Hydrodynamic fluctuations, broken symmetry, and correlation functions

344 Pages·1975·8.112 MB·English

by  Dieter Forster

Hydrodynamic Fluctuations,
Broken Symmetry, and
Correlation Functions
Dieter Forster
Temple University
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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