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978-1-107-15401-8 — Perspectives on Statistical Thermodynamics
Yoshitsugu Oono
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PerspectivesonStatisticalThermodynamics
Thisoriginaltextdevelopsadeep,conceptualunderstandingofthermalphysics,highlight-
ing the important links between thermodynamics and statistical physics, and examining
how thermal physics fits within physics as a whole, from an empirical perspective. The
first part of the book is devoted to elementary, mesoscopic topics such as Brownian
motion, which leads to intuitive uses of large deviation theory, one of the pillars of
modern probability theory. The book then introduces the key concepts behind statistical
thermodynamics and the final part describes more advanced and applied topics from
thermalphysicssuchasphasetransitionsandcriticalphenomena. Thisimportantsubject
ispresentedfromafreshperspectiveandinapedagogicalmanner,withnumerousworked
examples and relevant cultural side notes throughout, making it suitable as either a
textbook for advanced thermal physics courses or for self-study by undergraduate and
graduatestudentsinphysicsandengineering.
YoshitsuguOonoisaProfessorofPhysicsattheUniversityofIllinoisatUrbana-Champaign,
where he currently teaches the statisticalphysics courses. His research interests focus on
nonequilibriumstatisticalthermodynamicsandtheoryofbiologicalsystems.
© in this web service Cambridge University Press www.cambridge.org
Cambridge University Press
978-1-107-15401-8 — Perspectives on Statistical Thermodynamics
Yoshitsugu Oono
Frontmatter
More Information
Perspectives on Statistical
Thermodynamics
YOSHITSUGU OONO
UniversityofIllinoisatUrbana-Champaign
© in this web service Cambridge University Press www.cambridge.org
Cambridge University Press
978-1-107-15401-8 — Perspectives on Statistical Thermodynamics
Yoshitsugu Oono
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DOI:10.1017/9781316650394
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Names:Oono,Yoshitsugu,1948–author.
Title:Perspectivesonstatisticalthermodynamics/YoshitsuguOono,
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Description:Cambridge,UnitedKingdom;NewYork,NY:CambridgeUniversityPress,2017.|
Includesbibliographicalreferencesandindex.
Identifiers:LCCN2017014907|ISBN9781107154018(hardback;alk.paper)|
ISBN1107154014(hardback;alk.paper)
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Cambridge University Press
978-1-107-15401-8 — Perspectives on Statistical Thermodynamics
Yoshitsugu Oono
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TothememoryofRobertM.Clegg(1945–2012)
SharingcriticalpointsofviewwithBobwasextremelyencouraging...
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Cambridge University Press
978-1-107-15401-8 — Perspectives on Statistical Thermodynamics
Yoshitsugu Oono
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Contents
Preface pagexxi
Acknowledgements xxiii
Self-StudyGuide xxiv
ListofSymbols xxvi
PartI AtomicandMesoscopicViewpoint 1
1 Introduction 3
1.1 Atomisms,AncientandModern 4
1.2 WhatwasbeyondPhilosophers’Grasp? 4
1.3 HowNumerousareAtomsandMolecules? 5
1.4 WhyareMoleculessoSmall? 5
1.5 OurWorldisLawfultotheExtentofAllowingthe
EvolutionofIntelligence 6
1.6 MicroscopicWorldisUnpredictable 7
1.7 WhyourMacroscopicWorldisLawful:theLawofLargeNumbers 7
1.8 WeLiveinaRatherGentleWorld 8
1.9 Thermodynamics,StatisticalMechanics,andPhaseTransition 9
1.10 TheMesoscopicWorld 10
1.11 LargeDeviationandFluctuation 10
Problems 11
2 AtomicPictureofGases 12
2.1 AristotelianPhysicsandGalileo’sStruggle 12
2.2 Boyle:TheTruePioneerofKineticTheoryofHeat 13
2.3 DiscoveryofAtmosphericPressureandVacuum 13
2.4 DanielBernoulli’sModernDynamicAtomismanda100YearHiatus 14
2.5 BetweenBernoulliandMaxwell 14
2.6 DanielBernoulli’sKineticTheory 15
2.7 EquipartitionofKineticEnergy 17
Problems 18
3 IntroductiontoProbability 20
3.1 ProbabilityisaMeasureofConfidenceLevel 21
3.2 EventsandSets 21
vii
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978-1-107-15401-8 — Perspectives on Statistical Thermodynamics
Yoshitsugu Oono
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viii Contents
3.3 Probability 22
3.4 Additivity 22
3.5 ProbabilityMeasure 23
3.6 RelationtoCombinatorics 24
3.7 ObjectivityofSubjectiveProbability 24
3.8 ConditionalProbability 25
3.9 StatisticalIndependence 25
3.10 StochasticVariables 25
3.11 ExpectationValue 26
3.12 Variance 27
3.13 Indicator 27
3.14 IndependenceofStochasticVariables 28
3.15 IndependenceandCorrelation 28
3.16 StochasticProcess 29
Problems 30
Appendix3A RudimentsofCombinatorics 34
4 TheLawofLargeNumbers 38
4.1 HowcanweMeasureProbability? 38
4.2 PreciseStatementoftheLawofLargeNumbers 39
4.3 WhyistheLawofLargeNumbersPlausible? 40
4.4 Chebyshev’sInequalityandaProofofLLN 40
4.5 AlmostnoFluctuationofInternalEnergy,anExample 41
4.6 MonteCarloIntegration 42
4.7 CentralLimitTheorem 43
4.8 ThereisnoFurtherRefinementofLLN 44
Problems 44
5 Maxwell’sDistribution 47
5.1 DensityDistributionFunction 47
5.2 Maxwell’sDerivationoftheDistributionFunction 48
5.3 GaussianIntegral 49
5.4 GaussianDistributionFunctionandMaxwellDistribution 49
5.5 MomentGeneratingFunction 50
5.6 BernoulliRevisited 51
5.7 HowtoDerivetheBoltzmannFactor 52
5.8 ElementaryDerivationofMaxwell’sDistribution 53
Problems 54
6 DeltaFunctionandDensityDistributionFunctions 58
6.1 WhatAverageGivestheDensityDistribution? 58
6.2 Introducingtheδ-function 59
6.3 AFormalExpressionofDensityDistribution 60
6.4 DistributionFunction:The1DCase 60
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978-1-107-15401-8 — Perspectives on Statistical Thermodynamics
Yoshitsugu Oono
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ix Contents
6.5 DensityDistributionFunction:The1DCase 61
6.6 The1Dδ-function 61
6.7 KeyFormula1 62
6.8 KeyFormula2 63
6.9 SomePractice 63
6.10 HowtoComputeDensityDistributionFunctions 63
6.11 KineticEnergyDensityDistributionof2DIdealGas 64
Problems 64
Appendix6A ExtremeRudimentsofDistributions 66
7 MeanFreePathandDifusion 70
7.1 MeanFreePath 70
7.2 EffectofMolecularCollisions 71
7.3 SpatialSpreadofRandomWalks 73
7.4 MixingProcessofTwoDistinctGases:EachMoleculeWalksRandomly 74
7.5 CharacterizingtheEquilibriumMacrostate 75
7.6 EquilibriumMacrostateIsMostProbable 76
7.7 TheEquilibriumMacrostateIsStable 77
7.8 EntropicForce 77
7.9 HowwecanObserveEntropicForce 79
Problems 79
8 IntroductiontoTransportPhenomena 81
8.1 Whatisa(Linear)TransportPhenomenon? 81
8.2 Density 82
8.3 Flux 82
8.4 LinearTransportLaw 83
8.5 Gradient 84
8.6 ConservationLawandDivergence 84
8.7 LocalExpressionofConservationLaw 85
8.8 TheDiffusionEquation 86
8.9 TheMeaningoftheLaplacian 86
8.10 IntuitiveComputationofTransportCoefficient 87
8.11 TheDiffusionConstant 88
8.12 ShearViscosity 88
8.13 MaxwellEstimatedtheMolecularSize 90
8.14 ThermalConductivity 91
8.15 DimensionalAnalysisofTransportCoefficients 91
8.16 SignificanceofJ∝gradxˆ 92
Problem 92
9 BrownianMotion 94
9.1 HowMesoscopicParticlesBehave 94
9.2 BrownDiscoveredaUniversalMotion(nowCalledBrownianMotion) 95
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978-1-107-15401-8 — Perspectives on Statistical Thermodynamics
Yoshitsugu Oono
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x Contents
9.3 GeneralPropertiesofBrownianMotion 96
9.4 Langevin’sExplanationofBrownianMotion 97
9.5 RelationtoRandomWalk 98
9.6 Einstein’sTheoryofBrownianParticleFlux 99
9.7 Einstein’sFormula 100
9.8 TheEinstein–StokesRelation:DimensionalAnalysis 101
9.9 DisplacementofParticlesbyDiffusion 101
9.10 Einstein’sFundamentalIdea:Summary 102
9.11 SummaryoftheBoltzmannConstant,k 102
B
Problems 103
Appendix9A HowtoObtainEquation(9.23) 105
10 TheLangevinEquationandLargeDeviation 107
10.1 OverdampedLangevinEquation 108
10.2 LangevinNoise:QualitativeFeatures 108
10.3 LangevinNoise 109
10.4 SmoluchowskiEquation 109
10.5 RelationBetweentheLangevinNoiseandD 110
10.6 SmoluchowskiEquationandFluctuation–DissipationRelation 111
10.7 SignificanceoftheFluctuation–DissipationRelation 112
10.8 MesoscopicsandLargeDeviationPrinciple 112
10.9 Three“InfinitesimalTimes,”dt,δt,(cid:3)t 113
10.10 TimeAveragingtoGetMesoscopicResults 114
10.11 LangevinEquationAsaResultofLargeDeviationTheory 115
10.12 LangevinEquation:PracticalSummary 116
Problems 116
PartII StatisticalThermodynamics:Basics 121
11 MacrosystemsinEquilibrium 123
11.1 HowtoDescribeaMacroscopicSysteminMechanics 123
11.2 ConservationofMechanicalEnergyandtheFirstLaw
ofThermodynamics 124
11.3 WhatMechanics? 124
11.4 AdditivityofEnergy 125
11.5 WhyWeAreInterestedinAdditiveQuantities 126
11.6 TheFourthLawofThermodynamics 127
11.7 Time-ReversalSymmetryofMechanics 127
11.8 IrreversibilityfromMechanics? 128
11.9 TheBoltzmannEquation 129
11.10 WhatistheLesson? 129
11.11 PurelyMechanicalToyModelandArrowofTime 130
11.12 TheLawofLargeNumbersWins 131
Problem 131
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978-1-107-15401-8 — Perspectives on Statistical Thermodynamics
Yoshitsugu Oono
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xi Contents
12 ThermodynamicDescription 133
12.1 WhatisPhenomenology? 133
12.2 EquilibriumStates 135
12.3 ExternalDisturbance:Mechanicsvs.Thermodynamics 135
12.4 ThermodynamicCoordinates:aPrivilegedSetofVariables 136
12.5 ThermodynamicSpace 137
12.6 QuasistaticProcesses 138
12.7 StateQuantities/Functions 139
12.8 SimpleSystem 139
12.9 CompoundSystem 139
12.10 WhyThermodynamicsCanBeUseful 139
12.11 Partitioning–RejoiningInvariance 140
12.12 ThermalContact 141
12.13 TheZerothLawofThermodynamics 141
13 BasicThermodynamics 142
13.1 TheFirstLawofThermodynamics 142
13.2 SignConvention 143
13.3 VolumeWork 143
13.4 MagneticWork 144
13.5 ConventionforElectromagneticField 145
13.6 PrehistoryoftheSecondLaw 146
13.7 TheSecondLawofThermodynamics 147
13.8 Planck’sPrinciple,Kelvin’sPrinciple,andClausius’
PrincipleareEquivalent 148
13.9 ClausiusRecognizesEntropy 148
13.10 EntropyExists 148
13.11 EntropyStratifiestheThermodynamicSpace 149
13.12 EntropyandHeat 150
13.13 EntropyPrinciple 151
13.14 TheGibbsRelation 151
Problems 152
14 Thermodynamics:GeneralConsequences 156
14.1 SummaryofBasicPrinciples 156
14.2 EntropyMaximizationPrinciple 157
14.3 EntropyisConcave 158
14.4 InternalEnergyMinimizationPrinciple 159
14.5 InternalEnergyisConvex 160
14.6 ExtensiontoNon-AdiabaticSystems 160
14.7 Clausius’Inequality 161
14.8 EquilibriumConditionsbetweenTwoSystems 161
Problem 162
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