Statistical Physics Statistical Physics Second Revised and Enlarged Edition by Tony Guénault Emeritus Professor of Low Temperature Physics Lancaster University, UK A C. Catalogue record for this book is available from the Library of Congress. ISBN 978-1-4020-5974-2 (PB) ISBN 978-1-4020-5975-9 (e-book) Published by Springer, P. Box 17, 3300 AA Dordrecht, The Netherlands.com Printed on acid-free paper First edition 1988 Second edition 1995 Reprinted 1996, 2000, 2001, 2003 Reprinted revised and enlarged second edition 2007 All Rights Reserved © 1988, 1995 A.
Guénault © 2007 Springer No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Table of contents Preface ix 1 Basic ideas 1 1.3 The averaging postulate 3 1.5 The statistical method in outline 6 1.7 Statistical entropy and microstates 10 1.8 Summary 11 2 Distinguishable particles 13 2.1 The Thermal Equilibrium Distribution 14 2.2 What are α and β? 17 2.3 A statistical definition of temperature 18 2.4 The boltzmann distribution and the partition function 21 2.5 Calculation of thermodynamic functions 22 2.6 Summary 23 3 Two examples 25 3.2 Localized harmonic oscillators 36 3.3 Summary 40 4 Gases: the density of states 43 4.1 Fitting waves into boxes 43 4.2 Other information for statistical physics 47 4.3 An example – helium gas 48 4.4 Summary 49 5 Gases: the distributions 51 5.1 Distribution in groups 51 5.2 Identical particles – fermions and bosons 53 5.3 Counting microstates for gases 55 5.4 The three distributions 58 5.5 Summary 61 v vi Table of contents 6 Maxwell–Boltzmann gases 63 6.1 The validity of the Maxwell–Boltzmann limit 63 6.2 The Maxwell–Boltzmann distribution of speeds 65 6.3 The connection to thermodynamics 68 6.4 Summary 71 7 Diatomic gases 73 7.1 Energy contributions in diatomic gases 73 7.2 Heat capacity of a diatomic gas 75 7.3 The heat capacity of hydrogen 78 7.4 Summary 81 8 Fermi–Dirac gases 83 8.1 Properties of an ideal Fermi–Dirac gas 84 8.2 Application to metals 91 8.3 Application to helium-3 92 8.4 Summary 95 9 Bose–Einstein gases 97 9.1 Properties of an ideal Bose–Einstein gas 97 9.2 Application to helium-4 101 9.4 A note about cold atoms 109 9.5 Summary 109 10 Entropy in other situations 111 10.1 Entropy and disorder 111 10.2 An assembly at fixed temperature 114 10.3 Vacancies in solids 116 11 Phase transitions 119 11.1 Types of phase transition 119 11.2 Ferromagnetism of a spin- 12 solid 120 11.3 Real ferromagnetic materials 126 11.4 Order–disorder transformations in alloys 127 12 Two new ideas 129 12.1 Statics or dynamics? 129 12.2 Ensembles – a larger view 132 13 Chemical thermodynamics 137 13.1 Chemical potential revisited 137 13.2 The grand canonical ensemble 139 13.3 Ideal gases in the grand ensemble 141 13.4 Mixed systems and chemical reactions 146 Table of contents vii 14 Dealing with interactions 153 14.1 Electrons in metals 154 14.2 Liquid helium-3: A Fermi liquid 158 14.3 Liquid helium-4: A Bose liquid? 163 14.4 Real imperfect gases 164 15 Statistics under extreme conditions 169 15.1 Superfluid states in Fermi–Dirac systems 169 15.2 Statistics in astrophysical systems 174 Appendix A Some elementary counting problems 181 Appendix B Some problems with large numbers 183 Appendix C Some useful integrals 187 Appendix D Some useful constants 191 Appendix E Exercises 193 Appendix F Answers to exercises 199 Index 201 Preface Preface to the first edition Statistical physics is not a difficult subject, and I trust that this will not be found a difficult book. It contains much that a number of generations of Lancaster students have studied with me, as part of their physics honours degree work. The lecture course was of 20 hours’duration, and I have added comparatively little to the lecture syllabus.
A prerequisite is that the reader should have a working knowledge of basic thermal physics (i. the laws of thermodynamics and their application to simple substances). The book Thermal Physics by Colin Finn in this series forms an ideal introduction. Statistical physics has a thousand and one different ways of approaching the same basic results.
I have chosen a rather down-to-earth and unsophisticated approach, without I hope totally obscuring the considerable interest of the fundamentals. This enables applications to be introduced at an early stage in the book. As a low-temperature physicist, I have always found a particular interest in sta- tistical physics, and especially in how the absolute zero is approached. I should not, therefore, apologize for the low-temperature bias in the topics which I have selected from the many possibilities.
Without burdening them with any responsibility for my competence, I would like to acknowledge how much I have learned in very different ways from my first three ‘bosses’as a trainee physicist: Brian Pippard, Keith MacDonald and Sydney Dugdale. More recently my colleagues at Lancaster, George Pickett, David Meredith, Peter McClintock, Arthur Clegg and many others have done much to keep me on the rails. Finally, but most of all, I thank my wife Joan for her encouragement. Guénault 1988 Preface to the second edition Some new material has been added to this second edition, whilst leaving the organization of the rest of the book (Chapters 1–12) unchanged.
The new chapters aim to illustrate the basic ideas in three rather distinct and (almost) independent ways. Chapter 13 gives a discussion of chemical thermodynamics, including something about chemical equilibrium. Chapter 14 explores how some interacting systems can still be treated by a simple statistical approach, and Chapter 15 looks at two interesting applications of statistical physics, namely superfluids and astrophysics. ix x Preface The book will, I hope, be useful for university courses of various lengths and types.
Several examples follow: 1. Basic general course for physics undergraduates (20–25 lectures): most of Chapters 1–12, omitting any of Chapters 7, 10, 11 and 12 if time is short; 2. Short introductory course on statistical ideas (about 10 lectures): Chapters 1, 2 and 3 possibly with material added from Chapters 10 and 11; 3. Following (2), a further short course on statistics of gases (15 lectures): Chapters 4–6 and 8–9, with additional material available from Chapter 14 and 15.
For chemical physics (20 lectures): Chapters 1–7 and 10–13; 5. As an introduction to condensed matter physics (20 lectures): Chapters 1–6, 8–12, 14, 15. In addition to those already acknowledged earlier, I would like to thank Keith Wigmore for his thorough reading of the first edition and Terry Sloan for his considerable input to my understanding of the material in section 15. Guénault 1994 Preface to the revised and enlarged second edition This third edition of Statistical Physics follows the organization and purpose of the second edition, with comparatively minor updating and changes to the text.
I hope it continues to provide an accessible introduction to the subject, particularly suitable for physics undergraduates. Chapter summaries have been added to the first nine (basic) chapters, in order to encourage students to revise the important ideas of each chapter – essential background for an informed understanding of later chapters. Guénault 2007 Preface xi A SURVIVAL GUIDE TO STATISTICAL PHYSICS Chapter 1 Assembly of N identical particles volume VV, in thermal equilibrium at temperature T are the particles weakly interacting? YES NO gaseous particles? Chapter 12 or 14 could help orr read a thicker book orr give up! NO YES Chapters 2,3 Chapters 4,5 use Boltzmann statistics is occupation number of each state f << 1? N ( h2 ( 3/2 nj = exp (–εεj /kkBT) T N Z i. is A = << 1? V 2MkkBT partition function: Z = Σ (–εεj /kkBT) T all YES NO states Chapters 6,7 is Ψ A or S? use MB statistics F = –NkkBT ln Z N f (ε) = exp (–ε /kkBT) T Z A S 2MkkBT 3/2 Z=V (h2 ( Chapter 8 Chapter 9 for a monatomic gas use FD statistics use BE statistics 1 1 f (ε) = f (ε) = exp {(ε – ) /kkBT } + 1 B exp {ε/kkBT } – 1 N applies to zero or integral F = –kBT ln ( NZ ( applies to odd-half integral spin particles (electrons, 3He, nucleons, some cold spin particles (some cold atoms, 4He) Set B = 1 if no particle conservation atoms) (photons, phonons) 1 Basic ideas There is an obvious problem about getting to grips with an understanding of matter in thermal equilibrium.
Let us suppose you are interested (as a designer of saucepans?) in the thermal capacity of copper at 450 K. On the one hand you can turn to ther- modynamics, but this approach is of such generality that it is often difficult to see the point. Relationships between the principal heat capacities, the thermal expansion coefficient and the compressibility are all very well, but they do not help you to understand the particular magnitude and temperature dependence of the actual heat capacity of copper. On the other hand, you can see that what is needed is a micro- scopic mechanical picture of what is going on inside the copper.
However, this picture becomes impossibly detailed when one starts to discuss the laws of motion of 1024 or so copper atoms. The aim of statistical physics is to make a bridge between the over-elaborate detail of mechanics and the obscure generalities of thermodynamics. In this chapter we shall look at one way of making such a bridge. Most readers will already be familiar with the kinetic theory of ideal gases.
The treatment given here will enable us to discuss a much wider variety of matter than this, although there will nevertheless be some limitations to the traffic that can travel across the bridge.1 THE MACROSTATE The basic task of statistical physics is to take a system which is in a well-defined thermodynamic state and to compute the various thermodynamic properties of that system from an (assumed) microscopic model. The ‘macrostate’ is another word for the thermodynamic state of the system. It is a specification of a system which contains just enough information for its thermody- namic state to be well defined, but no more information than that. As outlined in most books on thermal physics (e.
Finn’s book Thermal Physics in this series), for the simple case of a pure substance this will involve: • the nature of the substance – e. natural copper; • the amount of the substance – e.5 moles; 1 2 Basic ideas • a small number of pairs of thermodynamic co-ordinates – e. pressure P and volume V ; magnetic field B and magnetization M ; surface tension and surface area, etc. Each of these pairs is associated with a way of doing work on the system.
For many systems only P − V work is relevant, and (merely for brevity) we shall phrase what follows in terms of P − V work only. Magnetic systems will also appear later in the book. In practice the two co-ordinates specified, rather than being P and V , will be those appropriate to the external conditions. For instance, the lump of copper might be at a specific pressure P (= 1 atm) and temperature T (= 450 K).
In this case the macrostate would be defined by P and T ; and the volume V and internal energy U and other parameters would then all be determined in principle from P and T. It is precisely one of the objectives of statistical physics to obtain from first principles what are these values of V , U , etc. (In fact, we need not set our sights as low as this. Statistical physics also gives detailed insights into dynamical properties, and an example of this is given in Chapter 12.) Now comes, by choice, an important limitation.