Principles of Statistical Mechanics [1 ed.] 071670319X, 9780716703198

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Principles of Statistical Mechanics

Principles of Statistical Mechanics THE INFORMATION THEORY APPROACH

Amnon Katz WEIZMANN INSTITUTE OF SCIENCE

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W. H. Freeman and Company SAN FRANCISCO AND LONDON

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©

Copyright 1967 by W. H. Freeman and Company

The publisher reserves all rights to reproduce this book, in whole or in part, with the exception of the right to use short quotations for review of the book. Library of Congress Catalog Card Number 67-12181 Printed in the United States of America

Preface

The criterion for acceptance of new ideas in physics is the experimental verifica¬ tion of new results which they may predict. Any conceptual and philosophical clarification that the new idea may bring about is considered merely as a benefit of secondary importance. For this reason, ideas that predict no new verifiable results, but only offer clarification of previous concepts, are themselves con¬ sidered of secondary importance. Their acceptance or rejection remains a matter of personal inclination. The information theory approach to statistical mechanics has not, to my knowledge, led to any concrete new results. For this reason it has remained “controversial,” the controversy being a matter of preference. Let me say a few words to explain my own bias in this matter. The aim of any theory is to predict verifiable results, to obtain new answers to the questions. But a vital stage in any theory is the one at which the questions are formulated. Indeed, a theory is considered complete when the rules tell us how to ask any question—for example, how to set up the Schrodinger equation for every set of interparticle potentials. No theory exists that enables us to answer all questions; we cannot, for example, actually solve the Schrodinger equation for a many-body system. Even when a theory is complete, we still have to struggle with various techniques and approximations in order actually to get at the answers. Information theory makes statistical mechanics a complete theory by setting up rules for asking any question. It contributes nothing, however, to the technical problem of getting the answer. Experts in statistical mechanics have long been able to see their way to asking questions, even without a fixed set of rules. Their efforts have been directed at extracting the answers. Since information theory is of no help in this, it has found little favor with them. In this book, as in the course of lectures on which it is based, 1 have put the pedagogic value of the information theory approach to statistical mechanics to

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vi

Preface

classroom use. 1 have tried to present to students rules for asking questions in statistical mechanics as a set of rules for guessing, which is simply superimposed on the (classical or quantum) canonical dynamical theory. In this 1 have tried to spare them the necessity of learning the mystical art of formulating questions in statistical mechanics. The unity of the whole of statistical mechanics (in and out of equilibrium), as well as the unity of all of physics, is stressed in this way. Chapter I is an introduction to statistical mechanics and the role information theory can play in it. Chapter II introduces the information theory concepts that are needed. Chapter III introduces the mechanics and statistics. Chapter IV completes the theory and fixes the rules for setting up any problem. Succeeding chapters are concerned with applications. Chapter V discusses approximations and idealizations used in actual problems to simplify the formalism. Chapters VI and VII make the connection to equilibrium thermodynamics, and Chapters VIII and IX deal with problems outside equilibrium. Classical and quantum mechanical treatments are developed in parallel and the analogy between them is stressed. It is possible to skip the quantum treat¬ ment and merely read the classical version, for which only a good knowledge of classical Hamiltonian dynamics is necessary. Knowledge of nonrelativistic quantum theory is assumed in the quantum treatment. The connections with phenomenological theory are not always complete; where this is true, the deficiencies are pointed out. In particular, phenomenologi¬ cal equations outside equilibrium have been obtained only for short periods of time, yet in practice they are used for long periods of time. I believe that such inconsistencies are still to be resolved, except possibly for linear equations deal¬ ing with infinitesimal deviations from equilibrium. The policy on the extent and scope of the treatment of each subject is to cover fairly fully the translation of the principles of the information theory approach into the equations specific to the problem at hand. It is broken off at the point where further appeal to these principles is no longer necessary. How far this point is from immediately applicable results, into which one can feed numerical data and obtain numerical answers with a slide rule, depends on the particular subject. For example, the concepts of heat and work are immediately applicable to ideal heat machines, and the general formalism of equilibrium calculations leads to explicit results in the model of ideal gases. Several illustrations are provided of numerical estimates based on these results. (A table of physical constants relevant to the subject matter of this book and necessary for these illustrations is provided in Appendix A.) A set of problems at the end of Chapter VII may guide the reader in extending the ideal-gas model beyond the monatomic case. In dealing with nonequilibrium statistical mechanics, however, the model of ideal gases is useless, and numerical calculations involve lengthy and complicated procedures. The solution of the Boltzmann equation is a subject of study in itself, and the formulas that express transport coefficients in terms of various correlation functions cannot be applied without knowledge of these correlation

Preface

vii

functions in systems of many interacting particles. The solution of the Boltzmann equation or the many-body problem is outside the scope of this book. The inter¬ ested reader should look up these points—as well as the more complicated equilibrium applications—in other works. The contribution of the present text is to show how all these diverse problems are guided by the same principles of statistical mechanics. The material presented here reflects the work of many authors over a long period of time. It is not possible for me to sort them all out and to distribute credit among them. For this reason there are no references, and names are used only where they have become associated with certain methods or formulas. (I should, however, mention E. T. Jaynes, who has pioneered the information theory approach to statistical mechanics.) Since even those parts I have worked out for myself may well be duplication of existing work, I can claim no credit for myself, other than for the logical presentation of the material. I am indebted to D. Bergman and Y. Frishman for going over parts of the manuscript and for many useful comments. I also owe thanks to J. Blatt for many comments and remarks that helped improve this manuscript. AMNON KATZ

Contents Notations and Conventions

i

Chapter 1 Introduction 1. What statistical mechanics is about

5

2. The problem of knowledge

5

3. Getting around the lack of initial data by separating a few degrees of freedom 4. Getting around the lack of initial data by guessing

7

8 11

5. The art and science of guessing

Chapter 2 Information 1. Definition of information 2. An illustration

14 17

3. Missing information when a 4. An infinite choice problem

18 20

5. A continuous choice problem

20

probability distribution is given

Chapter 3 Statistical Mechanics 1. Introduction 2. Classical mechanics 3. Canonical transformations 4. Classical statistics 5. The missing information 6. Quantum mechanics 7. Quantum statistics 8. The missing information

23 23 26 28 31 32 34 37

Chapter 4 Choice of Probabilities 1. 2. 3. 4. 5. 6.

The truth, the whole truth Nothing but the truth Maximum missing information—classical Maximum missing information—quantum mechanical Existence and uniqueness of the solution The Legendre transform

39 40 42 45 47 50

Chapter J General Considerations 1. 2. 3. 4.

The value of information and its age The concept of “Simple quantities” Past, future, and irreversibility The role of constants of the motion. Thermal equilibrium

52 53 55 57

Chapter 0 Equilibrium 1. Scope of our treatment 2. Choice of known constants of the motion 3. Equilibrium calculation 4. Mechanical and statistical Lagrange multipliers 5. Internal properties 6. The meaning of /3 1. (3 and temperature 8. The adiabatic theorem 9. Classical adiabatic theorem 10. Quantum mechanical adiabatic theorem 11. The most general equilibrium 12. Work and heat 13. Persistence of special equilibrium 14. Ideal heat machines

59 59 62 63 65 66 68 71 72 76 78 81 84 90

Chapter / Identical Particles 1. 2. 3. 4. 5. 6. 7. 8.

Identical particles A classical ideal gas of identical particles The number of particles as a dynamical quantity Grand canonical description of the classical ideal gas The quantum mechanical ideal gas Chemical equilibrium Degenerate quantum ideal gases Second quantization and the grand canonical approach

9. Problems

98 96 99 103 104 107 111 119 72/

Chapter O Small Deviations from Equilibrium 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

Introduction Small deviations from equilibrium Phenomenological equations Onsager coefficients and Onsager relations Diffusion The diffusion coefficient Connection with Einstein’s treatment of diffusion Dynamical disturbance of equilibrium Electric conductivity Alternating currents. The fluctuation-dissipation theorem Quantum mechanical modifications

122 122 124 126 132 134 137 139 142 144 145

Chapter V Nonequilibrium Equations 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

Introduction Density in the phase space of N particles Density in the phase space of one particle Equation of motion The phenomenological time derivative The Boltzmann equation Phenomenological increase of entropy The Maxwell-Boltzmann distribution of momenta The quantum case Projection operators as information Time development of the occupation probabilities The master equation Properties of the master equation as a differential equation 14. The master equation for long times

149 150 152 153 155 158 159 162 165 166 168 169 171 173

Appendix A: Constants and units

177

Appendix B: The classical analogue of second quantization

178

Notations and Conventions Three-dimensional vector notation. Ordinary vectors are denoted by q, x; in components, x = (xi, x2, x3). The components are also denoted xr, r — 1, 2, 3. The letters r, s, u are used as component indices and cover the range 1 to 3. The length of a vector x is denoted |x|. The scalar product of two vectors is denoted by a dot: x-y. When a vector is scalarly multiplied into itself, we write x2 == x x. The summation convention is used for repeated component indices. In this way x-y = xryr. A three-dimensional vector that is also a quantum operator is written q, p, etc. Vector differential operators. The gradient with respect to x is denoted d/dx. This means that for any function of space / = /(x), df/dx is a vector with com¬ ponents df/dxr; df/dx = (df/dxi, df/dx-i, df/dx/)- When the gradient d/dx is multiplied into itself, we obtain the laplacian, which we denote by A. We some¬ times write Ax to indicate that the laplacian acts on the variable x. In this way, 2 = d_ dx

d_ dx

Jacobians. The jacobian of &, . . .,

with respect torn,

, Vn is denoted by

d(£i, . . • , ^n)/d(rji, . . . , rjn).

Logarithms. Natural logarithms are denoted by log, decimal ones by log10; logio

x = log x/log 10.

Poisson brackets are denoted {u, v} and are defined as

Conjugations. Complex conjugation is denoted by a star (e.g. A*), Hermitian conjugation by a dagger (e.g. Af).

2

Notations and conventions

Kronecker 6. The Kronecker symbol 5rs is used. It equals one when r = s, and zero otherwise. It is also used for indices other than component indices. Dirac 6. A