Statistical Physics. Second Revised and Enlarged Edition. Tony Guénault Emeritus Professor of Low Temperature Physics Lancaster University, UK

Transcription

1 Statistical Physics

2 Statistical Physics Second Revised and Enlarged Edition by Tony Guénault Emeritus Professor of Low Temperature Physics Lancaster University, UK

3 A C.I.P. Catalogue record for this book is available from the Library of Congress. ISBN ISBN (ebook) DOI / 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.M. Guénault Originally published by Springer in 2007 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.

4 Table of contents Preface 1 Basic ideas The macrostate Microstates The averaging postulate Distributions The statistical method in outline A model example Statistical entropy and microstates Summary 11 2 Distinguishable particles TheThermal Equilibrium Distribution What are α and β? A statisticaldefinition of temperature The boltzmann distribution and the partition function Calculation of thermodynamic functions Summary 23 3 Two examples A Spin- 1 2 solid Localized harmonic oscillators Summary 40 4 Gases: the density of states Fitting waves into boxes Other information for statistical physics An example helium gas Summary 49 5 Gases: the distributions Distribution in groups Identical particles fermions and bosons Counting microstates for gases The three distributions Summary 61 v ix

5 vi Table of contents 6 Maxwell Boltzmann gases The validity of the Maxwell Boltzmann limit The Maxwell Boltzmann distribution of speeds The connection to thermodynamics Summary 71 7 Diatomic gases Energy contributions in diatomic gases Heat capacity of a diatomic gas The heat capacity of hydrogen Summary 81 8 Fermi Dirac gases Properties of an ideal Fermi Dirac gas Application to metals Application to helium Summary 95 9 Bose Einstein gases Properties of an ideal Bose Einstein gas Application to helium Phoney bosons A note about cold atoms Summary Entropy in other situations Entropy and disorder An assembly at fixed temperature Vacancies in solids Phase transitions Types of phase transition Ferromagnetism of a spin- 1 2 solid Real ferromagnetic materials Order disorder transformations inalloys Two new ideas Statics or dynamics? Ensembles a larger view Chemical thermodynamics Chemical potential revisited The grand canonical ensemble Ideal gases in the grand ensemble Mixed systems and chemical reactions 146

6 Table of contents vii 14 Dealing with interactions Electrons in metals Liquid helium-3: A Fermi liquid Liquid helium-4: A Bose liquid? Real imperfect gases Statistics under extreme conditions Superfluid states in Fermi Dirac systems 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

7 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.e. 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 statistical physics, and especially in how the absolute zero is approached. I should not, therefore, apologize for the low-temperature bias inthe 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 therails. Finally, but most of all,ithank my wife Joan for her encouragement. Preface to the second edition A.M. Guénault 1988 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 attwointeresting applications of statistical physics, namely superfluids and astrophysics. ix

8 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 addedfrom 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 15.2; 4. 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, 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 Preface to the revised and enlarged second edition A.M. Guénault 1994 This third edition of Statistical Physics follows the organization and purpose of the second edition, with comparativelyminor 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. A.M. Guénault 2007

9 ( Preface xi A SURVIVAL GUIDE TO STATISTICAL PHYSICS Chapter 1 Assembly of N identical particles volume V, in thermal equilibrium at temperature T are the particles weakly interacting? YES NO gaseous particles? Chapter 12 or 14 could help or read a thicker book or give up! NO YES Chapters 2,3 use Boltzmann statistics N n j = exp ( εε j /k B T) Z partition function: Z = Σ ( εε j /k B T) all states F = Nk B T ln Z Chapters 6,7 use MB statistics N f (ε) = exp ( ε /k B T) Z 2 Mk B T 3/2 Z = V ( h 2 for a monatomic gas Chapters 4,5 is occupation number of each state f << 1? h 2 3/2 N i.e. is A= V( <<1? 2 Mk B T YES ( ( A NO Chapter 8 use FD statistics 1 f (ε) = exp {(ε μ)/kb T } + 1 is Ψ A or S? S Chapter 9 use BE statistics 1 f (ε) = B exp {ε/kb T } 1 F = k B T ln ( Z N N applies to odd-half integral spin particles (electrons, 3 He, nucleons, some cold atoms) applies to zero or integral spin particles (some cold atoms, 4 He) Set B=1 if no particle conservation (photons, phonons)