Thermodynamics and Regulation of Biological Processes [Reprint 2019 ed.] 9783110861198, 9783110097894

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Thermodynamics and Regulation of Biological Processes

Thermodynamics and Regulation of Biological Processes Editors I. Lamprecht • A. I. Zotin

W DE G Walter de Gruyter • Berlin • New York 1985

Editors Prof. Dr. Ingolf Lamprecht Institute of Biophysics Free University Berlin D-1000 Berlin 33 Prof. Dr. A. I. Zotin Institute of Developmental Biology Academy of Science of the U.S.S.R. Moscow

Library of Congress Cataloging in Publication Data Main entry under title: Thermodynamics and regulation of biological processes. Translated from the Russian. Bibliography: p. Includes index. 1. Biophysics. 2. Thermodynamics. 3. Biological control systems. 4. Information theory in biology. 5. Evolution. 6. Ontogeny. I. Lamprecht, Ingolf, 1933. II. Zotin, A. I. (Aleksandr ll'ich), 1926QH505T394 1985 574.19'1 84-23302 ISBN 0-89925-007-6 (U.S.)

CIP-Kurztitelaufnahme der Deutschen Bibliothek Thermodynamics and regulation of biological processes / ed. Ingolf Lamprecht ; A. I. Zotin. - Berlin ; New York : de Gruyter, 1985. ISBN 3-11-009789-3 NE: Lamprecht, Ingolf [Hrsg.]

Copyright © 1984 by Walter de Gruyter& Co., Berlin 30. All rights reserved, including those of translation into foreign languages. No part of this book may be reproduced in any form by photoprint, microfilm or any other means nor transmitted nor translated into a machine language without written permission from the publisher. Printing: Gerike GmbH, Berlin. Binding: Dieter Mikolai, Berlin. Printed in Germany.

This monograph deals with thermodynamic aspects of control and regulation in biological systems; relationships between thermodynamics of irreversible processes, information and control theory; extremal principles and regulation in ecological systems; self-organisation, evolution and ontogenesis. The volume includes contributions from leading scientists in these fields from the Soviet Union and other countries, therefore it presents the most important trends in modern studies of the thermodynamics of biological processes which deal with control and regulation in living systems. The monograph will be of value to researchers in biophysics, physiology, developmental biology, as well as for physicists and mathematicians interested in modern thermodynamics, information and control theory, thermodynamics of biological processes and mathematical biology.

Preface

This book is the last in the series of monographs on the thermodynamics of biological processes which are the result of active cooperation between scientists of the USSR and many other countries. The first monograph of the series entitled 'Thermodynamics of Biological Processes' appeared in Russian in 1976 (Nauka publishers, Moscow) and in English in 1978 (de Gruyter, Berlin). It dealt mainly with thermodynamic aspects of developmental biology, dissipative structures, classification and evolution of organisms. The second monograph 'Thermodynamics and Kinetics of Biological Processes' appeared in Russian in 1980 (Nauka publishers, Moscow) and in English in 1982 (de Gruyter, Berlin). It was devoted to the kinetics of transition processes, the reaction of living systems to external influences, and to the adaptation of the organism to changing environmental conditions. Attention was paid mainly to purely thermodynamic problems concerned with the present state of phenomenological and statistical thermodynamics of nonequilibrium processes. This third monograph 'Thermodynamics and Regulation of Biological Processes' deals with the processes most specific to the living systems, such as regulation and control, self-organisation, development and evolution. Perhaps the discussion of these problems in terms of thermodynamics is not fully justified as yet, since the thermodynamics of organised systems, i. e. those systems in which regulation and control are realised, has not been adequately developed. However, recently, along with thermodynamics of nonequilibrium processes, there has appeared such promising trends as synergetics (Haken, 1978) and thermodynamics of information processes (Poplavsky, 1981), which makes us believe that in the very near future the thermodynamics of organised systems will achieve a satisfactory basis. The formation of such trends as the thermodynamics of information processes

VIII

seems most promising. Its main task is the thermodynamic analysis of such phenomena as: orderliness, organisation and control; energetic cost of information storage, transmission and usage; precision and uniformity in control systems. It is evident that such problems are of primary importance for biology, since a specific concern of biological objects is, primarily, the orderliness and organisation of all the processes proceeding in the living system. It is impossible to construct a thermodynamics of biological processes without taking into account these specific features of biological systems. Notwithstanding the above, it is too early to declare the beginning of a new era in the thermodynamics of biological processes. The fact is, that, these thermodynamic trends have not merged organically in order to form a satisfactory theoretical foundation for this field of science. For example, the thermodynamics of information processes is not sufficiently associated with the thermodynamics of non-equilibrium processes. This is partly due to historical reasons, since the thermodynamics of information processes emerged on the basis of the analysis of Maxwell demons (Maxwell, 1872), i. e. long before the appearance of a thermodynamics of irreversible processes (Onsager, 1931 a, b ) . Later these two trends developed independently, and a thermodynamics of non-equilibrium processes elaborated its own language and yielded some relationships which differed from those of classical thermodynamics, whilst the thermodynamics of information processes preserved the language and methodology of classical thermodynamics . As a result both trends have different languages, although they both deal with the same open non-equilibrium systems. Almost the same can be said about the interrelations between these trends and the science of selforganisation - synergetics. This separation is fully reflected in the present volume and will be clear to the readers on comparison of different chapters. It is hoped that the book will promote closer connections and, eventual, amalgamation of these trends in thermodynamics. Like the first two monographs of the series the present one has been published because of the active support of the Corresponding Member of the USSR Academy of Sciences T . M. Turpaev, Director of the Institute of Developmental Biology, USSR Academy of Sciences and with valuable

IX

assistance of I. A. Ilyuchenko, G. S. Lozovskaya, I. S . Nikolskaya, S . M. Malyghin, L. I. Radzinskaya and of E. V. Presnov especially. We thank all these and, of course, the contributors for supporting the idea of publication of this monograph, for their cooperation and effort in preparing the monograph. The English translation was again thoroughly checked by A. E. Beezer, London, to whom we are indebted very much.

Berlin, October 1984

I. Lamprecht A. I. Zotin

Contents

I. General Problems Information and Cybernetic Aspects of Biological Thermodynamics D. Leuschner Some Relations of Non-Equilibrium Thermodynamics of Open Systems R. L. Stratonovich

3

19

The Statistical Background of the Glansdorff-Prigogine Criterion F. Schlögl

35

Thermodynamic Analogies in Statistical Theory of Dynamic Systems A. A. Platonov

55

Stochastic Criterion in Dynamical Systems E. V. Presnov

73

Phases with Internal Degrees of Freedom and their Heterogeneous Transformation E. N. Eliseev

77

II. Information Theory Towards a Dynamic Information Theory H. Haken

93

On the Problem of the Valuability of Information R. L. Stratonovich

105

Relativistic Information and Biology G. Jumarie

121

XII

III. Thermodynamics of Information Processes Entropy and Information - The Supernatural Actions of Maxwell's Demon I . Lamprecht

139

Thermodynamics and Information R. P. Poplavsky

169

Entropy of Information Theory and Thermodynamic Entropy Z. A. Terentjeva

:

175

Optimal Thermodynamic Processes under Chemical Reaction and Diffusion L. I . Rozonoer

181

IV. Control and Regulation in Biological Systems Order and Disorder in Biological Control Systems R. Rosen

205

Simple Feedback System Used to Model Biomolecular Reactions in Drug-Receptor Populations A. Seese, P. P. Mager

213

Towards a Stochastic Theory of Photoreception J . Schnakenberg, W. Keiper

229

Temperature Homeostasis and Principles of its Regulation in the Living Organism K. P. Ivanov

243

XIII

V. Extremal Principles An Extremal Principle for the Biomass Diversity in Ecology D. Lurie, J . Wagensberg

257

Optimal Principle and Temperature Conditions for the Existence of Animals V. V. Lapkin

275

Optimal Heat Insulation of Homeotherms M. A. Khanin, O. G. Bat'

289

Thermodynamics of Processes and the Evolution of Simple Biothermodynamic Populations J . U. Keller

305

VI. Synergetics - The Problems of Seiforganisation Self-Organisation Processes and Their Modelling V. A. Vasiliev, Yu. M. Romanovskij

333

Stochastic Dissipative Structures A. S . Pikovsky, M. I. Rabinovich

343

Symmetry Breaking Instabilities in Chemical Systems G. Dewel, P. Borckmans, D. Walgraef

353

Dynamic Model of the Origin of Order in Controlled Macrosystems V. S. Lerner

383

VII. The Problems of Evolution Non-Equilibrium Thermodynamics of Hypercycles K. Ishida

401

XIV

New Information in Evolution N. M. Chernavskaya, D. S . Chernavsky

415

State Spread and Evolution in Chemical Systems L. A. Nikolaev

429

Stochastic Models of Evolutionary Processes R. Feistel, W. Ebeling

437

Bioenergetic Trends in the Evolutionary Progress of Organisms A. I. Zotin

451

VIII. Ontogenesis Continuous Phase Transitions and Morphogenesis J . L. Rius, B . C. Goodwin

461

Stability of Ontogenesis A. I. Zotin, T . A. Alekseeva

485

Analysis of Homeorhesis in Ontogenetic, Population and Evolutionary Aspects V. M. Zakharov

497

Thermodynamic Aspects of Plant Ontogenesis W. Zelawski, W. Galinski

509

Concluding Remarks

520

List of Contributors

522

References

525

Subject Index

565

General

Problems

Information and C y b e r n e t i c A s p e c t s of Biological

Thermodynamics

D. L e u s c h n e r

Monographs s u c h as those by Zotin (1976, 1980) and L a m p r e c h t , Zotin (1978, 1982), have shown t h e importance of thermodynamics in biology. This is t r u e also, f o r information t h e o r y (see e . g. Eigen, Winkler, 1973/74; Eigen, S c h u s t e r , 1977, 1978) and f o r c y b e r n e t i c s b y which is meant control t h e o r y or regulation (Drischel, 1973; H a s s e n s t e i n , 1967; Nicolis, P r i g o g i n e , 1977). T h e r e is n e i t h e r a complete t h e o r y n o r a conc r e t e distinction between biology and thermodynamics, nor between i n f o r mation t h e o r y and c y b e r n e t i c s . The aim of t h i s p a p e r is not to d e s c r i b e an e s s e n t i a l , c o r r e c t t h e o r y of biothermodynamics, bioinformation or bioc y b e r n e t i c s , n o r shall we attempt to c r e a t e s u c h a t h e o r y . We shall only i l l u s t r a t e a few of t h e i r i n t e r r e l a t i o n s between biology (Bio), t h e r m o d y namics ( T h ) , information t h e o r y ( I n f ) and c y b e r n e t i c s ( C y b ) . In this connection, t h e c e n t r a l subject of t h i s work is an outline of two new c o n c e p t s : Thermodynamic phoronomics ( T h P ) and thermodynamic taxonomy ( T h T ) . The f i r s t is t h e prologue to a t h e o r y of creation and p r o d u c t i o n of information, and the second to the c o n c e n t r a t i o n of information, b o t h , of c o u r s e , r e f e r to biology. F u r t h e r m o r e we give an example: p h o t o s y n t h e s i s and its i n t e r r e l a t i o n s to thermodynamics, information t h e o r y and c y b e r n e t i c s at d i f f e r e n t levels of organisation in the g r e e n plant and also include o n t o g e n y , p h y l o g e n y and g e n e t i c s .

Thermodynamics and Regulation of Biological and Processes © 1984 Walter de Gruyter & Co., Berlin • New York - Printed in Germany

4 General and specific interrelations

The structure of the interrelations between biology, thermodynamics, information theory and cybernetics is respresented in Fig. 1. The connectors are numbered ("

" ) and a short explanation for each number

appears in the t e x t . All levels of organisation in biology (molecular, cellular, organismic, ecologie) can be considered.

Fig. 1

Scheme of relationships between information theory ( I n f ) , cybernetics ( C y b ) , thermodynamics (Th) and biology ( B i o ) .

Thermodynamics and biology

("1")

The literature on this topic is large (see e. g. Zotin, 1974, 1976, 1982; Lamprecht, Zotin, 1978, 1980). Examples for different levels of organization have been discussed by Leuschner (1981 a ) . Consideration of examples was shown to be possible by Schrodinger (1951) who removed the apparent contradiction between the Second Law of Thermodynamics and phenomena associated with living matter. Biological phenomena do not exist in contradiction of this law; they exist on the basis of the Second Law which is also valid for open, living systems.

5 Modern biology includes study of the thermodynamics of equilibrium and of non-equilibrium (biological self-organisation see Ebeling, 1976; Ebeling, Feistel, 1981) states. The First Law can be written as cLE = c t A +

3-Q.

(1)

The change in internal energy E of the system (E is a state variable, and therefore dE a total differential) is equal to the sum of the work, c t A , and the heat ct^S

> O

and from this we obtain > CLE. + T c C ^ S , since = ct^EE. -

Ttit^S

and cLE =

+ cCxE. ,

ot^E=0

where F = free energy (subscripts e and i are analogous to those in ( 2 ) and T = absolute temperature). An organism is fed with free energy, ot^F, from the surroundings and the value of this free energy underlies ( 2 ) .

6 Information theory and biology

("2")

The term information as used in information t h e o r y is v e r y g e n e r a l . In (3) we find only (see below) probabilities and r e f e r e n c e to any phenomena (symbols, s t a t e s or q u a n t i t i e s ) . T h e r e f o r e s u c c e s s in t h e application of information t h e o r y in Eigen's (1971) t h e o r y of self-organization and evolution was possible. In t h i s r e s p e c t we n o t e : t h e information content of a DNA molecule and information creation (see (4) and Eigen, Winkler,

1973/74);

and von Neumann's t h e o r y of games, see e . g . Wentzel, 1964). It will be noticed t h a t information p o s s e s s e s importance not only in t h e s p h e r e of molecules, b u t also, f o r example, at t h e level of cells and of biocenoses. Two examples will illustrate this p o i n t . One is t h e g r e a t e r volume of a eucaryotic cell in comparison with a p r o c a r y o t i c cell as a c o n s e q u e n c e of the g r e a t e r information content of DNA in the former cell. The o t h e r is the information content in the manifold of a s t a t e of i n s e c t s . Both examples d e m o n s t r a t e t h e generality of information in biology, b u t t h e y also pose many q u e s t i o n s t h a t remain u n s o l v e d . For example: The volume of a d i n o s a u r is not equivalent to its analogous information c o n t e n t , and the information content of a s t a t e of i n s e c t s is p e r h a p s not b e s t d e s c r i b e d t h r o u g h equations s u c h as ( 3 ) . In spite of this we consider information to be t h e c e n t r a l theme of t h i s a r t i c l e . Without information t h e r e can be no thermodynamics, no c y b e r n e t i c s and no biology ( f o r t h e association between thermodynamics and information t h e o r y see e . g . Ebeling, 1976; Lamprecht, Zotin, 1978 and below; for t h e association between c y b e r n e t i c s and information t h e o r y see Laskowski, Pohlit, 1974 and below; l a s t , but not l e a s t , DNA is information a n d , without DNA t h e r e is no life, and no b i o l o g y ) .

Information t h e o r y i n v e s t i g a t e s t h e transmission of information along communication c h a n n e l s and t h e r e f o r e we must d i s c u s s , in a mathematical t e r m , t h e measure of information c o n t e n t , and its c h a n g e s along communication channels.

7

Information content (or more briefly, information; sometimes named Shannon's entropy; see Meyer-Eppler, 1959; Schwartz, 1963) is defined by I = - 2L ¿'A

'

'

,

'

^ ,

'

o

(3)

where p. = probabilities for i = 1, . . . , n arbitrary items (symbols, values of measure, objects, states and so on) and Id is the logarithim dualis (logarithim to base 2). We assume two different distributions of probabilities (p^, pg, • • . , p n ) and (q^, •••> Qjj) f ° r the same set of objects. The difference in the two values of and I for the p- and q- distribution, respectively, can be taken as a measure of the increase of information. This difference can be positive or negative and therefore a better measure of the information increase is (after Renyi, 1966))

1 ^ ] ^ ) =

¿c^ioUc^AtO.

(4)

Trans-information is the information possessed by both the transmitter and the receiver, if there is noise in the communication channel. When the pj are the probabilities of the transmitter, p. those of the receiver and p^ are the probabilities of a coupled event i and j in the transmitter and in the receiver the trans-information is T = X ( ^ ) + K ' y o p - Hi-p-cj)

(5)

with ~L(f>.c) = - 2_

1

=

• 5- \ r ^ i ^

,

•

The ideal case for the transmission of information (ideal coupling) without noise is that where T = I(pj) = I(p.) = I(pj.).

8 C y b e r n e t i c s and biology

("3")

Control p r o c e s s e s a r e v e r y important in p l a n t s and animals at all levels of o r g a n i z a t i o n . Without control s u c h dynamic s y s t e m s could not exist u n d e r t h e continuous p r e s s u r e from t h e i r s u r r o u n d i n g s . Because of t h e limited space available h e r e we can only note some t e r m s , d e s c r i b e some biological examples and give some important e q u a t i o n s . Firstly we h a v e to d i s t i n g u i s h a control chain (pilot c i r c u i t ) from a control circuit (control l o o p ) . An example of t h e f i r s t is t h e control of p h o t o s y n t h e s i s in g r e e n p l a n t s b y l i g h t , f o r t h e second t h e regulation of human body t e m p e r a t u r e . Secondly, t h e r e is regulation with c o n s t a n t command variation ( e . g . t h e body t e m p e r a t u r e ) , a c c o r d i n g to a program ( t h e ontogenesis of an o r g a n i s m ) , a c c o r d i n g to a set of u n k n o w n commands v a r y i n g in time ( t h e r h y t h m and d e e p n e s s of r e s p i r a t i o n u n d e r physical s t r e s s ) a n d , a c c o r d i n g to an optimum (it is diffcult to give biological examples because we do not know, f o r many c a s e s , what is t h e optimum and what a r e t h e conditions of t h e optimum s t a t e ; moreover many biological s y s t e m s do not work at t h e optimum s t a t e ) . In spite of o u r slight knowledge about optimum regulation we have to remember t h a t c y b e r n e t i c s alone h a s indicated what may be r e g a r d e d as optimization in t h e evolution of macromolecules (Eigen, 1971; Eigen, Winkler,

1973/74).

The g e n e r a l i t y of t h e following e q u a t i o n s is not as g r e a t as those of t h e r modynamics n o r those of information t h e o r y . F i r s t l y , we r e q u i r e a t r a n s f e r f u n c t i o n , f , between x E ( i n p u t ) and x A

(output): (6)

which, in t h e case of l i n e a r i t y , can be written

with K as t h e t r a n s f e r c o e f f i c i e n t . T h e q u a n t i t i e s Xg and x ^ can be absolute or d i f f e r e n c e v a l u e s . A special case of s u c h a t r a n s f e r c h a r a c t e r istic of a c o n t i n u o u s , l i n e a r , c o n s t a n t value controller is t h e relation b e t ween t h e control value x and t h e controlled value u ( d i f f e r e n c e between t h e command vaiable and the measured v a r i a b l e ) . T h i s is ( s e e J u n g e ,

1975),

9 for a proportional controller, X' -

K^/U.,

C8)

and for an integral controller, dy ciT

./ = K r ^ ,

(9)

and for a derivative controller *

(

1

0

)

Moreover for a proportional, two point controller we have an unavoidable control deviation, A ) ^ ,

of the controlled value Y (this is a measure of

the fluctuation around the command value)

where

eLY/dt

= velocity of change in the controlled value and

c

= dead

time (time from the registration of the "old" measured value to the establishment of the "new" measured value). A second relation in cybernetics concerns the co-work of several control loops ( e . g. the homeostat considered by Ashby, see Beier, 1968). In the case of linerarity the cooperation of these controllers (each controller has an influence on each control element) is described by a system of ordinary differential equations i*-

(12)

(x. = deviation of the controlled value of the control loop j from the command value, x. = velocity of change of such a deviation in the control circuit i ) . Consideration of the application of cybernetics in biology has an inverse aspect in bionics: biological experiences in the field of regulation can yield prototypes for application to physics and technology.

10 Cybernetics,

Through

information theory,

thermodynamics and biology

statistical p h y s i c s information t h e o r y can be r e l a t e d to t h e r m o -

dynamics ( " 4 " ) ; it is possible to calculate partition f u n c t i o n s in information t h e o r y . The association between partition f u n c t i o n s has been given b y Brostow (1972) and it is a well known fact t h a t t h e partition f u n c t i o n s of statistical p h y s i c s yield t h e thermodynamic p a r a m e t e r s (see e . g . Kittel, 1973; L e u s c h n e r , 1979). Association of information t h e o r y with t h e r m o d y n amics is also possible t h r o u g h e n t r o p y (see Ziesche, 1968) and t h i s association is important in biology. An u n d e r s t a n d i n g of living systems r e q u i r e s u n d e r s t a n d i n g of t h e i r motivation, t h e i r e x i s t e n c e and of t h e i r r e g u l a t i o n . The one yields thermodynamics and t h e o t h e r c y b e r n e t i c s . A s y n t h e s i s of b o t h , a p p r o p r i a t e to biology, is given in t h e following p a r t . Without information t h e r e is no r e g u l a t i o n . T h e r e f o r e we find information t h e o r y in t e x t b o o k s as p a r t of c y b e r n e t i c s and vice versa.

In information

t h e o r y we a r e i n t e r e s t e d in t h e t r a n s f e r of information from a t r a n s m i t t e r to a r e c e i v e r . Whereas, in c y b e r n e t i c s , we s t u d y t h e cooperation of s e v e r a l information c h a n n e l s and the utilization of t h e t r a n s f e r r e d information in r e g u l a t i o n . For a biological example see Eigen's h y p e r c y c l e (Eigen, S c h u s t e r , 1977, 1978). Eigen's t h e o r y of the prebiotic p h a s e of selforganization of macromolecules and of t h e i r evolution is t h e best illustration of t h e application of I n f , Cyb and Th to biology. It can be employed as a model from a physical point of view for t h e evolution of biological species (see Mayr, 1967). O u r knowledge on t h i s topic is v e r y slight and f o r t h e f u t u r e , the r e s u l t s of s u c h a t h e o r y h a v e to be clear in any s y n t h e s i s as is d i s c u s s e d in t h e last p a r t .

Thermodynamic

phoronomics

This c o n c e p t , established t h r o u g h consideration of e m b r y o g e n e s i s and P r i g o g i n e ' s principle of minimum e n t r o p y production ( L e u s c h n e r , 1981 b ) , is a contradiction of physical p r i n c i p l e s (dynamic and phoronomic o r k i n e matic a r e d i f f e r e n t ) . Thermodynamics has however c h a n g e d r e m a r k a b l y

11

from the time of Carnot's paper on the steam engine (1824) to Prigogine's paper on the thermodynamics of irreversible processes in open systems (1947). During this period at best we had thermostatics. It is an open question as to whether we have 'real' thermodynamics today. There is no term in thermodynamics like, or similar to, the term 'force' in mechanics (see also Leuschner, 1975). The forces in the. thermodynamics of irreversible processes are not forces in this sense. Therefore we refer to thermodynamic phoronomics, or thermodynamic kinematics. This requires a synthesis of thermodynamics, cybernetics and 'kinetics' with relevance for biology. Kinematics (this term is convenient in connection with the technology (see bionics) or phoronomics (this term is convenient in theoretical considerations) deals with the interrelations between, on the one hand, the trajectory and its curvature, and, on the other hand, velocity and acceleration (Lenk et al., 1972). In the case of ThP the trajectory is a trajectory in any space of parameters or characteristics (see e. g . Lerner, 1970; Leuschner, 1974 1974 a, b ) . Motion along this trajectory corresponds to a change. In biology this change can be physiological, ontogenetic or phylogenetic. The curvature of the trajectory is a measure of the 'strength' of such a change and we assume that there exists an interrelation between the kinematic quantities ( e . g. velocity, acceleration e t c . ) and the geometric quantities ( e . g . curvature). We therefore only investigate kinematic quantities, especially those quantities which refer to entropy, S. To this end we distinguish between processes and states. From the point of view of THP, in a system, a process takes place, when G^S/olt* *

0 .

(13)

On the other hand we speak about a state when

ct* S / d t z = O

(14)

12

This definition of a s t a t e includes alongside o L S / d t = C-

(c = c o n s t a n t )

(15)

t h e case of t h e s t e a d y s t a t e with C = O

(15 a)

and t h e case of equilibrium (S itself is a c o n s t a n t , namely t h e maximum value when r e f e r r e d to o t h e r s t a t e s ) . T h e s t e a d y s t a t e , as a specific, non-equilibrium s t a t e is also a s t a t e b y this definition, and s i t u a t i o n s of j q non-equilibrium with variable cvt a r e p r o c e s s e s in a real s e n s e . Moreover we d i s t i n g u i s h between optimum s t a t e s and optimocline p r o c e s s e s (those d i r e c t e d toward an optimum s t a t e ) . We give an example from p h y s i c s : T h e ' c o n s t r u c t i o n ' of a parallelogram of f o r c e s c o r r e s p o n d s to an optimocline p r o c e s s and t h e end point of t h e r e s u l t a n t is an optimum s t a t e in comparison to all o t h e r possible points in t h e p l a n e , Equilibrium and s t e a d y s t a t e a r e optimum s t a t e s in ThP and a optimocline p r o c e s s is a p r o c e s s d i r e c t e d toward a new optimum s t a t e . Non-linear p r o c e s s e s v e r y f a r from equilibrium can be r e g a r d e d as s u c h p r o c e s s e s . The r u l e s and c r i t e r i a for f u r t h e r c h a r a c t e r i z a t i o n of s t a t e s and p r o c e s s e s in t h e r m o d y n amic phoronomics can be g i v e n . F i r s t , we consider t h e decomposition of e n t r o p y p r o d u c t i o n as given b y Prigogine (see ( 2 ) ) cLS , , =- —n— cL-t act

ct-nS _, , c*-t

+

•s

(lb) J c

= e n t r o p y production in t h e s y s t e m ,

= e n t r o p y flux f r o m , or into

t h e s y s t e m ) . C o r r e s p o n d i n g to t h i s decomposition we write t h e following time d e r i v a t i v e s of e n t r o p y ( f i r s t , s e c o n d , t h i r d moment of e n t r o p y ) oLS C = S = dUt

•r p ~.

c

=

oLt

c

^ oU 5 cL~t

(17)

13

3 s

S =

ot^S cU*

dP di IS ©r>S — o l t = oTTT lt1

oL s S

1?

oLt 3

-

d Z p

oL?S oi-t 3

•R

cLO

¿\S

oCt

cU*

ott'

dt

3

(18)

(19)

At p r e s e n t t h e r e is no n e c e s s i t y to consider h i g h e r time d e r i v a t i v e s t h a n e n t r o p y jerk ( t h i r d time d e r i v a t i v e ) ; n e i t h e r from a t h e o r e t i c a l , n o r , from a practical point of view. What do we know about t h e s e q u a n t i t i e s ? T h e Second Law of Thermodynamics ( f o r a simple biological proof see L e u s c h n e r , 1981 c) s t a t e s P > O .

(20)

P r i g o g i n e ' s minimum principle in the linear region of the thermodynamics of i r r e v e r s i b l e p r o c e s s e s ( L e u s c h n e r , 1979 I . e . ) s t a t e s 3

p

^ O .

(21)

Prigogine and G l a n s d o r f f ' s principle in t h e n o n - l i n e a r region of t h e t h e r m o dynamics of i r r e v e r s i b l e p r o c e s s e s ( d ^ P = variation of the e n t r o p y p r o d u c t ion associated with variation in t h e thermodynamic f o r c e s , X, and not with t h a t of thermodynamic fluxes J ; G l a n s d o r f f , Prigogine, 1971) with c o n s t a n t b o u n d a r y conditions give d

x

P / d t

O.

(22)

T h e last t h r e e e q u a t i o n s a r e e i t h e r evolution c r i t e r i a (see Ebeling, 1976) or extremal p r i n c i p l e s . It is possible to complement t h e s e equations b y c r i t e r i a of stability which are s t a t e m e n t s about second o r d e r variation in e n t r o p y (see L e u s c h n e r ,

1979).

14

Of the other quantities C, B , R, R p , C r„ , B r_ , R r„ we only know that they can be greater, or less than zero or equal to zero. For example the statement (23)

C > O l. e. ci«.S oLt

d-.S oLt

cLe.S

oLt

< O

(24)

always means the possibility of the death of an organism. Further we are able to calculate the value of P P-Il^Kfe -K

ZLL^tCO^j. JL

(25)

(X^ = thermodynamic force, J ^ = thermodynamic flux) and to decompose the quantity

(see (22)) olP _ dt

dxP oLt

.

dbg-T oLt

(26)

with OLyF- Z ^ c L K f c

i")

(28)

Now we come back to the extremal principles mentioned above (Gyarmati's principle and Ziegler's principle are also of this type, Gyarmati, 1965; Zotin, 1976; Ziegler, 1977; Lamprecht, Zotin, 1978). Such principles have to be complemented by others in defining not only direction but also change in metabolism, in ontogenesis and in phylogenesis. For phylogenesis which is related with genetics, we briefly indicate the general idea. A certain d2 S value of the second entropy moment, g p - , in relation to other values determines 'selection, i. e. the 'survival of the fittest'. The third entropy d3S moment -tttdetermines a 'mutation' when a threshold is exceeded. The dt J

15

threshold can be zero, i. e. a thermodynamic or biological new quality could arise when

d3S dt J

t t t -

t

di, 3 S ^ 0 reflects internal dt J

0;

-r?s—

fluctuations,

and

d2S g p - external influences. These conditions are not necessary, but sufficient. This general 'selection-mutation law' is one of the centres of ThP, because it provides for the possibility of a creation of new optimocline processes and new optimum states. In conclusion the existence of dualities in thermodynamic extremal principles (Leuschner et a l . , in preparation) should be noted. Here we have a possibility for thermodynamic motivation of the term 'antiorganism' (see Soka, Sneath,

1963).

At present there is no understanding of the relationship between equations (20) to (22) and for example, Liebig's minimum law (see Stugren,

1974),

the bottle neck principle (see Romanovsky et a l . , 1974) and Zotin's principle of minimum external entropy production (Zotin, 1974; Zotin, Zotina, 1977; Zotin et al. , 1978) P^

in,

(29)

where total entropy production P is decomposed in P =

Pd.

< 30 >

(P^ = the proportion of the entropy production which flows into the surroundings and P u = the proportion of the entropy production which remaines within the system). We know that thermodynamic optimisation (represented by T in the following) is characterized by a certain value of the quantities in equations (17), (18), (19) and also in (22), (29) by reference to a constant value (for instance zero).

Thermodynamic

taxonomy

At the beginning of the last part we established the relationship between entropic velocities and accelerations on one hand side and the properties of the trajectories in a parameter space (for instance a taxonomic character

16 s p a c e ) on t h e o t h e r h a n d . However, the thermodynamic extremal p r i n c i p l e s of ThP a r e more important f o r ThT t h a n t h i s r e l a t i o n s h i p . T h e r e f o r e , f i r s t l y , we r e p r e s e n t t h e foundation of ThT via information t h e o r y and s e c o n d l y , we t a k e into account t h e importance of t h e thermodynamic o p t i misation f o r T h T . The second point c o r r e s p o n d s to consideration of t h e thermodynamic extremum p r i n c i p l e s of thermodynamic phoronomics in i t s relevance to thermodynamic taxonomy, i. e . we also h a v e , in o u r opinion, a relationship with c y b e r n e t i c s . The basis of thermodynamic taxonomy can be found in L e u s c h n e r (1981 d ) . In t h a t p a p e r we showed t h a t thermodynamic knowledge was important f o r t h e biological manifold and how to proceed in t h e application of t h e r m o dynamics to t h e systématisation of this manifold ( t h e taxonomy in t h e real s e n s e of t h i s t e r m ; see S n e a t h , Sokal, 1973). T h r e e s t a t e m e n t s a r e n e c e s sary - Thermodynamics and taxonomy have a common root in information t h e o r y ( f o r t h e l a s t , see Vôlz, 1982) - Thermodynamics and taxonomy have many common p r o p e r t i e s - Taxonomic similarity c o r r e s p o n d s to, t h e r e l a t e d , thermodynamic optimisation. We have already cited Browstow's concept of the deduction of t h e t h e r m o dynamics from information t h e o r y and we have also noted t h e g e n e r a l i t y of information t h e o r y ; especially in its u s e f u l n e s s in biology. The last point has been d e m o n s t r a t e d f o r taxonomy by L e u s c h n e r (1974). T h e r e f o r e , not only is a common root shown, b u t also t h e s y n t h e s i s of thermodynamics and taxonomy is made t h r o u g h thermodynamic taxonomy. Thermodynamics and taxonomy h a v e t h e following common c h a r a c t e r i s t i c s . T h e y a r e sciences - of macroscopic or phenomenological n a t u r e - of an axiomatic, b u t empirical t y p e - of u n i v e r s a l t y p e - with t h e effect of classification - with t h e possibility of microscopic foundation - with utilitarian aims

17

- dealing with complex systems - dealing with the creation of a natural system. Items referring to these points; to appropriate characteristics; to handling and processing in ThT ; and to examples ( e . g. Zotin's orderliness criterion) can be found in Leuschner (1981 d ) . We cannot and will not discuss the large number of measures of similarity (see Sokal, Sneath, 1963; Sneath, Sokal, 1973) however, we represent any such measure by S . Further for two taxonomic units ( e . g. two organisms) with thermodynamic optimal T^ and T^, respectively the taxonomic similarity S ^ of both these units is given by " A C T *

r\

(31)

where A is an operator, a function or a constant and r\ is the set-theoretical disjunction. If the relation ACT,

« A C O

*

A (TO

(32)

is valid, then both units are very similar.

Physics is concerned with thermodynamics and electromagnetism. The first includes (statistical) mechanics. Biology is concerned with physics, information theory and cybernetics. Is this all? We do not know. We could answer, if we knew what ordering is, especially in biology. But ordering is a very difficult problem. This is true for non-living matter (e. g. turbulence, Leuschner, 1978, 1981), is even more so for life. Zotin and his group (Konoplev, Zotin, 1975; Zotin, 1976, 1981; Zotin, Krivolucky, 1982) have made a start in attempts to define biological ordering and this is continued in this book. Thus our knowledge of biology will become more comprehensive through application of thermodynamics, information theory and cybernetics theory. But there remains a question. What is the relationship between thermodynamics of the life, in general, and especially of thought

(Kobozev, 1971)? We have to answer this question on the basis of

18

a definite criterion. A criterion that enables one to distinguish between real and chance coincidence between a biological phenomenon

and our thought.

Is the criterion of practice in this question necessary and sufficient? This last should not be theoretical, but practical. Energy? Raw materials? Food stuff? Environmental protection? Interplanetary travel? None of these is possible without biology; biology helps to solve problems. Partly, by means of biological laws which have their basis in thermodynamics (physics), information theory and cybernetics.

Some Relations of N o n - E q u i l i b r i u m

Thermodynamics

of Open Systems

R. L. Stratonovich

As may be seen from reviews (e. g . Nicolis, Prigogine, 1977; Lampreeht, Zotin, 1978), the application of the thermodynamics of non-equilibrium processes to strongly non-equilibrium, open systems is now of great interest. Such states are often encountered in physics, chemistry, and biology and no single view exists of the content of a thermodynamic of non-equilibrium processes and its application to such states. The purpose of non-equilibrium thermodynamics is the derivation of universal relations that describe the characteristics of macroscopic dissipative and fluctuative processes in complicated systems. The Onsager relations and the fluctuation-dissipation theorem are examples of such relations. The results of a non-equilibrium thermodynamic analysis of closed systems are a consequence of principles such as detailed equilibrium and microscopic time reversibility. The principle of detailed equilibrium we understand, in particular, as application of equilibrium distributions: Gibbs or microcanonical distributions. Non-equilibrium thermodynamic analysis of open systems is poorer, i. e. there are fewer results than arise from thermodynamic analysis of closed systems. The point is that in strongly non-equilibrium states one cannot effectively use the principle of the reversiblity of microscopic processes. In addition, far from equilibrium one cannot use canonical distributions, or, arising from them distributions of the form -FfBV/feT SCoVk cs ( 8 ) = con*!- 8;/u>CB) = cm^

t M

cB)1

d B

i. e . a functional of non-stationary distribution density ( w ( B ) / w c m ( B ) is the stationary distribution). The H-theorem under consideration here is closer to the conventional Second Law of Thermodynamics since this conventional entropy is not at all equal to the indicated functional.

The H-theorem Let us consider the non-equilibrium stationary state of an open system characterized by the non-equilibrium stationary distribution,

wcm

( B ) > which

cannot be represented by formula ( 1 ) . By writing ¿ ¿ a n ££>)= c c u t f - v * p r - Y C B i / a t ] > similarly to ( 1 ) , one can, however, introduce a quasi-free energy

(2) 'Y(B)

or a quasi-entropy S ( B ) = - ~ y ( B ) . The parameter Vt in (2) characterizes the intensity of fluctuations in the system. Equation (2) can be considered a non-equilibrium generalisation of equation ( 1 ) . The potential conjugated with " Y ( B ) :

also has the meaning of quasi-free energy or quasi-entropy, but, a s distinguished from " V ( B ) , it corresponds to fixed additional external forces />c, whereas ~ V ( B ) corresponds to fixed internal parameters B = ^ B ^ . It is easy to verify that the derivative, - Q ^ i x O / S ^ ^ other than the non-equilibrium mean,

of (3) is none

21

=

CLE/

which appears when the additional external forces X= are switched on. Higher derivatives opposite in sign are proportional to the correlators

which correspond to the biased distribution const exjo[-

.

(6)

To obtain the mean expectation value and correlators for the initial stationary distribution (2), one should assume in (4) and (5) x = 0. If now one introduces the function +

C - H o c a ^ A ) ,

(7)

which is a Legendre transformation of the potential (3), then, as can be easily verified, the dependence ^

r

- **

(8)

will be the reverse of the dependence dependence (4) at Aot=J.X

i. e. the

Now consider the time derivative of the function

(A) caused by change in time of the mean internal parameters A ( t ) . Substituting in (8) the equation d"M^(A)/dt = (d ^/©A)dA/d± .., we obtain dYCAUt^cUV/cLt

CWAVCJA •

Assume that time evolution of the non-equilibrium means A ^ = is described by the inertia-less, phenomenological equation

22

o L A o l / d ± = ($«*.CA) ,

dL- ' i j . - . j r .

(io)

Random values B ^ C t ) of the internal parameters satisfy some stochastic equation - the Langevin equation dlB^/clt»

SolCO ,

dL=-v.

Since the phenomenological equation (10) is intertia-less, the random process B(t) is a Markov process. The synchronous probability distribution density w^(B) satisfies the master equation 'SU^CB) _ y

*

,

The functions involved K ^

'd*

dL^B)

are

,

r

M

Mil

known to be determined by the

equations K.

d

03) =

( A t ) - A B ^ ... A B ,

where AB,*. is the increment in parameter

>B ,

(12)

B ^ f o r time A t ; the lower index

B implies that we take a conditional mean value corresponding to fixed values of the parameters B before the increase. Assuming in (12) A = A , we have I^olCB) =

< AB^/At >

s

.

(13)

The meaning of the derivative in (10) should be specified. Suppose that the derivative in (10) corresponds to averaging the derivative in (13) with biased distribution (6), where x is connected with A by the equality i. e by equation ( 4 ) . Then from (10) and (13) we have where

d A * . / d f c = A^Cx)

c>YCA)/SA;

X0L(X)=(KOLCBWXCB)OCBS-
O .

= O , K^-m

The

function

= 35°

a n d a local maximum

The difference between

the

by >

a p p r o a c h e s the critical point ©

a n d smaller a n d ,

3

^

o c

,

the point B

this d i f f e r e n c e becomes

smaller

= B ° becomes more a n d more

At a c e r t a i n moment the p a r a m e t e r B c h a n g e s h u g e l y a n d t h u s the is called f i r s t - o r d e r t r a n s i t i o n . characterized

b y the r e l a t i v e A=U

V v i a M

T h e d e g r e e of i n s t a b i l i t y of the point B °

TL-^jo

» say.

At the stability b o u n d a r y

is

difference

/ac = -

.

We c o n s i d e r the state B ° to b e s u f f i c i e n t l y s t a b l e if constant

unstable.

transition

vya -

/L

(44) exceeds

some

o.

we h a v e

/\, =

ryo

or

>

b y v i r t u e of

(44) (45)

T h e c o r r e l a t o r s of the i n t e r n a l p a r a m e t e r B n e a r the s t a b i l i t y b o u n d a r y b e o b t a i n e d b y means of the

distribution

can

32

^ ( B ) 4

^

^

~^ L X C B - B ^ ^ K C B -

S ^ ]

IB-£?K

IO and equation ( 4 5 ) . A simple analysis gives the following correlators

< £ , £ > =

( B J O ^ C - Z A ^ ^ ^ / C ^ A V O C A "

«

3

[3/(ZAH

1

) ]

j

OCA'1)].

From this it is seen that the order of smallness of these correlators in the parameter

St

is the same as .in the previous case. The fluctuations are

also relatively large and strongly non-Gaussian. c ) The case where 'M'i

+ 0

, i. e.

+ O

. I n this case we deal

with a first-order phase transition, but one that d i f f e r s from the previous case. At small

O

one can consider B - B ° small. The term "M^ CJ5 - B

0

)

4

has, here, a small e f f e c t . Discarding this term, we have the function •u-(s) =

CS-

which has a local minimum at the point B = B ° and a local maximum at the point B = B ° - ¿-"M'j /

. The excess of the maximum over the minimum

is now equal to

CH^)

^

• The degree of instability

of the point B ° is characterized by the relative excess A At fixed A

-

z c s a O " " ^

the quantity

C^)-

1 -

.

is a function of

C S ^ A / ^ C V a )

2 7 3

Using this equality, it is easy to find correlators in the critical region: fcgo4* < S,BlB> < B,

-

, B ,B > =

( V ^ r ^

U

+

CA

I > + O (/\~A)1 f

3

(

+

1

)]

>

j o( A-")] .

33 We see t h a t t h e e x p o n e n t s of t h e p a r a m e t e r i>e_ d i f f e r from t h e two p r e v ious c a s e s . Due to t h i s , t h e f l u c t u a t i o n s in t h e critical region are somewhat smaller, b u t all the same t h e y are anomalously l a r g e and s t r o n g l y n o n G a u s s i a n . This n o n - G a u s s i a n c h a r a c t e r d i s a p p e a r s with movement away from t h e critical p o i n t . T h e r e is no doubt t h a t phenomena similar to those d e s c r i b e d above also t a k e place in the case of multi-component p h a s e t r a n s i t i o n s , although t h e number of d i f f e r e n t t y p e s of p h a s e t r a n s i t i o n s will increase in these cases.

T h e Statistical B a c k g r o u n d of t h e C l a n s d o r f f - P r i g o g i n e

Criterion

F . Schlögl

Introduction

T h e development of thermodynamics in t h e last few d e c a d e s has been c h a r acterized b y t h e inclusion of n o n - l i n e a r p r o c e s s e s . Previously thermodynamics had been chiefly c o n c e r n e d with thermal equilibrium s t a t e s , and with n o n equilibrium p r o c e s s e s o c c u r r i n g ' n e a r ' an equilibrium, in t h e s e n s e t h a t t h e dynamical e q u a t i o n s can be linearized with r e s p e c t to t h e q u a n t i t i e s ( ' f o r c e s ' and ' f l u x e s ' ) which d e s c r i b e the deviation from this equilibrium. By r e s t r i c t ion to t h i s t y p e of non-equilibrium p r o c e s s , so called 'linear thermodynamics' is d e f i n e d . The field of thermodynamics which is c o n c e r n e d with n o n - e q u i librium p r o c e s s e s d e s c r i b a b l e b y macroscopic 'thermodynamics' q u a n t i t i e s , yet not o c c u r r i n g ' n e a r ' an equilibrium in t h e above s e n s e , is called ' n o n linear' thermodynamics. Most chemical r e a c t i o n s belong to this field and t h u s so do most of t h e thermodynamic p r o c e s s e s important f o r biology. In non-equilibrium thermodynamics 'open' s y s t e m s a r e c o n s i d e r e d which e x c h a n g e e n e r g y or matter or both with t h e i r e n v i r o n m e n t . Of p a r t i c u l a r i n t e r e s t a r e ' s t e a d y ' or ' s t a t i o n a r y ' s t a t e s , in which t h e system does not c h a n g e its macroscopic f e a t u r e s , whereas an e x c h a n g e of e n e r g y or matter o c c u r s s t e a d i l y . A typical example of s u c h a t y p e is a b u r n i n g flame which does not c h a n g e its shape b u t n e v e r t h e l e s s h a s a permanent input and o u t p u t of matter and e n e r g y . Such s t e a d y s t a t e s o f t e n show a distinct i n d i v i d ual s t r u c t u r e in space which r e s t o r e s itself if not d i s t u r b e d too much. We s p e a k of self-organization of 'dissipative' s t r u c t u r e s ( G l a n s d o r f f , P r i g o g i n e , 1971; Nicolis, P r i g o g i n e , 1977). S t a n d a r d examples of s u c h s t r u c t u r e s a r e t h e hydrodynamic p a t t e r n s of the Bénard flow and t h e Taylor flow which were known a l r e a d y in t h e time b e f o r e n o n - l i n e a r thermodynamics became a field of its own. T h e s e p a t t e r n s a r e t h e r e s u l t of a competition between t e m p e r a t u r e g r a d i e n t as d r i v i n g force f o r heat flow and g r a v i t a t i o n in t h e

Thermodynamics and Regulation of Biological and Processes © 1984 Walter de Gruyter & Co., Berlin • New York - Printed in Germany

36

Benard s y s t e m . T h e y a r e t h e r e s u l t of a competition between f r i c t i o n and c e n t r i f u g a l f o r c e in t h e Taylor s y s t e m . Friction as well as mixing h a s to b e c o n s i d e r e d as a typical thermodynamic phenomenon, even if not c o n n e c t e d with c h a n g e of t e m p e r a t u r e , b e c a u s e it is an i r r e v e r s i b l e p r o c e s s . Some s y s t e m s also show dissipative s t r u c t u r e s in time i n s t e a d of s p a c e . T h e y r e t u r n periodically to t h e same s t a t e s once t h e y h a v e r e a c h e d t h e so called 'limit c y c l e ' . In t h e following, h o w e v e r , we shall not be c o n c e r n e d with them, but with the s t e a d y s t a t e s . The development in time of macroscopically d e s c r i b e d thermal s t a t e s of t h e considered - in general open - system will be given b y t h e solutions of d i f f e r e n t i a l e q u a t i o n s in time f o r t h e r e s p e c t i v e macroscopic thermal v a r i a b l e s . These may be called the macroscopic dynamic e q u a t i o n s . T h e s t e a d y solutions are those f o r which t h e thermal v a r i a b l e s of t h e open system i t self do not c h a n g e in time. However, only t h e s t a b l e , s t e a d y solutions which r e s t o r e themselves a f t e r b e i n g p e r t u r b e d within c e r t a i n b o u n d s a r e practically realisable. A c h a r a c t e r i s t i c f e a t u r e of n o n - l i n e a r thermodynamics is the o c c u r r e n c e of u n s t a b l e s t e a d y solutions. And this is closely related to t h e formation of s t r u c t u r e s . A s t e a d y s t a t e belongs to c e r t a i n c o n s t r a i n t s of t h e e n v i r o n m e n t . It can h a p p e n t h a t an u n s t a b l e solution b e comes stable or vice v e r s a b y a continuous c h a n g e of a p a r a m e t e r c h a r a c t e r izing t h e c o n s t r a i n t s . A plot of t h e s t e a d y s t a t e s of t h e system in a diagram of t h e macrovariables is an analogue to a s t a t e diagram of equilibrium s t a t e s of a system in equilibrium thermodynamics. The c h a n g e from stability to instability can be compared, f o r i n s t a n c e , with t h e c h a n g e of t h e u n magnetized s t a t e of a f e r r o m a g n e t p a s s i n g with d e c r e a s i n g t e m p e r a t u r e t h r o u g h the critical t e m p e r a t u r e . Above t h e critical point this s t a t e is stable but becomes u n s t a b l e below. The c h a n g e i s , moreover, connected with a b i f u r c a t i o n into t h i s u n s t a b l e s t a t e and into stable s t a t e s with magnetization of d i f f e r e n t o r i e n t a t i o n . Similar b i f u r c a t i o n s o c c u r in t h e diagram of s t e a d y s t a t e s in n o n - l i n e a r thermodynamics, and t h e analogy with t h e situation in equilibrium s y s t e m s in yet f u r t h e r r e s p e c t s h a s led to t h e concept of so called 'non-equilibrium p h a s e - t r a n s i t i o n s ' . T h e s e are phenomena f a r away from equilibrium. As will not be explained in detail, t h e formation of p a t t e r n s i s , in a more complex way, connected with bif u r c a t i o n into competing s t a t e s . Here it is s u f f i c i e n t to s t r e s s t h a t c r i t e r i a

37 for t h e stability of a s t e a d y solution of t h e dynamical e q u a t i o n s a r e of real physical i n t e r e s t . Of c o u r s e t h e r e are v a r i o u s possibilities to check t h e stability of a s t e a d y s t a t e . One is t h e s t u d y of t h e solutions of t h e dynamics in t h e n e i g h b o u r hood of this s t a t e . More a d v a n t a g e o u s , h o w e v e r , a r e c r i t e r i a which can be applied without solving t h e dynamic e q u a t i o n s and which r e q u i r e t h e knowledge of t h e s e e q u a t i o n s o n l y . Such c r i t e r i a are given b y t h e Lyapunov t h e o r y . For a given system v a r i o u s d i f f e r e n t Lyapunov f u n c t i o n s can be f o u n d , d e p e n d e n t on individual f e a t u r e s of t h e s y s t e m . Glansdorff and Prigogine (1970), h o w e v e r , have given a c r i t e r i o n , which is b a s e d on a general thermodynamic q u a n t i t y t a k e n as L y a p u n o v f u n c t i o n (La Salle, Lifshitz, 1961), a n d , which t h u s is valid in t h e same form f o r all systems u n d e r c o n s i d e r a t i o n . So it can be called a thermodynamic t h e o r e m . The value of such a theorem is not only to give a direct answer to t h e original question c o n c e r n i n g s t a b i l i t y , b u t to allow g e n e r a l conclusions with general r e s u l t s which o f t e n a p p e a r relatively i n d e p e n d e n t of t h i s q u e s t i o n . Let t h i s be compared, for i n s t a n c e , with t h e p o s i t i v e n e s s of specific heat as a r e s u l t of thermostatic s t a b i l i t y . The c r i t e r i o n of Glansdorff and Prigogine h a s been given on t h e basis of macroscopic a r g u m e n t s . As it is a really new theorem, it was found in a r a t h e r intuitive and i n d u c t i v e way and not by deduction from known p r i n c i p l e s . T h u s it is t h e r e s u l t of a long d e velopment in which t h e exact conditions for validity had to be found b y d e g r e e s . This development s t a r t s with t h e formulation of t h e principle of minimum e n t r o p y production b y Glansdorff and Prigogine ( P r i g o g i n e ,

1947;

P r i g o g i n e , 1949) which - as we know today - d i s t i n g u i s h e s s t e a d y s t a t e s in the framework of linear thermodynamics if t h e system p o s s e s s e s O n s a g e r s y m m e t r y . Later is was s u s p e c t e d t h a t s t e a d y s t a t e s would be d i s t i n g u i s h e d in n o n - l i n e a r thermodynamics in a similar way b y a v a n i s h i n g ,

non-integrable

d i f f e r e n t i a l form which would have led to a general evolution c r i t e r i o n ( G l a n s d o r f f , P r i g o g i n e , 1964). The final s t e p was p e r f o r m e d b y Glansdorff and Prigogine (1970) who made clear t h a t t h e c r i t e r i o n d i s t i n g u i s h e s stable steady states.

38

In t h e following we shall show how this stability c r i t e r i o n , t h e ' G l a n s d o r f f Prigogine c r i t e r i o n ' which was given in t h e frame of macroscopic t h e r m o dynamics can be u n d e r s t o o d on t h e b a s i s of statistical mechanics (Schlógl, 1967, 1971). This derivation does not d e p e n d on individual f e a t u r e s of t h e system and shows in its g e n e r a l i t y t h a t t h e c r i t e r i o n in macroscopic t h e r m o dynamics is indeed an i n d e p e n d e n t theorem and not a c o n s e q u e n c e of alr e a d y known thermodynamic p r i n c i p l e s . It is t h e only theorem in n o n - l i n e a r thermodynamics which had not been d e d u c t e d earlier and is so g e n e r a l t h a t it can be called a thermodynamic t h e o r e m . In comparison, all o t h e r r e c e n t r e s u l t s in n o n - l i n e a r thermodynamics are valid f o r more special classes of systems. T h e self r e p r o d u c t i o n of an equilibrium s t r u c t u r e , f o r i n s t a n c e a c r y s t a l , is b a s e d on minimum e n e r g y in t h i s s t a t e , or more p r e c i s e l y , on minimum f r e e e n e r g y . T h e r e f o r e s u c h a s t r u c t u r e is s t a b l e . The self r e p r o d u c t i o n of a permanently fed dissipative s t r u c t u r e , like t h a t of a living o r g a n i s m , h o w e v e r , is of totally d i f f e r e n t c h a r a c t e r . It is t h e stability of t h e s t e a d y s t a t e f a r away from equilibrium t h a t is c h a r a c t e r i z e d b y t h e G l a n s d o r f f Prigogine c r i t e r i o n . The foundation of this t h e o r e m , on the basis of statistical mechanics, does not have an i n d u c t i v e c h a r a c t e r in the same s e n s e as in macroscopic p h y s i c s . T h e r e f o r e in t h e following t h i s will be d i s c u s s e d f i r s t , and t h e macroscopic formulation will t h e n be given as t h e r e s u l t .

S t a b i l i t y of s t o c h a s t i c

dynamics

Whereas in macroscopic thermodynamics only 'macrostates' o c c u r , i. e . thermal s t a t e s of a physical system which are d e s c r i b e d b y relatively few macroscopic thermal variables s u c h as t e m p e r a t u r e , p r e s s u r e ,

mass-densities

of chemical components, magnetization and o t h e r s , in statistical mechanics we u s e 'microstates' which a r e d e s c r i b e d b y all d e g r e e s of freedom of all particles and fields which form t h e s y s t e m . A c e n t r a l q u e s t i o n i s : What probability d i s t r i b u t i o n over t h e s e microstates c o r r e s p o n d s to a given macro-

39

state? For equilibrium states the answer is standard. In non-equilibrium states this is a question which we shall deal with later on. First we shall discuss a stability criterion for time dependent probability distributions in general. Let us consider an experiment which allows exactly n. different results. These results form a so called 'sample set' of events. This means, they exclude one another and it is certain that one of these events will occur. Let the events of the sample set be numbered by the subscript a, . We call the whole set of probabilities, these events the probability distribution

^o^

, of

. Later on these events will

be replaced by the microstates of a physical system and -yo will give the statistical description of a macrostate. We shall be concerned with the situation that ^o

is dependent on time, "fc , and that differential equations

of first order in t

determine -yo(t). We shall call these differential

equations a 'stochastic dynamics'. We can represent all possible distributions >p as points in a space of the ku positive rectangular coordinates s\pjL which lie in the subspace (hyper-plane) given by the normalization condition for all - p

: rv

JL

-CM

A .

1

(1)

We call this subspace the 'probability space'. A special solution of a given stochastic dynamics is called 'steady' or 'stationary' if it does not change with time. We are interested in the stability of such a steady solution and study therefore the behaviour of solutions i a U ) neighbourhood of

. We call

which start in the

stable, or more precisely 'Lyapunov

stable', if there exists an arbitrarily small neighbourhood of is never left by any ^ o ( t )

if once in the interior.

Let us consider a function that it vanishes for ^o

of the

equal to

^o

which has the features

, that it is positive for all other

and that, in the probability space all points at which value

which

I—

has a constant

(L

form a closed surface in such a way that all points with smaller

values of

CL lie in the interior of this surface whatever the value of CL .

We call such a function

i

L,

&.

of -yo monotonically increasing near f >

,

40

If now for a given stochastic dynamics there exists such a monotonic function L which never increases with time, that means that the time deriv.o ative L of L is never positive, L ^

O

(2)

in the i n t e r i o r n e i g h b o u r h o o d of l

O

Ap

in t h e probability space

^ d,

o)

t h e n a - / | p ( 0 , once in its i n t e r i o r , will s t a y t h e r e . Moreover this will t h e n hold for all smaller n e i g h b o u r h o o d s (4) with d'< C

. T h a t means t h a t t h e n •jo°

is L y a p u n o v s t a b l e .

L

is called

a 'Lyapunov f u n c t i o n ' . A special class of s u c h f u n c t i o n s ( C s i s z a r d , 1967) is given b y J(-p,f30)=Z-

f

/fU )

w h e r e , f o r positive U. , the function ^ f v - )

(5)

is c o n v e x ; which for o u r

p u r p o s e can be defined b y the positive n a t u r e of t h e second d e r i v a t i v e : f " ( u ) The v a r i a t i o n , the probabilities

cf 3

^O. , of

J

(6) r e s u l t i n g from small c h a n g e s , c f - f X

, of

-jo^ can be o r d e r e d by the powers of S - ^ j . : ¿ a

=

i-...

(7)

(8)

=

tZC^DfC^l)

(9)

41

The variation of second order,

, is always positive and thus J

is a convex function of ^o . The variation of first order yp

becomes equal to

vanishes if

p° because

2 1 d f o ; . =• O . Thus 3 cally near

as a function of

(10)

is minimum at ci

f ^ I •

The requirement that in this case 3 3 -

(ID becomes additive

3X + Ijr

(12)

leads = The corresponding 3

7x

is called the 'Kullback measure' (Kullback,

K C ^ - p " ) = 21 f ^ which is positive, vanishing only if Using K

(13)

, is identical to

1951): (14)

.

as Lyapunov functions we obtain the stability criterion in the

following form (Schlögl, 1971): If in the interior of a neighbourhood of (15)

42

and K C^-jo") ^ O

(16)

holds always and everywhere, then -|o will never leave this neighbourhood. This is a sufficient but not necessary condition for Lyapunov stability of the steady distribution -jo° . We shall show that this criterion yields the thermodynamic macroscopic criterion of Glansdorff and Prigogine. First, however, we shall discuss the role of the Kullback measure in information theory.

Relations to information

measures

In the 1950's when information theory was developed, a close association between information and entropy was revealed. Information theory uses information measures which can be based on the length of a communication. This length is expressed by the number of bits (yes-no decisions) which are necessary to convey the communication. If an observer already knows the probability distribution, /f> , over a sample set, then the length of the communication that the event does occur is (up to a numerical factor defining a unit) given by (17) Its negative measures the knowledge which the observer already had before the observation concerning the question of whether A- will occur. The average of -

is called the 'Shannon-information' (Shannon, Weaver,

1949) (18) and is a measure of the knowledge contained in the distribution

.

43

If, originally, the distribution

-p"

was known, and afterwards was

changed to /p by additional information, then this information can be measured by the mean value of the difference between correct distribution ^o

and Hy^ in the

. This is the 'Kullback measure' ( 1 4 ) .

It is also called 'information gain' or 'excess information' of /f* with respect to

.

The additivity for independent subsystems, which, in thermodynamics, is characteristic of extensive quantities, is considered also in information theory as a characteristic feature which has to be known for an information measure. This requirement, however, allows also an alternative to the Kullback measure, that is, the Renyi-information (Renyi, 1966).

with

•RC-jo,-^ 0 ) = (oc-/ii / l £o^

(19)

f(lx)=

(20)

oL > •A and

This is monotonic in -f 3 near

as well, and can also be used as a

Lyapunov function. For oL equal to one it becomes equal to the Kullback measure

K

. Also, for small deviations =

-p^ - f>Z




of quadratic order, it becomes equal to IX up to the numerical factor oL/Z . We can write K(f> ( "|o 0 )= A I - 2 L ( c > I ( f > ° ) / 3 f ^ ) A " ^

(22)

A X = X(f>) - X f - f u )

(23)

Apx,

(24)

where

and A-yo^ =

-

¿pi

44 This means K

is that part of the power expansion of A l l

to the variations A

with respect

which remains if the linear contribution is ignored.

It is the non-linear variation, A

, of X

in

•jo-space.

In macroscopic thermodynamics the states can be described uniquely by a set of variables which, in statistical theory, are mean values < of observables Vw-^

= ZL W - l f ^

(25)

, defined in the microstates. We shall call such quanti-

ties 'direct variables'. In the canonical distribution, for instance, energy is a direct variable, but temperature is not. Now, if -L is given as a function of the variables , the non-linear variation in Ap -space is equal to that in the space of the macrovariables

because finite

variations in both spaces are also connected linearly: ¿± = ZL wC^ A f V

•

(26)

So we can write also in terms of the macroscopic variables < K =

A

WL

v

X .

It should be stressed that this is true only if I

>

: (27)

is considered as a

function of direct variables (Schlögl, 1971). This restriction, however, is no longer important if the variations of -|o become infinitesimal, Thus for any kind of macroscopic variables K = ¿

< Z )

I

(28)

holds. This means that K

is then equal to the term of

c5 X

which is of second

order in the variation of the macrovariables used for the description (the second order variation of cf(1)t

X

).

O

(29)

then is sufficient for stability of the steady state if it holds for any infinitesimal deviation from this state.

45 Thermodynamic

probability

distributions

A fundamental question in statistical physics is, which probability distribution over the microstates of a physical system is adequate to characterize a given macroscopic state described by comparatively few thermodynamic variables? Comparatively few is meant in comparison to the enormously high number of variables which describe a microstate. In classical mechanics these microvariables are the canonical coordinates and their conjugate momenta. They have continuous ranges of values and form the high-dimensional Gibbs phase space. Let the whole set of these variables, describing a microstate by one point in this space, be called ^

, which takes over the role

of the subscript a, , in the preceding considerations but assumes, in contrast to x- , continuous values. Therefore a probability distribution in Gibbs phase space is described by a probability density A^C^) there corresponds the probability

of finding the system in a micro-

state which lies in a volume element with the volume of the phase space. The summation over gration over

. To /fix.

at the point ^

X has to be replaced by inte-

|

First we consider thermal equilibrium states. In macroscopic physics such a state is completely described by a set of direct variables which in statistical theory are mean values < Mv> = of phase space functions

/Lr(f)

(30)

. The corresponding phase space distrib-

ution is given by AJt(\)

The parameters

=

[ ( £ -

E. / l

•

v

(31)

/Ly > are implicitly determined by the mean values < M ^ > ,

and the parameter

(p

by the normalization of the distribution. The most

familiar case is that in which only one value

< M

>

for the mean value

of energy is given. This leads to the canonical distribution where the only occurring is, except for Boltzmann's constant, the reciprocal temperature. If, in addition for instance, the mean value of a particle number is given, we obtain the grandcanonical disbribution, where a second /l^ is the

46

n e g a t i v e of t h e chemical potential divided b y t e m p e r a t u r e and Boltzmann's c o n s t a n t . The general d i s t r i b u t i o n ( 4 . 2 ) will be called a 'generalized c a n o n ical d i s t r i b u t i o n ' . The macroscopic e n t r o p y of s u c h an equilibrium s t a t e i s , u p to a numerical f a c t o r , equal to t h e negative of t h e Shannon information, H , of t h e d i s t r i b ution 'W. This general connection of e n t r o p y with

was a l r e a d y known by

Boltzmann, long b e f o r e t h e i n t r o d u c t i o n of information m e a s u r e s . The canonical d i s t r i b u t i o n , t h e n , was d e f i n e d b y t h e r e q u i r e m e n t t h a t it c o r r e s p o n d s to maximum e n t r o p y u n d e r t h e r e s t r i c t i o n t h a t t h e mean value of e n e r g y be equal to t h e macroscopic thermal variable 'internal e n e r g y ' . More g e n e r a l l y , t h e generalized canonical d i s t r i b u t i o n c o r r e s p o n d s to maximum e n t r o p y with given values < m v >

. J a y n e s (1967) i n t e r p r e t e d t h i s p r e -

scription b y minimizing information, X

, to avoid any u n j u s t i f i e d p r e j u d i c e ,

and called t h e p r e s c r i p t i o n t h e method of 'unbiased g u e s s ' . The question of which d i s t r i b u t i o n s

c o r r e s p o n d to non-equilibrium s t a t e s

b r i n g s new problems. The macroscopic description of thermodynamic n o n equilibrium s t a t e s can be given b y local time d e p e n d e n t direct v a r i a b l e s . T h e y are mean values of p h a s e space f u n c t i o n s /»'¿'(Vj densities of q u a n t i t i e s

in three-dimensional space if . T h e y become

t i m e - d e p e n d e n t t h r o u g h the t i m e - d e p e n d e n c e of If we c o n s t r u c t ^

which a r e local

.

for given mean values of v ^ C in t h e same way as we

did f o r t h e equilibrium c a s e , we obtain t h e so called local equilibrium disbribution: (32) with t h e abbreviation for the summation o v e r t h e macroscopic local q u a n tities : (33) This local equilibrium d i s t r i b u t i o n s u p r e s s e s c e r t a i n c o r r e l a t i o n s in s p a c e . T h e r e f o r e such a d i s t r i b u t i o n is not suitable for an a d e q u a t e d e s c r i p t i o n of t r a n s p o r t phenomena. The r e a s o n is t h a t t h e method of u n b i a s e d g u e s s , if

47

b a s e d on t h e i n s t a n t a n e o u s mean v a l u e s , i g n o r e s t h e knowledge of earlier values which is contained in the dynamic f e a t u r e s of t h e s y s t e m . An a d e q u a t e thermodynamic description b y local variables holds only if t h e system shows a distinct separation into a microscopic and a macroscopic time scale. T h a t means t h a t t h e c h a n g e s of t h e macroscopic v a r i a b l e s a r e v e r y slow in comparison with t h e microscopic p r o c e s s e s between p a r t i c l e s which give r i s e to c h a n g e s of t h e m i c r o s t a t e s . For t h e s e s y s t e m s a c o n s t r u c t ion of d i s t r i b u t i o n s /Ur which c o r r e c t s t h e above shortcomings of

was

f i r s t given b y Mori (1959). We call them M o r i - d i s t r i b u t i o n s . T h e y have been given b y Mc Lennan (1963) and Zubarev (1965) too b y d i f f e r e n t a r g u m e n t s b u t with t h e same r e s u l t . For o u r p u r p o s e s it is not n e c e s s a r y to know t h e r e s u l t , b u t only t h e manner of c o n s t r u c t i o n . In this method /Co(-t) is obtained b y application of microscopic dynamics to x y ^ C t - x . ) over a time i n t e r v a l X

, which is l a r g e on t h e microscopic, and small on t h e

macroscopic, time scale. In this time i n t e r v a l , t h e mean values of t h e

m.

remain u n c h a n g e d b u t the d i s t r i b u t i o n c h a n g e s fundamentally b e c a u s e all c o r r e l a t i o n s which were s u p p r e s s e d in

a r e fully realised in AJ- b y t h e

r e v e r s i b l e microscopic dynamics d u r i n g t h e time T, h o w e v e r , do not c h a n g e t h e measure X

• T h e s e dynamics,

, b e c a u s e this measure h a s t h e

specific f e a t u r e of b e i n g i n v a r i a n t with r e s p e c t to r e v e r s i b l e motion. The c h a n g e s in t h e system due to the microscopic dynamics d u r i n g time x

are

essentially c h a n g e s in an isolated s y s t e m , b e c a u s e t h e influences of t h e s u r r o u n d i n g s do not become e f f e c t i v e in this short time except within t h e immediate s u r f a c e r e g i o n .

So, p r a c t i c a l l y , the c h a n g e from

t h e time ~c is r e v e r s i b l e . T h u s t h e measure XC/ur) ally equal to L C / ^ ) at t i m e t - T . And XC/GJ-^)

at time t

to Ay d u r i n g is e s s e n t i -

as a f u n c t i o n of macros-

copic variables c h a n g e s essentially only on t h e macroscopic scale. T h a t means, t h a t t h e measure (Schlögl,

is practically equal toX(/Cj^)at any time

1975).

It is possible to define t h e e n t r o p y of a non-equilibrium d i s t r i b u t i o n as b e i n g proportional to the n e g a t i v e of information with t h e same numerical f a c t o r as in t h e equilibrium c a s e . The numerical f a c t o r is fixed by t h e unit of e n e r g y and t e m p e r a t u r e . Total e n t r o p y h a s t h e form

48

G = ^oL^yr /i

(34)

M

where entropy density / > ( * ) is the same function of the local variables as in the equilibrium case. The entropy is correctly given by the local equilibrium distribution. It should be stressed that this is not true for other thermal quantities such as, for instance, transport coefficients.

T h e macroscopic criterion

Now we can give the macroscopic form of the stability criterion for systems with local variables. Any neighbourhood of a steady state in which the nonlinear part of the variation of entropy S

as a function of direct variables

fulfills (35)

(36)

will never be left because of thermodynamic changes. The Glansdorff-Prigogine criterion is concerned with infinitesimal variations and has the following form: A steady state is stable if the second order variation of entropy in this state always increases J

1

" S

à O

(37)

That is (38)

In linear thermodynamics Onsager's symmetry law is formulated by the introduction of 'forces'

49

= /Lv ~ /l^

(39)

describing the deviation from the neighbouring equilibrium state; and of 'fluxes' J

=

< Wa.v >

(40)

describing the dynamic response to these forces. It is not necessary to fix the equilibrium state uniquely to identify the variation state with S

of the steady

. Then (38) can be written in the standard form 3 .

i O.

(41)

(In the original work by Glansdorff and Prigogine this inequality is expressed by the deviation of the steady state values )C

from the moment-

ary state X , which gives the opposite sign in the inequality.) •

Often it is convenient to introduce local flow density vectors ^

and

source densities Co of the quantities with densities by the equation v

v

+ V - ^ = Co*.

(42)

If such a quantity is conserved, like energy, or the number of particles of a component not taking part in chemical reactions, then its source V ^ density Cj is zero. Generally, however, this is not the case and io

is the

local production of this quantity per unit time and volume. In general the flow densities

and source densities c o v are independent quantities of

different character. This can be demonstrated through the example of a many component mixture, in the interior of which, diffusion and chemical * ^ reactions take place. Flow densities

of particle numbers of different

components will in general be independent. The production rates

,

however, are dependent on one another through chemical reactions. They are linear functions of reaction rates \ A the number of which is equal to the number of independent reaction equations and smaller than the number of components: (43)

50

cL^

is the stochiometric coefficient of the V -th component in the /S -th

reaction. Equation

Z. V

-

introduces the affinities

A^

z

A^

A

(44)

. A generalisation of this definition is

possible for any association between rates

and independent rates

As we have to compare the steady state with non-equilibrium states in which the state of the surroundings is always the same, we admit only variations

cf

which vanish at the surface of the system. Therefore

a surface contribution vanishes when the integral in (38) is transformed by Gauss' theorem and we obtain ^ 0.

In traditional linear thermodynamics, the quantities present as 'forces' and the quantities

5

ar

>d -J-

(45)

and-VX"^ are as the corresponding

'fluxes' (de Groot, 1952; Meixner, Reitz, 1959) and thus we can also interpret this inequality in terms of the general form of ( 4 1 ) . The essential point is that the fluxes J

have to be independent. Only the corresponding

)C must be varied independently.

T h e p r i n c i p l e of minimum e n t r o p y

production

Using the shorthand of (41) we can write, generally, for entropy production "P == 3 . X" .

(46)

Its variation is given by ¿ P -

+

.

(47)

51

Glansdorff and Prigogine already pointed out that the expression which occurs in their criterion viz. = 2.6)(

(48)

relates to the contribution to i P forces only, i. e. the fluxes 3

which derives from the variation of are fixed. Let us call this expression

the Glansdorff-Prigogine variation. In linear thermodynamics the forces X" and fluxes 3 are connected by linear equations (de Groot, 1952; Meixner, Reitz, 1959), which are often called transport equations; even if no real transport in space occurs. Let these equations formally be written as

^ „ ^ u r - K h

(49)

r

with the so called Onsager matrix L

. The expression is formal as the

super- and subscripts like v~ are abbreviations for the pairs (v-, ">0 . The summation correspondingly contains integration in space. The matrix L independent of all )C and TS . Only its symmetric element,

L +

is

, enters

into P - Z L L ^ K j X -

SV =

X

( L j ) . y

.

(50)

(5D

We have to distinguish between systems with symmetric L , the so called 'Onsager symmetric' systems, and systems which show a so called 'Casimir symmetry'. In the latter L is not a symmetric matrix. Only in linear thermodynamics, i. e. in the neighbourhood of an equilibrium, and, only for systems with Onsager symmetry do we obtain, as a consequence of the Glandorff-Prigogine criterion, that

\

o (X-) = SL,

Q

a n d w h e r e F ( x ) is t h e g e n e r a l i s e d f r e e e n e r g y of t h e s y s t e m .

(8)

60

For the quantities

X. = X . ( x ) in the Gaussian approximation of (8) there

hold the known relations < ^ > = 6 ^

(9)

(10)

di)

In the stationary case, with account taken of (8) and ( 1 1 ) , then (7) takes the form =

d2)

Expanding the functions f j into a Maclaurin series and noting that moments of odd order are equal to zero, we obtain for (12) the expansion

- • z o ^ - o r ® - ) .

Thus at small

©

a non-linear dynamic system behaves, practically, as a

linear one, and there holds the equality S^o ( D R ) = - Z ©

SpA

(14)

where SpA, equivalent to Sp(DR), is the sum of diagonal elements of the matrices IIAll =

l(

\ lll)Rll . On the other hand, the latter relation is

a consequence of more rigid conditions i. e. of the existence of a stationary distribution. Substituting the Gaussian distribution into ( 2 ) , we obtain the relation

61

,,

(15)

which must hold at all X. This holds if the following matrix relations are valid

JD-T*. ^ T> V — Ote

^

+

i "F Vi

i Ç2"

; + u

u

(17) ;

Relation (16) mirrors (14) and (17) leads to the following

Krasovsky (1974) gives examples of the apllieation of relations (28) to (31) in the analysis of the statistical stability of dynamic systems. Relations of the type (16) to (19) are also useful both in solving identification problems and in investigating the synthesis of optimal control systems through noise. These relations are advantageous since they help simplify the corresponding Fokker-Planck-Kolmogoroff equation (because they are used to decrease the number of independent coefficients in the equations). Relations (14) to ( 1 9 ) , analogous to those presented in the paper by Krasovsky (1974) turn out to be close in many respects to the relations

62

describing the fluctuation-dissipation theorem and to the reciprocity relations in Markov version ( Stratonovich, 1967-1971). However, these relations are obtained here from the stationary conditions, an in non-equilibrium thermodynamics they are derived under condition of time reversibility of the processes in a system.

Time reversibility and thermodynamic t y p e relations for Markov dynamic systems

Time reversibility in dynamic systems is a little-studied problem and is not used practically in solving theoretical and applied problems in the theory of dynamic systems. At the same time, on the basis of time reversibility principle a whole number of important theoretical results with a wide range of application (the Onsager reciprocity relations, the fluctuation-dissipation theorem, their non-linear generalisation) have been obtained in non-equilibrium thermodynamics. Not all dynamic systems possess time reversibility, but it still takes place in a large number of cases. For example, in systems with Gaussian

fluctu-

ations of state variables. Obtaining thermodynamic type relations for such systems (close to the relations obtained by Stratonovich, 1967-1971) would make it possible to simplify the solution of many problems of optimal synthesis and control, and, in addition, to widen the analogies with the results of fluctuation theory and transport phenomena in statistical control theory. The relations obtained would also permit establishment of necessary conditions where the processes in Markov dynamic systems are reversible. This makes it possible to synthesize and study the behaviour of a wide range of dynamic systems which are time reversible.

63

Some difficulties in t h e s t u d y a r e due to t h e absence of a definition of time r e v e r s i b i l i t y f o r Markov n o n - s t a t i o n a r y p r o c e s s e s . Earlier s t u d i e s (Platonov, 1978, 1979, 1981) lead to t h e following general and c o n s t r u c t i v e definition: The Markov p r o c e s s , x ( t ) = { x ^ ( t ) ,

. . . , x n ( t ) ^ is called time r e v e r s i b l e

at t h e time moment t , i f , for any x-j and ~c2, at a given t , t h e r e holds t h e relation -fofX-^t+Tj, I Here p(X2> t x ^ ,

^ - c j = / p C C K j ; t - T z | £*A , t - T „ ) .

(20)

t ^ ) is t h e probability d e n s i t y of a t r a n s i t i o n from t h e

values x^ of t h e p r o c e s s at t h e moment t^ to t h e values X2 at a moment t 2 ; E*. =

+

1 f o r time-even components of t h e Markov p r o c e s s , and

f o r time-odd components (i = 1, . . . ,

= -1

n).

If condition (20) holds for any time moment, t h e n we call t h e p r o c e s s x ( t ) reversible. According to t h e l a t t e r definition, p e r f e c t time r e v e r s i b i l i t y of t h e p r o c e s s is possible only in t h e case of its s t a t i o n a r y r a t e . In t h e publications mentioned above we obtained FPK t y p e e q u a t i o n s , b u t for t h e t r a n s i t i o n probability d e n s i t y p ( x , 11 x ^, t^ )»(t ^ t ^) we have

p ^ ^ r M ^

which d i f f e r from t h e usual kinetic e q u a t i o n s of Marxov t h e o r y (see e . g . S t r a t o n o v i c h , 1970, 1971) in t h e form of t h e c o e f f i c i e n t s . The t r a n s f o r m e d kinetic coefficients a r e determined by the relations

64

where

.

x-, t )

a r e t h e usual i n t e n s i t y c o e f f i c i e n t s e n t e r i n g known

equations of t h e t y p e ( 2 ) ; Ki^ot

^

~ ^ol. ol

= f*. Cx,t) are

are drift coefficients;

diffusion coefficients, etc.

From definition ( 2 0 ) , u s i n g e q u a t i o n s (21) and ( 2 2 ) , we obtain t h e n e c e s s a r y and s u f f i c i e n t conditions f o r time r e v e r s i b i l i t y of t h e p r o c e s s : K

Relations (24) coincide with t h e analogous ones obtained b y S t r a t o n o v i c h (1970), b u t , h e r e , t h e y are d e r i v e d without t h e additional condition -jo o (x)

, which is r e d u n d a n t . If (24) h o l d s , t h e condition

t + T . ) = -f^COiC^i-z) can be shown to hold too. (24) e x t e n d s t h e time r e v e r s i b i l i t y conditions to t h e n o n - s t a t i o n a r y c a s e . The f l u c t u a t i o n - d i s s i p a t i o n relations of Markov non-equilibrium t h e r m o d y n amics are d e r i v e d d i r e c t l y from (24) ( S t r a t o n o v i c h , 1967-1971) by i n t e g r a t i n g with weight -j0 o (k ) evyo [ yr X"} and a s u b s e q u e n t t r a n s i t i o n to asymptotic b e h a v i o u r b y t a k i n g into account t h e n a r r o w n e s s of t h e e q u i librium d i s t r i b u t i o n p (*) ( d i s p e r s i o n = o 0 0 = c o n s t exp

(s'o)]

(26)

where S ( x ) is the entropy of a system, considered as a function of parameters

x (see Landau, Lifshitz, 1976, formulae 110.1).

As is usually done in non-equilibrium thermodynamics, let us introduce the conjugated (to the sequent internal parameters x ) 'thermodynamic forces' by means of the following definition:

K

where

a

=

S A S C K - )

-a *a-

S

X

M

*

( 2 7 )

A S ( * " ) =• S C * ) - S 0

SQ represents the entropy in an equilibrium state. Practically this means the replacement of variables and thus it allows us to rewrite the phenomenological relaxation equations for the thermodynamic parameters x ( t ) : (28)

66

in t h e form of

fluxes ¿*¿

=

f ¿ í * ( > c > } «

(29)

As is shown in t h e work of Stratonovich (1967-1970), relation (29) is an asymptotic b e i n g absolutely c o r r e c t for t h e linear f u n c t i o n s f ( x ) , it d i f f e r s only little from

= 0 ^

in t h e n o n - l i n e a r c a s e .

We assume, t h a t in t h e equilibrium s t a t e x = 0, x = 0, and X = 0. On movement away from t h e equilibrium s t a t e , u n d e r s t e a d y e x t e r n a l conditions , the thermodynamic system t u r n s back to t h e equilibrium s t a t e , which p r o c e s s is accompanied b y a growth of e n t r o p y . This means t h a t t h e entropy production, the function q r W

(30)

d u r i n g t h e relaxation p r o c e s s , has a positive value and is equal to zero in t h e limit. Note, t h a t )|=o.

o5)

The last relation determines the necessary conditions for the existence of a steady state in the system under consideration. For this reason, theoretically, it makes it possible to interprete the conjugated thermodynamic parameters X as definite kinds of forces, which are equal in a disturbed steady state to the real external forces, that affect the thermodynamic system and balance their influence. The production of entropy in the case of switched external forces may be written in the following form: B"C*,>u.)»

A^f^C*,^)

(36)

68

or c r ( X » =

n. H { x

^ J

3

,

(37>

where X, is connected as above, with x through relation ( 2 7 ) . Equation (32) is now transformed in the following way: ^

^

.

(38)

Further let us consider one of the fundamental principles in optimal control theory, the Pontryagin principle of maximum. This determines the necessary conditions for optimisation of u ( t ) , i. e. u ( t ) chosen among the 'possible') develops the evolution of phase coordinates x ( t ) of the dynamic system ( 39) in such a way, that the 'quality functional' t„ I = J f 0 ( * ( t ) , A^(±))cU (40) te reaches a minimum value, that this system passes over from point x ( ° ) = x ( t o ) to the point x^ 1 - 1 = x ^ ) . The solution of this problem (see Pontryagin et a l . , 1969) is based on consideration of the special Hamiltonian form :

¡ti-O

which allows us to write canonical equations for the variables X", =

=

: (42)

69

=

¿.i^.V,*

1

(43)

and (44) 'B XTp

= O.

The principle of maximum declares that, for u ( t ) and the trajectory x ( t ) to be optimal, the vector-function connected with functions u ( t ) and x ( t ) , must satisfy the conditions: 1. At any t e C t o , t . , l the Hamiltonian W (X ft), If ( I ) , /U.) considered as a function uf u reaches the maximum H m a x ( x ( t ) , ( t ) ) at the point u = u(t). 2. In the end of the time interval t^ tt^^tO.VM-O

(45)

should necessarily exist. It happens, that if lf/(t), x ( t ) , u ( t ) obey equations (42) and (43), then ( t ) and H m a x are constants, and the validity of (45) can be verified at any moment of time (not only at t^). The comparison between the equations adopted in optimal control and thermodynamic equations introduced above allows us to discover the following analogies: 1. There exists a formal equality of the dynamic (42) and relaxation (33) equations. 2. If the Hamiltonian (41) is determined as: n. = + , . (46) M-'i

70 then e q u a t i o n s ( 4 2 ) to (44) t a k e the form: (47)

= CT(

/U.) I

couv&t

.

(49)

3. Let v e c t o r u in ( 4 6 ) to ( 4 9 ) b e f i x e d . And let u s determine the f u n c tions

a

in the following way:

Then when

= -A

e q u a t i o n s (48) c o i n c i d e s with e q u a t i o n s

(38).

B u t the p a r a m e t e r s X ( x ) s a t i s f y e q u a t i o n s ( 3 8 ) for e v e r y thermodynamical s y s t e m . T h i s means that in the c a s e of a thermodynamic s y s t e m u n d e r fixed forces u , functions

, i n t r o d u c e d a c c o r d i n g to ( 5 0 ) ,

automatic-

ally s a t i s f y ( 4 8 ) . 4. T h e introduction of

=

- /^-j ) 14-0 — - A

into the Hamiltonian

(46) l e a d s to the following r e s u l t •H-Cx-.-Vp.-U.)-

+

= 0 . (51)

T h i s shows that the time e v o l u t i o n s of p r o c e s s e s in thermodynamic

systems

always d e v e l o p in s u c h a way that the Hamiltonian ( 4 6 ) r e a c h e s i t s maximum value

H m a x

= 0 at e v e r y moment of time. T h i s full e n t r o p y p r o d u c t i o n

along the t r a j e c t o r y I

=

j