Statistical mechanics: from first principles to macroscopic phenomena

297 Pages·2007·1.629 MB·English

by  J. Woods Halley

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STATISTICAL MECHANICS
FromFirstPrinciplestoMacroscopicPhenomena
Based on the author’s graduate course taught over many years in several physics
departments, this book takes a “reductionist” view of statistical mechanics, while
describing the main ideas and methods underlying its applications. It implicitly
assumes that the physics of complex systems as observed is connected to funda-
mentalphysicallawsrepresentedatthemolecularlevelbyNewtonianmechanicsor
quantummechanics.Organizedintothreeparts,thefirstsectiondescribesthefun-
damentalprinciplesofequilibriumstatisticalmechanics.Thenextsectiondescribes
applications to phases of increasing density and order: gases, liquids and solids;
it also treats phase transitions. The final section deals with dynamics, including a
carefulaccountofhydrodynamictheoriesandlinearresponsetheory.
Thisoriginalapproachtostatisticalmechanicsissuitablefora1-yeargraduate
course for physicists, chemists, and chemical engineers. Problems are included
followingeachchapter,withsolutionstoselectedproblemsprovided.
J. Woods HalleyisProfessorofPhysicsattheSchoolofPhysicsandAstro-
nomy,UniversityofMinnesota,Minneapolis.

STATISTICAL MECHANICS
From First Principles to Macroscopic Phenomena
J. WOODS HALLEY
UniversityofMinnesota
cambridge university press
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Cambridge University Press
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Published in the United States of America by Cambridge University Press, New York
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© J. Woods Halley 2007
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without the written permission of Cambridge University Press.
First published in print format 2006
isbn-13 978-0-511-25636-3eBook (EBL)
isbn-10 0-511-25636-1 eBook (EBL)
isbn-13 978-0-521-82575-7hardback
isbn-10 0-521-82575-Xhardback
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Contents
Preface pageix
Introduction 1
PartI Foundationsofequilibriumstatisticalmechanics 5
1 Theclassicaldistributionfunction 7
Foundationsofequilibriumstatisticalmechanics 7
Liouville’stheorem 14
Thedistributionfunctiondependsonlyonadditiveconstantsofthe
motion 16
Microcanonicaldistribution 20
References 24
Problems 24
2 Quantummechanicaldensitymatrix 27
Microcanonicaldensitymatrix 33
Reference 34
Problems 34
3 Thermodynamics 37
Definitionofentropy 37
Thermodynamicpotentials 38
Somethermodynamicrelationsandtechniques 42
Constraintsonthermodynamicquantities 46
References 49
Problems 49
4 Semiclassicallimit 51
Generalformulation 51
Theperfectgas 52
Problems 56
v
vi Contents
PartII Statesofmatterinequilibriumstatisticalphysics 57
5 Perfectgases 59
Classicalperfectgas 60
Molecularidealgas 62
Quantumperfectgases:generalfeatures 69
Quantumperfectgases:detailsforspecialcases 71
PerfectBosegasatlowtemperatures 74
PerfectFermigasatlowtemperatures 78
References 81
Problems 81
6 Imperfectgases 85
MethodIfortheclassicalvirialexpansion 86
MethodIIforthevirialexpansion:irreduciblelinkedclusters 95
Applicationofcumulantstotheexpansionofthefreeenergy 102
Clusterexpansionforaquantumimperfectgas(extensionof
methodI) 108
Gross–Pitaevskii–Bogoliubovtheoryofthelowtemperatureweakly
interactingBosegas 115
References 122
Problems 122
7 Statisticalmechanicsofliquids 125
Definitionsofn-particledistributionfunctions 126
Determinationofg(r)byneutronandx-rayscattering 128
BBGKYhierarchy 133
Approximateclosedformequationsforg(r(cid:3)) 135
Moleculardynamicsevaluationofliquidproperties 136
References 143
Problems 144
8 Quantumliquidsandsolids 145
FundamentalpostulatesofFermiliquidtheory 146
Modelsofmagnets 150
Physicalbasisformodelsofmagneticinsulators:exchange 150
ComparisonofIsingandliquid–gassystems 153
Exactsolutionoftheparamagneticproblem 153
HightemperatureseriesfortheIsingmodel 154
Transfermatrix 157
MonteCarlomethods 158
References 159
Problems 160
Contents vii
9 Phasetransitions:staticproperties 161
Thermodynamicconsiderations 161
Criticalpoints 166
Phenomenologyofcriticalpointsingularities:scaling 167
Meanfieldtheory 172
Renormalizationgroup:generalscheme 177
Renormalizationgroup:theLandau–Ginzburgmodel 181
References 189
Problems 189
PartIII Dynamics 193
10 Hydrodynamicsanddefinitionoftransportcoefficients 195
Generaldiscussion 195
Hydrodynamicequationsforaclassicalfluid 196
Fluctuation–dissipationrelationsforhydrodynamictransport
coefficients 199
References 214
Problems 214
11 Stochasticmodelsanddynamicalcriticalphenomena 217
Generaldiscussionofstochasticmodels 217
GeneralizedLangevinequation 217
Generaldiscussionofdynamicalcriticalphenomena 221
References 242
Problems 242
Appendix:solutionstoselectedproblems 243
Index 281