com METHODS OF STATISTICAL PHYSICS This graduate-level textbook on thermal physics covers classical thermodynamics, statistical mechanics, and their applications. It describes theoretical methods to calculate thermodynamic properties, such as the equation of state, specific heat, Helmholtz potential, magnetic susceptibility, and phase transitions of macroscopic systems. In addition to the more standard material covered, this book also describes more powerful techniques, which are not found elsewhere, to determine the correlation effects on which the thermodynamic properties are based. Particular emphasis is given to the cluster variation method, and a novel formulation is developed for its expression in terms of correlation functions.
Applications of this method to topics such as the three-dimensional Ising model, BCS superconductivity, the Heisenberg ferromagnet, the ground state energy of the Anderson model, antiferromagnetism within the Hubbard model, and propagation of short range order, are extensively discussed. Important identities relating different correlation functions of the Ising model are also derived. Although a basic knowledge of quantum mechanics is required, the mathe- matical formulation is accessible, and the correlation functions can be evaluated either numerically or analytically in the form of infinite series. Based on courses in statistical mechanics and condensed matter theory taught by the author in the United States and Japan, this book is entirely self-contained and all essential math- ematical details are included.
It will constitute an ideal companion text for graduate students studying courses on the theory of complex analysis, classical mechanics, classical electrodynamics, and quantum mechanics. Supplementary material is also available on the internet at http://uk.org/resources/0521580560/ T O M O Y A S U T A N A K A obtained his Doctor of Science degree in physics in 1953 from the Kyushu University, Fukuoka, Japan. Since then he has divided his time between the United States and Japan, and is currently Professor Emeritus of Physics and Astronomy at Ohio University (Athens, USA) and also at Chubu University (Kasugai, Japan). He is the author of over 70 research papers on the two-time Green’s function theory of the Heisenberg ferromagnet, exact linear identities of the Ising model correlation functions, the theory of super-ionic conduction, and the theory of metal hydrides.
Professor Tanaka has also worked extensively on developing the cluster variation method for calculating various many-body correlation functions.com METHODS OF STATISTICAL PHYSICS TOMOYASU TANAKA www.com Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge , United Kingdom Published in the United States of America by Cambridge University Press, New York www.org Information on this title: www.org/9780521580564 © Tomoyasu Tanaka 2002 This book is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2002 - ---- eBook (NetLibrary) - --- eBook (NetLibrary) - ---- hardback - --- hardback - ---- paperback - --- paperback Cambridge University Press has no responsibility for the persistence or accuracy of s for external or third-party internet websites referred to in this book, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.com To the late Professor Akira Harasima www.com Contents Preface page xi Acknowledgements xv 1 The laws of thermodynamics 1 1.1 The thermodynamic system and processes 1 1.2 The zeroth law of thermodynamics 1 1.3 The thermal equation of state 2 1.4 The classical ideal gas 4 1.5 The quasistatic and reversible processes 7 1.6 The first law of thermodynamics 7 1.7 The heat capacity 8 1.8 The isothermal and adiabatic processes 10 1.10 The second law of thermodynamics 12 1.11 The Carnot cycle 14 1.12 The thermodynamic temperature 15 1.13 The Carnot cycle of an ideal gas 19 1.14 The Clausius inequality 22 1.16 General integrating factors 26 1.17 The integrating factor and cyclic processes 28 1.19 Employment of the second law of thermodynamics 31 1.20 The universal integrating factor 32 Exercises 34 2 Thermodynamic relations 38 2.2 Maxwell relations 41 vii www.com Contents Preface page xi Acknowledgements xv 1 The laws of thermodynamics 1 1.1 The thermodynamic system and processes 1 1.2 The zeroth law of thermodynamics 1 1.3 The thermal equation of state 2 1.4 The classical ideal gas 4 1.5 The quasistatic and reversible processes 7 1.6 The first law of thermodynamics 7 1.7 The heat capacity 8 1.8 The isothermal and adiabatic processes 10 1.10 The second law of thermodynamics 12 1.11 The Carnot cycle 14 1.12 The thermodynamic temperature 15 1.13 The Carnot cycle of an ideal gas 19 1.14 The Clausius inequality 22 1.16 General integrating factors 26 1.17 The integrating factor and cyclic processes 28 1.19 Employment of the second law of thermodynamics 31 1.20 The universal integrating factor 32 Exercises 34 2 Thermodynamic relations 38 2.2 Maxwell relations 41 vii www.com Contents ix 5.6 The four-site reduced density matrix 114 5.7 The probability distribution functions for the Ising model 121 Exercises 125 6 The cluster variation method 127 6.1 The variational principle 127 6.2 The cumulant expansion 128 6.3 The cluster variation method 130 6.4 The mean-field approximation 131 6.5 The Bethe approximation 134 6.6 Four-site approximation 137 6.7 Simplified cluster variation methods 141 6.8 Correlation function formulation 144 6.9 The point and pair approximations in the CFF 145 6.10 The tetrahedron approximation in the CFF 147 Exercises 152 7 Infinite-series representations of correlation functions 153 7.1 Singularity of the correlation functions 153 7.2 The classical values of the critical exponent 154 7.3 An infinite-series representation of the partition function 156 7.4 The method of Padé approximants 158 7.5 Infinite-series solutions of the cluster variation method 161 7.6 High temperature specific heat 165 7.7 High temperature susceptibility 167 7.8 Low temperature specific heat 169 7.9 Infinite series for other correlation functions 172 Exercises 173 8 The extended mean-field approximation 175 8.1 The Wentzel criterion 175 8.2 The BCS Hamiltonian 178 8.4 The ground state of the Anderson model 190 8.5 The Hubbard model 197 8.6 The first-order transition in cubic ice 203 Exercises 209 9 The exact Ising lattice identities 212 9.1 The basic generating equations 212 9.2 Linear identities for odd-number correlations 213 9.3 Star-triangle-type relationships 216 9.4 Exact solution on the triangular lattice 218 www.5 Identities for diamond and simple cubic lattices 221 9.6 Systematic naming of correlation functions on the lattice 221 Exercises 227 10 Propagation of short range order 230 10.1 The radial distribution function 230 10.2 Lattice structure of the superionic conductor αAgI 232 10.3 The mean-field approximation 234 10.4 The pair approximation 235 10.5 Higher order correlation functions 237 10.6 Oscillatory behavior of the radial distribution function 240 10.7 Summary 244 11 Phase transition of the two-dimensional Ising model 246 11.1 The high temperature series expansion of the partition function 246 11.2 The Pfaffian for the Ising partition function 248 11.3 Exact partition function 253 11.4 Critical exponents 259 Exercises 260 Appendix 1 The gamma function 261 Appendix 2 The critical exponent in the tetrahedron approximation 265 Appendix 3 Programming organization of the cluster variation method 269 Appendix 4 A unitary transformation applied to the Hubbard Hamiltonian 278 Appendix 5 Exact Ising identities on the diamond lattice 281 References 285 Bibliography 289 Index 291 www.com Preface This book may be used as a textbook for the first or second year graduate student who is studying concurrently such topics as theory of complex analysis, classical mechanics, classical electrodynamics, and quantum mechanics. In a textbook on statistical mechanics, it is common practice to deal with two im- portant areas of the subject: mathematical formulation of the distribution laws of sta- tistical mechanics, and demonstrations of the applicability of statistical mechanics.
The first area is more mathematical, and even philosophical, especially if we attempt to lay out the theoretical foundation of the approach to a thermodynamic equilibrium through a succession of irreversible processes. In this book, however, this area is treated rather routinely, just enough to make the book self-contained.† The second area covers the applications of statistical mechanics to many ther- modynamic systems of interest in physics. Historically, statistical mechanics was regarded as the only method of theoretical physics which is capable of analyzing the thermodynamic behaviors of dilute gases; this system has a disordered structure and statistical analysis was regarded almost as a necessity. Emphasis had been gradually shifted to the imperfect gases, to the gas–liquid condensation phenomenon, and then to the liquid state, the motivation being to be able to deal with correlation effects.
Theories concerning rubber elasticity and high polymer physics were natural extensions of the trend. Along a somewhat sep- arate track, starting with the free electron theory of metals, energy band theories of both metals and semiconductors, the Heisenberg–Ising theories of ferromagnetism, the Bloch–Bethe–Dyson theories of ferromagnetic spin waves, and eventually the Bardeen–Cooper–Schrieffer theory of super-conductivity, the so-called solid state physics, has made remarkable progress. Many new and powerful theories, such as † The reader is referred to the following books for extensive discussions of the subject: R. Tolman, The Principles of Statistical Mechanics, Oxford, 1938, and D.
ter Haar, Elements of Statistical Mechanics, Rinehart and Co., New York, 1956; and for a more careful derivation of the distribution laws, E. Schrödinger, Statistical Thermodynamics, Cambridge, 1952.com xii Preface the diagrammatic methods and the methods of the Green’s functions, have been de- veloped as applications of statistical mechanics. One of the most important themes of interest in present day applications of statistical mechanics would be to find the strong correlation effects among various modes of excitations. In this book the main emphasis will be placed on the various methods of ac- curately calculating the correlation effects, i., the thermodynamical average of a product of many dynamical operators, if possible to successively higher orders of accuracy.
Fortunately a highly developed method which is capable of accomplish- ing this goal is available. The method is called the cluster variation method and was invented by Ryoichi Kikuchi (1951) and substantially reformulated by Tohru Morita (1957), who has established an entirely rigorous statistical mechanics foun- dation upon which the method is based. The method has since been developed and expanded to include quantum mechanical systems, mainly by three groups; the Kikuchi group, the Morita group, and the group led by the present author, and more recently by many other individual investigators, of course. The method was a theme of special research in 1951; however, after a commemorative publication,† the method is now regarded as one of the more standardized and even rather effec- tive methods of actually calculating various many-body correlation functions, and hence it is thought of as textbook material of graduate level.
Chapter 6, entitled ‘The cluster variation method’, will constitute the centerpiece of the book in which the basic variational principle is stated and proved. An exact cu- mulant expansion is introduced which enables us to evaluate the Helmholtz potential at any degree of accuracy by increasing the number of cumulant functions retained in the variational Helmholtz potential. The mathematical formulation employed in this method is tractable and quite adaptable to numerical evaluation by computer once the cumulant expansion is truncated at some point.10 a four-site approximation and in Appendix 3 a tetrahedron-plus-octahedron approximation are presented in which up to six-body correlation functions are evaluated by the cluster variation method. The number of variational parameters in the calculation is only ten in this case, so that the numerical analysis by any computer is not very time consuming (Aggarwal and Tanaka, 1977).
In the advent of much faster computers in recent years, much higher approximations can be carried out with relative ease and a shorter cpu time. Chapter 7 deals with the infinite series representations of the correlation func- tions.