EQUILIBRIUM STATISTICAL PHYSICS

Transcription

1 EQUILIBRIUM STATISTICAL PHYSICS 3rd Edition

2 This page is intentionally left blank

3 EQUILIBRIUM STATISTICAL 3rd Edition Michael Plischke Simon Fraser University, Canada Birger Bergersen University of British Columbia, Canada YJ? World Scientific NEW JERSEY LONDON SINGAPORE BEIJING SHANGHAI HONGKONG TAIPEI CHENNAI

4 Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore USA office: 27 Warren Street, Suite , Haekensack, NJ UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE Library of Congress Cataloging-in-Publication Data Plischke, Michael. Equilibrium statistical physics / Michael Plischke, Birger Bergersen.-3rd ed. p. cm. ISBN (alk. paper) - ISBN (pbk.: alk. paper) 1. Statistical physics-textbooks. 2. Critical phenomena (Physics)-Textbooks. I. Bergersen, Birger. II. Title. QC174.8.P dc British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Copyright 2006 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. Printed in Singapore by B & JO Enterprise

5 Contents Preface to the First Edition Preface to the Second Edition Preface to the Third Edition 1 Review of Thermodynamics State Variables and Equations of State Laws of Thermodynamics First law Second law Thermodynamic Potentials Gibbs-Duhem and Maxwell Relations Response Functions Conditions for Equilibrium and Stability Magnetic Work Thermodynamics of Phase Transitions Problems 24 xi xiv xv 2 Statistical Ensembles Isolated Systems: Microcanonical Ensemble Systems at Fixed Temperature: Canonical Ensemble Grand Canonical Ensemble Quantum Statistics Harmonic oscillator Noninteracting fermions Noninteracting bosons Density matrix 46 v

6 vi Contents 2.5 Maximum Entropy Principle Thermodynamic Variational Principles Schottky defects in a crystal Problems 54 3 Mean Field and Landau Theory Mean Field Theory of the Ising Model Bragg-Williams Approximation A Word of Warning Bethe Approximation Critical Behavior of Mean Field Theories Ising Chain: Exact Solution Landau Theory of Phase Transitions Symmetry Considerations Potts model Landau Theory of Tricritical Points Landau-Ginzburg Theory for Fluctuations Multicomponent Order Parameters: n-vector Model Problems Applications of Mean Field Theory Order-Disorder Transition Maier-Saupe Model Blume-Emery-Griffiths Model Mean Field Theory of Fluids: van der Waals Approach Spruce Budworm Model A Non-Equilibrium System: Two Species Asymmetric Exclusion Model Problems Dense Gases and Liquids Virial Expansion Distribution Functions Pair correlation function BBGKY hierarchy Ornstein-Zernike equation Perturbation Theory Inhomogeneous Liquids Liquid-vapor interface 164

7 Contents vii Capillary waves Density-Functional Theory Functional differentiation Free-energy functional and correlation functions Applications Problems Critical Phenomena I Ising Model in Two Dimensions Transfer matrix Transformation to an interacting fermion problem Calculation of eigenvalues Thermodynamic functions Concluding remarks Series Expansions High-temperature expansions Low-temperature expansions Analysis of series Scaling Thermodynamic considerations Scaling hypothesis Kadanoff block spins Finite-Size Scaling Universality Kosterlitz-Thouless Transition Problems Critical Phenomena II: The Renormalization Group The Ising Chain Revisited Fixed Points An Exactly Solvable Model: Ising Spins on a Diamond Fractal Position Space Renormalization: Cumulant Method First-order approximation Second-order approximation Other Position Space Renormalization Group Methods Finite lattice methods Adsorbed monolayers: Ising antiferromagnet Monte Carlo renormalization 272

8 viii Contents 7.6 Phenomenological Renormalization Group The e-expansion The Gaussian model The S 4 model Conclusion 290 Appendix: Second Order Cumulant Expansion Problems Stochastic Processes Markov Processes and the Master Equation Birth and Death Processes Branching Processes Fokker-Planck Equation Fokker-Planck Equation with Several Variables: SIR Model Jump Moments for Continuous Variables Brownian motion Rayleigh and Kramers equations Diffusion, First Passage and Escape Natural boundaries: The Kimura-Weiss model for genetic drift Artificial boundaries First passage time and escape probability Kramers escape rate Transformations of the Fokker-Planck Equation Heterogeneous diffusion Transformation to the Schrodinger equation Problems Simulations Molecular Dynamics Conservative molecular dynamics Brownian dynamics Data analysis Monte Carlo Method Discrete time Markov processes Detailed balance and the Metropolis algorithm Histogram methods Data Analysis Fluctuations 365

9 Contents ix Error estimates Extrapolation to the thermodynamic limit The Hopfield Model of Neural Nets Simulated Quenching and Annealing Problems Polymers and Membranes Linear Polymers The freely jointed chain The Gaussian chain Excluded Volume Effects: Flory Theory Polymers and the n-vector Model Dense Polymer Solutions Membranes Phantom membranes Self-avoiding membranes Liquid membranes Problems Quantum Fluids Bose Condensation Superfluidity Qualitative features of superfluidity Bogoliubov theory of the 4 He excitation spectrum Superconductivity Cooper problem BCS ground state Finite-temperature BCS theory Landau-Ginzburg theory of superconductivity Problems Linear Response Theory Exact Results Generalized susceptibility and the structure factor Thermodynamic properties Sum rules and inequalities Mean Field Response Dielectric function of the electron gas Weakly interacting Bose gas 475

10 x Contents Excitations of the Heisenberg ferromagnet Screening and plasmons Exchange and correlation energy Phonons in metals Entropy Production, the Kubo Formula, and the Onsager Relations for Transport Coefficients Kubo formula Entropy production and generalized currents and forces Microscopic reversibility: Onsager relations The Boltzmann Equation Fields, drift and collisions DC conductivity of a metal Thermal conductivity and thermoelectric effects Problems Disordered Systems Single-Particle States in Disordered Systems Electron states in one dimension Transfer matrix Localization in three dimensions Density of states Percolation Scaling theory of percolation Series expansions and renormalization group Rigidity percolation Conclusion Phase Transitions in Disordered Materials Statistical formalism and the replica trick Nature of phase transitions Strongly Disordered Systems Molecular glasses Spin glasses Sherrington-Kirkpatrick model Problems 565 A Occupation Number Representation 569 Bibliography 583 Index 603

11 Preface to the First Edition During the last decade each of the authors has regularly taught a graduate or senior undergraduate course in statistical mechanics. During this same period, the renormalization group approach to critical phenomena, pioneered by K. G. Wilson, greatly altered our approach to condensed matter physics. Since its introduction in the context of phase transitions, the method has found application in many other areas of physics, such as many-body theory, chaos, the conductivity of disordered materials, and fractal structures. So pervasive is its influence that we feel that it now essential that graduate students be introduced at an early stage in their career to the concepts of scaling, universality, fixed points, and renormalization transformations, which were developed in the context of critical phenomena, but are relevant in many other situations. In this book we describe both the traditional methods of statistical mechanics and the newer techniques of the last two decades. Most graduate students are exposed to only one course in statistical physics. We believe that this course should provide a bridge from the typical under-graduate course (usually concerned primarily with noninteracting systems such as ideal gases and paramagnets) to the sophisticated concepts necessary to a researcher. We begin with a short chapter on thermodynamics and continue, in Chapter 2, with a review of the basics of statistical mechanics. We assume that the student has been exposed previously to the material of these two chapters and thus our treatment is rather concise. We have, however, included a substantial number of exercises that complement the review. In Chapter 3 we begin our discussion of strongly interacting systems with a lengthy exposition of mean field theory. A number of examples are worked out in detail. The more general Landau theory of phase transitions is developed and used to discuss critical points, tricritical points, and first-order phase transitions. The limitations of mean field and Landau theory are described and the role of fluctuations is explored in the framework of the Landau-Ginzburg model. xi

12 Xll Preface Chapter 4 is concerned with the theory of dense gases and liquids. Many of the techniques commonly used in the theory of liquids have a long history and are well described in other texts. Nevertheless, we feel that they are sufficiently important that we could not omit them. The traditional method of viral expansions is presented and we emphasize the important role played in both theory and experiment by the pair correlation function. We briefly describe some of the useful and still popular integral equation methods based on the Ornstein-Zernike equation used to calculate this function as well as the modern perturbation theories of liquids. Simulation methods (Monte Carlo and molecular dynamics) are introduced. In the final section of the chapter we present an interesting application of mean field theory, namely the van der Waals theory of the liquid-vapor interface and a simple model of roughening of this interface due to capillary waves. Chapters 5 and 6 are devoted to continuous phase transitions and critical phenomena. In Chapter 5 we review the Onsager solution of the twodimensional Ising model on the square lattice and continue with a description of the series expansion methods, which were historically very important in the theory of critical phenomena. We formulate the scaling theory of phase transitions following the ideas of Kadanoff, introduce the concept of universality of critical behavior, and conclude with a mainly qualitative discussion of the Kosterlitz-Thouless theory of phase transitions in two-dimensional systems with continuous symmetry. Chapter 6 is entirely concerned with the renormalization group approach to phase transitions. The ideas are introduced by means of technically straightforward calculations for the one- and two-dimensional Ising models. We discuss the role of the fixed points of renormalization transformations and show how the theory leads to universal critical behavior. The original e-expansion of Wilson and Fisher is also discussed. This section is rather detailed, as we have attempted to make it accessible to students without a background in field theory. In Chapter 7 we turn to quantum fluids and discuss the ideal Bose gas, the weakly interacting Bose gas, the BCS theory of superconductivity, and the phenomenological Landau-Ginzburg theory of superconductivity. Our treatment of these topics (except for the ideal Bose gas) is very much in the spirit of mean field theory and provides more challenging applications of the formalism developed in Chapter 3. Chapter 8 is devoted to linear response theory. The fluctuation-dissipation theorem, the Kubo formalism, and the Onsager relations for transport coefficients are discussed. This chapter is consistent with our emphasis on equilibrium phenomena in the linear response approximation the central role is

13 Preface xni played by equilibrium correlation functions. A number of applications of the formalism, such as the dielectric response of an electron gas, the elementary excitations of a Heisenberg ferromagnet, and the excitation spectrum of an interacting Bose fluid, are discussed in detail. The complementary approach to transport via the linearized Boltzmann equation is also presented. Chapter 9 provides an introduction to the physics of disordered materials. We discuss the effect of disorder on the quantum states of a system and introduce (as an example) the notion of localization of electronic states by an explicit calculation for a one-dimensional model. Percolation theory is introduced and its analogy to thermal phase transitions is elucidated. The nature of phase transitions in disordered materials is discussed and we conclude with a very brief and qualitative description of the glass and spin-glass transitions. These subjects are all very much at the forefront of current research and we do not claim to be at all comprehensive in our treatment. In compensation, we have provided a more extensive list of references to recent articles on these topics than elsewhere in the book. We have found the material presented here suitable for an introductory graduate course, or with some selectivity, for a senior undergraduate course. A student with a previous course in statistical mechanics, some background in quantum mechanics, and preferably, some exposure to solid state physics should be adequately prepared. The notation of second quantization is used extensively in the latter part of the book and the formalism is developed in detail in the Appendix. The instructor should be forewarned that although some of the problems, particularly in the early chapters, are quite straightforward, those toward the end of the book can be rather challenging. Much of this book deals with topics on which there is a great deal of recent research. For this reason we have found it necessary to give a large number of references to journal articles. Whenever possible, we have referred to recent review articles rather than to the original sources. The writing of this book has been an ongoing (frequently interrupted) process for a number of years. We have benefited from discussion with, and critical comments from, a number of our colleagues. In particular, Ian Affleck, Leslie Ballentine, Robert Barrie, John Berlinsky, Peter Holdsworth, Zoltan Racz, and Bill Unruh have been most helpful. Our students Dan Ciarniello, Victor Finberg and Barbara Frisken have also helped to decrease the number of errors, ambiguities, and obscurities. The responsibility for the remaining faults rests entirely with the authors. MICHAEL PLISCHKE BIRGER BERGERSEN

14 Preface to the Second Edition During the five years that have passed since the first edition of this book was published, we have received numerous helpful suggestions from friends and colleagues both at our own institutions and at others. As well, the field of statistical mechanics had continued to evolve. In composing this second edition we have attempted to take all of this into account. The purpose of the book remains the same: to provide an introduction to state-of-the-art techniques in statistical physics for graduate students in physics, chemistry and materials science. While the general structure of the second edition is very similar to that of the first edition, there are a number of important additions. The rather abbreviated treatment of computer simulations has been expanded considerably and now forms a separate Chapter 7. We have included an introduction to density-functional methods in the chapter on classical liquids. We have added an entirely new Chapter 8 on polymers and membranes. In the discussion of critical phenomena, we have corrected an important omission of the first edition and have added sections on finite-size scaling and phenomenological renormalization group. Finally, we have considerably expanded the discussion of spin-glasses and have also added a number of new problems. We have also compiled a solution manual which is available from the publisher. It goes without saying that we have corrected those errors of the first edition that we are aware of. In this task we have been greatly helped by a number of individuals. In particular, we are grateful to Vinay Ambegaokar, Leslie Ballentine, David Boal, Bill Dalby, Zoltan Racz, Byron Southern and Philip Stamp. Michael Plischke Birger Bergersen Vancouver, Canada xiv

15 Preface to the Third Edition In the third edition we have added a significant amount of new material. There are also numerous corrections and clarifications throughout the text. We have also added several new problems. In Chapter 1 we have added a section on magnetic work, while in Chapter 2 we have added to the discussion of the maximum entropy principle, emphasizing the importance of the assumption that the entropy is extensive in a normal thermodynamic system. In Chapter 3 we have replaced the derivation of the Bragg Williams approximation from the density matrix, to a more intuitive one, stressing the mean field assumption of statistical independence of spins at different sites. We have also added a section on the Potts model. The sections on the Maier- Saupe model for liquid crystals, the Blume-Emery-GrifHths model for 3 He- 4 He mixtures and van der Waals fluid have been moved to a new chapter called "Applications of Mean Field Theory" that includes a section on an insect infestastion model in ecology and also includes a non-equilibrium system: the two species asymmetric exclusion model. This section illustrates the application of mean field theory outside the scope of equilibrium statistical mechanics. The new Chapters 5 and 6 only contain relatively minor changes to the old Chapters 4 and 5. In Chapter 7, the section on the epsilon expansion in the old Chapter 6 has been rewritten, and we have added a section on the Ising model on the diamond fractal. Because of the growing importance of the field we have added a new Chapter 8 on stochastic processes. We start with a description of discrete birth and death processes, and we return to the insect infestation model of Chapter 4. Most of the remainder of the chapter is concerned with the Fokker-Planck equation for both discrete and continuous processes. We apply the theory both to a genetics problems and diffusion of particles in fluids. Other applications involve the rate of excape from a metastable state and problems of heterogeneous diffusion. Finally we show how the Fokker-Planck equation can be transformed into a form similar to the Schrodinger equation, allowing the xv

16 XVI Preface application of techniques familiar from quantum mechanics. In Chapter 9 (old Chapter 7) we have rewritten the section on molecular dynamics and added a subsection on Brownian dynamics. There are only relatively minor changes to Chapters 10 and 11 (old Chapters 8 and 9) except that we have updated the references to the literature in view of important new developments in superconductivity and Bose condensation. A section on rigidity percolation has been added to Chapter 13 (old Chapter 11). Helpful comments and suggestions from Ian Affleck, Marcel Franz, Michel Gingras, Margarita Ifti, Greg Lakatos, Zoltan Racz, Fei Zhou and Martin Zuckermann are gratefully acknowledged. Updated information of interest to readers will be displayed on our website Michael Plischke Birger Bergersen Vancouver, Canada