Topology of Organized Chaos 9785891553385

Citation preview

Andrei MOLDAVANOV

TOPOLOGY OF ORGANIZED CHAOS

Moscow FIZMATKNIGA 2020

УДК 515.1 М75

MOLDAVANOV A. Topology of organized chaos. — М.: Fizmatkniga, 2020. — 70 p. ISBN 978-5-89155-338-5. Approach to simulation of evolution for open thermodynamic system based on mathematical model that employs randomized energy continuity equation is considered. Formalism of open model is achieved by use of unlimited number of energy links with external environment. Originality of suggested approach comes from utilization of the recent results in the field of evolutionary geometry. Found solution is characterized by a number of unusual effects, such as built-in phasing, dynamic reconfiguration for structure of energy exchange, regular drift of the range for random variations.

МОЛДАВАНОВ А. В. Топология организованного хаоса. — М.: Физматкнига, 2020. — 70 c. ISBN 978-5-89155-338-5. Рассмотрен подход к моделированию эволюции открытой термодинамической системы на основе математической модели, использующей рандомизированное уравнение непрерывности энергии. Приближение открытой модели достигается использованием неограниченного числа энергетических связей с внешней средой. Оригинальность подхода заключается в использовании последних результатов эволюционной геометрии. Найденные решения характеризуются рядом необычных эффектов, таких как встроенное фазирование, динамическое преобразование структуры энергообмена, закономерное изменение величины случайных вариаций.

ISBN 978-5-89155-338-5

c Молдаванов А. В., 2020 °

CONTENTS

INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

CHAPTER 1 FUNDAMENTALS OF RECE METHOD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.1. Approach substantiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2. System of ECE equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3. Phase space for RECE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.4. Energy interface function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.5. Time arrow of OTS evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 CHAPTER 2 RELATION TO ENTROPY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.1. Statistical entropy of OTS interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2. Probability density for random x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.3. Differential entropy of OTS interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.4. Entropy loop variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.5. Zonal structure of interface entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 CHAPTER 3 RELATION RECE WITH THERMODYNAMIC UNCERTAINTY . . . . . . . . . . 3.1. Thermodynamic uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Substantiation remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Finite interface factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5. Infinitesimal interface factor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6. General form of TUR/STUR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25 25 26 27 28 30 32 33

CHAPTER 4 GEOMETRY OF EVOLUTIONARY TRIANGLES . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Theorem on area ratio at k = 2p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Evolution theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Theorem on area ratio at any k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4. Distortion theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36 36 38 40 41 44

CHAPTER 5 EIGEN SOLUTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Boundary value problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Physical grounds of solution limitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. Leakage of unoccupied microstates to complex domain . . . . . . . . . . . . . . . . . . 5.4. Eigenvalue problem for OTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

46 46 48 49 50

4

CONTENTS

CHAPTER 6 EVOLUTION TOPOLOGY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. Evolutionary triangle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. Connection between topological and physical parameters . . . . . . . . . . . . . . . . . 6.3. Spectral harmonics as phase transition points. . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4. Dynamic decomposition of evolutionary triangle . . . . . . . . . . . . . . . . . . . . . . . . .

52 52 52 53 55

CHAPTER 7 DYNAMICS OF EVOLUTIONARY TRIANGLES . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1. Energy balance equation for evolving triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2. Triangular topology of energy exchange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3. Driving force of evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4. Individual and collective EEF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5. Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57 57 59 61 61 64

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

INTRODUCTION The origin of life cannot be readily explained without some idea, or better, definition of what ‘life’ actually means. Of course, to be comprehensive, to inspire more or less confidence, this explanation should include a profound biological and chemical contribution. However, paying a tribute to undisputable role of biology-chemical concepts in definition of life, we believe such definition will not be completed without clear understanding of physical, mainly energy, transformation accompanying all phases of natural life’s origin. Going through the human history, we find three major philosophical paradigms dealing with definition of natural life which have won the widest support in scientific community. In the retrospective review, we are talking about the views of Aristotle considering life as an animation, i. e. as a fundamental and irreducible property of nature; Descartes who has seen life as mechanism, when mechanistic interactions of matter compose life; and Kant who thought about life, first of all and primarily, as organization of different pieces and processes to provide support to some essential functions [1]. The last approach closely adjoins the idea of defining life as an emergent property of particular kind of complex systems [2]. These days, the traditional approach to reconcile two, possibly, the major scientific theories of the 19th century — the Second Law of Thermodynamics and Darwin’s theory of evolution within a single conceptual framework actually failed. As a result, “...the lack of such a theory impedes the natural merging of the biological and physical sciences, one that would place the biological sciences within a more general physicochemical framework.” [3]. “ As stress authors [4] “Above leads us directly to the problem of abiogenesis, the process by which life on earth is thought to have emerged. And to the relationship, if one at all exists, between the chemical process of abiogenesis and the biological process. One of indubitable questions, is what any scenario of the transition the non-living to living chemistry should include an “energy module” [5] or, more widely, “energy-information” module. Phenomenon of life, in general, and origin of life, in particular, requires energy flow to drive its production process. And what we observe as unique to living systems is the organized coupling of energetic interactions [6]. Acquiring information is unquestionably energy-consuming process and the availability of information means greater efficiency for energy consumption. That is why looks quite warranted to assign to information the status of a genuine thermodynamic entity kB T ln 2

6

INTRODUCTION

joules/bit [7]. Personally, we find as completely indispensable the suggestion to link the capacity of storing information to that of indefinitely increasing organization [8, 9]. Presence of positive feedback in this case, could provide a universal basis for the spontaneous rise of highly organized structures. This mechanism may ensure the robustness of life envisioned as a general system, by allowing it to accumulate and optimize microstate-reducing recipes, thereby giving rise to strong nonlinearity and decisional capacity [7]. Developing Kant’s approach, Schr¨odinger defined life within the context of living entities increasing their order by dissipating matter/energy gradients to maintain away from equilibrium [10]. Following this logic, the crucial issue here becomes the process of emergence of life from pre-biotic chemistry, emergence of evolutionary unit [11] or chemoton [12] concomitant with the emergence of function and information. Living entities then can be viewed as bounded, informed autocatalytic cycles feeding off matter/energy gradients, capable of growth, reproduction, and evolution [13]. However, as we see not only life requires energy for the integrity of its internal organization, not at all. It is well known that complex dynamical systems transform energy from the environment in maintaining themselves at a distance from equilibrium and can hold energy in non-linear relationships among system components supporting the self-amplifying [14]. That is why, acquisition of clear criteria to discriminate the living and non-living entities becomes even more important. So, we see from above that the genesis of complex systems, at least, deserves to be studied and discussed comparing with the abiotic formation of various organic substances. Not surprisingly then, that many authors are convicted that effective energy transformations should be associated with development of the open system, i.e.the system actively supporting the abovementioned dissipating of matter/energy gradients or, in other words, an energy exchange. Another key factor regularly emerged in the life related disputes, is the concept of the so-called minimal life. In a number of scientific reports [11, 15, 16, 17 and references there] is emphasized that staying within the reasonable scientific assumptions, we cannot ignore what even in its simplest manifestations, life comes in a form of some kind of cell. In other words, we are almost forced to accept that a model of minimum life should be that of a minimal cell. And suggest that such cell-like compartments have existed in prebiotic time.

INTRODUCTION

7

Based on above, we may think on a living minimal cell from physical point of view as an open system capable of self-maintenance, due to a process of permanent energy exchange using available chemical infrastructure within some boundary [15, 18]. To support emergence of above self-maintenance, such system should somehow be developed through appropriate processes, generally, accompanied with entropy decrease ∆S < 0. And evolution to living beings as autonomous systems with open-ended evolution capacities must include an energy transduction apparatus, i. e. set of energy currencies. We believe that another critical aspect of above evolution is necessity to account the both levels of it — individual and collective [18]. Chance to use the collective memory offers great evolutionary advantage comparing with individual experience. We are also agreed that original forms of life may have depended on not so efficient energy mechanisms that we may see in the living things these days. They could have been superseded in the course of evolutionary selection. Another important moment is what the early environment in which life originated on Earth is not known [12]. And significant part of this uncertainty, in many respects, comes from the ahistoric nature of the physico-chemical processes that must have led to complexification [19]. We do not know for sure, what exactly happened and how. That is why, the unprejudiced consideration of the origins of life should not depend on the assumption that substances and mechanisms, now seemingly essential, were essential initially. Instead, we should consider what might happen in the severe environment of young planets, when mineral surfaces with some bituminous material, were exposed to UV light in a reducing atmosphere and at the absence of predators [20]. And to firmly stay on the position, that we have no idea on the actual energy supplying mechanisms the primordial living entities used. Besides, we believe that the very universal and simple physical mechanisms should be accounted for accurate simulation of conditions on early Earth. Thus, the present work is based on the assumption that even the oldest known living cells [21] were very advanced and complex compared with earlier and more primitive forms that can still be considered as living entities. However, the fundamental attributes of life may be discerned already in those forms, which were presumably rather well organized biopolymeric assemblies, long before living cells as we know them came into being [22]. That is why, we should pay strong attention to the very basic, very universal physical relations

8

INTRODUCTION

we know to describe life’s origin and evolution in the most unbiased way possible. So, based on above, from biology and biophysics comes the construct of a minimal cell or evolutionary unit, the simplest possible form combining the bioenergetic and biosynthetic processes, ,capable of multiplication, heredity and heritable variation, that “...logically as well as historically precedes life” [11]. Also, it looks quite reasonable what the evolution of prebiotic and early biological systems are qualitatively different processes,in which a crucial role is played respectively by structural stability and by dynamical mechanisms of regulation and integration. These different features entail also “...distinct modalities of interaction between system and environment that need to be taken into consideration” [4]. Besides, as exact energetics on young Earth that could convoy origin and development of life is unknown [23], it is quite possible that earliest pre-living forms through its membrane transductions used as many energy sources as possible, such as light. thermal energy, pressure, touch, stretch, movement of water and air, gravity, electric and magnetic fields as well as a wide range of chemical substances [24, 25]. And, the forms featuring more energy supplies have definitely had more chances to survive. The transformation of inanimate matter to complex life is traditionally divided into two stages. The first, abiogenesis, involves the conversion of non-living material to simplest life, and the second, the biological phase, is the stage on which Darwinian evolution began to operate [26]. Further, we will follow this traditional classification. Also, we will operate with other convenient biological concepts, such as heredity and variability, and try to fill them in with some quantitative meaning.

CHAPTER 1

FUNDAMENTALS OF RECE METHOD

1.1. Approach substantiation One of the most general energy relations we typically deal with, explicitly or implicitly, is an energy conservation law in its integral or differential form. The latter is known as energy continuity equation (ECE) that in the non-relativistic approximation can be written as: ∂ε ∂t

= − div J,

(1.1)

where J is flux of energy, ε is an energy volume density, t is time, and div is a divergence operator. To the best of our knowledge, the platform of an energy conservation law in general, and ECE in particular, is a quite suitable for the purpose of evolution studies. Firstly, since ECE is of the unquestionably universal nature and it is very hard to imagine any physical scenario when (1.1) would not work. Secondly, ECE deals with the interface between open thermodynamic system (OTS) and energy flux by default. In this book, keeping (1.1) as a main semantic cell, we will not attribute flux J to any specific physical or chemical or any other process and restrict its variations in this way. To this end, as soon as some energy transportation occurs contributing to the change of OTS energy balance upon arrival, it falls under category for our consideration. Equally, density ε will be considered as the local total of those properties of OTS which are capable of the energy savingrelease properties. So, we assume that J is an aggregate of applicable energy fluxes exclusively coming to (Jin ) or exiting from (Jout ) the OTS, i. e. J = Jin ⊕ Jout

where ⊕ denotes exclusive disjunction and means that the non-zero Jin and Jout cannot coexist at the same time. In abstract mathematical terms, we believe that OTS features an infinite number of the energy channels (each channel takes care of only one specific energy transfer process) connected to the common

10

FUNDAMENTALS OF RECE METHOD

CH. 1

heat bath of unlimited capacity. Though at every single moment of time all energy channels are available for an energy transfer inwards or outwards OTS, however only one channel is actually picked up and this channel will take a role of the instant energy source (transfer inwards) or the energy sink (transfer outwards). In this sense, we see a definite analogy with an operation of the managing server which with a fixed rate sends a ping test to the sensors and receives signal back. Mathematically, above interaction between a single energy channel and OTS is done executing an ECE. So, the difference between classic ECE (1.1) and our approach (system (1.2) below) is that ECE at each moment of time deals with the predefined energy transfer through one fixed channel only, whereas (1.2) processes an infinite number of channels, and each single transfer is done on the random basis. Finally, we will assume uniform probability density distribution for all introduced random quantities.

1.2. System of ECE equations Now, based on (1.1) and abovesaid, introduce an infinite set of ECE  ∂ε 1  ∂t = − div J1 ,      ∂ε2 = − div J2 , ∂t (1.2) ..................    ∂εn    ∂t = − div Jn , ..................

Discussion on mathematical aspects of applicability for similar approach could be found in [27]. To make (1.2) solvable conduct some formal mathematical transformations. With this purpose, in (1.1) rid of the time variable. So, suppose the direction of the flux J is determined by the direction m, |m| = 1, m = (cos ϕx , cos ϕy , cos ϕz ), where cos ϕi =

Ji J

,

(1.3)

and Ji is i-th component of J. Let the radius-vector r be characterized by three numbers x, y , z in the Cartesian system. Also, define vector area dS = n · dS , where n is a unit normal oriented in the direction of incoming flux Jin .

1.2

SYSTEM OF ECE EQUATIONS

11

Perform the following transformation. Multiply the both sides of (1.1) by dV and use (1.3), so we have ∂U ∂t

= −dJ (m · dS) = −dJ (m · n) dS,

(1.4)

where dU = ε · dV , U and V is internal energy and volume of OTS, 3 P mi dSi , and m · n is a vector dot product correspondingly, m · dS = k=1

such that m · n = 1. Divide (1.4) by J · dS , and observe the value of the total instant energy Q = J · dS dt, so we obtain dU Q

=−

dJ J

x.

(1.5)

where x = cos ϕ, ϕ is an angle between the direction of an instantaneous change of resultant energy flux and the unit normal. Here and further, we believe that the flux J is the deterministic quantity while interface-factor x is the random one in the range [−1, 1]. The key role in (1.5) plays the cosine term in the right part. In the trivial case (cos dϕ = 1), (1.5) is a regular ECE asserting precise match between the amount of energy brought to the OTS interface and that of finally acquired by the OTS, dU = δQ. In the non-trivial case (volatile cos dϕ), the amount of energy δQ which could be transferred through the OTS interface and the one which is actually being acquired by the OTS is not mandatorily the same, dU 6= δQ. Now, consider n energy links between the common heat reservoir and the OTS. Assume that only one of these n links can be open at each given time t. Let transfer of energy along each link is obeyed to (1.5) with appropriate (dU/Q)n . Then, at each t holds (dU/Q)N = −dJp /J = −dJ/J · cos dϕn , where Jp = J · cos dϕn is the normal component of J , dϕn is some fixed dϕ. Then, at n → ∞, the support range for (dU/Q)n and cos dϕn tends to the continuous one, so (dU/Q)n and cos dϕn is being became the continuous random quantities (dU/Q)R and (cos dϕ)R , and (1.2) becomes equivalent to (1.5) where dU/Q and cos dϕ are now the random values. Further we will not use the random notation in (1.5) however imply it. So, we believe that (1.5) is an accurate approximation for the coherent description of the general energy exchange of OTS with infinite number of energy sources and sinks irrelevant to its actual physical nature.We will call (1.5) as the randomized energy continu-

12

FUNDAMENTALS OF RECE METHOD

CH. 1

ity equation (RECE) and consider it as the major energy evolution relation. Then, the ratio dU/Q serves as the instant snapshot of the bidirectional energy exchange flux, i. e. as the indicator of the state of an energy interface in OTS.

1.3. Phase space for RECE Based on (1.5), it is possible to introduce an energy interface function as ZZ ZZ dU dy Υ= =− x (1.6) Q

M

y

M

where y (energy exchange rate) = J/J0 , normalizing constant J0 > 0, M ⊆ R2 is the phase space (manifold) for all possible states of δΥ (Fig. 1.1).

Fig. 1.1. Configuration space M for all possible states of OTC. In the plot, by an abscissa axis a unitless energy rate y = J/J0 is indicated, by an ordinate axis the instant efficiency of energy exchange δΥ. Plot area is divided into 5 stages: (1) “Non-System” (y < 0); (2) “Pre-System” (0 6 y 6 1/e); (3) “System Growth” (1/e 6 y 6 1); (4) “System Decay” (1 6 y 6 e); (5) ”PostSystem” (y > e) with the dissimilar physical meaning. Also, two closed loops with the opposite direction of circumnavigation C1 (clockwise) and C2 (counterclockwise) are shown. The meaning of above stages and δΥ loops will be discussing further. The space M is marked by a light-grey color

1.4

ENERGY INTERFACE FUNCTION

Then, taking Riemann integral on dy , we obtain Z Z Z dy Υ=− x = − δ Υ(x, y), y

13

(1.7a)

M

where δ Υ(x, y) = x ln y

(1.7b).

The non-integrable structure of (1.7a) warrants usage of the phase (energy exchange) space where random δΥ = δΥ(x ln y) : Φ → ∆Υ

is a measurable function from the set of possible outcomes x · ln y to some set δΥ, with Φ as a probability space and δΥ is a measurable space [28].

1.4. Energy interface function From (1.7a) applying integration of a pointwise product [29]

Υ(y) = −

Z

x · ln y = −

Z1

dx ·

−1

Zy

ln y dy = −by (ln y − 1)

(1.8)

0

where b normalization factor. Note, that M possesses an axial symmetry in respect of line x = 0 and the logarithmic one with respect of the line y = 1. It means that for any non-zero x0 and y1 ∈ [0, 1], y2 ∈ [0, e − 1], 1+y Z 2

dy

Zx0

δΥ(x, y) dx = 0,

−x0

1−y1

where y1 and y2 are related as the roots of Υ(y1 ) = Υ(y2 ). In particular, for the whole M , Ze 0

dy

Za

−a

δΥ(x, y) dx = 0,

14

so that

FUNDAMENTALS OF RECE METHOD

Z1 0

dy

Za

−a

δΥ(x, y) dx =

Ze 1

dy

Za

CH. 1

δΥ(x, y) dx

−a

which means that the total energy exchange occurring through an interface at 0 6 y 6 1 precisely matches the one at 1 6 y 6 e. Find the solution satisfying condition Υ(y) = 0. Then, resolving (1.8) in respect of y ³ h ´i Υ y1 = exp 1 + WL −1, − = 0; e

³ h ´i Υ y2 = exp 1 + WL 0, − =e e

where WL is a Lambert function, WL (0) is the upper branch and WL (−1) is the lower one [30].

Fig. 1.2. Integral efficiency of energy exchange Υ. In the plot, designation of the horizontal axis is the same as in the Fig. 1.1, by the ordinate axis an integral efficiency of energy exchange Υ is indicated. Two major zones are shown – the “Flux Area” and the “System Area”. In the “Flux Area”, there is no signatures of OTS while in the “System Area” there is an interaction of the energy flux y with the volume potentially (subject of attained rate y) leading to emergence of OTS. The phase transition points GRP (y = 1/e), SP (y = 1), T P (y = e) are also shown, where e is Euler number. Meaning of these points will be discussed later. Area with dU > 0 designates y -range where in average the growth of internal energy U is observed, whereas dU < 0 means opposite case

1.5

TIME ARROW OF OTS EVOLUTION

15

Apply the following condition Υ(y = 1) = 1

(1.9)

Υ(y) = y − y ln y

(1.10)

to (1.8), then b = 1 and shown in Fig. 1.2. Note that (1.10) takes a rate of energy exchange through OTS interface y and converts it into an integral characteristic of an interface. In this sense, (1.10) could be considered as an operator equation for y .

1.5. Time arrow of OTS evolution As it is known, the concept of classical and quantum chaos assumes that trajectory or wave function description loses their operational meaning, incorporating a time-symmetry breaking [31,32]. Below, we demonstrate suitability of this fact to our model. From (1.10) and since Υ does not directly depend on t, dΥ dt

= − ln y

dy dt

.

(1.11)

Now, keep dt > 0 and consider the sign of the product ln y · dy/dt. Firstly, let the right part in (1.5) be negative, i. e. x

dy y

6 0.

(1.12)

Then, x ln y 6 0 (1.13) since result of Riemann integration keeps the sign of its integrand [33]. Multiply the both parts of (1.12) by the positive y/dt, so x

dy dt

6 0.

(1.14)

Finally, multiplying (1.13) by (1.14), we have dy dt

since x2 = cos2 dϕ > 0.

ln y > 0

(1.15)

16

FUNDAMENTALS OF RECE METHOD

CH. 1

Reiterating (1.13), (1.14) for the case x

dy y

>0

we obtain the same result (1.15). Then, finally from (1.11), δΥ dt

60

(1.16)

for any sign of x dy/y . Note that the sign of equality in (1.16) holds for the quazistationarity points dy/dt = 0 and stationarity point P0 (y = 1). Obviously that δΥ/dy in contrast to (1.16) changes its sign at y = 1, i. e. at y < 1 δΥ >0 dy

and at y > 1 δΥ dy

6 0.

CHAPTER 2

RELATION TO ENTROPY

2.1. Statistical entropy of OTS interface In M , the largest possible number of microstates with a definite energy W [34] is δΥ = ∂Υ/∂y dy , then using (1.11) and Gibbs’ definition of statistical entropy ∆S = LnW = Ln |dΥ| = ln | ln y|.

(2.1)

It is worthwhile to note that (2.1) coincides with the main part of differential entropy (2.11) discussed later. From (2.1), based on inequality (1.15), the total entropy for interface of OTS grows at any time ³ ´ dS 1 dy = ln y >0 (2.2) 2 dt

y ln y

dt

regardless the sign of dy/dt. The latter, on the one hand, demonstrates compliance with the 2nd thermodynamics law [35], on the other hand, it affirms validity of our approach used for proof of (1.16). Also, from (2.1) d2 S dy 2

=−

ln y + 1 y 2 ln2 y

.

(2.3)

At y = 1/e, d2 S/dy 2 changes its sign bringing entropy derivative dS/dy to maximum. Also, from (2.1) at y = 1/e and y = e, ∆S = 0

(2.4)

2.2. Probability density for random x Let g(x)to be the probability density function for random x, i. e. g(x) > 0 for x ∈ [−1, 1] g(x) = 0 at x ∈ / [−1, 1]

(2.5)

18

RELATION TO ENTROPY

CH. 2

then from (1.7b) at each given y f y (x) =

g(x)

| ln y|

,

x ∈ [−1, 1]

(2.6)

[37], where | ln y| is taken to keep f y (x) > 0 and y is a parameter. Calculation of f y (x) is done in assumption that x is the random at each given deterministic y . Then, normalization condition for f y (x) is Zln y

f y (x) dx = 1.

(2.7)

− ln y

Applying the well-known Leibnitz rule for differentiation under the integral sign [38], we obtain that (2.7) is equivalent to 1

g(x) = . 2

Note that (2.7) at | ln y| < 1 could be rewritten as Z1

y

f (x) dx =

−1

Zln y

y

f (x) dx + 2

Z1

f y (x) dx

(2.8)

ln y

− ln y

where is accounted that ¯ ¯ ¯¯ −1 ¯ ln y ¯ ¯ ¯ Z ¯Z ¯ ¯ ¯ ¯ ¯ y ¯ f y (x) dx¯ = ¯ f (x) dx ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ 1

− ln y

and, equally,

¯ ¯ ¯ ¯ ¯ Z1 ¯ ¯ Z−1 ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ f y (x) dx¯ = ¯ f y (x) dx¯ . ¯ ¯ ¯ ¯ ¯ ¯ln y ¯ ¯− ln y ¯

Important detail of (2.8) is what the physical meaning¯of two terms¯ ¯ R1 ¯ in the right part is not the same. At | ln y| < 1, integrals ¯¯ f y (x) dx¯¯ ln y ¯ −1 ¯ ¯ ¯ R y f (x) dx¯¯ deal with x-microstates which are formally left and ¯¯ − ln y

2.3

DIFFERENTIAL ENTROPY OF OTS INTERFACE

19

unoccupied. So, integral quantity in the [ln y, 1] lacks the equiphasic behavior with the quantity within the range [0, ln y] and, therefore could not be described in the same way. To support this difference, further we will treat quantities for unoccupied microstates as imaginary ones. Based on that, rewrite (2.8) as Z1

y

f (x) dx =

−1

Zln y

y

f (x) dx + i · 2

Z1

f y (x) dx.

(2.9)

ln y

− ln y

2.3. Differential entropy of OTS interface Although differential entropy cannot be considered as the mathematically rigorous extension of statistical entropy because of the known limitation lim ln(∆) → −∞ [36], we consider dynamics of its main ∆→0

part yet believing that it may provide useful auxiliary information, here ∆ is the size of discretizing for probability density. Moreover, we are only interested in the difference of differential entropy ∆S y between adjacent states which obviously lacks above shortcoming. So, we will consider dynamics of ∆S y , taking value of S y at y = 0 as the anchor one. So, consider an entropy probability distribution for instantaneous random interface function δΥ y

∆S = −

Z1

−1

¡ ¢ f y (x) ln f y (x) dx.

(2.10)

Insert (2.6) to (2.10) and simplify, then ∆S y = Hx + ln | ln y|,

(2.11)

where Hx is entropy probability distribution for x on the compact support [−1, 1] Z1 ¡ ¢ Hx = − g(x) ln g(x) dx. −1

Observe from (2.11) that ∆S y is the two-valued function with the roots ¡ ¢ yc± = exp ± exp (−Hx ) (2.12)

where “+” fits to the larger root and “−“ to the lesser one.

20

RELATION TO ENTROPY

CH. 2

Now, from (2.10) y

∆S = −

Z1

Ay (x) dx

(2.13)

−1

where probabilistic anisotropy of interface à ! f

Ay (x) = ln

out fout

−fin

(2.14)

fin

and fin = f (y = yin ), fout = f (y = yout ), yin = Jin /J0 , yout = Jout /J0 . Then, the roots ∆S y = 0 correspond to ´ ³ −fout fin = WL fout (2.15) with the plot in Fig. 2.1.

Fig. 2.1. Dependence of probability density fin for inward energy flow yin on fout for outward energy flow yout at ∆S y = 0 through OTS interface, fin = −fout = WL (fout ). In the plot, by the abscissa axis density fout is indicated, by the ordinate axis density fin is indicated. fin = f (y = yin ), fout = f (y = yout ), yin = Jin /J0 , yout = Jout /J0

2.4

21

ENTROPY LOOP VARIATION

In the above physical context and assuming that functions fin and fout are regular and smooth, we initialize fin and fout so that ∆S y > 0 mean ´ ³ −fout (2.16a) fin > WL fout while ∆S y 6 0 does

³ ´ −fout fin 6 WL fout .

(2.16b)

2.4. Entropy loop variation Consider circular integral of δΥ over the sample loops C1 = 1 → 2 → → 3 → 4 → 1 and C2 = 5 → 6 → 7 → 8 → 5 with the opposite sense of circumnavigation as shown in Fig. 1.1. Note that the integrals taken over the horizontal paths ± ln y nullify each other and the non-zero contribution is provided only by the vertical paths 2 ln yk at dyk = 0, where yk is the y -coordinate of the nodes in the loop. From (2.1), µ +¶ ln yS + − ∆S(y) = S1 − S2 = ln (dΥ1 ) − ln (dΥ2 ) = ln (2.17) − ln yS

where δΥ+ is the increment of Υ over the vertical path of loop directed from the less probable state of δΥ to the higher probable one, and δΥ− otherwise (Fig. 1.1). The loop C1 is in the y -range [1/e < y < 1] where (2.16b) holds. Then, path 2 → 3 occurs from the less probable state yin to the higher probable state yout , i.e. along the direction of the increasing of probability and, thus, entropy. In contrast, path 4 → 1 takes place in the opposite direction, from the higher probable state yout to the lower one yin , i.e. along the direction of the decreasing of probability and entropy. Contribution of the path 4 → 1 overweighs the one for the path 2 → 3, i.e. in total ln(ys+ ) < ln(ys− ), so from (2.17), change of entropy for OTS interface is negative, i.e. ∆S(y) < 0. In the range [1 < y < e], (2.16b) holds either. So, the path 5 → 6 from the higher probable state yout to the less probable state yin will lead to decrease of entropy S . In contrast, the path 7 → 8 will increase entropy . The contribution of the path 7 → 8 overweighs the one for the path 5 → 6, then ln (ys+ ) > ln (ys− ) so ∆S(y) > 0. Applying above reasoning for interval [0 < y < 1/e] (Fig. 2.2), we could obtain that ∆S > 0.

22

RELATION TO ENTROPY

CH. 2

So, in the considering range [0 < y < e] change of entropy is positive with exception of the interval (1/e < y < 1), where entropy drops. Note that in the light of above, in vicinity of y = 1/e there are favor conditions for fluctuations of entropy. To demonstrate that, consider the two random loops, L1 = 6 → 2 → 3 → 5 → 6 and L2 = 1 → 6 → 2 → 3 → 5 → 4 → 1 (Fig. 2.2). For L1 , the paths 5 → 6 and 2 → 3 directed towards the same side, hence when they are subtracted, ln(ys+ ) ≈ ln(ys− ), and change ∆S is small. For L2 , formal subtraction of oppositely directed paths 2 → 3 and 4 → 1 actually sums up them thereby making resultant change of ∆S significantly higher, ln (ys+ ) ≫ ln (ys− ). As selection of the loops L1 and L2 is random, it could cause fluctuations of entropy ∆S around the point 1/e.

Fig. 2.2. Mechanism of fluctuations in vicinity of y = 1/e(GRP ). Designations of the axes are the same as in the Fig. 1.1. Two sample loops L1 = = 6 → 2 → 3 → 5 → 6 and L2 = 1 → 6 → 2 → 3 → 5 → 4 → 1 in vicinity of the point y = GRP are shown. Note that integrals taken over the horizontal paths ± ln y nullify each other and the non-zero contribution is provided only by the vertical paths 2 ln yk at dyk = 0, where yk is the y-coordinate of the nodes in the loop (points 1, 2, . . . 6). If δΥ takes the loop L1 , then the paths 5 → 6 and 2 → 3 directed towards the same side, hence when they are subtracted, ln(ys+ ) ≈ ln(ys− ), and change ∆S is also small. However, if system takes the loop L2 , then subtraction of oppositely directed paths 2 → 3 and 4 → 1 sums them up thereby making resultant change of ∆S significantly higher

2.5

ZONAL STRUCTURE OF INTERFACE ENTROPY

23

2.5. Zonal structure of interface entropy As it is seen from (2.11), interface entropy includes the term Hx , which is an entropy probability distribution for the compact support x ∈ [−1, 1] Hx = −

Z1

−1

¡ ¢ g(x) ln g(x) dx,

(2.18)

Hx is non-negative value in the range Hx min 6 Hx 6 Hx max ,

(2.19)

where bounds Hx min and Hx max depend on the base of logarithm z [86]. Since Hx is limited, it leads to the limited range for y − S variations Hx min 6 S − ln | ln y| 6 Hx max

(2.20)

¡ ¢ yc± = exp ± exp (S − Hx )

(2.21)

or rewriting,

where “+” fits to the larger root of S and “−“ to the lesser one. function, where the leftObserve from (2.21) that¡ S is the two-valued ¢ hand branch is y = exp − exp (S − H ) and the right-hand branch x ¡ ¢ is y = exp exp(S − Hx ) . Then, the full set of admissible values y − S consists of the two ranges. The left-hand ¡range is [yL− , yL+ ] and ¢ − + − [y , y ] , where y = exp − exp(S − H ) , the right-had one is x min R R L ¡ ¢ ¡ ¢ − + yL+ = exp − exp (S − H ) , y = exp exp (S − H ) , y = x max x max R R ¡ ¢ = exp exp (S − Hx min ) . Conveniently, minimum dissipation fits ¡ Hx = 0. ¢Then, the pair of the outmost entropy curves is y = exp ± exp(S) which at S = 0 + yields the roots of entropy S equal to yL− = 1/e and yR = e. To calculate the innermost border curves, we should know value of Hx max which physically corresponds to the state of the highest disorder possible when a random variate x has uniform distribution [86], which within given model means that density g(x) = 1/2, then − Hx max = ln 2 and the roots yL+ = (1/e)1/2 and yR = (1/e)−1/2 .

24

RELATION TO ENTROPY

CH. 2

Now, based on found entropy roots, we may write = exp (±1);

(2.22a)

³ 1´ lim S0 = exp ±

(2.22b)

Hx min + ln | ln y| 6 S 6 Hx max + ln | ln y|

(2.23)

lim S0 Hx → min Hx → max

2

where S0 denotes the set of the roots of entropy S . So, the feasible solutions of entropy are localized inside the area bound by the curves

as is shown in Fig. 2.3. It is important to note that y−roots of entropy are equal to the first and second harmonic of discrete spectrum evolving OT C considered in the chapter 5.

Fig. 2.3. Dependence of differential entropy ∆S(Hx , y) for OTS interface. In the plot, by an abscissa axis a unitless energy exchange rate y = J/J0 is indicated, by an ordinate axis entropy S. Two pairs of the curves are presented. One pair is for the outmost boundary for y − S, another pair is for the innermost boundary for y − S. So, allowable y − S values are located within the dashed area. Limitation stems from the bounded range for Hx at random interface-factor x. Shown y-roots of entropy S match the first and second harmonic of the discrete spectrum evolving OTS as discussed in the Chapter 5.

CHAPTER 3

RELATION RECE WITH THERMODYNAMIC UNCERTAINTY

3.1. Thermodynamic uncertainty Energy budgeting of a macro- and microscale OTS plays the critical role for its existence in the classic and quantum context as the both theoretical and experimental research show. During recent decade, numerous publications emerged targeting above agenda under an angle of the dependence between energy and parameters of OTS, to name just a few of them refer to [39–50]. On the theoretical side, [39] employed a nano-thermodynamical approach to perform numerical calculation and confirm correlation between the binding energy and the size of nano-particles, in [40] it is shown that the energy properties of a gas in nano-domains considerably differ from those in macro domains. In [41], the role of accurate temperature model for the thermal control of OTS is highlighted, while in [42] the validity of thermodynamic uncertainty relation (TUR) for classical and quantum nanoscale OTS is discussed. On the experimental side, it is demonstrated clear dependence of the heat transfer rate on the nano-particles concentration and the method for its encapsulation [43, 44, 45, 46]. Measurement of energy for nanosystem is being discussed in dependence on nano-particles temperature [45], and presence of a nanomechanical resonator [47]. In [48], is noted that a heat transport in nano-structures is dominated by the non-equilibrium effects and ratio between measured change of temperature and energy parameters plays important role to optimize performance of a heat nano-engine, while [49] presented conclusion on impact of presence of nano materials on thermal performance of a heat pipe. Finally, [50] confirmed validity of thermodynamic uncertainty relation (TUR) ³1´ ∆E · ∆ > kB (3.1) T

in the part of the quantum analogue for zeroth law in the form of saturated Schr¨odinger uncertainty relation, where kB denotes the Boltz-

26

RELATION RECE WITH THERMODYNAMIC UNCERTAINTY

CH. 3

mann constant, and arguments ∆E and ∆(1/T ) could be, generally, replaced with some functions of these arguments [51]. All above works, in direct or indirect form, refer to dependence between the energy change ∆E and the temperature one ∆T occurring in nanosystems. For macroscale objects, discussion on existence in classic physics of some general relation between ∆T and ∆E that restricts their simultaneous measurement as it occurs in quantum physics for position and momentum, was initiated by Bohr and Heisenberg [52]. Elaborated consideration to derive TUR for macrosystems was given in a number of articles, for example, [53–59]. Recently, progress in this field brought the new limiting condition shaped in nonequilibrium thermodynamics for total entropy rate in Markov jump processes [60, 61] dealing with the state arbitrary far from an equilibrium, which could be written as S˙ t 2kB

>

hJ i2 hδJ i2

(3.2)

where S˙ t is total entropy rate, hJi is an average flux, hδJi is variance of flux fluctuations. Further in this paper, to call (3.2), we will use STUR. A convenient methodology employed in above reasoning for evidencing of (3.1) and (3.2) in less or more extent is based on the mathematical apparatus of probability theory. The alternative approach [62–65] for obtaining of an uncertainty principle (3.1), (3.2) by default utilizes another fundamental physical concept — a paradigm of continuity that was utilized earlier (1.1). Further, based on (1.1) we will substantiate the model for finite and infinitesimal interface-factor cos (dϕ) to verify its applicability for deriving of TUR and STUR.

3.2. Substantiation remarks Taking into consideration presence of an energy interface between the flux and the OTS, (1.1) could be brought to the form dU

δQ

= cos (dϕ)

(3.3a)

where cos (dϕ) is the interface factor with, generally dϕ = ∠(dJ, n) 6= 0, n is an outward normal, dU , δQ are changes of OTS internal energy,

3.3

ASSUMPTIONS

27

and heat transfer between OTS and its surroundings, correspondingly [85]. Obviously, in the trivial case cos (dϕ) = 1, and (3.3a) comes to its convenient form dU

δQ

= 1.

(3.3b)

For current research it is critical to recognize difference in the scope of (3.3a) and (3.3b). While (3.3b) treats the point of local contact with an energy flow in the mathematical sense, (3.3a) does it for some physical vicinity of the contact. In the other words, replacing (3.3b) with (3.3a) we do step aside and replace the predefined approach (3.3b) with the multi-optional one (3.3a). It allows to apply the system-based concepts normally utilized in meso- and nanophysics [66], like temperature or internal energy. We believe that suitable verification for above suggestion to use (3.3a) as the model for general energy exchange would be considering its consequences against some well-known canonic energy relations and compare how far these consequences are from the canonic ones. Based on the said in Introduction section, we see that validity of the both TUR (3.1) and STUR (3.2) is actually limited by the effective size of the consideration vicinity. It assumes that the both (3.1) and (3.2) work the better the lesser effective size of the OTS is, which allows us to consider (3.1) and (3.2) for the nanoscale OTS. So, we will use (3.1) and (3.2) as the canonic relations to compare energy-temperature consequences of (3.3a) with. Doing this, we will consider notation dU , d(1/T ) in (3.1), and dS , δJ in (3.2), not as the statistical fluctuation variances [51] but the differentials (for infinitesimal dϕ) and differences (for finite ∆ϕ) in its classical sense, though as it is known, for ex. [67], in the infinitesimal approximation the stochastic variables can be interpreted as differentials anyway. We will use the statistical notation only to demonstrate the extent of similarity of our results with the ones previously reported.

3.3. Assumptions Bearing above in mind, consider a classic time-continuous closed (no transfer of matter) thermodynamical system that can exchange energy with its surroundings. We will analyze changes occurred at permanent coupling with a thermal bath supporting regular energy exchange between OTS

28

RELATION RECE WITH THERMODYNAMIC UNCERTAINTY

CH. 3

and its surroundings and keep out the after-action effects caused by impulse energy impact to the system. Let internal energy U and temperature T interrelated so, U =g

T T0

kB T

(3.4)

where g = g(T /T0 ) is an unitless normalized quantity, T0 is some reference temperature. Introduce compressibility factor A

∆X = g . U

(3.5)

In the section 3.4, we will believe that our OTS is in the compressible state so dX ∼ 1 (3.6a) while in the section 3.5, it is in the quasi-incompressible one with but δX 6= 0.

dX ≪ 1

(3.6b)

Further, we will employ energy conservation law in its thermodynamic form ∆U A =1− (3.7) Q

Q

and the Clausius inequality [68] δQ 6 T ∆S.

(3.8)

As a final remark, we will discuss the links (3.3a) → (3.1), (3.3a) → (3.2) using the same conceptual apparatus but not exactly the same methods, so we will split our consideration between the separate sections.

3.4. Finite interface factor 3.4.1. Link between ECE and STUR. Employing (3.4), (3.5), (3.7), and (3.8), from (3.3a) we have ∆S kB ∆X

>

1 1

− cos (∆ϕ)

(3.9)

3.4

29

FINITE INTERFACE FACTOR

Now, as 1 − cos (∆ϕ) = 2 sin2 (∆ϕ/2) 6 2 sin2 (∆ϕ), let sin (∆ϕ) = = ∆J/J , where J is a module of an energy flux vector J and ∆J is the tangential component of vector J , then dS kB

>

∆X J 2 2 ∆J 2

.

(3.10)

At ∆X > 1, (3.10) is equivalent to dS kB

>

J2 2∆J 2

(3.11a)

.

Consider case ∆X 6 1. Since from inequality y > ax at a 6 1 follows that x > y , then applying that to (3.10), we obtain the inequality with the opposite sign that using (3.9) could be written as dS kB

6

1 1

(3.11b)

− cos(∆ϕ)

where a > 0 is arbitrary constant, x, y are variables. Note that having averaged (3.11a) on the time interval τ ∼ ∆t, and denoting S˙ t = = ∆S/∆t, we have solution that up to a constant multiple coincides with STUR (3.2). Highlight here that (3.11b) does not violate (3.10) and, therefore, the Clausius inequality (second thermodynamic law). Also, pay attention that in its physical meaning, quantity ∆X deals with the significance of compression effects or, the relative ratio of physical work A and internal energy U . Unless condition ∆X > 1 (A > U/g ) compromises energy integrity of OTS, the both physical scenarios, (3.11a) and (3.11b) have equal probability to occur. In the next section, we will use (3.11b) in our calculations. 3.4.2. Link between ECE and TUR. Write ∆U = (∆U/∆T )∆T then, (3.3a) could be expressed as (∆U/∆T )∆T Q

= cos (∆ϕ).

(3.12)

Multiplying (3.3a) and (3.12) and using (3.8), we have ∆U ∆(1/T ) 6

∆S 2 ∆U/∆T

where ∆U ∆(1/T ) =

cos (∆ϕ)2

∆U ∆T T2

.

(3.13)

30

RELATION RECE WITH THERMODYNAMIC UNCERTAINTY

CH. 3

In (3.13), represent ∆U/∆T as ∆U ∆T

= ∆U ∆(1/T ) ·

T2 ∆T 2

and find the square root from the both parts of resultant relation, then we obtain 1 cos (∆ϕ)

∆U ∆(1/T ) ·

T kB ∆T

6

∆S kB

.

(3.14)

Using (3.11b), replace ∆S/kB in (3.14), so kB > L · ∆U ∆(1/T )

where unitless L=

T ∆T

µ

1 cos(∆ϕ)

(3.15)

¶

−1 .

(3.16)

As at cos (∆ϕ) = 1, L = 0, and at cos (∆ϕ) → 0, L → ∞, we may think of L as an indicator of what how significant is the deviation of an actual energy exchange pattern from an ideal one (3.3b). Then, employing reasoning we did (3.4.1), we conclude that at L < 1, from (3.15) we have (3.1). Solving L 6 1, we obtain that at fixed T /∆T , cos (∆ϕ) ∈ [1/2, 1).

3.5. Infinitesimal interface factor 3.5.1. Link between ECE and STUR. Using (3.7) in its differential form, (3.4), (3.8), and Taylor expansion for cos (dϕ) [69], Eq. (3.3a) could be written as β2

dS kB dX

> 1,

(3.17)

where infinitesimal β2 =

∞ X (−1)k+1 (dϕ)2k

(2k)!

k=1

dϕ2 ∼ . = 2

In the infinitesimal approximation, dϕ ∼ sin (dϕ) ≈ dJ/J , hence we obtain dS 2kB dX

>

J2

dJ 2

.

(3.18)

3.5

INFINITESIMAL INTERFACE FACTOR

31

Then formally, based on (3.6b) and the reasoning in 3.4.1, we can write for the range [dX, dX + dα] dS kB

62

J2 dJ 2

=

2 sin2 (dϕ)

(3.19)

where dα ∼ dX . We will use (3.19) in the section 3.5.2. 3.5.2. Link between ECE and TUR. Following logic of 3.4.2, replace the differences ∆ with the differentials d, then avoiding repetition, and using (3.19), we have kB > M dU d(1/T )

where unitless M=

sin2 (dϕ) T 2 cos (dϕ) dT

.

(3.20) (3.21)

Fig. 3.1. Dependence of factors L and M (vertical axis) on interfacefactor cos β at constant T /∆T . The zones where valid |L| 6 1 (zone III) and |M | 6 1 (zone I and II). Zone III formally fits 1/2 6 cos β 6 +∞ (at complex β), zone I fits 1/ϕ 6 cos β 6 ϕ, zone II fits −ϕ 6 cos β 6 −1/ϕ. One can see that zone I and III overlap, and |L| 6 1 does not have negative roots. Width of zone fits to an amplitude of random factor, at which we may obtain TUR in the finite (L) and infinitesimal approximation (M ). So, at not so small interface factor | cos β| > 1/ϕ, presented theory yields acceptable results. Curve for L shown in red, for M in blue.

32

RELATION RECE WITH THERMODYNAMIC UNCERTAINTY

CH. 3

For quantity M , we can repeat above said for L. Indeed, at cos (∆ϕ) = 1, sin (∆ϕ) = 0, and M = 0. In the opposite case, at cos (∆ϕ) → 0, sin (∆ϕ) → 1, and M → ∞. So, like L, quantity M characterizes the extent in which existing pattern of energy exchange is close to an ideal case (3.3b). So, if M < 1, then from (3.20) we may have (3.1). Solving M 6 1, we obtain that at fixed T /(2∆T ), cos (∆ϕ) ∈ ∈ [0, 1/ϕ), where ϕ is a golden ratio (Fig. 3.1).

3.6. General form of TUR/STUR Generalizing (3.14), we may write ∆Z ∆Y

6 ∆Sk

(3.22)

where the OTS factor ∆Z = ∆U ∆(1/T ) ·

T kB ∆T

the interface factor ∆Y = cos(∆ϕ)

and the entropy transfer factor ∆Sk =

∆S kB

.

Let us consider what particular cases (3.22) actually contains. So, at L 6 1 (3.16), we have TUR (3.1), while at L > 1, we have the inequality opposite to TUR. At ∆Z > 1, we obtain STUR (3.2), whereas at ∆Z 6 1, we do opposite. As at all real ϕ, quantity ∆Y 6 1, then we have an equality (∆Z = ∆Sk ) only if ∆Y = 1, and an inequality otherwise. It means that any reconfiguration in the OTS may match the change in an entropy (energy) transfer only for the case (3.3b) leaving ∆Z < ∆Sk in the generic case (3.3a). We see that the cases discussed earlier in the sections 3.4, 3.5 are the particular cases of (3.22), so, we could call (3.22) as a general thermodynamic uncertainty relation (GTUR) for finite and infinitesimal interface-factor of OTS.

3.7

DISCUSSION

33

3.7. Discussion Our research has shown that combining ECE (3.3a) with the Clausius inequality (second thermodynamic law) (3.8), we can get the canonic form for TUR (3.1) and STUR (3.2). The major difference of our approach from the used before is that we employ the modified ECE with the changeable interface factor cos (dϕ). Physically, this quantity is an instantaneous indicator to the pattern of energy exchange flows between OTS and its surroundings. On the other hand, quantity cos (dϕ) is meaningful only for the physical system possessing the finite size. We have demonstrated that the generalized form (3.22) could contain not only the physical scenarios that are described by (3.1) and (3.2) but it does include some intermediate patterns as well. Underline here that while we did obtain the solution family containing canonic STUR (3.2), we did not prove in general case validity of TUR (3.1). What we did is to discover conditions when TUR is valid. Generalization of thermodynamic uncertainties was considered in a number of articles. As it is shown in [54], in a limit of very high energies it looks feasible to introduce into the standard thermodynamic uncertainty relation an additional term proportional to the interior energy that leads to existence of the lower limit for inverse temperature. In principle, eq. (3.3a) also could be considered as the result of applying to (3.3b) some extra uncertainty (energy) which leads to (3.3a) and, finally, comes in as limitation for entropy. In [70], authors analyze an ensemble in which energy E , temperature T and multiplicity N can fluctuate and with the help of non-extensive statistics they propose a relation connecting all fluctuating variables that generalizes known Lindhard’s TUR [57]. Assuming that the non-extensive statistics does not employ traditional Boltzmann-Gibbs statistical mechanics, this result seems promising especially for the research of complicate and yet unexplored objects of different spatial scale. Results [58] assume that an indeterminacy in the fundamental description enables the use of information theory framework. In this sense, it looks reasonable to find proofs of uncertainty relations based on the results of information theory. We believe that it is in compliance with our reasoning in the section 2 about the “uncertainty” meaning of (3.3a) in contrast to quite determined (3.3b). Note that physical meaning of (3.22) is well complied with the modern views on a nature of energy-entropy coupling occurring at evolution of OTS at interaction with its surroundings.

34

RELATION RECE WITH THERMODYNAMIC UNCERTAINTY

CH. 3

Indeed, from the information standpoint, we can interpret (3.22) as an indication to what the value of uncertainty of our knowledge about the OTS related to deviation ∆Z , cannot in any case exceed that of energy flow related to ∆Y multiplied by additional uncertainty ∆Sk related to an interfacing between OTS and energy flow. As a result, ∆Z never matches ∆Y with an exemption of the special case when ∆Sk = 1. In the thermodynamical sense, the OTS change ∆Z does not match the flow change ∆Y due to the entropy flow ∆Sk emerged on the interface which is in a perfect agreement with the second thermodynamic law. Following this logic, zero entropy flow ∆Sk could occur only at ∆X/∆Z = 0 assuming no coupling between OTS and energy flow at all. Comparing the violation conditions for (3.1) and (3.2), we may, in a simplified way, say that STUR fails at the modest or small extent of compressibility and small deviations of energy flow (∆J < J ), while TUR does it at small ∆T and cos (∆ϕ) ∼ 1. At high ∆S , (3.1) and (3.2), by and large, can be valid at the same time, while at low ∆S , (3.2) can break leaving the validity for (3.1) as an option. Taking a retrospective look, we could say that ultimately, the limitations for this model come from the ones for (1.1). This model can work for description of the processes characterizing by high ∆T , ∆J , ∆U and small cos (dϕ), i. e. occurring far away from equilibrium. This model could be applicable for broad class of the situations where the permanent energy exchange between the OTS of interest and external environment takes place. It is valuable to note that this model works both for the one source-one sink case (eq. (3.3b)) and for the multiple sources-multiple sinks one, ideally an infinite number of the source-sink pairs (eq. (3.3a)). However, there is an important difference between these two cases. We mean that in the case (3.3b), we rather deal with an equilibrium case and the thermal fluctuation is the essential reason causing the uncertainty. But, if we use (3.3a), the driving force for an uncertainty is the permanent reproduction of the non-equilibrium conditions when at any given moment of time OTS has the set of options (source-sink pairs) with the non-zero probability to pick up. So, the badly we break (3.3b), the wider range for uncertainty we should claim. In this sense, emergence of uncertainty for (3.3a) could be considered as natural amendment of (3.3b) to get to the higher accuracy of the results. That is why, all uncertainty relations above include the value of deviation ∆T , ∆J , ∆U and ∆dϕ. In this sense, our model

3.7

DISCUSSION

35

provides desired accuracy within a quite wide range of time/space limits at only one condition of using suitable uncertainty relation. So, thermodynamic uncertainty relations (3.1), (3.2) can be obtained on the same platform of ECE (3.3a) which allows considering them as consequences of ECE (3.3a). TUR and STUR assume generalization (GTUR) covering more possible physical scenarios at interaction between system, system-energy flux interface, and energy flux.

CHAPTER 4

GEOMETRY OF EVOLUTIONARY TRIANGLES

4.1. Theorem on area ratio at k = 2p In 2-dimensional Euclidean space, consider arbitrary right-angle ∆ABC such that ratio AC/BC = k , where k = 2p , p = ±1.

Fig. 4.1. Evolutionary triangles. Shown evolutionary triangles ABC, AF C T (area S F = S C + 2S T ), AC1 C (area S C ), CC ´ (area ³ 1 B´ (area S ³), CBF S T ). For ∆AF C, the following ratio is valid

T 2Sn C Sn

p

SF

p

n = γ(k) , where T ³ T ´p ³2SnF ´p 2S S γ(k) = (4/k 2 )p , p = ±1. At k = 2, above becomes S Cn = 2SnT = ϕp , n n where ϕ number of golden ratio.

Mark off distance from the vertex B √as BC1 = BC and BF = BC . Since √ AC = kBC , then AB = BC k 2 + 1, AC√1 = AB − BC1 = = BC( k 2 + 1 − 1) and AF = AB + BC = BC( k 2 + 1 + 1). For convenience, in this section denote S1 the area of ∆AC1 C , S2 the area of ∆CC1 B , S3 the area of ∆BCF , S1 + S2 + S3 = S , the golden ratio [71] as ϕ, and √ fk =

AF

AC

=

AC

AC1

=

k2 + 1 + 1 k

.

(4.1)

Then, Theorem 1. If in right-angle ∆ABC with leg ratio AC/BC = 2p , where p = ±1, to mark off from the vertex B along hypotenuse AB

37

THEOREM ON AREA RATIO AT k = 2P

4.1

the line segments BC1 and BF with the length equal to length of BC, then the ratio µ ¶p µ ¶p ³ ¶p µ ´ S S 2 + S3 AF p AC = = = = ϕp . (4.2) S2 + S3

S1

AC

AC1

Firstly, prove some lemmas. Lemma 1. Area

S2 = S 3 =

BC 2 2

r 1−

1 k2 + 1

.

(4.3)

√ Proof. Since cos ∠ABC = − cos ∠CBF = 1/ k√2 + 1, then for ∆CC1 B using law of cosines C1 C 2 = 2BC 2√ (1 − 1/ k 2 + 1). In the 2 2 same way, for ∆BCF , CF = 2BC (1 + 1/ k 2 + 1). Then, employing Heron’s formula for area [72], obtain (4.3).

Lemma 2. Area S1 =

BC 2 2

r µ k− 1−

1 k2 + 1

¶

.

(4.4)

Proof. The area of ∆ABC = kBC 2 /2, then from (4.3), obtain (4.4). Lemma 3. If k = 2, the ratio S 2 + S3 S1

= ϕ.

(4.5)

Proof. From (4.3) and (4.4), r r ¶ µ S2 + S3 1 1 2 =√ =2 1− 2 : k− 1− 2 = 2 S1 k +1 k +1 k +1−1 √ 2

k

k2 + 1 + 1

·

k

=

2

k

· fk .

(4.6)

So, at k = 2, using (4.1), ratio. (S2 + S3 )/S1 = fk =

√

5 + 1 2

= ϕ.

Lemma 4. If k = 2, the ratio S S 2 + S3

= ϕ.

(4.7)

38

GEOMETRY OF EVOLUTIONARY TRIANGLES

CH. 4

Proof. Modify (4.7) as S S2 + S3

=1+

S1 S2 + S3

= fk .

(4.8)

From (4.5), S1 S2 + S 3

=

1 k fk 2

.

Then (4.8) becomes 1+

1 k fk 2

= fk

(4.9)

and at k = 2, the positive root of (4.9) fk = ϕ. Obviously, all reasoning in Lemma 3 and 4 is valid for p = −1 if we replace k = 2 with k = 1/2, and rename the vertexes A and B . Now, we can prove Theorem 1. Proof of Theorem 1. In ∆ABF , from (4.1), (4.5), and (4.7), immediately follows (4.2). Corollary 1.1. From (4.2), (SS1 )p = (S2 + S3 )2p .

(4.10)

4.2. Evolution theorem For the rest of section 4.2, we will use p = 1. So, in ∆AB1 C1 , from (4.2), AC1 = AC/ϕp . From the point C1 raise a perpendicular to AB , then on this perpendicular mark off the line segment B1 C1 equal to BC/ϕ and connect the points B1 and A. Since AB1 = AB/ϕp then ∆AB1 C1 is similar to ∆ABC as having three proportional edges and, obviously, the area of ∆AB1 C1 is S/ϕ2p . Continuing this way, at an arbitrary n > 1, in ∆ABn Cn , AC/ACn = ϕpn and the area Sn of ∆ABn Cn is S/ϕ2pn . Further, we will call similar triangles ∆AFn Cn as the full ones with area SnF , ∆ACn+1 Cn as the core ones with area SnC in contrast to the transfer triangles, Cn Cn+1 Bn and Cn Fn Bn with area SnT . Then, F (S F ) of the first immediate folTheorem 2. The full area Sn+1 n lower (precursor) in the chain of similar triangles is exactly equal C ) of the core triangle in its immediate precursor to the area SnC (Sn+1 (follower), i. e. ¡ F ¢ F C Sn+1 = SnC Sn = Sn+1 . (4.11)

4.2

39

EVOLUTION THEOREM

Proof. For the case ∂Sn /∂n < 0, where Sn is area of the triangle ABn Cn , n is the number of evolution step, F Sn

F Sn+1 =

ϕ2

.

(4.12)

From (4.7), ¡

F Sn

¢2

¢ T 2 4 Sn ¡

= ϕ2 .

(4.13)

Insert (4.13) into (4.12) and apply (4.10) for arbitrary n, which converts (4.13) to ¡ T ¢2 ¡ T ¢2 F S

F Sn+1 = 4 n¡

Sn

F Sn

¢2

=4

Sn

F Sn

=

F SC Sn n F Sn

= SnC .

For the case ∂Sn /∂n > 0, using the same logic as above, ¡ F ¢2 ¡ F ¢2 S

C Sn+1 = ϕ2 SnC = SnC ¡ n ¢2 = SnC T 4 Sn

Sn

F SC Sn n

= SnF

F Theorem 3. At ∂Sn /∂n < 0, the full area Sn+1 of the first immediate follower in the chain of similar triangles is exactly equal to the area SnF of its immediate precursor decremented by the double area SnT of transfer triangle, i. e. F Sn+1 = SnF − 2SnT

(4.14a)

while at ∂Sn /∂n > 0, incremented by the double area SnT of transfer triangle multiplied by ϕ + 1, i. e. F Sn+1 = SnF + 2(ϕ + 1)SnT .

(4.14b)

Proof. For the case ∂Sn /∂n < 0, obviously, SnC = SnF − 2SnT .

(4.15)

F , and Then, based on Theorem 2, replace in (4.15) SnC with Sn+1 obtain (4.14a). T . Since S C C F F + 2Sn+1 = Sn+1 For the case ∂Sn /∂n > 0, Sn+1 n+1 = Sn T and Sn+1 = ϕ2 SnT = (ϕ + 1)SnT , we obtain (4.14b).

40

GEOMETRY OF EVOLUTIONARY TRIANGLES

CH. 4

4.3. Theorem on area ratio at any k Now, let AC/BC = k , where k is any real number. Write some relations, which we obtained earlier, we will use it further √ k2 + 1 + 1

fk =

SnC

= SnT

SnF

=

k

2 Bn Cn

2

=

r (k − 1 −

2 B n Cn

2

2 Bn Cn

2

r 1−

;

(4.16)

1 k2 + 1 1

k2 + 1

r (k + 1 −

);

(4.17)

;

1 k2 + 1

(4.18) ).

(4.19)

Theorem 4. If in arbitrary right-angle ∆ABn Cn with fixed leg ratio ACn /Bn Cn = k , where k is any real number, to mark off from the vertex Bn along hypotenuse ABn the line segments Bn Cn+1 and Bn Fn with the length equal to length of Bn Cn , then irrelevant to the sign of ∆Sn , for any integer n > 1 µ T ¶p µ F ¶p 2Sn Sn (a) = γ(k) (4.20) C T Sn

2Sn

where γ(k) = (4/k 2 )p , p = ±1; (b) the full area SFn+1 (SnF ) of the first immediate successor (predecessor) in the chain of similar triangles at similarity coefficient C matching f k is exactly equal to the area SC n (Sn+1 ) of the core triangle in its immediate predecessor (successor), i.e. η=

C Sn F Sn+1

= 1.

(4.21)

Proof. (a) Relation (4.20) immediately follows √ from definition of ACn = kBn Cn , AFn√= ABn + Bn Cn = Bn Cn k 2 + 1 + 1 and ACn+1 = ABn − Bn Cn = k 2 + 1 − 1. Calculate γ(k) using (4.16)–(4.19) as µ T ¶p F 4fk2 2Sn Sn 4 p : ( )p = ( 2 )p ) = ( γ(k) = C T 2 Sn

2Sn

2kfk + k

k

4.4

41

DISTORTION THEOREM

Now, prove (4.21), using Fig. 4.2.

Fig. 4.2. Consecutive changes in family of right angle triangles at planar similarity transformation. Figure is family of similar right angle triangles with arbitrary leg’s ratio k in 2-dimensional Euclidean space. It is proved SF

that at similarity coefficient matching fk , (see below), holds η = Sn+1 = 1, C n F F where Sn+1 (Sn ) is full area of the first immediate successor (predecessor), C SnC (Sn+1 ) area of core triangle in its immediate predecessor (successor).

For specificity, consider case p = 1, the case p = −1 is proved by simple reversing of all the ratios. Using (4.16), obtain F Sn+1

=

F Sn

fk2

=

SnF

·

Ã√

k2 + 1 + 1

then apply (4.19) which yields r µ 2 Bn Cn F k− 1− Sn+1 = 2

k

1 k2 + 1

¶

!−2

= SnC ,

that proves (4.21).

4.4. Distortion theorem So far, we believed that Bn Cn = Bn Cn+1 and proved that at that condition, relation (4.21) is identity. Now, let the length α of BCn+1 be random, i. e. generally BCn+1 6= Bn Cn and evaluate the measure of C ). From that, random F and SnC (SnF and Sn+1 distortion β between Sn+1

42

GEOMETRY OF EVOLUTIONARY TRIANGLES

CH. 4

√ α ∈ [− k√2 + 1, 1] with origin in the vortex Bn and the full range R = = AF = k 2 + 1 + 1√with the subrange R− = k for the random event ∆Sn 6 0 and R+ = k 2 + 1 + 1 − k for the random event ∆Sn > 0 (Fig. 4.2). Assuming uniform probability density distribution in R, calculate appropriate probability density function R−

g − (k) =

and

R+

+

g (k) =

=√

R

R

=

√

k

(4.22a)

k2 + 1 + 1

k2 + 1 + 1

√

−k

k2 + 1 + 1

(4.22b)

.

Theorem 5. If to supplement Theorem 1 by requirement of random BCn+1 = α (random selection of the point Cn+1 ) at condition of uniform probability √ density distribution, where α is random number in the range [− k 2 + 1, 1] with origin in the vortex Bn then (a) F (S F ) of successor (predecessor) and distortion between area Sn+1 n C ) of predecessor (successor) SnC (Sn+1 β (α, k) = 1 − η

(4.23)

achieves minimum β = 0 at α = ±1 at any k ; (b) at ACn+1 = k , the image of rotation-similarity transformation degenerates to circle with β 6= 0 at any k : ACn = AC1 exp [g (ϕ)bθ],

(4.24)

where AC1 is assumed to be constant, b = 2π/ arctan (1/k). Proof. (a) Rewrite (4.17) assuming α to be random as 2 Bn Cn

SnC =

2

(1 − √

α k2 + 1

).

(4.25)

Also, write F Sn+1 =

F Sn

fk2

=

2 Bn Cn

2

·√

k2 + 1

− α2 k 2 + 1 k 2 ( k 2 + 1 + α) 1

·

√

Inserting (4.25), (4.26) to (4.23), we have β=

F Sn+1

− SnC

F Sn+1

=1−

k2 + 1

− α2

k2

.

(4.26)

4.4

DISTORTION THEOREM

43

Function β(α, k) has minimum β = 0 at α = ±1 (Bn Cn = = Bn Cn+1 ) for any k . Plot β(α, k) for different k is shown in Fig. 4.3. So, invariance relations (4.21) holds at α = ±1 only. At α 6= ±1, we should account distortion factor 1 − β, i. e. η = 1 − β (α, k)

Fig. 4.3. Dependence measure of distortion β(α, k) on α for three different k. There is no any distortion at α = ±1, it is modest at −1 6 α 6 1, and drastically grows at |α| > 1. Also, at α = 0 (α = position of vertex B), all three curves demonstrate local maximum β

(b) If random ACn+1 = k , |ACn | = |ACn+1 |, so (4.24) obviously describes a circle, where |z| denotes module of number z . In ∆ABn Cn , random Bn Cn+1 = α = ABn − ACn+1 , i. e. p α = Bn Cn+1 = k 2 + 1 − k = 1. (4.27)

Solution (4.27) is k = 0, which means that α can never be ±1, so β 6= 0 for any k . It also directly follows from the well-known triangle inequality declaring that any side (Bn Cn ) of a triangle is greater than the difference (ABn − ACn ) between two other sides. Above, we believed that α ∈ [−AC, 1], i. e. Bn Cn+1 > 1 which, generally, assumes the high bias ∆z = AC − 1 in α-distribution. To fix it, restrict bias within the range [−R − ∆z, R] with the center (α = 0) at the vertex C , where now bias ∆z < AC − 1. Consider parent triangle ABC (Fig. 4.4). Let B1 be the new random position of B at the parent-offspring transformation. Define ∆AB1 = = AB − AB1 > 0. Then,

44

GEOMETRY OF EVOLUTIONARY TRIANGLES

CH. 4

Theorem 6. In assumption of uniform probability distribution within the support range [−R − ∆z, R] in parent-offspring transformation, at ∆z 6 ∆AB , the probability of random event ∆AB1 > 0 exceeds the one for ∆AB1 < 0. Proof. At uniform distribution, the sign of preferable AB1 change is determined by the length of the appropriate support range. Then, we have two options (Fig. 4.4). The random event ∆AB1 < 0 has the support range DB1 with the length R − ∆AB1 + ∆z . The random event ∆AB1 > 0 has the support range B1 F with the length R + ∆AB . Compare segments DB1 and B1 F 1 + ∆AB > 1 − ∆AB1 + ∆z.

(4.28)

∆AB1 > 0.

(4.29)

Accounting above requirement ∆z 6 ∆AB , we obtain So, B1 F > DB1 holds at any position of the point B1 within DF . Then, the probability of random event ∆AB1 > 0 exceeds the one ∆AB1 < 0, i. e. transformation goes in the direction to support P (∆AB1 > 0) > P (∆AB1 < 0).

(4.30)

Fig. 4.4. Triangle ABC is shown. Position of B1 is random within the range DF , where DR = ∆z 6= 0 is the bias of distribution for random B1 . The center of the range [−R, R[ is in the vertex C. The random ∆AB1 = AC − AB1 . As B1 F > DB1 , the probability of the random event ∆AB1 > 0 exceeds the one for ∆AB1 < 0.

4.5. Conclusions We have shown that in the right-angle triangle with arbitrary leg’s ratio k , the ratio of built-in triangles is obeyed (4.20). In particular, at k = 2p , where p = ±1, above ratio matches the number of

4.5

CONCLUSIONS

45

golden ratio. Being under planar similarity transformation the family of right-angle triangles demonstrates consistent internal restructuring in the successor/predecessor chain at any transformation step n with measure of distortion from exact similarity obeyed to (4.23). At similarity coefficient equal to fk (4.16), the exact equality between the full area of the successor (precursor) triangle and the area of the built-in core triangle in its precursor (successor) is observed (η = 1), i. e. quantity η is being become invariant under that transformation irrelevant to the sign of the area change. Transformation goes in the direction to support incremental development of parameters in the parent triangle.

CHAPTER 5

EIGEN SOLUTIONS

5.1. Boundary value problem Function of integral efficiency for energy exchange Υ (1.10) by its construction does not have physical meaning beyond the y -range [0, e]. In this sense, solution Υ(y) is bound within above range and, formally, we have a two-point boundary value problem on a finite interval [0, e], i. e. Υ(0) = Υ(e) = 0. From (1.10), to meet above boundary conditions should hold 1 − | ln y| = 0

(5.1)

1 = ±n ln y

(5.2)

or, in discrete form so, at | ln y| 6 1, i. e. in the range [1/e, e] h 1i yn = exp ± n

(5.3)

and appropriate harmonics

Υn = y(1 − n ln y)

(5.4)

where n = 1, 2, . . . Location of harmonics Υn relative to the first harmonic Υ1 is shown in (Fig. 5.1). Using hyperbolic notation, we can also write Υ=

2 e

+2

∞ X ¡

n=2

¢ cosh (1/n) − sinh (1/n) .

In considering model, Υ and entropy S is related (2.1), so appropriate limitation could also be applied for S ∆S (y = 1/e) = 0;

(5.5a)

∆S(y = e) = 0.

(5.5b)

5.1

BOUNDARY VALUE PROBLEM

47

Fig. 5.1. Discreteness of spectrum for integral efficiency of energy exchange Υ. The discrete spectrum for Υn formed at evolution of OTS is shown. In the plot, by an abscissa axis, a unitless energy exchange rate y = J/J0 is indicated, by an ordinate axis the integral efficiency of energy exchange Υ (a). A few first harmonics of Υn , are schematically shown by thick vertical segments in the range 0 6 y 6 e.

Applying (2.1) we obtain that (5.5a,b) are valid at condition (5.1), (5.2) only, then formally, entropy also acquires the quazi-discrete character as is shown in Fig. 2.3. Using (2.11) and (5.3) ln ln y + nHx = 0

(5.6)

At 0 6 Hx 6 Hmax , exp (±1/2) 6 y 6 exp (±1), therefore ∆Sn > 0 at n 6 2, whereas at n > 2, ∆Sn < 0, which is also demonstrated in Fig. 2.3. From (5.6) follows that maximum ∆Sn is at maximum Hx (maximum dissipation and number of loops) and n = 1 (fits to the first harmonic of discrete spectrum y1 at OTS evolution). According to principle of maximum entropy production [14] this scenario fits to the most probable and the fastest path of evolution. In turn, minimum ∆Sn is at minimum Hx (minimum dissipation and number of loops) and n > 1. Then, following logic of [14] this scenario corresponds to the least probable and the slowest path of evolution. Also,

48

EIGEN SOLUTIONS

CH. 5

(5.6) indicates that the growth of n can overweight maximum Hx , it happens at n = 2, i. e. at the second harmonic of discrete spectrum y2 . So, the most probable (maximum randomness) evolutionary scenario assumes realization of evolution process within the y -range limited by the first harmonic y1 [1/e, e]. In the contrast, the least probable (minimum randomness) evolutionary scenario assumes realization of evolution process within the y -range limited by the second harmonic y2 [e−1/2 , e1/2 ]. The remaining scenarios take the niche in between above two by decreasing probability of realization from y1 to y2 . Finally, one can state that those evolution scenarios which do not have the positive phase ∆Sn > 0 are the least probable. Pay special attention that the found discrete solution exists in the range [1/e, e] only whereas the spectrum in the range [0, 1/e] is still continuous.

5.2. Physical grounds of solution limitation As we saw above, limitation imposed on Υ within range [0, e] mathematically leads to seizing of solution within the range [1/e, e]. Consider possible physical reason of that. At y < 1/e, |δΥ| > 1 (Fig. 5.2, dark-grey area) so the support range df for f y (x) (2.6) exceeds the support one dg for g(x) [−1, 1] (3.5), i. e. df > dg , and there is no any correlation between df and dg . However, at y > 1/e, |δΥ| < 1 and physical situation changes. Now df < dg which means that the portion of dg range still being available formally remains unoccupied (Fig. 5.2, light grey area). We denote its value as dgu . In mathematical terms, it means that ln y and the occupied portion dga (or unoccupied portion dgu ) are become correlated (not x and ln y ). Because Υ and entropy S depends on dga (dgu ), it will impose limitation on their variation. That is why solution is localized between two boundary points where the range of unoccupied microstates exists, i. e. y = 1/e and y = e. So, to meet this limitation, (5.1) should be changed to (5.2) and our solution acquires the discrete nature. As it will be shown later, presence of above correlation causes emergence of predetermination in changing between adjacent evolutionary states. These matters are discussed in the next chapters, where we look at development of discreteness under more formal angle and demonstrate mechanism for emergence of discrete twodimensional energy exchange form (EEF).

5.3

LEAKAGE OF UNOCCUPIED MICROSTATES TO COMPLEX DOMAIN

49

Fig. 5.2. Mechanism for forming of discreteness in energy spectrum of evolving OTS at 1/e 6 y 6 e. In the plot, designation of axes is the same as in Fig. 1.1. Zone | ln y| > 1 at 0 6 y < 1/e is shown in the dark grey. The adjacent zone with | ln y| 6 1 at 1/e 6 y 6 e where discretization occurs, is shown in the light grey. Discretization mechanism is based on appearance of correlation between ln y and occupied portion dga (or unoccupied portion dgu ) of support range for density g(x).

5.3. Leakage of unoccupied microstates to complex domain Now, we try to quantify abovementioned process for appearance of unoccupied portion in support range dgu . Recall that earlier we have already introduced association of unoccupied portion dgu with an imaginary domain (2.9). So, any increase in the number of microstates in the imaginary domain dgu proportional to the length of segment between curve − ln y and 1 can come only at expense of the whole domain [0, 1] (Fig. 5.2). Then, accounting an orthogonality between the real domain dga proportional to the length of segment between curve −lny and 0 and complex one dgu (Fig. 5.3), one could be written that

or

¡ ¢2 1 = ln2 y + 1 − | ln y| ¡ ¢ 2 ln y | ln y| − 1 = 0.

(5.7)

50

EIGEN SOLUTIONS

CH. 5

Ignoring trivial solution ln y = 0, we obtain that (5.7) is valid at the conditions (5.2) only. Thus, energy transfer between real and complex domain is possible at the discrete points y = yn only.

Fig. 5.3. Leakage of energy exchange microstates from real to complex domain. Triangle formed by real (dga ) and imaginary (dgu ) portion of the support range dg for random interface-factor x. In the plot, by an abscissa axis, the real portion dga (Re d) is indicated, by an ordinate axis the imaginary portion dgu (Im d). Condition for the points y = yn where an actual transfer of energy between complex and real domain can occur is Re2 dg + + Im2 dg = 1.

5.4. Eigenvalue problem for OTS Write an eigenvalue problem as L(y) ± κy = 0

(5.8)

where operator of interface comes from (1.10) and k is some number. Replace in (5.8) L(y) with (1.10), so we obtain y − y ln y ± κy = 0

and κ = ±(ln y − 1)

(5.9)

Since y -range for discrete solution is limited within 1/e 6 yn 6 e, then from (5.9) 0 6 kn 6 2. (5.10)

5.4

EIGENVALUE PROBLEM FOR OTS

51

As kn takes an infinite counting number of values in the range [0, 2], therefore it could be thought of as the eigenvector of operator L(y). Below, we will show that eigenvector kn plays critical role in our model as it connects thermodynamic and topology parameters of solution.

CHAPTER 6

EVOLUTION TOPOLOGY

6.1. Evolutionary triangle In this chapter, we apply results from the previous chapters with the purpose of the detailed analysis of OT C evolution. In this connection, we consider family of arbitrary triangles ∆ABn Cn composed by the legs Υ and y for dependence Υ(y) (Fig. 5.1). In further text, such triangles are called the evolutionary ones. As will be shown in the next chapters, the main meaning in the quantitative balance of an energy exchange in OTC has area of triangles ABn Cn described in the Chapter 4. The reason of such statement stems from the following. As is shown in the Chapter 1, function Υ(y) is the generalized characteristics of an average efficiency for the process of energy transportation through OTC interface at given rate of energy exchange y . Then, area ABn Cn , (numerically equal to the product yΥ) is the weighted efficiency for an interaction between external environment and OTC. Keeping this in mind, evolutionary triangles ABn Cn will be also called energy exchange forms (EEF).

6.2. Connection between topological and physical parameters On the one hand, (5.4) could be written as Υn = kn yn ,

(6.1)

where kn is determined from (5.9). On the other hand, as is clear from Fig. 6.1, designating BC = Υn and AC = yn , tan Bn ACn = kn =

Υn yn

.

(6.2)

6.3

SPECTRAL HARMONICS AS PHASE TRANSITION POINTS

53

Compared (6.1) with (6.2) we come to conclusion that thermodynamic and topological parameters in our model are linked through the factor kn which is an eigenvector for interface operator (1.10).

6.3. Spectral harmonics as phase transition points Below, we study evolutionary meaning of spectral harmonics y = yn (5.3). 6.3.1. Golden Ratio Point. In Fig. 6.1, ∆ABC defined by legs [0, 1/e] in OX axis and [0, 2/e] in OY axis as it directly follows from (1.10), so Υ(e−1 ) = 2e−1 . Then, k = Υ/y = 2 and according to (4.20), we have classic relation of golden ratio µ T ¶p µ F ¶p 2Sn Sn = = ϕp . (6.3) C T Sn

2Sn

Fig. 6.1. Evolutionary triangles formed by curve of integral efficiency Υ. Designation of the axes are the same as in the Fig. 1.2. Three evolutionary triangles ABC, AB1 C1 , AB2 C2 are shown. These triangles are defined by the legs AC, AC1 , и AC2 in y and the legs BC, B1 C1 , and B2 C2 , in Υ, appropriately. ∆ABC is defined by leg y = AC(GRP ) and Υ = BC(2GRP ) which fits to the state of dynamic balance between, on the one hand, the area 2S T — double area of ACD, and on the other hand, the area of S C — area of BCD. ∆AB1 C1 is defined by leg y = B1 C1 (GRE) and Υ = AC1 (4/3GRE). At this point we observe dynamic balance between energy exchange process in the real and complex domain (6.5). The state of dynamic balance assumes that the ratio of appropriate quantities is equal to the number of golden ratio.

54

EVOLUTION TOPOLOGY

CH. 6

That is why we call the point y = y1 = 1/e as Golden Ratio Point (GRP). Following the notations used in the Chapter 4, in Fig. 6.1, SnT = = ∆ACD, SnC = ∆CDB , SnF = ∆CDB + 2∆ACD. Then, we may say that at GRP, values SnC and 2SnT are in the state of dynamic balance [75]. This assertion will be discussing in detail below. Besides, GRP has the following properties. In this point: — spectrum changes from continuous to discrete (5.3). — ∆S = 0 both for differential (2.11) (at Hx = 0) and statistical (sect. 2.1) definition of entropy. — dS/dy takes maximum both for differential ((2.11) (at Hx = 0)) and statistical (sect. 2.1) entropy. — probabilistic predominance of flux y = yin is replaced with y = = yout (2.16a,b). — probability for fluctuations of entropy ∆S experiences increase (sect. 2.4, Fig. 2.2). — forming of domain for unoccupied energy exchange microstates starts (sect. 5.2). — derivative z = dΥ/dy − 1 changes sign from positive to negative. So, GRP separates two physically dissimilar y -regions when for evolving OTS the transition from one operational mode to another is observed. Then, to the best of our knowledge we could classify GRP as a phase transition point. Based on above, we will call two opposite y -ranges of OTS evolution as agenesis (y < GRP ) and genesis (y > GRP ). 6.3.2. Point of least possible scenario of OTS’ evolution. Point y = y2 is the maximum (for y < 1) y -root of entropy possible (Fig. 2.3) and, at the same time, as it was discussed, it corresponds to the least possible scenario of OTS evolution. The latter indicates to the boundary meaning of this point which allows to consider y = y2 as another phase transition point. 6.3.3. Point for dynamic balance of fluxes. The next point is the third harmonic of spectrum y = y3 . In its vicinity holds 3S C = 2S T

(6.4)

Also, at this point (Fig. 5.2) the following 1

− | ln y| =2 | ln y|

(6.5)

6.4

DYNAMIC DECOMPOSITION OF EVOLUTIONARY TRIANGLE

55

is valid. In appropriate right-angle triangle with the leg | ln y| and 1 − | ln y| which describes energy exchange between real and complex domain, the ratio of built-in area is equal to the golden ratio, i. e. takes place the state of dynamic balance between suitable parameters of an exchange process. So, further y = y3 will be referred to as the point of Golden Ratio Exchange (GRE). As at y = y3 = GRE the qualitative change of OTS functioning happens, GRE could also be considered as the phase transition point. 6.3.4. Other points. From (5.3), all discrete y -points can be expressed as the terms of convergent series 1

yn = √ n yn =

√ n

e

e

at y < 1; at y > 1

with the lim yn = 1. n→∞

Also, for the absolute value of natural logarithm it could be presented as the terms of divergent harmonic series 1

ln yn = .

(6.6)

n

Then lim

n→∞

³ ´ ¡ ¢n 1 n 1+ = lim 1 + | ln yn | = lim knn = e. n

n→∞

n→∞

(6.7)

Meaning of (6.7) is that at n → ∞ and suitable approaching of k to 1 both from the left and from the right, impact of the random factor (swing for x-variations) goes down attaining zero at stationarity point, when OTC interface is in the pure deterministic state. So, spectral harmonics y = yn of an evolving OTS demonstrate the features normally attributed to the phase transition points. In this sense, an OTS has infinite number of phase transition points. It assumes that at each new yn OTS experiences the more or less significant leap in the quality of its operation.

6.4. Dynamic decomposition of evolutionary triangle As is highlighted in the sect. 6.1, product S tr = 12 yΥ carries the meaning of an average change for energy circulating through OTS interface at given rate y .

56

EVOLUTION TOPOLOGY

CH. 6

On the other hand, in the k -basis, i. e. ignoring actual value of y and Υ, on each evolution step n Sntr = SnT + SnC = k

(6.8)

i. e. S C and S T share evolutionary meaning of S tr . Focus on the differences between S C and S T . With this purpose, return to triangle ABC depicted in Fig. 6.1. In the terms of 6.1, triangles ACD and BCD are adjacent and, therefore, interdependent. Nevertheless, underline that S T essentially depends on AC and AD (instant y ), whereas S C does on BC (integral Υ). Look at this problem under the other angle. Combining (4.20) and (4.21), we obtain at similarity coefficient (4.16) ¡ T ¢2 C Sn+1 =

4

Sn

γ(k) SnC

(6.9)

which means that each area SnC -th is the quantity which is passed from the node n (parent/offspring) to/from the node n + 1 (offspring/parent). Mechanism of such passing is coupling of the quanC with the same quantity (transferred energy) 2SnT . tities SnC and Sn+1 Then, we can suggest that S T and S C are the portions of change in average energy circulating through OTS interface at given rate y , such that in above process S C addresses the node (structure) of EEF and S T the transfer (variability) between EEFs. So, the partitioning of EEF or, in other words, the fine structure of spectral harmonic, plays critical role in evolution of OTS interface. In this sense, (6.3) which declares the dynamic balance between conservation (S C ) and changeability (S T ) is necessary condition for the quality jump in OTS evolution. And, in the same way, (6.9) indicates to the dynamic balance between characteristics of the energy transfer as another necessary condition for the quality change in evolution process.

CHAPTER 7

DYNAMICS OF EVOLUTIONARY TRIANGLES

7.1. Energy balance equation for evolving triangle Introduce decomposition ratio for area of triangle Sntr C Sn T Sn

= ηn .

(7.1)

Reduce (4.20) to the form µ

C Sn T 2Sn

¶2

+

C Sn T 2Sn

=

4 2 kn

.

(7.2)

Then, based on (7.1) ηn (ηn + 2) − kn2 = 0

(7.3)

which provides equation for an energy balance of OTS interface at fixed node n. From (7.3) follows ∆k ∆η

=

ηn+1 + ηn + 2 kn+1 + kn

(7.4)

where ∆k = kn+1 − kn , ∆η = ηn+1 − ηn , n and n + 1 refer to к n and n + 1 step of evolution, appropriately. Compared (7.3), ratio (7.4) deals with energy balance at transition between n and n + 1. Using (4.17), (4.18), (7.1) comes as p η = k 2 + 1 − 1. (7.5)

So, (7.3) deals with the predetermination in appearance of EEF for fixed η and k as it affirms that to survive at given kn , y -mode should get the most effective ratio S C and S T possible i. e. the ratio

58

DYNAMICS OF EVOLUTIONARY TRIANGLES

CH. 7

which is determined by (7.5). Eq. (7.4) takes the next step and provides the predeterminate distortion ∆k/∆η between the consecutive evolutionary steps. Hence, evolution manifests by a mixture of the quickly depressing modes at y 6= yn and the stable modes (ordered EEFs) at y = yn . As to the latter, we could say about existence of energy evolution infrastructure in the development of OTS interface. Of course, use of this infrastructure is subject of attainment of suitable energy exchange rate y which is not assured. Geometry interpretation of (7.3) is shown in Fig. 7.1. As per said in the chapter 1, every single evolution step δΥ is defined by joint

Fig. 7.1. Geometry interpretation of energy evolution equation (7.3) for OTC interface. Evolutionary triangles E1 AO, EAO, F1 BO, F BO for consecutive evolutionary steps are shown, point O — center of circle. Length of the tangent AE and AE1 , AF and AF1 correspond to the current value of k, the length of secant through the center of circle to the value of ηn + 2 and ηn+1 + 2, appropriately, where ηn and ηn+1 is the outer part of these secants, radii F O = F1 O and EO = E1 O correspond to current value of y = 1 in k-basis. The secants through point O correspond to the discrete values of energy spectrum with y = yn , whereas other secants BB1 and BB2 (dotted lines) fit to y 6= yn which are being suppressed at forming of spectrum

contribution of deterministic y (F OF , F1 OF , EOE , E1 OE in Fig. 7.1) multiplied by random x (1.7b). Then, all possible options for δΥ are within the circle with center O. At that, the most probable solutions

7.2

TRIANGULAR TOPOLOGY OF ENERGY EXCHANGE

59

fitting to y = yn lie on the secant going through the center O (making an innermost secant part equal to a diameter of circle, i. e. equal 2 in the k -basis). All other secants fit to the less probable and, therefore, the depressed solutions. Then, difference between the secant through the center O and any other one is that the former features the maximum ratio of its innermost part to the length of whole secant. In physical terms, it means that the surviving (discrete) modes have the maximum variability (topologically coincides with length of diameter) possible. Otherwise, the more predetermination in picking up of actual value and sign of δΥ exists, the less evolution potential of this mode is and the less chances to survive it has. In this sense, there exists physical relation between the concrete means for realization of division of EEF (fine structure of spectral harmonic) and its ability to survive and be the active part of discrete energy spectrum. Finally, as follows from (7.5) η monotonically grows with k (reduces with y ) in the whole y -range [1/e, e]. It quite agrees with the fact that in energy evolution the role of energy structuring and preserving (S C ) reduces and the role of variability and optimal energy transferring (S T ) grows to support efficient development. Under this angle, factor η expresses dynamic division of EEF into portions to support the most surviving means of energy utilization in OTS evolution.

7.2. Triangular topology of energy exchange As it was demonstrated in the chapter 4, the vacant energy microstates in zone 1 − ln y (Fig. 5.2) could be re-taken forming a discrete spectrum at y > GRP . Below, we formalize process of energy exchange between real and complex domain and show how triangular EEF in a virtual space y − Υ could appear. Energy circulated through OT S interface at exchange with real domain which is equivalent to area under the curve Υ(y) (grey and white area) in Fig. 7.2

Sln y =

Zy

1

Υ(z)dz = y 2 (3 − 2 ln y). 4

0

In turn, energy circulated through OTS interface at exchange with complex domain which is equivalent to area under the curve Υ(y) − y

60

DYNAMICS OF EVOLUTIONARY TRIANGLES

CH. 7

(white area in Fig. 7.2) S1−ln y =

Zy

1

(Υ(z) − z) dz = y 2 (1 − 2 ln y) 4

(7.6)

0

Note that (7.6) has meaning in genesis stage only, i. e. at y > 1/e. Keeping this in mind, find the difference 1

∆S = Sln y − S1−ln y = y 2 . 2

(7.7)

Multiplying the both parts of (7.7) by eigenvector kn , we have area of ∆ABC (Fig. 7.2) kn (Sln y − S1−ln y ) = SEEF

(7.8)

as SEEF = 21 kn yn2 is area of triangular EEF with y = yn and k = kn .

Fig. 7.2. Curvilinear topology of energy exchange between real and complex domain at y < 1. Designations of the axes are the same as in the Fig. 6.1. Curvilinear figure AF BCA defined by legs AC, BC, arcs AF B, ADC which area corresponds to an averaged energy circulated through OT S interface at exchange with real domain. Curvilinear figure ADCA defined by leg AC and arc ADC which area corresponds to an averaged energy circulated through OT S interface at exchange with complex domain. Relation kn (SAF BCA − SADCA ) = SEEF shows procedure for creation of triangular EEF , i. e. ∆ABC.

7.3

DRIVING FORCE OF EVOLUTION

61

Forming of the triangular EEF at y > 1 takes the same way (not shown). Hence, we may conclude that triangular EEF is coined in genesis stage due to interaction between two energy exchange fluxes, one related to the real domain and second to the complex domain.

7.3. Driving force of evolution As follows from (1.16), at completion of each single loop, the pathdependent function Υ experiences the non-zero change δΥ. In topological terms, it assumes that appearance of leg y will be inevitably accompanied by emergence of another leg δΥ. As a result, considering model acquires one more degree of freedom thereby becoming the two-dimensional one and contributing to appearance of twodimensional EEF . Will focus on the most probable EEF only for now, then the range of possible random outcomes narrows down to the range R1 R2 on hypothenuse AC (рис. 4.4). As before, we assume uniform probability distribution, then the direction of evolution is determined in accordance with what segment has the biggest length. Then, all preconditions of the theorem 6 hold and we can jump down to its conclusions. So, assign AB1 the meaning of the energy exchange rate y , then we can state that at any y , driving force of OT S evolution supports (4.30), i. e. P (∆y > 0) > P (∆y < 0).

(7.9)

So, driving force of OTS evolution essentially depends on presence of the non-zero δΥ which leads to appearance of asymmetry in the support range for random outcomes in the sign of ∆y . Note that the asymmetry (7.9) between the support range for back and forward energy development gradually decays becoming zero at y = e.

7.4. Individual and collective EEF We investigated vicinity of y = GRP and discovered a number of dramatic changes in behavior of evolution parameters as was highlighted earlier. In this point, OTS experiences deep internal energy reconstruction which manifests itself through appearance of new quality features. In this sense, genesis logically and timely follows agenesis. Under this angle, it appears that a major essence of agenesis stage is to supply energy (continuous spectrum of Υ) and prehistory

62

DYNAMICS OF EVOLUTIONARY TRIANGLES

CH. 7

(δΥ loops) in an amount sufficient to support the more complicate (discrete) mode for operation of OTS at the stage of genesis. We demonstrated that GRP has a bunch of breaking features and, in our opinion, deserves to be thought as a hallmark separating two very dissimilar phases in OTS evolution. Perhaps, accounting evident change in topology, GRP could be called a dynamic topological phase transition point as it was discussed in [76, 77]. Since at agenesis OTS encounters strong influence of incoming energy flow yin , maintenance of its energy structure requires the highly efficient energy-saving mechanisms (factor S C ) leaving significantly less energy on the variability endeavors (factor S T ), i. e. holding S C ≫ S T . Further, outcoming flux yout gradually becomes bigger, finally reaching some parity with yin in vicinity of GRP which is compliant with the state of dynamic balance. In this sense, effective management of outward energy streaming is regarded one of the main merits of orderly advanced system. Staying compliant with abovesaid on eminent segregation between two stages of OTS evolution, we think that it would be reasonable and physically warranted to assume existence in different evolutionary stages the different kind of an energy utilization mode. Namely, the collective mode in agenesis and the individual one (EEF) in genesis. Above assertion acquires more sense if we acknowledge that collectivity in energy organization is relied on the low accuracy of copying between adjacent forms because of the infinitesimal y−difference between them. Then, it looks like that collective form is the only acceptable solution to evolve in an agenesis. On the other hand, the higher accuracy of copying in genesis naturally matches the finite y -distance between the separated EEFs. We think that an evolution meaning of (1.16) for function Υ could be understood in the terms of some accumulative or collective memory. Thinking this way, we mean that collective memory, like Υ, “writes” and “keeps” all the attempts, successful and unsuccessful ones. With each consecutive even unsuccessful attempt, OTS becomes more and more ready to contribute to fastening of some preferences thereby moving it to the next level of complexity. Especially stress that on the one hand, the most stable EEFs (spectral modes) should have the maximum ratio S T /S C . On the other hand, the highest accuracy of copying (S C ) between the predecessor and the successor EEF is achieved at the y -values matching the eigen values of energy exchange yn . So, emerged EEF is a result of compromise and balance between necessity of the highest accuracy of coping and the highest variability.

7.4

INDIVIDUAL AND COLLECTIVE EEF

63

Stationarity point (SP) at yn→∞ = 1 (point Cn in Fig. 7.3) keeps its usual thermodynamic meaning and corresponds to OTS state in a complete equilibrium with its environment [78] when energy of system does not change, however entropy and energy fluxes can change. The terminal point (TP) at y = e designates the very last y -point behind which solution does not exist. It comes that approaching TP the disordering processes in OTS prevail leading to breaking of EEF structure and, finally, termination of OTS as a separate energy entity.

Fig. 7.3. Family of EEF for n = 1, 2, 3 and n → ∞. Designations of the axes are the same as in the Fig. 7.2. It is seen that triangular EEFs are not similar, instead they are obeyed to equation of OTS energy balance (7.3) claiming predetermined character of changes.

Following this way, it is possible to comment out an existence of some distance (in y -terms) from SP as essential requirement for OTS functioning. The maximum distance is SP − GRP (at y < 1) and T P − SP (at y > 1), but the minimum one could not be defined exactly. Existence of such minimum stems from the fact that staying around SP, evolutionary states of OTS lack its unique identity. It happens due to decreasing of separating distance between individual EEFs along y (separation distance approaching 0 at n → ∞) still keeping the high accuracy of copying. All this makes individual EEFs practically indiscernible.

64

DYNAMICS OF EVOLUTIONARY TRIANGLES

CH. 7

7.5. Final remarks In this report, we presented analytical model for energy evolution of OTS based on the infinite number of the energy links with a thermal bath. We knowingly established an infinite number of the links, i. e. energy sources and sinks considering that such condition is critical in evolution process. We did it because, in our opinion, the number of energy connections, all sorts of different nature between an object which potentially could become less or more advanced OTS and an external world is what makes evolving entities different from the stagnating ones. Hence, we think that for evolving entities the number of such links should be incommensurably higher than for the non-evolving ones. That is why we replaced classic ECE with the system (1.2) and, finally, RECE (1.5). Doing this, we demonstrated possible way how organization and evolution can happen as a consequence of the flow of energy through matter [79]. And what undisputable role in scenario of transition from the non-evolutionary to evolutionary physics belongs to “energy module” [5]. Also, we showed how the reduction and optimizing of the microstate volume could favor to nonlinearity (forming of resembling but nor similar EEF) and, thereby, an evolution capacity as discussed in [7]. In this model, we used ECE as mathematical model for energy link, which is sufficiently simple. Though, it cannot be ruled out that the class of link equations which allows to obtain convenient analytical description for OTS evolution is actually wider. However, we will not discuss this question here. Our approach is being essentially built around function Υ which (a) by construction, is naturally integrated with the properties of both OTS and total energy exchange flux between OTS and outside world; (b) does depend on the physical path between the successive states in the OTS-surroundings development; (c) related to the direction of OTS evolution. We found that interface function Υ can be considered as a marker in a lifetime cycle for energy development of OTS. Based on above, we exhibited how multiple energy recharges (looping) of function Υ may activate evolution potential of OTS. As we have already underlined, an evolution ability of discussed model, in particular, is based on the probabilistic anisotropy of OTS interface for energy fluxes yin and yout (2.14). Observe that (2.14) looks identical in its mathematical form to the results [80] where term ln [π (II → I)/π (I → II)] establishes macroscopic compliance with the second law of thermodynamic, and the quantities π (II → I)

7.5

FINAL REMARKS

65

and π (I → II) determined as the reverse and forward transition probabilities. Our results support position [81–83] that system’s native ability to adapt to changes in environment goes through suppressing big fluctuations around critical points (phase transition points y = yn in this model) and dissipating abundant energy (yout in this model) and reflect the well-known property of OTS to operate close to the phase transition point keeping away from equilibrium [10]. So, the driving force of evolution we discussed earlier, creates general probabilistic background which makes complexification and evolution at some conditions unavoidable. Author [83] comes to close conclusion stating that the adaptation feature of systems “...may be embedded more deeply in the thermodynamics of complex systems”. It is worth to note that in this model genesis is different from agenesis, first of all, by emergence of new qualities in functioning of OTS (listed in the section 6.3.1 for GRP) that were not presented before. It is quite aligned with [6] where is stressed that emergence of quality differences for OTS is identifiable not by the new parts but novelty of operation. As fairly noted in [84], a detailed quantitative understanding of evolution would flesh out the balance between evolvability and robustness (S T and S C in this model). Personally, we strongly support abovesaid, and are confident about the critical importance of keen insight into this balance for attainment meaning of evolution. Also, highlight here that all distinguishing differences of a genesis compared agenesis, listed above can come only altogether, as one united ensemble. If we take off any of these features, it immediately breaks a whole pattern of this stage. Exactly the same should be said for the stage of agenesis.

REFERENCES

1. Gayon J. Defining Life: Synthesis and Conclusions. Origins of Life. 2010. Vol. 40. Iss. 2. P. 231-244. 2. Cleland C.E. and Chyba C.F. Defining “Life”. Origins of Life. 2002. Vol. 32. Iss. 4. P. 387-393. 3. Pross A. How Does Biology Emerge From Chemistry? Origins of Life. 2012. Vol.42. Iss. 5. P. 433-444. 4. Dibrova D.V., Chudetsky M.Y., Galperin M.Y. et al. The Role of Energy in the Emergence of Biology from Chemistry. Origins of Life. 2012. Vol. 42. Iss. 5. P. 459-468. 5. Adami C. Information-Theoretic Considerations Concerning the Origin of Life. Origins of Life. 2015. Vol. 45. Iss. 3. p. 309-317. 6. Fleischaker G.R. Origins of Life: An Operational Definition. Origins of Life. 1990. Vol. 20. Iss. 2. P. 127. 7. Michel D. Life is a Self-Organizing Machine Driven by the Informational Cycle of Brillouin. Origins of Life. 2013. Vol. 43. Iss. 2. p. 137-150. 8. Brillouin L. Am. Sci. 1950. Vol. 38. p. 594. 9. Szilard L. On the decrease of entropy in a thermodynamic system by the intervention of intelligent beings. Behav Sci. 1964. Vol. 9. Iss.4. p. 301-310. 10. Schrodinger E. What is Life (Cambridge University Press, Cambridge,1944). 11. Smith J.M. The Problems of Biology (The Problems of Science). Oxford University Press: Oxford. 1986. 144 pp. 12. Ganti T. The Principles of Life/ 2003. Oxford University Press, Oxford. 201 pp. 13. Weber B.H. What is life? Defining life in the context of emergent complexity. Origins of Life. 2010. Vol. 40. Iss.2 P. 221-229. 14. Swenson, R. 1989, Emergent attractors and the law of maximum entropy production: Foundations to a theory of general evolution. Systems Research 6 (No. 3), 187–197. 15. Luisi, P. L.: 2014, The Minimal Autopoietic Unit Origins of Life, 2014. Vol. 44. Iss. 4 P. 335-338. 16. Luisi P.L, Ferri F., Stano P. Approaches to semi-synthetic minimal cells: a review. Naturwissenschaften, 2006. Vol. 93. P. 1-13. 17. Marvelli F., Murazo K.R. Question 8: Bridging the Gap Between In Silico and In Vitro Approaches to Minimal Cells. Origins of Life, 2007. Vol. 37. Iss. 4-5 P. 455-458. 18. Murazo K.R., Pereto J., and Moreno A. A. Universal Definition of Life: Autonomy and Open-Ended Evolution. Origins of Life, 2004. Vol. 34. Iss. 3 P. 323346. 19. Pross A. On the Emergence of Biological Complexity: Life as a Kinetic State of Matter. Origins of Life. 2005. Vol. 35. Iss. 2. P. 151-166. 20. Pine N.W. Fact and assumption in studies on the origins of life. Origins of Life. 1985. Vol. 15. P. 207. 21. Schopf, J. W.: 1983, Earth’s Earliest Biosphere, Its Origin and Evolution, Princeton University Press, Princeton, New Jersey. 22. Lahav, N. FThe synthesis of primitive ‘living’ forms: Definitions, goals, strategies and evolution synthesizers. 1985. Vol. 16. P. 129. 23. Muller A.W., Schulze-Makuch D. Thermal Energy and the Origin of Life Origins of Life. 2006. Vol. 36. Iss. 2. P. 177-189.

REFERENCES

67

24. Viswanath, V., Story, G.M., Peier, et al. Nature 2003, Vol. 423, P. 822-823. 25. Schmidt-Nielsen, K. 1990 Animal Physiology: Adaptation and Environment, 4th ed., Cambridge U.P., Cambridge, p. 521. 26. Luisi P.L. 2006 The Emergence of Life: From Chemical Origins to Synthetic Biology, 1st ed., Cambridge U.P., Cambridge, p. 332. 27. Quang A. D. A. and T. D. Hung. Method of Infinite System of Equations for Problems in Unbounded Domains. Journal of Applied Mathematics / 2012, 17 pp. 28. Fristedt B., Gray L. A modern approach to probability theory.Boston: Birkh¨auser, 1996. 29. Sobolev V.I. Convolution of functions, in Hazewinkel, Michael, Encyclopedia of Mathematics,Springer, 2001, ISBN 978-1-55608-010-4. 30. R. Roy, F.W.J. Olver. "Lambert W function", in F.W.J. Olver. D.M. Lozier, R.F.W. Boisvert, NIST Handbook of Mathematical Functions, Cambridge University Press, 2010, ISBN 978-0521192255,MR 2723248. 31. Prigogine, I. From Poincar´e’s divergences to quantum mechanics with broken ¨ Naturforschung 52a, 37 – 47, 1997. time symmetry, Zeitschrift fur 32. Petrosky, T. and Rosenberg, M. Microscopic non-equilibrium structure and dynamical model of entropy flow, Foundations of Physics 27 (2), 239-259, 1997. 33. Levermore D., Advanced Calculus: MATH: MATH 410 Riemann Integrals and Integrability, 2010, http://www.terpconnect.umd.edu/∼lvrmr/2010-2011F/Classes/MATH410/NOTES/Riemann.pdf. 34. Gibbs J.W., Elementary Principles in Statistical Mechanics Dover Publication, New York, 1960. 35. Bazarov I.P. Thermodynamics. 5 ed. in Russian. — Lany: Sankt-Peterburg — M. — Krasnodar. — 2010. — 384 pp. 36. Marsh C., Introduction to Continuous Entropy, Department of Computer Science, Princeton University, [email protected], 37. Levin B.P. Theoretical basics of statistical radio engineering. 3-rd revised edition, Moscow.: Radio and communication, 1989. 656 pp. 38. Protter M.H., Morrey C.B. Jr. (1985). Differentiation under the Integral Sign. Intermediate Calculus, 2nd ed. New York: Springer. pp. 421–426. 39. Yaghmaee MS., and B. Shokri. 2007. Effect of size on bulk and surface cohesion energy of metallic nano-particles. Smart Materials and Structures. 16, (2): 349-354. 40. Firat G, and A. Sisman. 2013. Quantum forces of a gas confined in nanostructures. Physica Scripta. 87 (4), 045008. 41. Guo W, and C. Zhu. 2006. Isothermal atomistic simulations of nanoelectromechanical systems. Nanotechnology. 17 (5): 1370-1375. 42. Miller HJD., and J. Anders. 2018. Energy-temperature uncertainty relation in quantum thermodynamics. https://arxiv.org/pdf/1801.08057.pdf. 43. Kim S., A. Haque, J.Kim 2017 et al. Laminar heat transfer with alumina nanofluid in cu tube with different diameters. Journal of Physics: Conference Series 885, conference 1. pp.1-7. 44. Beemkumar N, Karthikeyan A., Saravanakumar B. et al. 2017. Performance Improvement of Energy Storage System with nano-additivesin HTF. IOP Conference Series: Materials Science and Engineering. 197 (1): 1-6. 45. Maurer H.R, and H. Kersten. 2017. On the heating of nano- and microparticles in process plasmas. Journal of Physics D: Applied Physics. 44 (17), 174029. 46.Wingert P.A, H. Mizukami, A.E. Ostafin. 2007. Enhanced chemiluminescent resonance energy transfer in hollow calcium phosphate nanoreactors and the detection of hydrogen peroxide. Nanotechnology, 18 (29), 295707.

68

REFERENCES

47. Jacobs K, AN. Jordan, EK. Irish. 2008. Energy measurements and preparation of canonical phase states of a nano-mechanical resonator. EPL. 82 (1), 18003. 48. Szukiewicz B, U. Eckern, KI. Wysokinski. 2016. KI. Optimisation of a three-terminal nonlinear heat nano-engine. New Journal of Physics, 18, 023050. 49. Krishna J., PS. Kishore, B. Solomon. 2017. Experimental Study of Thermal Energy Storage Characteristics using Heat Pipe with NanoEnhanced Phase Change. Materials, IOP Conf. Ser.: Mater. Sci. Eng. 225, 012058. 50. Artamonov A., and E. Plotnikov. 2017. Thermodynamic and relativistic uncertainty relations. International Conference on Particle Physics and Astrophysics, IOP Publishing, Conf. Series: Journal of Physics: Conf. Series. 798 012098, doi:10.1088/17426596/798/1/012098. 51. Uffink J., and van JH Lith-van. 1999. Thermodynamic uncertainty relations. Found. Phys. 29 (5): 655-692. 52. Bohr N. 1985. Collected Works, edited by J, North-Holland, Amsterdam, 6, pp. 316–330, 376–377. 53. Gilmore R. 1985. Uncertainty relations of statistical mechanics. Phys. Rev. A. 31(5): 3237–3239. 54. Shalyt-Margolin A. E., and A. Ya. Tregubovich. 2004. Deformed Density Matrix and Generalized Uncertainty Relation in Thermodynamics. Mod. Phys. Lett. A. 19(1): 71–81. https://doi.org/10.1142/S0217732304012.812. 55. Cooper A. 1984. Protein fluctuations and the thermodynamic uncertainty principle. Progress in Biophysics and Molecular Biology. 44 (3):181-214. 56. Wilk G., and Z. Wiodarczyk. 2002. Application of nonextensive statistics to particle and nuclear physics. Physica A: Statistical Mechanics and its Applications. 305 (1-2): 227-233. 57. Lindhard J. 1986. Complementarity’ between energy and temperature, in The Lesson of Quantum Theory. edited by de Boer J, Dal E, Ulfbeck O., 99-112. North-Holland, Amsterdam. 58. Faigon A. 2007. Uncertainty and Information in Classical Mechanics Formulation. Common Ground for Thermodynamics and Quantum Mechanics. https://arxiv.org/abs/quant-ph/0311153. 59. Proesmans K., and C. van den Broeck. 2017. Discrete-time thermodynamic uncertainty relation, EPL. 119 (2): 20001. 60. Gingrich TR., JM. Horowitz, N. Perunov at al. 2016. Dissipation Bounds All Steady-State Current Fluctuations. Phys. Rev. Lett. 116 (12): 120601. DOI: 10.1103/PhysRevLett.116.120601 61. Pietzonka, P., A. Barato, and U. Seifert. 2016. Universal bounds on current fluctuations Phys. Rev. E 93, 052145. 62. Nalewajski, R. 2016. On entropy/information continuity in molecular electronic states. Molecular Physics. 114 (1): 1225-1235. 63. Scardigli F. 1999. Generalized Uncertainty Principle in Quantum Gravity from Micro-Black Hole Gedanken Experiment. https://pdfs.semanticscholar.org/0dc7/1bf6ee77d5971c5f4c233c3c726d84bc0301. 64. Fannes M. 1973. Continuity property of the entropy density for spin lattice systems. Commun. Math. Phys. 31 (4): 291–294. 65. Audenaert KMR. 2007. A sharp continuity estimate for the von Neumann entropy. J. Phys. A Math. Theor. 40 (28): 8127-8137. 66. Haug, H, and S. Koch. 1994. Quantum Theory of the Optical and Electronic Properties of Semiconductors. World Scientific Pub Co Inc. 492 pp., ISBN-10: 9810218648, ISBN-13: 978-9810218645. 67. Oksendal, B. 2003. Stochastic Differential Equations: An Introduction with Applications. 377 pp. Berlin: Springer. ISBN 3-540-04758-1

REFERENCES

69

68. Hsieh, J. S. 1975. Principles of Thermodynamics. Washington, D.C.: Scripta Book Company. 69. Hazewinkel, M. Taylor formula. Encyclopedia of Mathematics. Springer Science+Business Media B.V. / Kluwer Academic Publishers, 2001, ISBN 978-1-55608010-4. 70. Wilk G., and Z. Wiodarczyk. 2002. Application of nonextensive statistics to particle and nuclear physics. Physica A: Statistical Mechanics and its Applications. 305 (1-2): 227-233. 71. I Bogomolny, A. "Golden Ratio in Geometry." http://www.cut-theknot.org/do_you_know/GoldenRatio.shtml. 72. W. Dunham, Journey through Genius: The Great Theorems of Mathematics. New York, Wiley, pp. 113-132, 1990. 73. Moldavanov A. Classical Right-Angle Triangles and Golden Ratio, Forum Geometricorum, 17 (2017) 433—437. 74. Moldavanov A. Non-Trivial Invariance of Predecessor-Successor Ratio Under Planar Rotation-Similarity Transformation. Int. Electron. J Geom. 2018. Vol. 7. Iss. 2. p. 66-71. 75. de Vries M. The Whole Elephant Revealed: Insights Into the Existence and Operation of Universal Laws and the Golden Ratio. John Hunt Publishing, 2012. 403 pp. 76. Santos F.A.N., Raposo E.P., Coutinho-Filho M.D., Copelli M., et al. Topological phase transitions in functional brain networks. Phys. Rev. E 89, 2019. 77. Vaikuntanathan S., Gingrich T.R.„ Geissler P.L. Dynamic phase transitions in simple driven kinetic networks. Phys. Rev. E 89, 2014. 78. Bejan A. Life and evolution as physics. Commun Integr Biol. 2016. Vol. 9. Iss. 3. 79. Fox H., Bolton W. Mathematics for Engineers and Technologists. Butterworth-Heinemann, 2002, p. 30. ISBN 978-0-08-051119-1. 80. England J.L. Statistical physics of self-replication. J. Chem. Phys. 2013. Vol. 139. p. 121923. 81. Haken H. Synergetics, an Introduction: Nonequilibrium Phase Transitions and Self-Organization in Physics, Chemistry, and Biology", 3rd ed. — 1983. — New York: Springer-Verlag. — pp. 356. 82. Mora T., Bialek W. Are Biological Systems Poised at Criticality? Journal of Statistical Physics. 2011. Vol. 144. Iss. 2. pp. 268-302. 83. England J.L. Dissipative adaptation in driven self-assembly. Nature Nanotechnology.2015. Vol. 10. Iss. 11. pp. 919-923. 84. Lenski R., Barrick J., and Ofria C. Balancing Robustness and Evolvability. PLoS Biol. 2006. Vol. 4. p.2190. 85. Andrei Moldavanov, Randomized Continuity Equation Model of Energy Transport in Open System, DEStech Transactions on Engineering and Technology Research 2017. Proceedings of the 2017 2nd International Conference on Applied Mathematics, Simulation and Modelling (AMSM2017). 86. Park, Sung Y.; Bera, Anil K. Maximum entropy autoregressive conditional heteroskedasticity model, Journal of Econometrics (Elsevier) 150 (2): 219–230. 2009, doi:10.1016/j-jeconom.2008.12.014.

Научное издание

MOLDAVANOV Andrei TOPOLOGY OF ORGANIZED CHAOS

МОЛДАВАНОВ Андрей Васильевич ТОПОЛОГИЯ ОРГАНИЗОВАННОГО ХАОСА Верстка выполнена в издательстве «Физматкнига» Оператор А. В. Елфимов Издательство «Физматкнига». 141701, Московская область, г. Долгопрудный, ул. Первомайская, д. 3А. Тел. (499) 390-51-38. Подписано в печать 15.07.2020. Формат 60×90/16. Бумага офсетная. Печать цифровая. Усл. печ. л. 4,25. Уч.-изд. л. 4,4. Тираж 100 экз. Отпечатано в АО «T8 Издательские технологии» 109316, г. Москва, Волгоградский пр-кт, д. 42, корп. 5