Thermodynamic Bases of Biological Processes: Physiological Reactions and Adaptations 9783110849974, 9783110114010

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Thermodynamic Bases of Biological Processes Physiological Reactions and Adaptations

Α. I. Zotin

Thermodynamic Bases of Biological Processes Physiological Reactions and Adaptations

W DE G_ Walter de Gruyter · Berlin · New York 1990

Author Professor Dr. A. I. Zotin Institute of Developmental Biology Academy of Science of the U.S.S.R. 26, Vavilov Street Moscow 117 334 U.S.S.R.

Library of Congress Cataloging-in-Publication

Data

Zotin, A. I. (Aleksandr Il'ich), 1926 — Thermodynamic bases of biological processes : physiological reactions and adaptations / A. I. Zotin. Introd. Includes bibliographical references. ISBN 3-11-011401-1 : - ISBN 0-89925-383-0 1. Thermodynamics. 2. Biochemistry. I. Title. QP517.T48Z68 1990 574.19Ί6 —dc20 89-25854 CIP

Deutsche Bibliothek Cataloging in Publication Data

Zotin, Aleksandr I.: Thermodynamic bases of biological processes : physiological reactions and adaptations / A. I. Zotin. — Berlin ; New York : de Gruyter, 1990 ISBN 3-11-011401-1

® Printed on acid free paper. © Copyright 1990 by Walter de Gruyter & Co., D-1000 Berlin 30.

-

All rights reserved, including those of translation into foreign languages. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. Printed in Germany. Typesetting: Arthur Collignon G m b H , Berlin — Printing: Gerike GmbH, Berlin — Binding: Lüderitz & Bauer GmbH, Berlin.

Dedicated to my wife, Zotina Rimma Sergeevna. I am thankful for her assistance and decisive participation.

Remark

This is the first book in a series of monographs, where we endeavour to use the 'Thermodynamics of Irreversible Processes' as the bases for theoretical biology. It includes the description of the steady state of organisms and their reactions to external effects and internal signals. The principle of minimum energy dissipation serves as the basis for a discussion of phenomenological thermodynamics in this book, including linear and nonlinear processes and the processes proceeding in organized systems and living organisms. The basal and standard metabolism of organisms and their dependence on body mass, their taxonomic position and different environmental factors, as well as the problems concerning homeostasis and homeorhesis stability and reliability are considered in the book. Transition and adaptation processes as the response to external effects and under pathological processes, as well as active and maximal metabolism are discussed in detail. Special consideration is given to the problem of the discrepancy between data from direct and indirect calorimetry in ontogenesis, when doing work, when reacting to a damaging effect, diseases, etc. The monograph will be useful for biophysicists, physiologists and developmental biologists, as well as for physicists who take an interest in modern thermodynamics and its applications in biology. It may also be useful for those occupied in medicine and for research into problems of labour and sports.

Preface

Wide use of thermodynamic concepts and methodology for the description of biological phenomena dates back to Helmholtz and Boltzmann (see Lazarev, 1945; Geifer, 1973; Broda, 1975). However classical thermodynamics cannot serve as a basis for understanding biological processes because it deals with equilibrium systems and situations only; biological processes are, to a large extent, nonequilibrium processes. The emergence of a thermodynamics of nonequilibrium processes, therefore, provided a great impetus for the introduction of thermodynamics into biology. It was used to describe certain physiological processes as well as more general biological phenomena related to organism development and evolution (Katchalsky, Curran, 1965; Antonov, 1968; Zotin, 1972, 1974; Nikolaev, 1976; Caplan, Essig, 1983; Rubin, 1984; Brooks, Wiley, 1986, etc.). This monograph is devoted to consideration of the thermodynamic basis of such physiological phenomena as stability and reliability of organisms, their responses to external and internal signals, their adaptation to the changing environment and to diseases. We would like to underline that it is thermodynamic basis of physiological processes this monograph deals with and not the proper thermodynamics of these phenomena. The creation of a thermodynamic theory of physiological change would be presumptious. The point is that thermodynamics in itself is not yet well enough developed as a science to allow a thermodynamic theory of physiological processes. At present, in addition to classical thermodynamics, only the thermodynamics of linear irreversible processes, i. e. processes in systems close to equilibrium, has been developed. The thermodynamics of nonlinear irreversible processes is in an early stage of development and is abundant with unsolved problems. Also, living organisms are not only considerably nonlinear, but are also organized systems, i.e. systems with regulation and control processes. Despite the fact that a new trend in thermodynamics, known as the thermodynamics of information processes, has come into being (Poplavsky, 1981) this is still a far way from creating a thermodynamic basis for describing biological phenomena. We also must not forget that living organisms are characterized as open thermodynamic systems that have many properties that have been thoroughly studied. This book is aimed at demonstrating that even though living organisms specifically react to external and internal effects, what actually lies behind these reactions is their property as open systems. Eventually all specific qualities of their reactions can be confined to slight modification of a typical open system reaction subject to the change of external or internal parameters. Some cases permit a thermodynamic solution to be found for purely physiological problems as, e. g., an account for the

X

Preface

influence of temperature on energy metabolism in Poikilothermie animals, or the reasons for the discrepancy between data from direct and indirect calorimetric studies of organism growth and development, consideration of physiological reactions and pathological processes. But as a whole the thermodynamic theory of physiological processes is far from being complete. The attempt by the author to examine the thermodynamic basis of some physiological reactions and processes refers mainly to the use of methods and relations accepted in phenomenological thermodynamics of nonequilibrium processes. This book does not view (or only to a very small extent) the problems of statistical thermodynamics and synergetics. This does not mean that these scientific trends are less important for theoretical biology — the point is that they have some complicated problems of their own which might obscure a more clear and simple approach of phenomenological thermodynamics. Along with this I would like to warn readers that the phenomenological thermodynamics of nonequilibrium processes, especially when it comes to describing nonlinear irreversible processes and processes in organized systems, is at present so undeveloped that every author is compelled to take his own subjective view-point; this will also be referred to this monograph. This monograph will be useful for those acquainted with thermodynamics or having some experience in reading books on thermodynamics. It is essential at the same time to bear in mind that this book was written by a biologist, therefore the material is presented in a manner which will be most convenient to suit the real task, that of describing biological processes. This might cause difficulties with readers not equipped with a profound knowledge of biology. The only way to overcome these difficulties is to read widely and around every problem raised in this book. I, on my part, have done my best to survey the largest possible amount of literature on the biological problems touched upon in this book. As the literature on the subjects discussed in this book is huge I can not guarantee that I have not missed an important and probably even decisive research paper. I would like to apologize to the authors of these important researches, as well as to the readers for my involuntary omissions. In 1974,1 was invited by Professor I. Lamprecht to the Institut für Biophysik of the Freie Universität in Berlin to deliver a few lectures on the thermodynamics of biological processes. Since that time I. Lamprecht, Β. Schaarschmidt and myself have discussed the problems raised in this book more than once in the course of my work in the Institute in 1974 and again in 1980 and during their stay in Moscow and Tbilisi. As a result of our friendship and cooperation a few articles and three collective monographs on the thermodynamics of biological processes (Lamprecht, Zotin, 1978, 1983, 1985) have been published. I am very thankful to my friends from this Institute and especially to Prof. Lamprecht for the spirit of amiability which permeated all our meetings and work together. In the course of my work on this book I have been constantly discussing the problems brought up in it with my friends and colleagues from the Laboratory of

Preface

XI

Developmental Biophysics of the Institute of Developmental Biology, Academy of Sciences of the USSR — T. A. Alekseeva, V. V. Gavrilov, S. Yu. Kleimenov, I. S. Nikolskaya, N. D. Ozernyuk, E. V. Presnov, E. A. Prokofiev, L. I. Radzinskaya and I. G. Vladimirova. In the preparation of the manuscript special help was rendered by I. G. Vladimirova. I am grateful to Β. N. Manukhin and R. P. Poplavsky who made valuable comments which, for most part, I tried to take into account in writing the final version of this book. I am also much indebted to Prof. T. M. Turpaev, director of Institute of Developmental Biology, for his generous support of the research work carried out in our Laboratory and for willing discussion of many problems brought up in this monograph. Translated from Russian by A. I. Mazurova.

Contents

Designation

XVII

1 Thermodynamics of Nonequilibrium Processes 1.1 Linear irreversible processes 1.1.1 Basic notions and relations Phenomenological equations Rate of entropy production Stationary state 1.1.2 Variational principles 1.1.3 Transition processes Evolution criterion Kinetics of relaxation processes 1.2 Nonlinear irreversible processes 1.2.1 Nonlinear phenomenological equations 1.2.2 Dissipation functions 1.2.3 Principle of minimum energy dissipation 1.2.4 Transition processes Unsteady transition processes Steady transition processes 1.3 Irreversible processes in organized systems 1.3.1 Orderliness and organization 1.3.2 Maxwell's demon and negentropy effects Negentropy principle of information Information processes 1.3.3 Organized systems Principle of minimum energy dissipation for organized systems Steady state of organized systems Stability of irreversible processes in organized systems

1 2 2 3 5 9 11 14 14 16 19 21 23 27 31 33 38 41 42 44 45 47 50 50 52 54

2 The Stable State of Organisms 2.1 Energy metabolism 2.1.1 Basal and standard metabolic rates Standard metabolic rate rhythms Torpor and hibernation metabolic rate Negentropy effects

57 58 59 60 63 65

XIV

Contents

2.1.2 Standard metabolic rate and size of organisms Allometric relations Rubner-Richet's surface rule 2.1.3 Influence of temperature on standard metabolic rate Poikilothermie animals Homeothermic animals 2.2 Homeostasis and homeorhesis 2.2.1 Stability of organisms and homeostasis reliability Homeostasis reliability criteria Reliability of thermoregulation 2.2.2 Biological process stability and homeorhesis reliability Criteria of homeorhesis reliability New method for the determination of optimal conditions for rearing animals Reliability of development and growth processes 2.3 Stable state evolution 2.3.1 Criteria of evolutionary progress Orderliness criterion Criteria of organization 2.3.2 Energy metabolism and animal taxonomy Standard metabolic rate in protozoa Standard metabolic rate in mammals 2.3.3 Directions of evolutionary process Aromorphosises and progressive evolution Idioadaptations and morpho-physiological regress 3 Transition Processes and Adaptations 3.1 Organism reactions to external affects 3.1.1 Inducible pulsed reactions Reactions to external signals Reactions to external affects 3.1.2 Inducible adaptive processes Adaptation to temperature Adaptation to hypoxia Adaptation to physical work 3.1.3 Adaptation mechanisms of energy metabolism Inducible pulsed reactions Inducible adaptive processes 3.2 Organism reactions to internal signals 3.2.1 Active metabolism Energetic cost of movements

69 70 77 80 81 86 98 99 100 103 106 106 109 112 114 117 117 120 122 124 126 131 132 138 141 143 144 144 147 155 156 166 169 170 170 173 176 177 179

Cost of excitation and control

185

Physical and mental work

189

Contents 3.2.2 Maximal metabolism Maximal metabolism and body mass Metabolic diapason 3.2.3 Pathological processes and diseases 3.3 Bound dissipation function and physiological processes 3.3.1 Development process Animal development and growth Insect metamorphosis Plant development and growth 3.3.2 Physiological reactions and pathological processes Physiological reactions Affect of environmental factors Diseases 3.3.3 Psiumetry — a new method of analyzing the physiological state of organisms Physical sense of the psiufunction Field of psiumetry application

XV 192 192 194 202 204 207 207 214 217 220 220 223 227 230 230 233

4 Concluding remarks

237

5 References

239

6 Subject Index

289

Designation

Thermodynamics : S U H G Τ V nk μ* vr ξΓ Ar hr r®V d¡ S de S Xj Ij ïj Ly t ρ (α) Pst σ υ ψ ψ8( yd ψα ψΟΓ ψΝ v|/(b) ψ(ί) ψ(3)

— entropy — internal energy — enthalpy — Gibbs free energy — absolute temperature — volume — moles amount of the substance k — chemical potential — stoichiometric coefficient — extent of reaction completeness — chemical affinity — partial molar enthalpy — heat effect of β reaction — entropy production — entropy flow — thermodynamic force — thermodynamic flow — specific thermodynamic flow — phenomenological coefficients — time — probability density of deviation in parameter α from their equilibrium value — probability density in stationary state of the system — local rate of entropy production — Onsager-Machlup function — specific dissipation function — specific dissipation function in stationary state — external dissipation function (psidfunction) — bound dissipation function (psiufunction) — external dissipation function of organized system — dissipation function linked with negentropy effect — dissipation function in steady state of organism (standard metabolism) — dissipation function of inducible pulsed reaction — dissipation function of inducible adaptive process

XVIII \|/(k) A (t) Δ SN Δ ST η

Designation — dissipation function of constitutive process (process of development and growth) — generalized measure of system aging — entropy change linked with the useful control effect — entropy change in the medium (thermostat) — entropy efficiency coefficient.

Energy metabolism: Mb n Ν

—

body mass of animals number of measurements number of species heat production rate heat production intensity oxygen consumption rate

j)]xj = Σ j - ^ [ σ (jj> xj)1 xj -

[Φ C)j, jk)]Xj| 5 j k .

As follows from (1.28) and (1.32) 8σϋ,χ) _ v . — xkj

8(pQ,j)

0Jk

0Jk

_ — Xk,

therefore δ[σΰ,χ) -

j)] = max

where υ is the Onsager-Machlup function (Machlup, Onsager, 1953; Tisza, Manning, 1957). Gyarmati (1970) arrived at the conclusion that Onsager's principle of the least dissipation of energy is equivalent to Prigogine's principle (theorem) of minimum entropy production in a stationary state. This assumption in fact is also present in a series of other works (Wergeland, 1951; Haase, 1963; Zhuravlev, 1979). Let the flow of entropy through the surface of a closed system be equal to s* = - J

IqdQ

14

1. Thermodynamics of Nonequilibrium Processes

where I Q is the heat flow through the surface element Ω. Considering (1.27) and (1.28) we may write g = S*.

(1.38)

In a stationary state the variational principle (1.34) takes the following form: [s* - (p(j,j)] = max. The heat flow through the surface of the system in a stationary state is constant j I q d Q = const Ω

and this is why from (1.38) we draw the condition of a stationary state as 0 V dt j=i

15

(1-40)

where Ij are specific thermodynamic flows. In accordance with Prigogine's theorem then

dt

(1.41)

The inequality (1.41) is an evolution criterion for a thermodynamic system as it shows the direction of change in the system. This inequality means that in the process of system relaxation to the stationary or equilibrium state the specific dissipation function is constantly decreasing, and in a stationary state, reaches its minimum, namely ψ = min

(1-42)

i. e. in a stationary state the variational principle holds good: δψ = 0.

(1.43)

The variational principle (1.43) follows from the evolution criterion (1.41) which in turn is based upon inequality (1.9). Thus, the Second Law of thermodynamics as formulated by Prigogine (1.9) is equivalent to Onsager's principle of least dissipation and to Prigogine's theorem. Therefore, not only (1.9) but also the variational principle (1.43) can be accepted as generalized forms of the Second Law of thermodynamics. Earlier we introduced the notion of flows and forces (1.23) by considering a system which deviated from the equilibrium state. In this case these notions are justified only for isolated systems. To characterize the stationary state of an open system it is essential to use statement (1.28) or (1.40). Then there question arises of defining the forces and flows for open systems which deviate from the stationary state. A number of authors have shown that the ideas of flows and forces can be introduced rather strictly for any dissipative system, including nonlinear open systems (Edelen, 1972, 1973, 1974 a, b; Presnov, 1973, 1978; Bataille et al., 1983). The work of Bataille et al. (1983) has a special interest in this respect for it demonstrates that presenting thermodynamic flows in the form of (1.1) follows from inequality (1.28) or (1.40) and is a particular case of a more general definition of flows. We shall not repeat here these rather complicated mathematical computations, but shall resort to some simpler explanations which are more pertinent to this work. We may get the expression of flows and forces for the open systems in the same way as was done for isolated systems, i.e. repeating again conclusions (1.22) to (1.24) and substituting Δ ψ instead of AS. The meaning of flows and forces will be somewhat different because in the case of system deviation from a stationary state

16

1. Thermodynamics of Nonequilibrium Processes

it will no longer be the entropy difference that plays the role of the potential, but the difference in the dissipation function. In this case forces characterize a strain which emerges in the system as a result of deviation from the stationary and nonequilibrium state. In other words the open system which deviates from the stationary state is not only under the influence of forces which tend to bring the system closer to equilibrium but is also under the influence of forces tending to bring it into a stationary state. The existence of two types of flows and forces analogous to the above clearly follows from the work of Bataille et al. (1978, 1980). The two time scales described by Prigogine (1959, 1967) as a condition for the emergence of a stationary state are therefore linked through the existence of two groups of forces affecting an open system.

Kinetics of relaxation processes Expressions (1.41) and (1.42) are criteria for evolution of a thermodynamic system. At the same time they do not enable us to quantitatively describe transition relaxation process kinetics, but only indicate the direction of thermodynamic system evolution. Hence we need some other conditions to establish kinetic equations for open system changes. The description of relaxation process in linear systems follows from rather simple considerations (Zotin, 1972). Thus A. F. Ioffe considering the problem of the correspondence of entropy and time in 1911 put forward an idea that "extraordinary events vanish the sooner the more extraordinary they are" (see Pokrovsky, 1914). The deviation of a thermodynamic system from the stationary state is a case of an extraordinary event as mentioned by Ioffe. That is, the more the system deviates from the stationary the more the rate of elimination of this deviation should be. In other words -

dt

= λΔψ

(1.44)

where Δ ψ = ψ — ψ8„ ψ5( is a specific dissipation function in a stationary state. Integrating (1.44) results in ψ = ψ81[1 + C e x p ( - À t ) ] .

(1.45)

Equation (1.45) can be drawn from more traditional thermodynamic considerations (Beier, 1962; Katchalsky, Curran, 1965; Rubin, 1984). Again in this case the exponential relaxation of the system to a stationary state is justified only where the thermodynamics of linear irreversible processes can be applied. Equation (1.45) may be also derived starting from the statistical meaning of the dissipation function (Zotin et al., 1975; Zotin, Zotina, 1977; Zotina, Zotin, 1983). It is a well-known fact (Glansdorff, Prigogine, 1971; Zalevski, 1973; Gurov, 1978)

1.1 Linear irreversible processes

17

that the emergence of fluctuation in an isolated equilibrium system is described by Einstein's equation: p = Cexp— k

(1.46)

where ρ (a,,..., a n ) is the probability density of deviation in parameters oci,...,a n from their equilibrium values a ? , . . α £ ; Δ S = S0 — S; S 0 the value of entropy at the equilibrium k the Boltzmann constant. If we transform (1.46) and take the time derivative, we shall have dAS dt

k dp ρ dt

(1.47)

In some cases Einstein's equation may be used to describe fluctuations in nonequilibrium open systems in a stationary state (Nicolis, Babloyantz, 1969; Nicolis, Prigogine, 1971; Glansdorff, Prigogine, 1971; Nicolis, 1972). In an open system the general entropy change is linked with entropy production inside the system and with entropy flow coming into the system or going out of it. In using equation (1.47) for open systems we must not forget that it may be referred only to entropy production inside the system d A S = d¡AS, i.e. in this case

Ψ

"

Τ djS V dt

Τ djS,t _ Τ d A S V dt ~ V dt

or considering (1.47): Ψ = Ψ* + — 4 f ρ dt

(1-48)

where k' = kT/V. The use of Einstein's equation in nonequilibrium systems has met with some objections (see the discussion Nicolis, 1972; Glansdorff et al., 1974; Keizer, Fox, 1974; Nicolis et al., 1974; Nitzan, Ross, 1974; Kuramoto, 1975; Landauer, 1975). Without dwelling upon the detail of this discussion we must say that in more than one case Einstein's equation is applicable to nonequilibrium systems (Glansdorff, Prigogine, 1971; Nicolis, Prigogine, 1971; Nicolis, 1972; Chakrabarti, 1974, 1975; Kuramoto, 1975). In addition we may arrive at an equation analogous to Einstein's equation if we accept that the process of reaching a stationary state by a nonequilibrium open system is a stochastic Markovian process (Presnov, Zotina, 1978; Zotina, 1986). In this case all the problems connected with the discussion of the applicability of Einstein's equation to nonequilibrium systems are no longer important, but new difficulties arise from the necessity of substituting Α ψ instead of A S in (1.46) and this means introducing new flows and forces which are different from (1.3).

18

1. Thermodynamics of Nonequilibrium Processes

On arriving at (1.47) the probability density was supposed to be the only time function. Equally in Einstein's equation (1.46) this probability function is a function of thermodynamic parameters only. The reason for this is that Einstein's equation describes the fluctuation already present and the probability density in this case is not a function of time. In the case under discussion it is the problem of fluctuation relaxation and the probability density is only a function of time. It is logical to assume (Zotin, Zotina, 1977; Zotina, Zotin, 1983) that 4 Τ = f (P) dt

(1-49)

where function f(p) is infinitely differentiable in the vicinity of the point p st (i.e. in the vicinity of the system stationary state). Consequently we think that the change of probability density is a Markovian process. Applying a Taylor expansion to function (1.49) we have

dt

= ffeO + ^

1 !

(p - PsO + ^

2!

(p - PsO2 + ....

(1.50)

As with ρ = Pst, d p/d t = 0, so f (pst) = 0. If a system has deviated far from a stationary state then in expansion (1.50) may be restricted to a linear member and we then have 4f dt where

= ai(Ps."P)

(1.51)

= -f'(pst).

Introducing (1.51) into (1.48) we have Ψ = Ψ„ + b,

-

l)

(1.52)

where bi = a ( k T/V. The solution for (1.51) takes the form Ρ

=

Pst[l

- B e x p ( - a , t)]

and combining this with (1.52) we have ψ = ψΜ + b,

Bexp(—a^) 1 - Bexp(—att)

As the system is not far from the stationary state then t » 0 and Β exp ( — a, t) « 1. Hence ψ = ψ„[1 + A exp (—ait)]

(1.53)

where A = bj Β/ψ 3ΐ . Equation (1.53) is identical with (1.45). They both demonstrate that a thermodynamic system approaching an equilibrium or stationary state is accompanied by

1.2 Nonlinear irreversible processes

19

an exponential decrease in a specific dissipation function of the system. In the area of the application of the thermodynamics of linear irreversible processes not only will the dissipation function change exponentially but so, in accordance with (1.1) and (1.3), will other parameters of the system.

1.2 Nonlinear irreversible processes The field of application of the thermodynamics of linear irreversible processes is limited to systems close to equilibrium or to a stationary state for which linear laws (1.1) and Onsager's reciprocity relations (1.2) are true. Along with this the bulk of real irreversible processes take place in systems far from equilibrium. That is why now, as well as earlier, efforts have been made to create a theory capable of extending thermodynamic methods to any phenomena including considerably nonlinear phenomena (see de Groot, Mazur, 1962; Prigogine, 1967; Gyarmati, 1970; Glansdorff, Prigogine, 1971; Zotin, 1972,1974; Bakhareva, 1976; Nicolis, Prigogine, 1977; Presnov, 1978; Rubin, 1984). These researches do not always agree with one another and each is developed in its own way. One of the most popular theories nowadays is that which gives primary importance to further development of thermodynamic research into so called dissipative structures (Glansdorff, Prigogine, 1971; Ebeling, 1976; Nicolis, Prigogine, 1977; Polak, Mikhailov, 1983). Dissipative structures present a phenomenon of space or time periodic organization in considerably nonlinear systems. In contrast to orderly equilibrium structures, e. g. crystals, dissipative structures are closely linked with the environment and are kept up at the expense of energy inflow into the system or energy redistribution in the system. As similar situations arise in considerably dissipative systems and are accompanied with intensive energy dissipation they have been given the name of dissipative structures (Prigogine, Nicolis, 1967). Notions analogous to dissipative structures have been introduced into biology previously: the deformed proteins of Bauer (1935), the molecular constellations of Gurwitsch (1945), the dynamic structures of Nikolaev (1976). But the theory of these structures of a similar kind has, however, not been thermodynamically substantiated and nor was there an adequate mathematical apparatus until the work of Prigogine and Nicolis (Prigogine, 1969, 1975; Nicolis, 1971, 1975; Glansdorff, Prigogine, 1971; Nicolis, Prigogine, 1977) following which it has got universal recognition (Ebeling, 1976; Lamprecht, 1978 b; Vasiliev, 1978 a, b; Belinzev, 1983; Polak, Mikhailov, 1983; Romanovsky et al., 1984; Rubin, 1984, etc.). Usually many examples of dissipative structures found in chemistry, physics and biology are described (Lamprecht, 1978 b). Bénard's cells which arise in a thin layer of a viscous fluid warmed from below are typical example of a spacial dissipative

20

1. Thermodynamics of Nonequilibrium Processes

structure. The appearance of hexagonal structures in this case is a result of the transition of random thermal motion to a macroscopically ordered flow of the viscous liquid maintained by the temperature gradient. The Belousov — Zhabotinsky reaction (Zhabotinsky, 1974) is an example of a time-periodic dissipative structure. This reaction consists mainly of the oxidation of malonic acid by potassium bromate under the influence of eerie ions. Oscillations arise in the system which persist for some time and are accompanied by heat fluctuations (Lamprecht, 1978 b; Lamprecht, Schaarschmidt, 1978). Bromate is evidently the source of energy as well as the limit link of the reaction. All the examples of dissipative structures taken from biology may be refered to a molecular level (time fluctuation of glycolysis in non-cellular systems for example), at a cellular level (oscillations in the form of induction and repression in the Jacob and Monod model), at a supracellular level (circadian rhythms in organisms), at a population level (oscillations in the number of individuals in a predator-prey system) (Lamprecht, 1978 b). This list shows that the notion of dissipative structures has lost clarity and to a certain extent its thermodynamic meaning. What unites them at present is only the mathematical apparatus used for their description and referred to the field of the qualitative theory of differential equations. In fact the whole of this research field has formed the now separate branch of science called synergetics by Haken (1973,1978). And though the theory of dissipative structures can evidently contribute much to the consideration of selforganization and pattern formation it can not be referred to the field of phenomenological thermodynamics. Within the framework of phenomenological thermodynamics efforts have been made to create a theory of nonlinear irreversible processes. The main direction of research are the extension of the principle of local equilibrium to nonlinear processes (Prigogine, 1967; Glansdorff, Prigogine, 1971; Nicolis, 1971, 1972; Nicolis, Prigogine, 1977; Rubin, 1984; Schlögl, 1985); creation of variational principles for nonlinear systems (Biot, 1955, 1970; Ono, 1961; Ziegler, 1963; Bakhareva, 1968, 1971, 1976; Gyarmati, 1968, 1969, 1970, 1974; Lavenda, 1974, 1978; Zhuravlev, 1979; Virodov, 1983; Bistriy, 1986); the use of network thermodynamic ideas (Oster, Desoer, 1971; Oster et al., 1971, 1973; Schnakenberg, 1981), use of fluctuationdissipative theorems (Stratonovich, 1967, 1969, 1984, 1985; Platonov, 1983, 1985), use of global characteristics (Edelen, 1972, 1974 a, b; Presnov, 1973, 1978; Zotin, 1974; Lamprecht, Zotin, 1978; Bataille et al., 1983; Presnov, Malyghin, 1983), stochastic consideration of nonequilibrium processes (Bakhareva, Biryukov, 1969, 1970, 1974; Presnov, Zotina, 1975, 1978; Bakhareva, 1976; Timonin, Zotina, 1983; Zotina, Zotin, 1983) etc. The future development of a phenomenological theory of nonlinear irreversible processes is not clear and this is why it is possible to have different approaches in this field including the so called global approach (Presnov, 1978) represented below.

1.2 Nonlinear irreversible processes

21

1.2.1 Nonlinear phenomenological equations Nonlinear phenomenological equations and other nonlinear correlations of thermodynamics can be drawn from Hadamard's lemma (Presnov et al., 1973; Presnov, 1973, 1978). Consider this lemma. If f = f(xi, ...,x n ) is a differentiable function of the second order of class C 2 in the convex vicinity V of the point 0 in the space IR n then there are functions of the class C1 g, = g¡(xi, ...,x n ) defined in V and corresponding to ί ^ , . , . , χ , Ο - f(0) = Σ g¡x¡

(1.54)

where

|^(τχ1>...,τχη) ...,X n ) = Lji (Χι, ..., Χ η ).

(1.60)

Mathematical formalism including nonlinear phenomenological equations and generalized reciprocity relations is also to be found in the work of Edelen and others (Edelen, 1973, 1974 a, b; Bataille et al., 1983; González-Fernández et al., 1984). A similar, though less strict, approach to the problem of nonlinear phenomenological thermodynamics has been worked out earlier by Gyarmati (1968, 1969, 1970, 1977; Lengyel, Gyarmati, 1981) and Vojta (1967). Equality (1.60) is true for any point of space and this means that also f(0) = 0 and f ( X f , ...,Χη) Φ 0 (the point 0, ...,0 = 0 corresponds to the equilibrium state of the system, the point X f , ...,X„ = X st corresponds to the stationary state). Consequently Onsager's reciprocity relation (1.2) is true when the system is close to a stationary state as well as to an equilibrium state. This is a very important

1.2 Nonlinear irreversible processes

23

conclusion for it means that linear laws (1.1) are true not only close to equilibrium but also to a stationary state. Another important conclusion which will be used further is that in the course of the system approaching equilibrium or a stationary state nonlinear phenomenological equations must switch over to linear equations for which Onsager's reciprocity relation is true. This conclusion, based upon the principle of supplement, can serve as one of the criteria of validity of the nonlinear equations. The system (1.59) is the most general form of nonlinear phenomenological equations. It is maximally nonlinear as the phenomenological coefficients depend on all the forces participating in the processes: L y = L¡j(Xi, ...,X n ). It is evident that other levels of nonlinearity are associated with the situation when the phenomenological coefficients do not depend on all forces (Zotin et al., 1978 b). The first level is that in which coefficients do not depend on forces (the linear case) and in this case equations (1.1) and Onsager's reciprocity relation (1.2) are true. The second level (quasilinear case) is where the nonlinear phenomenological equations take the form: η Ii = Lii(X)Xi + Σ LijXj j=l

( i * j ; i = l,...,n)

(1.61)

and the reciprocity relation (1.2) is true. The third level (a nonlinear system close to a linear system is given by: η Ii = Li;(X¡)Xi + Σ L¡j(Xi,Xj)Xj

(i * j; i = l , . . . , n )

(1.62)

j=i The generalized reciprocity relation (1.60) holds good in the form Lij (X„ Xj) = Lji (Xj, Xj) (1.63) The fourth level (nonlinear system) is where nonlinear phenomenological equations (1.59) and the generalized reciprocity relation (1.60) are true. And finally it is pertinent to distinguish the fifth level (a strong nonlinear system) which appreciably differs from the above mentioned approximation level by the fact that dissipation function fluctuations take place in stationary state of the system. This case is the least worked out and probably has its own level of approximation.

1.2.2 Dissipation functions Prigogine's principle of local equilibrium is the theoretical basis of the thermodynamics of linear irreversible processes. In accordance with this principle the rate of entropy production in a thermodynamic system with volume V is equal to Ρ = — dt

= f odv ^ 0 i

(1.64)

where ν is a local volume, σ the local rate of entropy production equal to (1.28).

24

1. Thermodynamics of Nonequilibrium Processes

As entropy is represented by local entropy production (1.28) in the thermodynamics of linear irreversible processes so the energetic representation is ψ = Τ σ (Zhuravlev, 1979) [where ψ is the local dissipation function of the system]. This dissipation function is equivalent to the local appearence of dissipation heat. Similar dissipation functions φ (j,j) and ψ(], χ) are used in the variational principles of thermodynamics (1.39) and (1.42). In the so called global approach to thermodynamics of nonequilibrium processes (Presnov, 1978) it is not the local but the specific dissipation function (1.40) that is used. The specific dissipation function has the same form as the local function (1.31) only it has another meaning. The difference is as follows. Firstly, it can be easily measured experimentally because it is equal to the total heat production of the system divided by the volume or mass of the system. Secondly, it is not so strictly linked to the Gibbs equation as a local dissipation function and allows mathematical modelling of phenomenological equations (1.1) and (1.59). In contrast to classical thermodynamics where intensive and extensive factors have only one meaning, in the thermodynamics of nonequilibrium processes, as has been mentioned, there is relative freedom in the choice of flows and forces. However the use of the Gibbs equation as one of the basic correlations in the thermodynamics of linear irreversible processes considerably limit this freedom. In fact all limitations caused by use of the Gibbs equation are so great that the thermodynamics of nonequilibrium processes, based upon the principle of local equilibrium, is, in practice, not much different from classical thermodynamics and is only applied close to equilibrium. To be able to consider systems far from equilibrium it is necessary to break away from the Gibbs equation and from classical parameters and turn to mathematical modelling. In part this is the case in the thermodynamics of linear irreversible processes because Fourier's, Fick's and Ohm's laws are mathematical models of the processes of heat transfer, diffusion and electrical phenomena. The global approach allows a freer use of mathematical modelling in drawing phenomenological equations. This makes it possible to consider a broader class of phenomena inaccessible to the thermodynamics of linear irreversible processes. Only the equations (1.3) and (1.58) require limitations in modelling flows and forces in the framework of the global approach. The global approach is also based upon the principle of minimum energy dissipation and the dissipation function characterizing the intensity of energy dissipation and the degree of process irreversibility therefore plays a very important role in the global approach when the system behaviour is characterized. Apart from dissipation functions (1.3) and (1.58) two more functions are used: the function of external dissipation and that of bound dissipation (Zotin, 1974; Lamprecht, Zotin, 1978; Zotina, Zotin, 1983; Zotin, Lamprecht, 1982). The necessity of introducing these functions is justified as follows. According to the Second Law of thermodynamics as formulated by Prigogine (1967) all irreversible processes are accompanied by dissipation entropy production, i. e. heat dissipation. In an isolated system this leads to an increase of entropy in

1.2 Nonlinear irreversible processes

25

the system taken as a whole. If the system is not adiabatically isolated, the heat dissipation can leave the system and the total entropy of the system can remain constant or even decrease. In the case when the open system is close to equilibrium and irreversible processes proceed in it slowly enough the emerging dissipation heat can leave the system completely. However, if the system is far from equilibrium and the rate of dissipation heat production in it is large, not all dissipation heat will leave the system: part of it can be used to realize some irreversible processes. In other words not all dissipation heat in this case will be dissipation heat in the direct sense of this term. Consequently for thermodynamic systems far from equilibrium the specific dissipation function may be divided into two parts (Zotin, 1974): ψ = Ψά + ψυ

(1.65)

where Τ d¡ Sd =

ψυ =

Τ di S u "ν ~ d F

y d is the external dissipation function (psidfunction); the bound dissipation function (psiufunction). A similar subdivision has been suggested by some other authors (Landauer, 1973; Tykodi, 1974; Wangsness, 1975; Lurié, Wagensberg, 1979 a, b). Subdivision (1.65) is entailed by the physical sense of energy dissipation in an open system. In a particular case with the use of Gibbs equation it may be understood in more traditional thermodynamics ways (Zotin, Zotina, 1977; Zotin et al., 1978 a; Lurié, Wagensberg, 1979 a, b; Zotina, Zotin, 1983). In fact it is presented in a concealed form in the work of Prigogine (1967) and Rubin (1984) where the authors discuss the possibility of measuring the rate of entropy production in chemical systems by the heat effects of the reactions. Statistical justification of the possibility of a subdivision similar to (1.65) is mentioned in the work of Landauer (1973) and Wangsness (1975). Here we draw attention to the proof given for purely chemical systems (Zotin, Zotina, 1977). For such systems equation (1.21) is true, that is why we may write for a dissipation function: 1 n Ψ = — Σ AQve V β=ι where v0 = d í,e/d t is the rate of reaction ρ. It is known (Prigogine, 1967) that

where G is the Gibbs free energy; H the enthalpy. Introducing this expression into (1.66) we have

(1.66)

26

1. Thermodynamics of Nonequilibrium Processes

1 A /9H\ Τ A 8S λ - Γ ι - p,tV qΰ + — Σ V V ρ=ι \δξ6

Ψ = -T7

ρ,τ v e .

Now introducing the symbols = - 4 Σ ( ! f LΡ T V . 4 Ì rgV ve = q V 0-i \ 0 ξ β / " * Ve-,

(1.67)

and

we arrive at the subdivision (1.65). Here Γρ0)τ is the heat effect of reaction q, and q the specific rate of heat production. The physical sense of the external dissipation function (1.67) is clear — it is equal to the intensity of the system heat production. The physical sense of the bound dissipation function results from the equation (1.68): in purely chemical systems the psiufunction depends on the temperature, the volume of the system, the rate of chemical reactions and the change of entropy associated with the degree of the reaction extent. Subdivision (1.65) may be described not only for chemical systems but for any other systems where the Gibbs equation is applicable (Zotina, Zotin, 1983). Thus according to Haase (1963) the specific dissipation function can be expressed in the form 1 dWd V dt

.

1 ν a V e=i

where d W d is dissipation work, which in the general case may be considered equal to dWd =

Σ

Ml,

where 1¡ are coordinates of the work of extensive type; Ly coordinates of the work of intensive type. Following the development similar to that used above we may describe the values for psid- and psiufunction in the following form (Zotina, Zotin, 1983) ¥d = q and Ψ, =

Τ γ

V

ídS\

ν

ί

s s

(1.69)

1.2 Nonlinear irreversible processes

27

Thus, in systems where, apart from chemical reactions, other types of work are performed, the psiufunction depends on the rates of processes and on the entropy change and is related to changes in extensive parameters. If the system is close to equilibrium all the processes proceed at a low rate. The values (0 S/3 lj)PjT and (0 S/0 ξ β ) ρ ,τ tend to zero and the psiufunction in such a system is negligably small. In this case (1.65) transforms into (1.40). For purely chemical systems the relation (1.40) holds good in all cases when the heat effects of the reactions exceed by far the entropy change associated with the extent of reaction (Prigogine, 1967; Rubin, 1984), i.e. under the condition (1.70) Consequently the dissipation function is not equal to the intensity of heat production when the system is far from equilibrium and the inequality (1.70) ceases to be true. It is also easy to derive a similar inequality for these systems in which, apart from chemical reactions, other types of physico-chemical processes proceed. If the psiufunction is not equal to zero it can be determined via (1.65), (1.40) and (1.67) and using the formula: η \|/u =

Σ i=

Ii X i

- q.

(1.71)

l

This formula permits an experimental check on whether the subdivision (1.65) is justified.

1.2.3 Principle of minimum energy dissipation In the thermodynamics of linear irreversible processes the degree of process irreversibility is determined by the value of the specific dissipation function. It is evident that the greater the degree of irreversibility of the process the further from equilibrium the thermodynamic system is and the less is the probability of such a system existing. Onsager's principle of the least dissipation energy (1.39), Prigogine's theorem and principle of minimum entropy production (1.42) are, from this viewpoint, an expression of a more general law according to which any physical system evolves in the direction from a less probable to a more probable state. This can be seen, for example, from equation (1.52). If the system is far from equilibrium we must not restrict ourselves to only a linear member in expansion (1.50) as was done in the process of developing equation (1.52). This however will not change the general conclusion of the reverse dependence between the energy dissipation intensity in the system and the probability of its state (Zotin, Zotina, 1977; Zotina, Zotin, 1983). Let us by way of example take two members of a Taylor power series (1.50), i. e. consider the equation:

28

1. Thermodynamics of Nonequilibrium Processes

(1.72) where a 2 = Ρ (p st )/2 !. Introducing this equation into (1.48) we shall have Pst

ψ = (b, + b2Pst) — + ψ8ί - bj + b 2 (p Ρ

2p st )

(1.73)

where b2 = a 2 k — . γ Comparing (1.73) with subdivision (1.65) we shall have now (1.74)

¥d = (bi + b 2 p st ) — Ρ ψπ = ψ8ι - b! + b 2 (p -

2Pst).

(1.75)

It is evident that if ρ = p st , the equality \|/d = ψ5( must be true, and thus (1.74) may be written as follows (1.76)

¥d = ¥st — Ρ

where \|/st = b t + b 2 p s t . Compatibility of (1.74) and (1.75) results from the following considerations. Under p—>pst, which follows from (1.76), then vj/d—>ψ8ι, which agrees with the general concept of psiufunction behaviour. On the other hand, under p—>pst we have bi — b 2 (p — 2p s t ) —* fy + b 2 p s t = ψ5( and ψυ—>0, which, in its turn, is consistent with psiufunction behaviour near the stationary state. All other comparisons of (1.65) and (1.73) will not satisfy these properties of v|/d- and \|/ u -functions. It is easy to show that relations of the type (1.74) to (1.76) will also be valid if all the members of the Taylor series (1.50) are taken into account. For example, it is true for three members of the Taylor series. In this case v|/st = bi + b 2 p s t + b3 Ps,. For four members of Taylor series ψ5ΐ = bj + b2 pst + b3 p,t + b4 ps3t, etc. The above mentioned considerations entail two important conclusions which both concern equation (1.76). Firstly, it requires that ψάΡ = vfstPst = ζ where Ζ may be a universal constant. Secondly, taking the time derivative of (1.76), we have d v|/d dt ~

ζ p2

dp dt

(1.77)

1.2 Nonlinear irreversible processes

29

and, knowing that when the system approaches a stationary state the probability density is not diminished we arrive at the evolution criterion for the system far from equilibrium: (1.78)

dt

One of the trends in nonequilibrium thermodynamics holds that the further development of a nonlinear theory of irreversible processes must be closely connected with the introduction of a new postulate similar to the Second Law of thermodynamics. It was proposed, for example, to extend Prigogine's principle of minimum entropy production to any thermodynamic system and to call it a new thermodynamic law (Zotin, 1971a). This suggestion has not met approval as Prigogine's theorem is justified for systems close to equilibrium or to the stationary state only and is not true in the nonlinear field. More recently a differently formulated principle of least external dissipation was suggested (Zotin, 1974) based upon inequality (1.78) and this is, in fact, a variational principle for a stationary state ψ ä 2 and bj > b2). In the system transition from the state I to state II relation (1.85) hold true. This relation contains two exponential members, and thus a transition process can be described by a small number of types of curves which depend on the values of constants C n , Cj 2 , C 2 i, C22 (Fig. 1.3). For system (1.83) transition from state I to state II the curves of changes in substance concentration will vary: 1) smoothly, gradually reaching the concentration value in the second stationary state by the exponent (Fig. 1.3 b); 2) will overshoot (Burton, 1939) with the following return to the first stationary concentration and motion towards the second stationary state (Fig. 1.3 c, d); 3) will pass the minimum, which lies below the final stationary state, through the so called false start (Fig. 1.3 a). Overshoot and false start can also appear in the case of a system transition from a stationary state with low concentration A and Β to a stationary state with a higher concentration of these substances. If reactions proceed in the system Y

A

k,

k, k-(„-i)

Ν

Ζ

(1.87)

they will be consistent with the equations

di dt

dt

(1.88) = k z (n z - n) + k n _! (η - 1) -

k_ ( n _ 1 ) n

1.2 Nonlinear irreversible processes

35

Δα Δ6

Figure 1.3 Transition processes between stationary states of an open system (1.83) from the initial, I, to the final II states: a) at C u = - 4 0 , b) C„ = 5, c) C„ = 20, d) C u = 50 [see equation (1.85)] (Denbigh et al., 1948).

the general solution for which taking the form a = a + C u e ^ ' 1 + Ci2e-^

+ ... +

Clne~x'1

x

b = b + C 2 1 e" ·' + C 2 2 e " ^ + ... + C 2 n e ~ M η = ïï + C„i ε~ λ ι ' + C n 2 e - ^ + ... +

(1.89)

C^e"^.

The possibility of establishing a stationary state in system (1.87) depends on the value of constants λ·,. If λ, are not imaginary and λ, > 0, the system can reach a stationary state. If λ; are real, but negative, then (1.89) means that concentrations a,b, ...,n will grow with time and the system will move away from the stationary state determined by the concentrations of substances â,b, ...,ñ. In the case of combinations of real and imaginary values of λ ; then fading or diverging oscillations of substances concentrations may emerge in the system (Moore, 1949; Lotka, 1956; Pasynsky, 1963).

36

1. Thermodynamics of Nonequilibrium Processes

Thus even in a simple open chemical system oscillation of substance concentrations may emerge under a stationary state if the values of constants λ( in (1.89) are imaginary. The more so for the case of different autocatalytic reactions, the kinetics of which is described by a system of nonlinear differential equations. The possibility of oscillating processes in the course of successive autocatalytic reactions was demonstrated long ago (Lotka, 1925; Volterra, 1931) and has been justified by a great number of researches (see Salnikov, 1948; Moore, 1949; Hearon, 1952; Bak, 1961; Spangler, Snell, 1961; Salnikov, Volter, 1963; Zhabotinsky, 1967, 1974; Romanovsky et al., 1975, 1984; Rubin et al., 1977; Ivanitsky et al., 1978; Rubin, 1984, etc.). These phenomena though refer to the category of dissipative structures and autowave processes, i. e. to the field of synergetics and will not be considered here. Thermodynamic analysis of transition processes, described by kinetic methods, is, as yet, difficult to perform. We have every reason to suppose, though, that the behaviour of dissipation function under a transition process is analogous to the change of substance concentrations in an open chemical system of type (1.83) or (1.87). This statement is based upon the fact that the flows (1.84) or (1.88) in a thermodynamic description of an open system must be included in (1.40) or (1.58) and therefore the behaviour of substance concentrations of the type (1.85) and (1.89) must result in corresponding changes in the dissipation functions, including false start and overshoot. Thermodynamic approach. The process of the system reaching equilibrium or a stationary state can be compared with the process of fluctuation relaxation (Onsager, 1931 b; Gurov, 1978). From this point of view the theory of fluctuation emergence and relaxation can contribute much to the understanding of transition process kinetics. In particular, as is shown above (1.53) a transition process in an open thermodynamic system close to equilibrium is accompanied by exponential change in the dissipation function. For a nonlinear case then Taylor's series (1.50) must be used in its full form and this makes solution of the derived equations much more difficult. Therefore we confine ourselves only to the second order member of the Taylor's series (Zotin, Zotina, 1977; Zotina, Zotin, 1983). In this case equation (1.72) takes the form:

dt

= ap2 + b p + c

(1.90)

where a = a, ; b = - ( a j + 2a 2 p s t ); c = aip s t + a 2 pâEquation (1.90) is called the Riccati equation (see Kamke, 1959). Solving it we obtain Ρ = Pst +

ai exp(—att) a 2 [C + e x p ( - a , t ) ] '

Substituting (1.91) into (1.74) and (1.75) we obtain

(1.91)

1.2 Nonlinear irreversible processes

Ψιι = . ^ J * , . · 1 + Cexp(ait)

37

(1.93)

As follows from these correlations under the constraint t—> oo, i|/d—>ψ5, and \|/u—>0, which is in conformity with the properties of these functions. To compare the kinetics of changes in \|/d and i|/u let us define the signs of the constants a¡ and a 2 . In equation (1.50) Γ (p st ) < 0 because the change in the rate of probability density decreases as the system transfers to the stationary state and this is why ai > 0 (since a, = — f (p st )). Analogous to this f'(p s t ) < 0 and also a 2 < 0, since a 2 = f'(p st )/2!. This means that functions vyd and ψ„, which are defined by equations (1.92) and (1.93) will continuously decrease with time. If we assume that all the constants except ψ 5[ in equations (1.92) and (1.93) are equal to 1 then Vfd = Vfst(l + e l );

Ψ»

1 1 + e