OTHER TITLES IN THE SERIES IN NATURAL PHILOSOPHY Vol. 1. DAVIDOV—Quantum Mechanics Vol. 2. FOKKER—Time and Space, Weight and Inertia Vol. 3. KAPLAN—Interstellar Gas Dynamics Vol. 4. ABRIKOSOV, GOR'KOV and DZYALOSHINSKII—Quantum Field Theoretical Methods in Statistical Physics Vol. 5. OKUN'—Weak Interaction of Elementary Particles Vol. 6. SHKLOVSKH—Physics of the Solar Corona Vol. 7. AKHIEZER et al.—Collective Oscillations in a Plasma Vol. 8. KIRZHNITS—Field Theoretical Methods in Many-body Systems Vol. 9. KLIMONTOVICH—The Statistical Theory of Non-equilibrium Processes in a Plasma Vol. 10. KURTH—Introduction to Stellar Statistics Vol. 11. CHALMERS—Atmospheric Electricity (2nd Edition) Vol. 12. RENNER—Current Algebras and their Applications Vol. 13. FAIN and KHANIN—Quantum Electronics, Volume 1 -Basic Theory Vol. 14. FAIN and KHANIN—Quantum Electronics, Volume 2-Maser Amplifiers and Oscillators Vol. 15. MARCH—Liquid Metals Vol. 16. HORI—Spectral Properties of Disordered Chains and Lattices Vol. 17. SAINT JAMES, THOMAS and SARMA—Type II Superconductivity Vol. 18. MARGENAU and KESTNER—Theory of Intermodular Forces Vol. 19. JANCEL—Foundations of Classical and Quantum Statistical Mechanics Vol. 20. TAKAHASHI—An Introduction to Field Quantization Vol. 21. Y VON—Correlations and Entropy in Classical Statistical Mechanics Vol. 22. PENROSE—Foundations of Statistical Mechanics Vol. 23. VISCONTI—Quantum Field Theory, Volume 1 Vol. 24. FURTH—Fundamental Principles of Theoretical Physics Vol. 25. ZHELEZNYAKOV—Radioemission of the Sun and Planets Vol. 26. GRINDLAY—An Introduction to the Phenomenological Theory of Ferroelectricity Vol. 27. UNGER—Introduction to Quantum Electronics Vol. 28. KOGA—Introduction to Kinetic Theory Stochastic Processes in Gaseous Systems Vol. 29. GALASIEWICZ—Superconductivity and Quantum Fluids Vol. 30. CONSTANTTNESCU and MAGYARI—Problems in Quantum Mechanics Vol. 31. KOTKIN and SERBO—Collection of Problems in Classical Mechanics Vol. 32. PANCHEV—Random Functions and Turbulence Vol. 33. TALPE—Theory of Experiments in Paramagnetic Resonance Vol. 34. TER HAAR—Elements of Hamiltonian Mechanics (2nd Edition) Vol. 35. CLARK and GRAINGER—Polarized Light and Optical Measurement Vol. 36. HAUG—Theoretical Solid State Physics, Volume 1 Vol. 37. JORDAN and BEER—The Expanding Earth Vol. 38. TODOROV—Analytic Properties of Feynman Diagrams in Quantum Field Theory Vol. 39. SITENKO—Lectures in Scattering Theory Vol. 40. SOBEL'MAN—Introduction to the Theory of Atomic Spectra Vol. 41. ARMSTRONG and NICHOLLS—Emission, Absorption and Transfer of Radiation in Heated Atmo­ spheres Vol. 42. BRUSH—Kinetic Theory, Volume 3 Vol. 43. BOGOLYUBOV—A Method for Studying Model Hamiltonians Vol. 44. TSYTOVICH—An Introduction to the Theory of Plasma Turbulence STATISTICAL MECHANICS by R. K. PATHRIA Department of Physics, University of Waterloo, Ontario, Canada PERGAMON PRESS Oxford · New York · Toronto Sydney · Braunschweig Pergamon Press Ltd., Headington Hill Hall, Oxford Pergamon Press Inc., Maxwell House, Fairview Park, Elmsford, New York 10523 Pergamon of Canada Ltd., 207 Queen's Quay West, Toronto 1 Pergamon Press (Aust.) Pty. Ltd., 19a Boundary Street, Rushcutters Bay, N.S.W. 2011, Australia Vieweg & Sohn GmbH, Burgplatz 1, Braunschweig Copyright © 1972 R. K. Pathria All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of Pergamon Press Ltd. First edition 1972 Library of Congress Catalog Card No. 73-181690 Printed in Hungary 08 016747 0 TO PROFESSORS F. C. AULUCK AND D. S. KOTHARI who initiated me into the study of this subject PREFACE THIS book has arisen out of the notes of lectures that I have given to the graduate students at the McMaster University (1964-5), the University of Alberta (1965-7), the University of Waterloo (1969-71) and the University of Windsor (1970-1). While the subject matter, in its finer details, has changed considerably during the preparation of the manuscript, the style of presentation remains the same as followed in these lectures. Statistical mechanics is an indispensable tool for studying physical properties of matter "in bulk" on the basis of the dynamical behavior of its "microscopic" constituents. Founded on the well-laid principles of mathematical statistics on one hand and hamiltonian mechanics on the other, the formalism of statistical mechanics has proved to be of immense value to the physics of the last 100 years. In view of the universality of its appeal, a basic knowledge of this subject is considered essential for every student of physics, ir­ respective of the area(s) in which he may be planning to specialize. To provide this knowledge, in a manner that brings out the essence of the subject with due rigor but without undue pain, is the main purpose of this work. The fact that the dynamics of a physical system is represented by a set of quantum states and the assertion that the thermodynamics of the system is determined by the multiplicity of these states constitute the basis of our treatment. The fundamental connection between the microscopic and the macroscopic descriptions of a system is uncovered by investigating the conditions for equilibrium between two physical systems in thermodynamic contact. This is best accomplished by working in the spirit of the quantum theory right from the beginning; the entropy and other thermodynamic variables of the system then follow in a most natural manner. After the formalism is developed, one may (if the situation permits) go over to the limit of the classical statistics. This message may not be new, but here I have tried to follow it as far as is reasonably possible in a textbook. In doing so, an attempt has been made to keep the level of presentation fairly uniform so that the reader does not encounter fluctuations of too wild a character. The text is confined to the study of the equilibrium states of physical systems and is in­ tended to be used for a graduate course in statistical mechanics. Within these bounds, the coverage is fairly wide and provides enough material for tailoring a good two-semester course. The final choice always rests with the individual instructor; I, for one, regard Chapters 1-9 (minus a few sections from these chapters plus a few sections from Chapter 13) as the "essential part" of such a course. The contents of Chapters 10-12 are relatively advanced (not necessarily difficult); the choice of material out of these chapters will depend entirely on the taste of the instructor. To facilitate the understanding of the subject, the xi xii Preface text has been illustrated with a large number of graphs; to assess the understanding, a large number of problems have been included. I hope these features are found useful. I feel that one of the most essential aspects of teaching is to arouse the curiosity of the students in their subject, and one of the most effective ways of doing this is to discuss with them (in a reasonable measure, of course) the circumstances that led to the emergence of the subject. One would, therefore, like to stop occasionally to reflect upon the manner in which the various developments really came about; at the same time, one may not like the flow of the text to be hampered by the discontinuities arising from an intermittent addition of historical material. Accordingly, I decided to include in this account an Historical Introduc­ tion to the subject which stands separate from the main text. I trust the readers, especially the instructors, will find it of interest. For those who wish to continue their study of statistical mechanics beyond the confines of this book, a fairly extensive bibliography is included. It contains a variety of references —old as well as new, experimental as well as theoretical, technical as well as pedagogical. Hopefully, this will make the book useful for a wider readership. Waterloo, Ontario, Canada R. K. P. ACKNOWLEDGMENTS THE completion of this task has left me indebted to many. Like most authors, I owe consider­ able debt to those who have written on the subject before. The bibliography at the end of the book is the most obvious tribute to them ; nevertheless, I would like to mention, in particular, the works of the Ehrenfests, Fowler, Guggenheim, Schrödinger, Rushbrooke, ter Haar, Hill, Landau and Lifshitz, Huang and Kubo, which have been my constant reference for several years and which have influenced my understanding of the subject in a variety of ways. As for the preparation of the text, I am indebted to Robert Teshima who drew most of the graphs and checked most of the problems, to Ravindar Bansal, Vishwa Mittar and Surjit Singh who went through the entire manuscript and made several suggestions that helped me unkink the exposition at a number of points, to Mary Annetts who typed the manuscript with exceptional patience, diligence and care, and to Fred Hetzel, Jim Briante and Larry Kry who provided technical help during the preparation of the final version. As this work progressed I felt increasingly gratified towards Professors F. C. Auluck and D. S. Kothari of the University of Delhi with whom I started my career and who initiated me into the study of this subject, and towards Professor R. C. Majumdar who took keen interest in my work on this and every other project that I have undertaken from time to time. I am grateful to Dr. D. ter Haar of the University of Oxford who, as the general editor of this series, gave valuable advice on various aspects of the preparation of the manu­ script and made several useful suggestions towards the improvement of the text. I am thank­ ful to Professors J. W. Leech, J. Grindlay and A. D. Singh Nagi of the University of Waterloo for their interest and hospitality that went a long way in making this task a pleasant one. The final tribute must go to my wife whose cooperation and understanding, at all stages of this project and against all odds, have been simply overwhelming. Waterloo, Ontario, Canada R. K. P. HISTORICAL INTRODUCTION STATISTICAL mechanics is a formalism which aims at explaining the physical properties of matter in bulk on the basis of the dynamical behavior of its microscopic constituents. The scope of the formalism is almost as unlimited as the very range of the natural phenomena, for in principle it is applicable to matter in any state whatsoever. It has, in fact, been applied, with considerable success, to the study of matter in the solid state, the liquid state or the gaseous state, matter composed of several phases and/or several components, matter under extreme conditions of density and temperature, matter in equilibrium with radiation (as, for example, in astrophysics), matter in the form of a biological specimen, etc. Further­ more, the formalism of statistical mechanics enables us to investigate the nonequilibrium states of matter as well as the equilibrium states; indeed, these investigations help us to understand the manner in which a physical system that happens to be "out of equilibrium" at a given time t approaches a "state of equilibrium" as time passes. In contrast with the present status of its development, the success of its applications and the breadth of its scope, the beginnings of statistical mechanics were rather modest. Barring certain primitive references, such as those of Gassendi, Hooke, etc., the real work started with the contemplations of Bernoulli (1738), Herapath (1821) and Joule (1851) who, in their own ways, attempted to lay foundation for the so-called kinetic theory of gases—a discipline that finally turned out to be the forerunner of statistical mechanics. The pioneering work of these investigators established the fact that the pressure of a gas arose from the motion of its molecules and could be computed by considering the dynamical influence of the mole­ cular bombardment on the walls of the container. Thus, Bernoulli and Herapath could show that, if the temperature remained constant, the pressure P of an ordinary gas was inversely proportional to the volume V of the container (Boyle's law), and that it was essentially independent of the shape of the container. This, of course, involved the explicit assumption that, at a given temperature, the (mean) speed of the molecules is independent of both pressure and volume. Bernoulli even attempted to determine the (first-order) correction to this law, arising from the finite size of the molecules, and showed that the volume V appearing in the statement of the law should be replaced by (V—b) where b is the "actual" 9 volume of the molecules.* Joule was the first to show that the pressure Pis directly propor­ tional to the square of the molecular speed c, which he had assumed to be the same for all the molecules. Krönig (1856) went a step further. Introducing the "quasi-statistical" assump­ tion that, at any time t, one-sixth of the gas molecules could be assumed to be flying in * As is well known, this "correction" was correctly evaluated, much later, by van der Waals (1873) who showed that, for large V, b is equal to four times the "actual" volume of the molecules; see Problem 1.4. l 2 Statistical Mechanics each of the six "independent" directions, namely +JC, —x, +y —y, +z and —z, he de­ 9 rived the equation P = \nmc\ (1) where n is the number density of the molecules and m the molecular mass. Krönig, too, assumed the molecular speed c to be the same for all the molecules; of course, from (1), he concluded that the kinetic energy of the molecules should be directly proportional to the absolute temperature of the gas. Krönig justified his method in these words: "The path of each molecule must be so irregular that it will defy all attempts at calculation. However, according to the laws of probability, one could assume a completely regular motion in place of a completely irregular one!" It must, however, be noted that it is only because of the special form of the summa­ tions appearing in the calculation of the pressure that Krönig's model leads to the same result as the one following from more refined models. In other problems, such as diffusion, viscosity or heat conduction, this is no longer the case. It was at this stage that Clausius entered into the field. First of all, in 1857, he derived the ideal-gas law under assumptions far less stringent than Krönig's. He discarded both of the leading assumptions of Krönig and showed that eqn. (1) was still true; of course, c2 now became the mean square speed of the molecules. In a later paper (1859), Clausius introduced the concept of the mean free path and thus became the first to analyze the transport phe­ nomena. It was in these studies that he introduced the famous "Stosszahlansatz"—the hy­ pothesis on the number of collisions (among the molecules)—which had to play, later on, a prominent role in the monumental work of Boltzmann.* With Clausius, the introduction of the microscopic and statistical points of view into physical theory was definitive, rather than speculative. Accordingly, Maxwell, in a popular article entitled "Molecules", written for the Encyclopedia Britannica, has referred to him as the "principal founder of the kinetic theory of gases", while Gibbs, in his Clausius obituary notice, has called him the "father of statistical mechanics".1" The work of Clausius attracted Maxwell to the field. He made his first appearance with the great memoir "Illustrations in the dynamical theory of gases" (1860), in which he went much ahead of his predecessors by deriving his famous law of "distribution of molecular speeds". This derivation was based on the elementary principles of probability and was clearly inspired by the Gaussian law of "distribution of random errors". A derivation based on the requirement that "the equilibrium distribution of molecular speeds, once acquired, should remain invariant under molecular collisions" appeared in 1867. This led Maxwell to establish what is known as Maxwell's transport equation which, if skilfully used, leads to the same results as the ones following from the more fundamental equation due to Boltzmann. t * For an excellent review of this and related topics, see P. and T. Ehrenfest (1912). t For further details, refer to Montroll (1963) where an account is also given of the pioneering work of Waterston (1846, 1892). Î This has been demonstrated in Guggenheim (1960) where the coefficients o fviscosity, thermal conduc­ tivity and diffusion of a gas of hard spheres have been calculated on the basis of Maxwell's transport equa­ tion. Historical Introduction 3 Maxwell's contributions to the subject diminished considerably after his appointment, in 1871, as the Cavendish Professor at Cambridge. By that time Boltzmann had already made his first strides. In the period 1868-71 he generalized Maxwell's distribution law to polyatomic gases, also taking into account the presence of external forces, if any; this gave rise to the famous Boltzmann factor exp ( — βε), where ε denotes the total energy of a molecule. These investigations also led to the equipartition theorem. Boltzmann further showed that, just like the original distribution of Maxwell, the generalized distribution (which we now call the Maxwell-Boltzmann distribution) is stationary with respect to molecular collisions. In 1872 came the celebrated H-theorem which provided a molecular basis for the natural tendency of physical systems to approach, and stay in, a state of equilibrium. This established the fundamental connection between the microscopic approach (which characterizes statistical mechanics) and the phenomenological approach (which characterized thermodynamics) much more transparently than ever before; it also provided a direct method of computing the entropy of a given physical system from a purely micro­ scopic standpoint. As a corollary to the H-theorem, Boltzmann showed that the Maxwell- Boltzmann distribution is the only distribution that stays invariant under molecular collisions and that any other distribution, under the influence of molecular collisions, ultimately goes over into a Maxwell-Boltzmann distribution. In 1876 Boltzmann derived his famous trans­ port equation which, in the hands of Chapman and Enskog (1916-17), has proved to be an extremely powerful tool for investigating the macroscopic properties of systems in nonsteady states. Things, however, proved quite harsh for Boltzmann. His H-theorem, and the consequent irreversible character of physical systems, came under heavy attack, mainly from Loschmidt (1876-7) and Zermelo (1896). While Loschmidt wondered how the consequences of this theorem could be reconciled with the reversible character of the basic equations of motion of the molecules, Zermelo wondered how these consequences could be made to fit with the quasi-periodic behavior of closed systems (which arose in view of the so-called Poincaré cycles). Boltzmann defended himself against these attacks with all his might but could not convince his opponents of the correctness of his work. At the same time, the energeticists, led by Mach and Ostwald, were criticizing the very (molecular) basis of the kinetic theory,* while Lord Kelvin was emphasizing the "nineteenth-century clouds hovering over the dynamical theory of light and heat".1^ All this left Boltzmann in a state of despair and induced in him a persecution complex.! He wrote in the introduction to the second volume of his treatise Vorlesungen über Gas­ theorie (im):* I am convinced that the attacks (on the kinetic theory) rest on misunderstandings and that the role of the kinetic theory is not yet played out. In my opinion it would be a blow to science if the contempo­ rary opposition were to cause the kinetic theory to sink into the oblivion which was the fate suffered by * These critics were silenced by Einstein whose work on the Brownian motion (1905b) established atomic theory once and for all. t The first of these clouds was concerned with the mysteries of the "aether", and was dispelled by the theory of relativity. The second was concerned with the inadequacy of the "equipartition theorem", and was dispelled by the quantum theory. % Some people attribute Boltzmann's suicide on September 5,1906 to this cause. § Quotation from Montroll (1963).