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Hydrodynamic Fluctuations, Broken Symmetry, and Correlation Functions Dieter Forster Temple University 3a ‘v‘x 45 EJ 2 as it ¢(Wod *m= 1975 THE BENJAMIN/CUMMINGS PUBLISHING COMPANY, INC. ADVANCED BOOK PROGRAM Reading, Massachusetts London e Amsterdam e Don Mills, Ontario e Sydney e Tokyo CODEN: FRHPA Firstprinting, 1975 Secondprinting, 1980 Library of Congress Cataloging in Publication Data Forster, Dieter, 1938- Hydrodynamic fluctuations, broken symmetry, and correlation functions. (Frontiers in physics ; 47) Bibliography: p. 1. Correlation (Statistics) 2. Fluid dynamics. 3. Symmetry (Physics) 4. Spin waves. I. Title. II. Series. QC20.7.C6F67 530.1'5'9537 75-16294 Reproduced by W. A. Benjamin, Inc., Advanced Book Program, Reading, Massachusetts,fromcamera-readycopypreparedbytheauthor. Copyright © 1975by W. A. Benjamin, Inc. Published simultaneously inCanada. Allrightsreserved. Nopartofthispublicationmaybereproduced,stored inaretrieval system, or transmitted, in any form or by any means, electronic, mechanical, photo- copying,recording,orotherwise,withoutthepriorwrittenpermissionofthepublisher, W. A. Benjamin, Inc.,Advanced BookProgram,Reading,Massachusetts01867,U.S.A. Manufactured in the United StatesofAmerica BCDEFGHIJK-MA-89876543210 To Martha and Willi Forster CONTENTS Editor's Foreword xiii Preface xvii CHAPTER 1. INTRODUCTION 1 CHAPTER 2. A SIMPLE EXAMPLE--SPIN DIFFUSION 7 2.1 Hydrodynamic Description 9 2.2 Spin Correlation Function (Roughly) 12 2.3 Magnetic Neutron Scattering 16 2.4 The Static Susceptibility 18 2.5 Linear Dynamical Response 20 2.6 Hydrodynamics and Correlation Function 24 2.7 The Fluctuation-Dissipation Theorem 26 2.8 Positivity of wy"(kw) 30 2.9 Sum Rules 31 2. 10 Relaxation Time Approximation 33 2.11 Dispersion Relation Representation 38 CHAPTER 3. FORMAL PROPERTIES OF CORRELATION FUNCTIONS 44 3.1 Linear Dynamical Response 45 3.2 Symmetry Properties 48 3.3 Positivity ofwy"(kw) andDissipation 52 3.4 Sum Rules 53 3.5 The Fluctuation-Dissipation Theorem 54 CHAPTER 4. THE NORMAL FLUID 60 4. 1 The Equations of Fluid Dynamics 63 4.2 Solution of the Hydrodynamic Equations 67 4.3 Thermodynamic Sum Rules 7] 4.4 The Hydrodynamic Correlation Functions 75 4.5 Light Scattering 79 4.6 Kubo Expressions for the Transport Coefficients 83 4.7 Free Particle Behavior 87 4.8 Sum Rule Calculations 88 Contents CHAPTER 5. THR mEmMORY FUNCTION FORMALISM 95 tn- Projectors and Memory Functions 96 nN Memory Function Matrices 103 & nQ Extension to Quantum Mechanics 105 OG n& Spin Diffusion Revisited 109 t CHAPTER 6. BROWNIAN MOTION 121 o- The Momentum Autocorrelation Function 121 oh The Generalized Langevin Equation 131 0G Diffusion ofa Heavy Particle 137 0h The Fokker-Planck Equation 139 A CHAPTER 7. BROKEN SYMMETRY 146 N- Long-Ranged Correlations and Slow Modes 149 NN Broken Symmetry in a Ferromagnet 154 NG The Bogoliubov Inequality 161 NR The Goldstone Theorem 163 NO Some Additional Considerations 166 No Hydrodynamic Goldstone Modes 171 O CHAPTER 8. HYDRODYNAMIC SPIN WAVES IN FERROMAGNETS 180 C- Symmetry Properties 181 oN Undamped Spin Waves 183 c oG Damped Spin Waves 185 t oA Sum Rules and Kubo Formulae 189 c &U Corrections 192 oo Longitudinal Fluctuations 193 cO CHAPTER 9. HYDRODYNAMIC SPIN WAVES IN ANTIFERROMAGNETS 195 os- Long-Ranged Order 197 oN Symmetry Considerations 200 s oG Undamped Spin Waves 203 s oB Spin Wave Damping 206 0n Paro-, Ferro-, and Antiferromagnets 210 U CHAPTER 10. SUPERFLUIDS 214 10.1 Superflow and Long-Ranged Correlations 215 10.2 Broken Gauge Symmetry 223 10.3 Bose Condensation and the Order Parameter 226 10.4 Long-Ranged Phase Coherence 232 10.5 Hydrodynamics without Dissipation 235 10.6 Dissipation 240 10.7 Kubo Relations 246 10.8 The Phenomenological Two-Fluid Equations 248 10.9 Absence of Bose Condensation in 2 Dimensions 251 Contents xi CHAPTER 11. NEMATIC LIQUID CRYSTALS 255 11.- The Free Energy of Distortion 257 11.N The Order Parameter 260 11.t The Light Scattering Intensity 264 11.A Transverse Hydrodynamic Fluctuations 266 11.r Longitudinal Hydrodynamic Fluctuations 274 O 11.O The Phenomenological Equations of Motions 278 11.N What is a Solid? 281 CHAPTER 12. SUPERCONDUCTORS 288 12.1 Bose Condensation in a FermiSystem 288 12.2 Superconductivity and Meissner Effect 293 2.3 Coulomb Forces 296 12.4 Linear Response ofCharged Systems 298 APPENDIX 311 p- Electron Scattering 311 >h Neutron Scattering 313 p >o Magnetic Neutron Scattering 314 t >j Light Scattering 315 References 319 Index 323 EDITOR'S FOREWORD Theproblemofcommunicating in acoherentfashion the recentdevelopments in themostexcitingand activefieldsofphysicsseemsparticularly pressingtoday. Theenormous growth in the numberofphysicistshastended to makethefamiliarchannelsofcommuni- cation considerably lesseffective. Ithasbecome increasinglydifficultforexpertsin agiven field to keep upwith the current literature;the novicecan only beconfused. What isneeded is both aconsistentaccountofafield and the presentation ofadefinite"pointofview" concerning it. Formal monographscannotmeetsuch aneed in arapidlydeveloping field, and, perhapsmore important,the reviewarticleseemstohavefallen intodisfavor. Indeed, itwould seem thatthepeoplemostactivelyengaged indevelopingagiven field arethe people least likelytowriteatlength about it. FRONTIERS IN PHYSICShasbeenconceived in an effortto improvethe situation in several ways. First, totakeadvantageofthe factthatthe leading physiciststoday frequentlygivea seriesof lectures,agraduateseminar, oragraduatecourse in theirspecial fieldsof interest. Such lecturesserveto summarizethe presentstatusof arapidlydeveloping field and maywell constitutetheonly coherentaccountavailable atthetime. Often, noteson lecturesexist (prepared bythe lecturerhimself, bygraduatestudents,or by postdoctoral fellows) and aredistributed in mimeographed form on a limited basis. Oneofthe principal purposesofthe FRONTIERS IN PHYSICS Seriesisto makesuch notesavailableto awider audienceofphysicists. It should be emphasizedthat lecturenotesare necessarily rough and informal, both in styleand content, and those in the Serieswill prove no exception. This isasitshould xiii xiv Editor's Foreword be.The pointoftheSeriesistooffer new, rapid, more informal, and, it ishoped, more effectivewaysfor physiciststo teach oneanother.Thepoint is lost ifonlyelegant notes qualify. Anotherway isto improvecommunication invery activefieldsofphysicsbythe publicationofcollectionsofreprintsofrecentarticles. Such collectionsarethemselvesuseful topeopleworking in thefield. Thevalueofthereprintswould, however,seem much enhanced ifthecollectionwereaccompanied by an introduction of moderate length,whichwould serve totiethecollectiontogetherand, necessarily, constituteabriefsurveyofthe presentstatusof thefield. Again, it isappropriatethatsuch an introduction be informal, in keepingwiththe activecharacterofthefield. A third possibilityfortheSeriesmight becalled an informal monograph,to connotethefactthat itrepresentsan intermediatestep between lecture notesand formal monographs. Itwould offertheauthoran opportunity to present hisviewsof afieldwhich has developed tothepointwhereasummation might proveextraordinarily fruitful, butforwhich a formal monograph might notbefeasibleordesirable. Afourth mannerofpresentation isthecontemporaryclassics--papersorlectures which constitutea particularlyvaluableapproach totheteachingand learningof physics today. Hereonethinksoffieldsthat lieattheheartof much ofpresent-day research, but whoseessentialsareby nowwell understood, such asquantumelectrodynamicsor magnetic resonance. In such fields, some of the best pedagogical material is not readily available, either because it consistsof paperslongoutofprintor lecturesthathavebeverbeen published. Theabovewords,written in August, 1961,continuetobe applicable. Duringthe pastdecade,condensed matterphysicshasundergone a period ofsubstantial growth, asboth experimentalistsand theoristshave madeuseofawidevariety of sophisticated techniques in