PrefacetotheFirstEdition
HabeMuthdichdeineseigenenVerstandeszubedienen. (Havethecouragetothinkforyourself.)
ImmanuelKant,in BeantwortungderFrage:WasistAufklärung?
Thedisciplinesofstatisticalmechanicsandthermodynamicsareverycloselyrelated, althoughtheirhistoricalrootsareseparate.Thefoundersofthermodynamicsdeveloped theirtheorieswithouttheadvantageofcontemporaryunderstandingoftheatomic structureofmatter.Statisticalmechanics,whichisbuiltonthisunderstanding,makes predictionsofsystembehaviorthatleadtothermodynamicrules.Inotherwords, statisticalmechanicsisaconceptualprecursortothermodynamics,althoughitisa historicallatecomer.
Unfortunately,despitetheirtheoreticalconnection,statisticalmechanicsandthermodynamicsareoftentaughtasseparatefieldsofstudy.Evenworse,thermodynamicsis usuallytaughtfirst,forthedubiousreasonthatitisolderthanstatisticalmechanics.All toooftentheresultisthatstudentsregardthermodynamicsasasetofhighlyabstract mathematicalrelationships,thesignificanceofwhichisnotclear.
Thisbookisanefforttorectifythesituation.Itpresentsthetwocomplementary aspectsofthermalphysicsasacoherenttheoryofthepropertiesofmatter.Myintention isthatafterworkingthroughthistextastudentwillhavesolidfoundationsinboth statisticalmechanicsandthermodynamicsthatwillprovidedirectaccesstomodern research.
GuidingPrinciples
InwritingthisbookIhavebeenguidedbyanumberofprinciples,onlysomeofwhich aresharedbyothertextbooksinstatisticalmechanicsandthermodynamics.
• Ihavewrittenthisbookforstudents,notprofessors.Manythingsthatexpertsmight takeforgrantedareexplainedexplicitly.Indeed,studentcontributionshavebeen essentialinconstructingclearexplanationsthatdonotleaveout‘obvious’stepsthat canbepuzzlingtosomeonenewtothismaterial.
• Thegoalofthebookistoprovidethestudentwithconceptualunderstanding,and theproblemsaredesignedintheserviceofthisgoal.Theyarequitechallenging, butthechallengesareprimarilyconceptualratherthanalgebraicorcomputational.
• Ibelievethatstudentsshouldhavetheopportunitytoprogrammodelsthemselves andobservehowthemodelsbehaveunderdifferentconditions.Therefore,the problemsincludeextensiveuseofcomputation.
• Thebookisintendedtobeaccessibletostudentsatdifferentlevelsofpreparation. Idonotmakeadistinctionbetweenteachingthematerialattheadvancedundergraduateandgraduatelevels,andindeed,Ihavetaughtsuchacoursemanytimes usingthesameapproachandmuchofthesamematerialforbothgroups.Asthe mathematicsisentirelyself-contained,studentscanmasterallofthematerialeven iftheirmathematicalpreparationhassomegaps.Graduatestudentswithprevious coursesonthesetopicsshouldbeabletousethebookwithself-studytomakeup foranygapsintheirtraining.
• Afterworkingthroughthistext,astudentshouldbewellpreparedtocontinuewith morespecializedtopicsinthermodynamics,statisticalmechanics,andcondensedmatterphysics.
PedagogicalPrinciples
Theover-archinggoalsdescribedaboveresultinsomeuniquefeaturesofmyapproach totheteachingofstatisticalmechanicsandthermodynamics,whichIthinkmeritspecific mention.
TeachingStatisticalMechanics
• Thebookbeginswith classical statisticalmechanicstopostponethecomplications ofquantummeasurementuntilthebasicideasareestablished.
• Ihavedefinedensemblesintermsofprobabilities,inkeepingwithBoltzmann’s vision.Inparticular,thediscussionofstatisticalmechanicsisbasedonBoltzmann’s 1877definitionofentropy.Thisisnotthedefinitionusuallyfoundintextbooks,but whatheactuallywrote.TheuseofBoltzmann’sdefinitionisoneofthekeyfeatures ofthebookthatenablesstudentstoobtainadeepunderstandingofthefoundations ofbothstatisticalmechanicsandthermodynamics.
• Aself-containeddiscussionofprobabilitytheoryispresentedforbothdiscreteand continuousrandomvariables,includingallmaterialneededtounderstandbasic statisticalmechanics.Thismaterialwouldbesuperfluousifthephysicscurriculum weretoincludeacourseinprobabilitytheory,butunfortunately,thatisnotusually thecase.(Acourseinstatisticswouldalsobeveryvaluableforphysicsstudents— butthatisanotherstory.)
• Diracdeltafunctionsareusedtoformulatethetheoryofcontinuousrandom variables,aswellastosimplifythederivationsofdensitiesofstates.Thisisnot thewaymathematicianstendtointroduceprobabilitydensities,butIbelievethatit isbyfarthemostusefulapproachforscientists.
• Entropyispresentedasalogicalconsequenceofapplyingprobabilitytheoryto systemscontainingalargenumberofparticles,insteadofjustanequationtobe memorized.
• Theentropyoftheclassicalidealgasisderivedindetail.Thisprovidesan explicitexampleofanentropyfunctionthatexhibitsallthepropertiespostulated inthermodynamics.Theexampleissimpleenoughtogiveeverydetailofthe derivationofthermodynamicpropertiesfromstatisticalmechanics.
• ThebookincludesanexplanationofGibbs’paradox—whichisnotreallyparadoxicalwhenyoubeginwithBoltzmann’s1877definitionoftheentropy.
• Theapparentcontradictionbetweenobservedirreversibilityandtime-reversalinvariantequationsofmotionisexplained.Ibelievethatthisfillsanimportantgap inastudent’sappreciationofhowadescriptionofmacroscopicphenomenacan arisefromstatisticalprinciples.
TeachingThermodynamics
• ThefourfundamentalpostulatesofthermodynamicsproposedbyCallenhavebeen reformulated.Theresultisasetofsixthermodynamicpostulates,sequencedsoas tobuildconceptualunderstanding.
• Jacobiansareusedtosimplifythederivationofthermodynamicidentities.
• Thethermodynamiclimitisdiscussed,butthevalidityofthermodynamicsand statisticalmechanicsdoesnotrelyontakingthelimitofinfinitesize.Thisis importantifthermodynamicsistobeappliedtorealsystems,butissometimes neglectedintextbooks.
• Mytreatmentincludesthermodynamicsofnon-extensivesystems.Thisallowsme toincludedescriptionsofsystemswithsurfacesandsystemsenclosedincontainers.
OrganizationandContent
TheprinciplesIhavedescribedaboveleadmetoanorganizationforthebookthat isquitedifferentfromwhathasbecomethenorm.Aswasstatedabove,whilemost textsonthermalphysicsbeginwiththermodynamicsforhistoricalreasons,Ithinkit isfarpreferablefromtheperspectiveofpedagogytobeginwithstatisticalmechanics, includinganintroductiontothosepartsofprobabilitytheorythatareessentialto statisticalmechanics.
Topostponetheconceptualproblemsassociatedwithquantummeasurement,the initialdiscussionofstatisticalmechanicsinPartIislimitedtoclassicalsystems.The entropyoftheclassicalidealgasisderivedindetail,withaclearjustificationforeverystep. AcrucialaspectoftheexplanationandderivationoftheentropyistheuseofBoltzmann’s 1877definition,whichrelatesentropytotheprobabilityofamacroscopicstate.This definitionprovidesasolid,intuitiveunderstandingofwhatentropyisallabout.Itismy experiencethatafterstudentshaveseenthederivationoftheentropyoftheclassical
idealgas,theyimmediatelyunderstandthepostulatesofthermodynamics,sincethose postulatessimplycodifypropertiesthattheyhavederivedexplicitlyforaspecialcase.
ThetreatmentofstatisticalmechanicspavesthewaytothedevelopmentofthermodynamicsinPartII.WhilethisdevelopmentislargelybasedontheclassicworkbyHerbert Callen(whowasmythesisadvisor),therearesignificantdifferences.Perhapsthemost importantisthatIhavereliedentirelyonJacobianstoderivethermodynamicidentities. Insteadofregardingsuchderivationswithdread—asIdidwhenIfirstencountered them—mystudentstendtoregardthemasstraightforwardandrathereasy.There arealsoseveralotherchangesinemphasis,suchasaclarificationofthepostulatesof thermodynamicsandtheinclusionofnon-extensivesystems;thatis,finitesystemsthat havesurfacesorareenclosedincontainers.
PartIIIreturnstoclassicalstatisticalmechanicsanddevelopsthegeneraltheory directly,insteadofusingthecommonroundaboutapproachoftakingtheclassicallimit ofquantumstatisticalmechanics.Achapterisdevotedtoadiscussionoftheapparent paradoxesbetweenmicroscopicreversibilityandmacroscopicirreversibility.
PartIVpresentsquantumstatisticalmechanics.Thedevelopmentbeginsbyconsideringaprobabilitydistributionoverallquantumstates,insteadofthecommon adhoc restrictiontoeigenstates.Inadditiontothebasicconcepts,itcoversblack-bodyradiation, theharmoniccrystal,andbothBoseandFermigases.Becauseoftheirpracticaland theoreticalimportance,thereisaseparatechapteroninsulatorsandsemiconductors. ThefinalchapterintroducestheIsingmodelofmagneticphasetransitions.
Thebookcontainsaboutahundredmulti-partproblemsthatshouldbeconsideredas partofthetext.Inkeepingwiththelevelofthetext,theproblemsarefairlychallenging, andanefforthasbeenmadetoavoid‘plugandchug’assignments.Thechallengesinthe problemsaremainlyduetotheprobingofessentialconcepts,ratherthanmathematical complexities.Acompletesetofsolutionstotheproblemsisavailablefromthepublisher.
Severaloftheproblems,especiallyinthechaptersonprobability,relyoncomputer simulationstoleadstudentstoadeeperunderstanding.InthepastIhavesuggested thatmystudentsusetheC++programminglanguage,butforthelasttwoyearsIhave switchedtoVPythonforitssimplicityandtheeasewithwhichitgeneratesgraphs.An introductiontothebasicfeaturesofVPythonisgivenininAppendixA.Mostofmy studentshaveusedVPython,butasignificantfractionhavechosentouseadifferent language—usuallyJava,C,orC++.Ihavenotencounteredanydifficultieswithallowing studentstousetheprogramminglanguageoftheirchoice.
TwoSemestersorOne?
Thepresentationofthematerialinthisbookisbasedprimarilyonatwo-semester undergraduatecourseinthermalphysicsthatIhavetaughtseveraltimesatCarnegie MellonUniversity.Sincetwo-semesterundergraduatecoursesinthermalphysicsare ratherunusual,itsexistenceatCarnegieMellonforseveraldecadesmightberegarded assurprising.Inmyopinion,itshouldbethenorm.Althoughitwasquitereasonable toteachtwosemestersofclassicalmechanicsandonesemesterofthermodynamicsto
PrefacetotheFirstEdition xv undergraduatesinthenineteenthcentury—thedevelopmentofstatisticalmechanicswas justbeginning—itisnotreasonableinthetwenty-firstcentury.
However,evenatCarnegieMellononlythefirstsemesterofthermalphysicsis required.Allphysicsmajorstakethefirstsemester,andabouthalfcontinueontothe secondsemester,accompaniedbyafewstudentsfromotherdepartments.WhenI teachthecourse,thefirstsemestercoversthefirsttwopartsofthebook(Chapters1 through18),plusanoverviewofclassicalcanonicalensembles(Chapter18)and quantumcanonicalensembles(Chapter22).Thisgivesthestudentsanintroduction tostatisticalmechanicsandaratherthoroughknowledgeofthermodynamics,evenif theydonottakethesecondsemester.
Itisalsopossibletoteachaone-semestercourseinthermalphysicsfromthisbook usingdifferentchoicesofmaterial.Forexample:
• Ifthestudentshaveastrongbackgroundinprobabilitytheory(whichis,unfortunately,fairlyrare),Chapters3and5mightbeskippedtoincludemorematerialin PartsIIIandIV.
• Ifitisdecidedthatstudentsneedabroaderexposuretostatisticalmechanics,but thatalessdetailedstudyofthermodynamicsissufficient,Chapters14through17 couldbeskimmedtohavetimetostudyselectedchaptersinPartsIIIandIV.
• Ifthestudentshavealreadyhadathermodynamicscourse(althoughIdonot recommendthiscoursesequence),PartIIcouldbeskippedentirely.However,even ifthischoiceismade,studentsmightstillfindChapters9to18usefulforreview.
OnepossibilitythatIdonotrecommendwouldbetoskipthecomputationalmaterial. Iamstronglyoftheopinionthattheundergraduatephysicscurriculaatmostuniversities stillcontaintoolittleinstructioninthecomputationalmethodsthatstudentswillneedin theircareers.
Acknowledgments
ThisbookwasoriginallyintendedasaresourceformystudentsinThermalPhysics I(33–341)andThermalPhysicsII(33–342)atCarnegieMellonUniversity.Inan importantsense,thosestudentsturnedouttobeessentialcollaboratorsinitsproduction. Iwouldliketothankthemanystudentsfromthesecoursesfortheirgreathelpin suggestingimprovementsandcorrectingerrorsinthetext.Allofmystudentshave madeimportantcontributions.Evenso,Iwouldliketomentionexplicitlythefollowing students:MichaelAlexovich,DimitryAyzenberg,ConroyBaltzell,AnthonyBartolotta, AlexandraBeck,DavidBemiller,AlonzoBenavides,SarahBenjamin,JohnBriguglio, ColemanBroaddus,MattBuchovecky,LukeCeurvorst,JenniferChu,KuntingChua, CharlesWesleyCowan,CharlesdelasCasas,MatthewDaily,BrentDriscoll,Luke Durback,AlexanderEdelman,BenjaminEllison,DanielleFisher,EmilyGehrels,Yelena Goryunova,BenjaminGreer,NilsGuillermin,AsadHasan,AaronHenley,Maxwell
Hutchinson,AndrewJohnson,AgnieszkaKalinowski,PatrickKane,KamranKarimi, JoshuaKeller,DeenaKim,AndrewKojzar,RebeccaKrall,VikramKulkarni,Avishek Kumar,AnastasiaKurnikova,ThomasLambert,GrantLee,RobertLee,JonathanLong, SeanLubner,AlanLudin,FlorenceLui,ChristopherMagnollay,AlexMarakov,Natalie Mark,JamesMcGee,AndrewMcKinnie,JonathanMichel,CoreyMontella,Javier Novales,KenjiOman,JustinPerry,StephenPoniatowicz,ThomasPrag,AlisaRachubo, MohitRaghunathan,PeterRalli,AnthonyRice,SvetlanaRomanova,ArielRosenburg, MatthewRowe,KaitlynSchwalje,OmarShams,GabriellaShepard,KarpurShukla, StephenSigda,MichaelSimms,NicholasSteele,CharlesSwanson,ShaunSwanson, BrianTabata,LikunTan,JoshuaTepper,KevinTian,EricTurner,JosephVukovich, JoshuaWatzman,AndrewWesson,JustinWinokur,NanfeiYan,AndrewYeager,Brian Zakrzewski,andYuriyZubovski.Someofthesestudentsmadeparticularlyimportant contributions,forwhichIhavethankedthempersonally.Mystudents’encouragement andsuggestionshavebeenessentialinwritingthisbook.
YutaroIiyamaandMariliaCabralDoRegoBarroshavebothassistedwiththegrading ofThermalPhysicscourses,andhavemadeveryvaluablecorrectionsandsuggestions.
ThelaststagesinfinishingthemanuscriptwereaccomplishedwhileIwasaguestat theInstituteofStatisticalandBiologicalPhysicsattheLudwig-Maximilians-Universität, Munich,Germany.IwouldliketothankProf.Dr.ErwinFreyandtheothermembersof theInstitutefortheirgracioushospitality.
Throughoutthisproject,thesupportandencouragementofmyfriendsandcolleagues HarveyGouldandJanTobochnikhavebeengreatlyappreciated.
IwouldalsoliketothankmygoodfriendLawrenceErlbaum,whoseadviceand supporthavemadeanenormousdifferenceinnavigatingtheprocessofpublishinga book.
Finally,Iwouldliketothankmywife,Roberta(Bobby)Klatzky,whosecontributions arebeyondcount.Icouldnothavewrittenthisbookwithoutherlovingencouragement, sageadvice,andrelentlesshonesty.
Mythesisadvisor,HerbertCallen,firsttaughtmethatstatisticalmechanicsand thermodynamicsarefascinatingsubjects.Ihopeyoucometoenjoythemasmuchas Ido.
RobertH.Swendsen Pittsburgh,January2011
1Introduction 1
1.1ThermalPhysics1
1.2WhataretheQuestions?2 1.3History2
1.4BasicConceptsandAssumptions4 1.5PlanoftheBook6
PartIEntropy
2TheClassicalIdealGas 11
2.1IdealGas11
2.2PhaseSpaceofaClassicalGas12
2.3Distinguishability13
2.4ProbabilityTheory13
2.5Boltzmann’sDefinitionoftheEntropy14
2.6 S = k log W 14
2.7IndependenceofPositionsandMomenta15 2.8RoadMapforPartI15 3DiscreteProbabilityTheory 16
3.1WhatisProbability?16
3.2DiscreteRandomVariablesandProbabilities18
3.3ProbabilityTheoryforMultipleRandomVariables19
3.4RandomNumbersandFunctionsofRandomVariables21
3.5Mean,Variance,andStandardDeviation24
3.6CorrelationFunction25
3.7SetsofIndependentRandomNumbers25
3.8BinomialDistribution27
3.9GaussianApproximationtotheBinomialDistribution29
3.10ADigressiononGaussianIntegrals30
3.11Stirling’sApproximationfor N ! 31
3.12BinomialDistributionwithStirling’sApproximation34
3.13MultinomialDistribution35 3.14Problems36
4TheClassicalIdealGas:ConfigurationalEntropy 43
4.1SeparationofEntropyintoTwoParts43
4.2ProbabilityDistributionofParticles44
4.3DistributionofParticlesbetweenTwoIsolatedSystemsthatwere PreviouslyinEquilibrium45
4.4ConsequencesoftheBinomialDistribution46
4.5ActualNumberversusAverageNumber47
4.6The‘ThermodynamicLimit’48
4.7ProbabilityandEntropy48
4.8TheGeneralizationto M ≥ 2Systems51
4.9AnAnalyticApproximationfortheConfigurationalEntropy52 4.10Problems53
5ContinuousRandomNumbers 54
5.1ContinuousDiceandProbabilityDensities54
5.2ProbabilityDensities55
5.3DiracDeltaFunctions57
5.4TransformationsofContinuousRandomVariables61
5.5Bayes’Theorem63
5.6Problems65
6TheClassicalIdealGas:EnergyDependenceofEntropy 70
6.1DistributionfortheEnergybetweenTwoSubsystems70
6.2Evaluationof p 72
6.3DistributionofEnergybetweenTwoIsolatedSubsystemsthatwere PreviouslyinEquilibrium75
6.4ProbabilityDistributionforLarge N 76
6.5TheLogarithmoftheProbabilityDistributionandthe Energy-DependentTermsintheEntropy78
6.6TheGeneralizationto M ≥ 2systems79
7ClassicalGases:IdealandOtherwise 81
7.1EntropyofaCompositeSystemofClassicalIdealGases81
7.2EquilibriumConditionsfortheIdealGas82
7.3TheVolume-DependenceoftheEntropy85
7.4AsymmetricPistons87
7.5IndistinguishableParticles87
7.6EntropyofaCompositeSystemofInteractingParticles89
7.7TheSecondLawofThermodynamics95
7.8EquilibriumbetweenSubsystems95
7.9TheZerothLawofThermodynamics97 7.10Problems97
8Temperature,Pressure,ChemicalPotential,andAllThat 99
8.1ThermalEquilibrium99
8.2WhatdoweMeanby‘Temperature’?100
8.3DerivationoftheIdealGasLaw101
8.4TemperatureScales105
8.5ThePressureandtheEntropy106
8.6TheTemperatureandtheEntropy106
8.7EquilibriumwithAsymmetricPistons,Revisited107
8.8TheEntropyandtheChemicalPotential108
8.9TheFundamentalRelationandEquationsofState109
8.10TheDifferentialFormoftheFundamentalRelation109
8.11ThermometersandPressureGauges110
8.12Reservoirs110
8.13Problems111
PartIIThermodynamics
9ThePostulatesandLawsofThermodynamics 115
9.1ThermalPhysics115
9.2MicroscopicandMacroscopicStates117
9.3MacroscopicEquilibriumStates117
9.4StateFunctions118
9.5PropertiesandDescriptions118
9.6TheEssentialPostulatesofThermodynamics118
9.7OptionalPostulatesofThermodynamics120
9.8TheLawsofThermodynamics123
10PerturbationsofThermodynamicStateFunctions 124
10.1SmallChangesinStateFunctions124
10.2ConservationofEnergy125
10.3MathematicalDigressiononExactandInexactDifferentials125
10.4ConservationofEnergyRevisited128
10.5AnEquationtoRemember129
10.6Problems130
11ThermodynamicProcesses 132
11.1Irreversible,Reversible,andQuasi-StaticProcesses132 11.2HeatEngines133
11.3MaximumEfficiency135 11.4RefrigeratorsandAirConditioners136 11.5HeatPumps137
11.6TheCarnotCycle137 11.7AlternativeFormulationsoftheSecondLaw139 11.8PositiveandNegativeTemperatures140 11.9Problems146
12ThermodynamicPotentials 148
12.1MathematicalDigression:TheLegendreTransform148
12.2HelmholtzFreeEnergy152
12.3Enthalpy153 12.4GibbsFreeEnergy155
12.5OtherThermodynamicPotentials155 12.6MassieuFunctions156 12.7SummaryofLegendreTransforms156 12.8Problems157
13TheConsequencesofExtensivity 159
13.1TheEulerEquation160 13.2TheGibbs–DuhemRelation161 13.3ReconstructingtheFundamentalRelation162 13.4ThermodynamicPotentials163
14ThermodynamicIdentities 165
14.1SmallChangesandPartialDerivatives165 14.2AWarningaboutPartialDerivatives165 14.3FirstandSecondDerivatives166 14.4StandardSetofSecondDerivatives168 14.5MaxwellRelations169 14.6ManipulatingPartialDerivatives170 14.7WorkingwithJacobians174 14.8ExamplesofIdentityDerivations176 14.9GeneralStrategy179 14.10Problems180
15ExtremumPrinciples 184
15.1EnergyMinimumPrinciple185
15.2MinimumPrinciplefortheHelmholtzFreeEnergy188
15.3MinimumPrinciplefortheEnthalpy190 15.4MinimumPrinciplefortheGibbsFreeEnergy191
15.5Exergy192
15.6MaximumPrincipleforMassieuFunctions193 15.7Summary194 15.8Problems194
16StabilityConditions 195
16.1IntrinsicStability195
16.2StabilityCriteriabasedontheEnergyMinimumPrinciple196
16.3StabilityCriteriabasedontheHelmholtzFreeEnergyMinimum Principle199
16.4StabilityCriteriabasedontheEnthalpyMinimizationPrinciple200
16.5InequalitiesforCompressibilitiesandSpecificHeats201 16.6OtherStabilityCriteria201 16.7Problems203
17PhaseTransitions 205
17.1ThevanderWaalsFluid206 17.2DerivationofthevanderWaalsEquation206 17.3BehaviorofthevanderWaalsFluid207 17.4Instabilities208
17.5TheLiquid–GasPhaseTransition210 17.6MaxwellConstruction212 17.7CoexistentPhases212 17.8PhaseDiagram213 17.9HelmholtzFreeEnergy214 17.10LatentHeat216
17.11TheClausius–ClapeyronEquation217 17.12Gibbs’PhaseRule218 17.13Problems219
18TheNernstPostulate:TheThirdLawofThermodynamics 223
18.1ClassicalIdealGasViolatestheNernstPostulate223 18.2Planck’sFormoftheNernstPostulate224 18.3ConsequencesoftheNernstPostulate224
18.4CoefficientofThermalExpansionatLowTemperatures225 18.5TheImpossibilityofAttainingaTemperatureofAbsoluteZero226 18.6SummaryandSignposts226 18.7Problems227
PartIIIClassicalStatisticalMechanics
19EnsemblesinClassicalStatisticalMechanics 231 19.1MicrocanonicalEnsemble232
19.2MolecularDynamics:ComputerSimulations232
19.3CanonicalEnsemble234
19.4ThePartitionFunctionasanIntegraloverPhaseSpace237 19.5TheLiouvilleTheorem238
19.6ConsequencesoftheCanonicalDistribution240
19.7TheHelmholtzFreeEnergy241
19.8ThermodynamicIdentities242
19.9BeyondThermodynamicIdentities243
19.10IntegrationovertheMomenta244 19.11MonteCarloComputerSimulations245 19.12FactorizationofthePartitionFunction:TheBestTrickin StatisticalMechanics249 19.13SimpleHarmonicOscillator250 19.14Problems252 20ClassicalEnsembles:GrandandOtherwise
20.1GrandCanonicalEnsemble258 20.2GrandCanonicalProbabilityDistribution259
20.3ImportanceoftheGrandCanonicalPartitionFunction261 20.4 Z (T , V , μ)fortheIdealGas262 20.5SummaryoftheMostImportantEnsembles263 20.6OtherClassicalEnsembles264 20.7Problems264
22.1WhatNeedstobeExplained?271 22.2TrivialFormofIrreversibility272 22.3Boltzmann’sH-Theorem272
22.4Loschmidt’s Umkehreinwand 272
22.5Zermelo’s Wiederkehreinwand 273 22.6FreeExpansionofaClassicalIdealGas273 22.7Zermelo’s Wiederkehreinwand Revisited278 22.8Loschmidt’s Umkehreinwand Revisited278
22.9Whatis‘Equilibrium’?279 22.10Entropy279 22.11InteractingParticles281
PartIVQuantumStatisticalMechanics
23QuantumEnsembles 285
23.1BasicQuantumMechanics286 23.2EnergyEigenstates287
23.3Many-BodySystems290
23.4TwoTypesofProbability290 23.5TheDensityMatrix293
23.6UniquenessoftheEnsemble294
23.7ThePlanckEntropy295
23.8TheQuantumMicrocanonicalEnsemble296
24QuantumCanonicalEnsemble 297
24.1DerivationoftheQMCanonicalEnsemble297
24.9Two-LevelSystems309
25Black-BodyRadiation 322
25.7TotalBlack-BodyRadiation329
25.8SignificanceofBlack-BodyRadiation329 25.9Problems330
26.1ModelofaHarmonicSolid331
26.2NormalModes332
26.3TransformationoftheEnergy336
26.4TheFrequencySpectrum338
26.5AlternateDerivation:EquationsofMotion340 26.6TheEnergyintheClassicalModel341 26.7TheQuantumHarmonicCrystal342 26.8DebyeApproximation343 26.9Problems348
28.5Bose–EinsteinCondensation373 28.6BelowtheEinsteinTemperature373 28.7EnergyofanIdealGasofBosons375 28.8WhatAbouttheSecond-LowestEnergyState?376 28.9ThePressurebelow T < TE
28.10TransitionLinein P -V Plot378
28.11ANumericalApproachtoBose–EinsteinStatistics378 28.12Problems380
29Fermi–DiracStatistics 386
29.1BasicEquationsforFermions386
29.2TheFermiFunctionandtheFermiEnergy387
29.3AUsefulIdentity388
29.4SystemswithaDiscreteEnergySpectrum389
29.5SystemswithContinuousEnergySpectra390
29.6IdealFermiGas391
29.7FermiEnergy391
29.8CompressibilityofMetals392
29.9SommerfeldExpansion393
29.10GeneralFermiGasatLowTemperatures396
30.1Tight-BindingApproximation404 30.2Bloch’sTheorem406
30.3Nearly-FreeElectrons408
30.4EnergyBandsandEnergyGaps410
30.5WhereistheFermiEnergy?411
30.6FermiEnergyinaBand(Metals)412
30.7FermiEnergyinaGap412
30.8IntrinsicSemiconductors416
31.1TheIsingChain424
31.2TheIsingChaininaMagneticField( J = 0)425
31.3TheIsingChainwith h = 0,but J = 0426
31.4TheIsingChainwithboth J = 0and h = 0428
31.5Mean-FieldApproximation432
31.6CriticalExponents436
31.7Mean-FieldExponents437
31.8AnalogywiththevanderWaalsApproximation438
31.9LandauTheory439
31.10BeyondLandauTheory440 31.11Problems441
