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Advanced Statistical Mechanics 1122441144 99778899881111224422114444 ttpp..iinndddd 11 1133//1100//2211 33::5522 PPMM Recommended Titles in Related Topics Lectures on Statistical Mechanics by Berthold-Georg Englert ISBN: 978-981-122-457-7 ISBN: 978-981-122-554-3 (pbk) Thermodynamics and Statistical Mechanics by Richard Fitzpatrick ISBN: 978-981-122-335-8 ISBN: 978-981-122-423-2 (pbk) Introduction to Statistical Mechanics by John Dirk Walecka ISBN: 978-981-4366-20-5 ISBN: 978-981-4366-21-2 (pbk) YYoonnggQQii -- 1122441144 -- AAddvvaanncceedd SSttaattiissttiiccaall MMeecchhaanniiccss..iinndddd 11 1133//1100//22002211 44::4411::4400 ppmm Advanced Statistical Mechanics JIAN-SHENG WANG NATIONAL UNIVERSITY OF SINGAPORE, SINGAPORE World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI • TOKYO 1122441144 99778899881111224422114444 ttpp..iinndddd 22 1133//1100//2211 33::5522 PPMM Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. ADVANCED STATISTICAL MECHANICS Copyright © 2022 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. ISBN 978-981-124-214-4 (hardcover) ISBN 978-981-124-215-1 (ebook for institutions) ISBN 978-981-124-216-8 (ebook for individuals) For any available supplementary material, please visit https://www.worldscientific.com/worldscibooks/10.1142/12414#t=suppl Typeset by Stallion Press Email: [email protected] Printed in Singapore YYoonnggQQii -- 1122441144 -- AAddvvaanncceedd SSttaattiissttiiccaall MMeecchhaanniiccss..iinndddd 22 1133//1100//22002211 44::4411::4400 ppmm October18,2021 19:13 AdvancedStatisticalMechanics-9inx6in b4388-fm pagev To Esther, Francis, and Aloysius v B1948 Governing Asia TTTThhhhiiiissss ppppaaaaggggeeee iiiinnnntttteeeennnnttttiiiioooonnnnaaaallllllllyyyy lllleeeefffftttt bbbbllllaaaannnnkkkk BB11994488__11--AAookkii..iinndddd 66 99//2222//22001144 44::2244::5577 PPMM October18,2021 19:13 AdvancedStatisticalMechanics-9inx6in b4388-fm pagevii Preface Statistical mechanicsis a fundamental subjectthateverygraduatestudent ofphysicsneedstolearn. Itisfundamentalasphysicalsystemscannotexist inisolation,eitherbecauseofsmallbutdefiniteinteractionswithotherparts of the world or interactions among similar entities, such as a collection of gas molecules. This book arises out of many years of teaching of a course known as “Advanced Statistical Mechanics.” This book, parallel to the taught course, will begin by reviewing the basics of equilibrium statistical mechanics. The materials should already have been learned in an under- graduate statistical mechanics class. Thus, the coverage will be somewhat formalandbrief. Wewillthengoontointeractingsystems,suchasthevan derWaalsgasandtheIsingmodel,withmean-fieldtheoriesofphasetransi- tions, andcertainexactmethods. Thesecondpartofthebookwillcontain moreadvancedtopicsofnonequilibriumstatisticalmechanics,whichcovers Brownian motion, Langevin equations, Fokker-Planck equations, Kubo’s linear response theory, Boltzmann equations, and the Jarzynski equality. The book, as a whole, offers a semester’s worth of contents — about 13weeksoflectures. Ihavealsocollectedagoodnumberofproblemswhich should be helpful to students in reinforcing fundamental concepts. I like to take this opportunity to thank the Department of Physics of National University of Singapore for giving me the privilege to teach this course. Agooddeal ofinteractionsoverthe yearswith the students taking the course sharpened my own understanding of the subject. I also thank ProfessorRobertH.Swendsenforintroducingmetotheworldofstatistical mechanics and for exchanges over the concept of entropy lately. Finally, I thank my family members for their love and continued support. vii B1948 Governing Asia TTTThhhhiiiissss ppppaaaaggggeeee iiiinnnntttteeeennnnttttiiiioooonnnnaaaallllllllyyyy lllleeeefffftttt bbbbllllaaaannnnkkkk BB11994488__11--AAookkii..iinndddd 66 99//2222//22001144 44::2244::5577 PPMM October18,2021 19:13 AdvancedStatisticalMechanics-9inx6in b4388-fm pageix Contents Preface vii 1. Thermodynamics 1 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Callen’s postulates . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Definition of temperature . . . . . . . . . . . . . . . . . . 5 1.4 Some formal thermodynamic relations . . . . . . . . . . . 7 1.5 Supplementary reading — differential, total differential, and line integral . . . . . . . . . . . . . . . . . . . . . . . 9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2. Foundation of Statistical Mechanics, Statistical Ensembles 13 2.1 Classical dynamics . . . . . . . . . . . . . . . . . . . . . . 13 2.2 Statistical description . . . . . . . . . . . . . . . . . . . . 15 2.2.1 Liouville’s equation . . . . . . . . . . . . . . . . . 16 2.3 Microcanonical ensemble. . . . . . . . . . . . . . . . . . . 17 2.3.1 Boltzmann’s principle . . . . . . . . . . . . . . . . 18 2.4 Sackur-Tetrode formula . . . . . . . . . . . . . . . . . . . 19 2.4.1 Thermodynamic limit . . . . . . . . . . . . . . . . 20 2.4.2 Equations of states . . . . . . . . . . . . . . . . . 21 2.5 From microcanonical ensemble to canonical ensemble . . . 22 2.5.1 Connection to thermodynamics . . . . . . . . . . 24 2.5.2 Harmonic oscillators . . . . . . . . . . . . . . . . . 25 2.6 Grand canonical ensemble . . . . . . . . . . . . . . . . . . 27 ix