input parameters (ts, te, N e, δ N e) as equation ‰49, that is: ⎛ m(t)⎝1 −



δ 2N e N e2

⎞ t−ts te−ts +1 − 1⎠.

(10)

In conclusion, we have shown how to “pass from just a curve, that in reality is a mean value curve, to a lognormal stochastic process that, in addition to the mean value curve, also has the two upper- and lower-standard deviation curves”. Now please look carefully at Fig. 5 again. You will notice that the upper and lower standard deviation curves differ nowadays (i.e. at t = 0) from the mean value by the same amount of a thousand units in the vertical scale. In other words, we have assumed δ N e = 1000.

(11)

So, δ N e is the most important free parameter, that one may choose at will. It really characterizes the stochastic process “amplitude” around the mean value! And, if you let δ N e = 0 into all the above equations, the two upper- and lowerstandard deviation curves collapse just onto the mean value curve m(t). Please check that by a glance to (8) and (10) right now!!!

3 A Strong Upper Bound upon the Standard Deviation in the Number of Species …

829

Fig. 5 Molecular Clock straight-line (blue, central straight line) plus and minus the two standard deviation curves above and below (red and green lines, respectively) under the two assumptions that N e = 50 million Species and δ N e = 10 million Species. These two hypotheses are just examples to show how the diagram “behaves”: they are NOT the actual numeric values to fit the real Molecular Clock

3 A Strong Upper Bound upon the Standard Deviation in the Number of Species Living Today: Delta Ne N s. But then (16) reveals a much stronger inequality too:

(18)

3 A Strong Upper Bound upon the Standard Deviation in the Number of Species …

√ N e N e2 N e2 N e2 δ N e