Nonlinear Nonequilibrium Thermodynamics I: Linear and Nonlinear Fluctuation-Dissipation Theorems

375 Pages·1992·27.1 MB·English

by  Professor Dr. Rouslan L. Stratonovich (auth.)

Springer Series in Synergetics
Editor: Hermann Haken
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Linear and Nonlinear Fluctuation-Dissipation Theorems By R. Stratonovich
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Rouslan L. Stratonovich
Nonlinear
Nonequilibrium
Thermodynamics I
Linear and Nonlinear
Fluctuation-Dissipation Theorems
With 26 Figures
Springer-Verlag
Berlin Heidelberg NewY ork
London Paris Tokyo
Hong Kong Barcelona Budapest
Professor Dr. Rouslan L. Stratonovich
Physics Department
Moscow State University
Lenin Hills
119899 Moscow, Russia
Series Editor:
Professor Dr. Dr. h. c. Hermann Haken
Institut mrTheoretische Physik und Synergetik der Universitiit Stuttgart,
Pfaffenwaldring 57/IV,7000 Stuttgart 80, Fed. Rep. of Germany, and Center
for Complex Systems, Florida Atlantic University, Boca Raton, FL 33431, USA
ISBN-13 :978-3-642-77345-7 e-ISBN-13 :978-3-642-77343-3
DOl: 10.1007/978-3-642-77343-3
Library of Congress Cataloging-in-Publication Data
Stratonovich, R. L.
Nonlinear non equilibrium thermodynamics I: linear and nonlinear
fluctuation-dissipation theorems / Rouslan Stratonovich.
p. cm. --(Springer series in synergetics ; v. 57)
Includes bibliographical references and index.
ISBN -13 :978-3-642-77345-7
1. Nonequilibrium thermodynamics. 2. Nonlinear theories.
I. Title: Nonlinear nonequilibrium thermodynamics 1.
II. Title: Nonlinear nonequilibrium thermodynamics one. III. Series.
QC318.I7S664 1992 536'.7--dc20 92-27990 CIP
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Preface
This book gives the first detailed coherent treatment of a relatively young branch of
statistical physics - nonlinear nonequilibrium and fluctuation-dissipative thermo
dynamics. This area of research has taken shape fairly recently: its development
began in 1959. The earlier theory -linear nonequilibrium thermodynamics - is in
principle a simple special case of the new theory. Despite the fact that the title of
this book includes the word "nonlinear", it also covers the results of linear
nonequilibrium thermodynamics. The presentation of the linear and nonlinear
theories is done within a common theoretical framework that is not subject to the
linearity condition.
The author hopes that the reader will perceive the intrinsic unity of this
discipline, and the uniformity and generality of its constituent parts. This theory has a
wide variety of applications in various domains of physics and physical chemistry,
enabling one to calculate thermal fluctuations in various nonlinear systems.
The book is divided into two volumes. Fluctuation-dissipation theorems (or
relations) of various types (linear, quadratic and cubic, classical and quantum) are
considered in the first volume. Here one encounters the Markov and non-Markov
fluctuation-dissipation theorems (FDTs), theorems of the first, second and third
kinds. Nonlinear FDTs are less well known than their linear counterparts.
Advanced problems are considered in the second volume. Among them the
reader will find (1) the non-Markov generating equation from which FDTs of all
degrees of nonlinearity can easily be obtained; (2) nonequilibrium thermodynamics
of open systems; (3) FDTs for coherent waves interacting with a physical body
(these are sometimes called the Kirchhof-type FDTs); and (4) the problem of
obtaining the Markov process and Markov FDT from Hamiltonian dynamics.
The connection and interdependence of the material in the various chapters are
illustrated in the diagram on the next page.
The first volume of the book begins with two introductory chapters. Chapter 1
gives some preliminary concepts of nonequilibrium thermodynamics and a historical
outline of its development. Chapter 2 presents some useful information, in particu
lar a number of useful concepts of probability theory and some important facts of
equilibrium thermodynamics. The reader might possibly find the presentation in
this chapter somewhat unusual. The important parts of the theory are the non
equilibrium thermodynamics of Markov systems, i.e. systems with no after-effect
(Chaps. 3 and 4), and the nonequilibrium thermodynamics of arbitrary systems in
which an after-effect is allowed (Chaps. 5 and 6). In the absence of an after-effect a
system is described by phenomenological equations that can be reduced to a set of
memoryless first-order equations. These equations are the starting point for the
VI Preface
r----- ----
I
------------------- I
-~
Vol. D
Ch.4
Markov theory. A set of functions - admittances or impedances -, which describe
the response of the system to variable external input forces, are the starting point
for the non-Markov, i.e. more general nonequilibrium thermodynamics.
The level of rigor and mathematical techniques in the book are those generally
accepted in theoretical physics (see, e.g., Course of Theoretical Physics by L.D.
Landau and E.M. Lifshits).
The book will appeal to theoretical physicists and applied scientists. The latter
may wish to ignore derivations of some universal relations and simply utilize them
to handle problems of interest to them. The theoretical treatment is supported by
numerous illustrative examples and applications of the general results to a variety
of electrical, thermal, mechanical and chemical systems. It is shown that the most
complete analysis of specific systems is achieved by the fusion of the Markov and
non-Markov techniques of nonequilibrium thermodynamics.
The bibliography, especially in relation to linear nonequilibrium thermo
dynamics and the nonequilibrium thermodynamics of open systems, does not
claim to be exhaustive.
Chapters 1-5 were translated by Mrs. V.V. Stratonovich, and Chap. 6 was
translated by Mr. A.P. Repjev.
Preface VII
I would like to thank Professor H. Haken for his suggestion to publish this
book in the Springer Series in Synergetics. I am grateful to the first reader of the
book Professor Yu. Klimontovich for his useful comments. I also thank Angela
Lahee for her careful editing of the manuscript.
Moscow, June 1992 R.L. Stratonovich
Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 What Is Nonlinear Nonequilibrium Thermodynamics? . 1
1.1.1 Foundations of Nonequilibrium Thermodynamics 3
1.1.2 What Nonequilibrium Results Are Discussed in This Book? 4
1.1.3 Distinguishing Features of Nonlinear Nonequilibrium Thermody-
namics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 5
1.2 Early Work on Nonlinear Nonequlibrium Thermodynamics . . . . . . . .. 6
1.3 Some Particular Problems and Their Corresponding FDRs: Historical
Aspects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 7
1.3.1 Einstein's Problem: Determination of the Diffusion Coefficient of a
Brownian Particle ............................ 7
1.3.2 A Second Problem: Determination of the Intensity of a Random Force
Acting on a Brownian Particle . . . . . 8
1.3.3 The More General Linear Markov FDR 9
1.3.4 Onsager's Reciprocal Relations .... 11
1.3.5 Nyquist's Formula . . . . . . . . . . . 13
1.3.6 The Callen-Welton FDT and Kubo's Formula 16
1.3.7 Mori's Relation. . . . . . . . . . . . . . . . 18
1.3.8 Thermal Noise of Nonlinear Resistance: The Markov Theory 19
1.3.9 Thermal Noise of Nonlinear Resistance: The Non-Markov Theory 22
2. Auxiliary Information Concerning Probability Theory and Equilibrium Ther-
modynamics . . . . . . . . . . . . . . . . . . . . . . 25
2.1 Moments and Correlators. . . . . . . . . . . . . . . . . 25
2.1.1 Moments and the Characteristic Function. . . . . 25
2.1.2 Correlators and Their Relationship with Moments 26
2.1.3 Moments and Correlators in Quantum Theory . . 29
2.2 Some Results of Equilibrium Statistical Thermodynamics 30
2.2.1 Entropy and Free Energy . . . . . . . . . . . . . 30
2.2.2 Thermodynamic Parameters. The First Law of Thermodynamics . 31
2.2.3 The Second Law of Thermodynamics . . . . . . . . . . . . . . 33
2.2.4 Characteristic Function of Internal Parameters and Free Energy 33
2.2.5 Thermodynamic Potential r (a) . . . . . . . . . . . . . . . . . 35
X Contents
2.2.6 Conditional Entropy ......................... 36
2.2.7 Formulas Determining the Equilibrium Probability Density of Inter-
nal Parameters ............................ 38
2.2.8 Conditional Thermodynamic Potentials and the First Law of Ther
modynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 40
2.2.9 The Functions S (B) and F (B) and the Second Law of Thermody-
namics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 43
2.2.10 The Case in which Energy Is an Argument of Conditional Entropy . 44
2.2.11 Formulas of Quantum Equilibrium Statistical Thermodynamics 45
2.3 The Markov Random Process and Its Master Equation. . . . . . . . 46
2.3.1 Definition of a Markov Process. . . . . . . . . . . . . . . . . 46
2.3.2 The Smoluchowski-Chapman Equation and Its Consequences 47
2.3.3 The Master Equation . . . . . . . . . . . . . . . . . 49
2.3.4 The Fokker-Planck Equation and Its Invariant Form .. 50
2.3.5 The Stationary Markov Process ............. 51
2.4 Infinitely Divisible Probability Densities and Markov Processes 52
2.4.1 Infinitely Divisible Probability Density. . . . . . . . . . 52
2.4.2 Stationary Markov Process with Independent Increments 54
2.4.3 Arbitrary Markov Processes 56
2.5 Notes on References to Chapter 2 ............... 57
3. The Generating Equation of Markov Nonlinear Nonequilibrium Thermody
namics. . . . . . . . . . . . . . . . . . 59
3.1 Kinetic Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 59
3.1.1 Definition of Kinetic Potential . . . . . . . . . . . . . . . . . . . .. 59
3.1.2 Relation Between the Kinetic Potential and the Free Energy: Asymp
totic Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 61
3.1.3 Example: Kinetic Potential for a System with Linear Relaxation and
Quadratic Free Energy . . 62
3.1.4 Kinetic Potentiallmage .............. 63
3.1.5 Modified Kinetic Potential . . . . . . . . . . . . . 65
3.1.6 Properties of the Kinetic Potential and of Its Image 67
3.2 Consequences of Time Reversibility . . . . . . . . . . . 69
3.2.1 Time-Reversal Symmetry of the Hamiltonian and of the One-Time
Probability Density . . . . . . . . . . . . . . . . . . . . . . . . . .. 69
3.2.2 Conditions Imposed on Transition Probabilities by Time Reversibility. 71
3.2.3 Time-Reversal and the Markov Operator . . . . . . . . . . . 72
3.2.4 Restrictions Imposed on the Kinetic Potential and on Its Image . . .. 74
3.2.5 The Modified Generating Equation . . . . . . . . . . . . . . . . .. 76
3.3 Examples of the Kinetic Potential and of the Validity of the Generating
Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 77
Contents XI
3.3.1 Consequences of the Generating Equation for a System with Linear
Relaxation and Quadratic Free Energy. . . . . . . . . . . . . . . .. 77
3.3.2 Diode Model of a Nonlinear Resistor: Relaxation Equation . . . . .. 77
3.3.3 Diode Model: Explanation of the Paradox Related to Detection of
Thermal Fluctuations. . . . . . . . . . . . . . . . . . . . . . . . .. 80
3.3.4 Diode Model: The Kinetic Potential and Its Image. . . . . . . . . .. 82
3.3.5 Poisson Model of Nonlinear Resistor: Construction of the Markov
Operator Using Current-Voltage Characteristics 83
3.3.6 Gupta's Formulas .............. 86
3.4 Other Examples: Chemical Reactions and Diffusion 90
3.4.1 Chemical Reactions and Reaction Equations. 90
3.4.2 Chemical Potentials ............. 91
3.4.3 The Kinetic Potential Corresponding to (3.4.5) 93
3.4.4 Extent of Reaction and the Corresponding: Kinetic Potential. 95
3.4.5 Chemical Reactions as Spatial or Continuum Fluctuational Processes 97
3.4.6 Diffusion of a Gaseous Admixture in a Homogeneous Gas. 98
3.4.7 Chemical Reactions with Diffusion. . . . . . . . 101
3.5 Generating Equation for the Kinetic Potential Spectrum . 102
3.5.1 The Kinetic Potential Spectrum 102
3.5.2 The Generating Equation . 103
3.5.3 Examples of Spectra . . . . 105
3.6 Notes on References to Chapter 3 107
4. Consequences of the Markov Generating Equation . 108
4.1 Markov FDRs. . . . . . . . . . . . . . . . . . 108
4.1.1 Relations for Images of Coefficient Functions 108
4.1.2 Basic Fluctuation-Dissipation Relations . . . 110
4.1.3 Modified FDRs . . . . . . . . . . . . . . . . 112
4.1.4 Generalization of FDRs to the Case of an External Magnetic Field or
Other Time-Odd Parameters . . . . . . . . . . . 113
4.1.5 The Functions R+ and R_ and their Relationship 113
4.1.6 Another Form of Many-Subscript Relations . . . 114
4.2 Approximate Markov FDRs and Their Covariant Form 116
4.2.1 Approximate Relationship Between the Coefficient Function and Its
Image. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
4.2.2 Markov FDRs in Zeroth and First Orders in k . . . . . . . . . . 118
4.2.3 The Covariant Form of One-Subscript and Two-Subscript FDRs 120
4.2.4 The Covariant Form of Quadratic FDRs . . . . . . . . . . . . . 121
4.2.5 The Covariant Form of Cubic FDRs . . . . . . . . . . . . . . . 123
4.3 Application of FDRs for Approximate Determination of the Coefficient
Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126