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StatisticalMechanics R.K.Pathria
DepartmentofPhysics
UniversityofCaliforniaSanDiego LaJolla,CA,UnitedStates
PaulD.Beale
DepartmentofPhysics
UniversityofColoradoBoulder Boulder,CO,UnitedStates
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Prefacetothefourtheditionxiii
Prefacetothethirdeditionxv
Prefacetothesecondeditionxix
Prefacetothefirsteditionxxi
Historicalintroductionxxiii
1.Thestatisticalbasisofthermodynamics1
1.1.Themacroscopicandthemicroscopicstates1
1.2.Contactbetweenstatisticsandthermodynamics: physicalsignificanceofthenumber (N,V,E) 3
1.3.Furthercontactbetweenstatisticsandthermodynamics6
1.4.Theclassicalidealgas9
1.5.TheentropyofmixingandtheGibbsparadox16
1.6.The“correct”enumerationofthemicrostates20 Problems22
2.Elementsofensembletheory25
2.1.Phasespaceofaclassicalsystem25
2.2.Liouville’stheoremanditsconsequences27
2.3.Themicrocanonicalensemble30
2.4.Examples32
2.5.Quantumstatesandthephasespace35 Problems37
3.Thecanonicalensemble39
3.1.Equilibriumbetweenasystemandaheatreservoir40
3.2.Asysteminthecanonicalensemble41
3.3.Physicalsignificanceofthevariousstatisticalquantities inthecanonicalensemble50
3.4.Alternativeexpressionsforthepartitionfunction53
3.5.Theclassicalsystems54
3.6.Energyfluctuationsinthecanonicalensemble: correspondencewiththemicrocanonicalensemble58
3.7.Twotheorems–the“equipartition”andthe“virial”62
3.8.Asystemofharmonicoscillators65
3.9.Thestatisticsofparamagnetism70
3.10.Thermodynamicsofmagneticsystems: negativetemperatures77 Problems83
4.Thegrandcanonicalensemble93
4.1.Equilibriumbetweenasystemandaparticle–energy reservoir93
4.2.Asysteminthegrandcanonicalensemble95
4.3.Physicalsignificanceofthevariousstatisticalquantities97 4.4.Examples100
4.5.Densityandenergyfluctuationsinthegrandcanonical ensemble:correspondencewithotherensembles105
4.6.Thermodynamicphasediagrams107
4.7.PhaseequilibriumandtheClausius–Clapeyronequation111 Problems113
5.Formulationofquantumstatistics117
5.1.Quantum-mechanicalensembletheory: thedensitymatrix118
5.2.Statisticsofthevariousensembles122
5.3.Examples126
5.4.Systemscomposedofindistinguishableparticles131
5.5.Thedensitymatrixandthepartitionfunctionofa systemoffreeparticles136
5.6.Eigenstatethermalizationhypothesis142 Problems151
6.Thetheoryofsimplegases155
6.1.Anidealgasinaquantum-mechanical microcanonicalensemble155
6.2.Anidealgasinotherquantum-mechanicalensembles160
6.3.Statisticsoftheoccupationnumbers163
6.4.Kineticconsiderations166
6.5.Gaseoussystemscomposedofmoleculeswith internalmotion169
6.6.Chemicalequilibrium184 Problems187
7.IdealBosesystems195
7.1.ThermodynamicbehaviorofanidealBosegas196
7.2.Bose–Einsteincondensationinultracoldatomicgases207
7.3.Thermodynamicsoftheblackbodyradiation216
7.4.Thefieldofsoundwaves221
7.5.Inertialdensityofthesoundfield228
7.6.ElementaryexcitationsinliquidheliumII231 Problems239
8.IdealFermisystems247
8.1.ThermodynamicbehaviorofanidealFermigas247
8.2.MagneticbehaviorofanidealFermigas254
8.3.Theelectrongasinmetals263
8.4.UltracoldatomicFermigases274
8.5.Statisticalequilibriumofwhitedwarfstars275
8.6.Statisticalmodeloftheatom280
9.Thermodynamicsoftheearlyuniverse291
9.1.ObservationalevidenceoftheBigBang291
9.2.Evolutionofthetemperatureoftheuniverse296
9.3.Relativisticelectrons,positrons,andneutrinos298
9.4.Neutronfraction301
9.5.Annihilationofthepositronsandelectrons303
9.6.Neutrinotemperature305
9.7.Primordialnucleosynthesis306
9.8.Recombination309
9.9.Epilogue311 Problems312
10.Statisticalmechanicsofinteractingsystems: themethodofclusterexpansions315
10.1.Clusterexpansionforaclassicalgas315
10.2.Virialexpansionoftheequationofstate323
10.3.Evaluationofthevirialcoefficients325
10.4.Generalremarksonclusterexpansions331
10.5.Exacttreatmentofthesecondvirialcoefficient336
10.6.Clusterexpansionforaquantum-mechanicalsystem341
10.7.Correlationsandscattering347 Problems356
11.Statisticalmechanicsofinteractingsystems: themethodofquantizedfields361
11.1.Theformalismofsecondquantization361
11.2.Low-temperaturebehaviorofanimperfectBosegas371
11.3.Low-lyingstatesofanimperfectBosegas377
11.4.EnergyspectrumofaBoseliquid382
11.5.Stateswithquantizedcirculation386
11.6.Quantizedvortexringsandthebreakdown ofsuperfluidity392
11.7.Low-lyingstatesofanimperfectFermigas395
11.8.EnergyspectrumofaFermiliquid:Landau’s phenomenologicaltheory401
11.9.CondensationinFermisystems408 Problems410
12.Phasetransitions:criticality,universality,andscaling417
12.1.Generalremarksontheproblemofcondensation418
12.2.CondensationofavanderWaalsgas423
12.3.Adynamicalmodelofphasetransitions427
12.4.Thelatticegasandthebinaryalloy433
12.5.Isingmodelinthezerothapproximation436
12.6.Isingmodelinthefirstapproximation443
12.7.Thecriticalexponents451
12.8.Thermodynamicinequalities454
12.9.Landau’sphenomenologicaltheory458
12.10.Scalinghypothesisforthermodynamicfunctions462
12.11.Theroleofcorrelationsandfluctuations465
12.12.Thecriticalexponents ν and η 472
12.13.Afinallookatthemeanfieldtheory476 Problems479
13.Phasetransitions:exact(oralmostexact)results forvariousmodels487
13.1.One-dimensionalfluidmodels487
13.2.TheIsingmodelinonedimension492
13.3.The n-vectormodelsinonedimension498
13.4.TheIsingmodelintwodimensions504
13.5.Thesphericalmodelinarbitrarydimensions524
13.6.TheidealBosegasinarbitrarydimensions535
13.7.Othermodels542 Problems546
14.Phasetransitions:therenormalizationgroupapproach555
14.1.Theconceptualbasisofscaling556
14.2.Somesimpleexamplesofrenormalization559
14.3.Therenormalizationgroup:generalformulation568
14.4.Applicationsoftherenormalizationgroup575
14.5.Finite-sizescaling586 Problems595
15.Fluctuationsandnonequilibriumstatisticalmechanics599
15.1.Equilibriumthermodynamicfluctuations600
15.2.TheEinstein–Smoluchowskitheoryofthe Brownianmotion603
15.3.TheLangevintheoryoftheBrownianmotion609
15.4.Approachtoequilibrium:theFokker–Planckequation619
15.5.Spectralanalysisoffluctuations: theWiener–Khintchinetheorem625
15.6.Thefluctuation–dissipationtheorem633
15.7.TheOnsagerrelations642
15.8.Exactequilibriumfreeenergydifferencesfrom nonequilibriummeasurements648 Problems653
16.Computersimulations659
16.1.Introductionandstatistics659
16.2.MonteCarlosimulations662
16.3.Moleculardynamics665
16.4.Particlesimulations668
16.5.Computersimulationcaveats672 Problems673 Appendices675
A.Influenceofboundaryconditionsonthe distributionofquantumstates675
B.Certainmathematicalfunctions677
C.“Volume”and“surfacearea”ofan n-dimensional sphereofradius R 684
D.OnBose–Einsteinfunctions686
E.OnFermi–Diracfunctions689
F.ArigorousanalysisoftheidealBosegasand theonsetofBose–Einsteincondensation692
G.OnWatsonfunctions697
H.Thermodynamicrelationships698
I.Pseudorandomnumbers706 Bibliography711 Index727
Prefacetothefourthedition Thethirdeditionof Statisticalmechanics waspublishedin2011.Thenewmaterialadded atthattimefocusedonBose–EinsteincondensationanddegenerateFermigasbehavior inultracoldatomicgases,finite-sizescalingbehaviorofBose–Einsteincondensates,thermodynamicsoftheearlyuniverse,chemicalequilibrium,MonteCarloandmoleculardynamicssimulations,correlationfunctionsandscattering,thefluctuation–dissipationtheoremandthedynamicalstructurefactor,phaseequilibriumandtheClausius–Clapeyron equation,exactsolutionsofone-dimensionalfluidmodels,exactsolutionofthetwodimensionalIsingmodelonafinitelattice,pseudorandomnumbergenerators,dozens ofnewhomeworkproblems,andanewappendixwithasummaryofthermodynamicassembliesandassociatedstatisticalensembles.
Thenewtopicsaddedtothisfourtheditionare:
• Eigenstatethermalizationhypothesis :MarkSrednicki,JoshuaDeutsch,andothersdiscoveredthatitispossiblefornonintegrableisolatedmacroscopicquantummany-body systemstoequilibrate.Thisoverturnedthedecades-longpresumptionthatequilibriumbehaviorofisolatedmany-bodysystemswasprecludedbecauseoftheunitary timeevolutionofpurestates.Eventhoughanisolatedsystemasawholewillnotequilibrate,mostmacroscopicmany-bodysystemswilldisplayequilibriumbehaviorfor localobservables,withthesystemasawholeservingasthereservoirforeachsubsystem.Thisbehavioristhequantumequivalenttoergodicbehaviorinclassicalsystems. Theexceptionstothisareintegrablesystemsandstronglyrandomsystemsthatdisplay many-bodylocalization.
• Exactequilibriumfreeenergydifferencesfromnonequilibriummeasurements :ChristopherJarzynskiandGavinCrooksshowedthattheaverageofthequantity exp( βW) along nonequilibrium paths,where W istheexternalworkdoneonthesystemduring thetransformation,dependsonlyon equilibrium freeenergydifferences,independentofthenonequilibriumpathchosenorhowfaroutofequilibriumthesystemis driven.Thispropertyisnowusedtomeasureequilibriumfreeenergydifferencesusing nonequilibriumtransformationsinexperimentsonphysicalsystemsandincomputer simulationsofmodelsystems.
•WehaverewrittenSection5.1onthedensitymatrixincoordinate-independentform usingHilbertspacevectorsandDiracbra–ketnotation.
•WehaveexpandedAppendixHtoincludebothelectricandmagneticfreeenergiesand haverewrittenequationsinvolvingmagneticfieldsthroughoutthetexttoexpressthem inSIunits.
•Wehaveensuredthatalloftheeditsandcorrectionswemadeinthe2014“secondprinting”ofthethirdeditionwereincludedinthisedition.
•Wehaveaddedover30newend-of-chapterproblems.
•Wehavemademinoreditsandcorrectionsthroughoutthetext.
R.K.P.expressedhisindebtednesstomanypeopleatthetimeofthepublicationofthe firstandsecondeditionsso,atthistime,hesimplyreiterateshisgratitudetothem.
P.D.B.wouldliketothankhisfriendsandcolleaguesattheUniversityofColorado Boulderforthemanyconversationshehashadwiththemovertheyearsaboutphysics researchandpedagogy,manyofwhomassistedhimwiththethirdorfourthedition:AllanFranklin,NoelClark,TomDeGrand,JohnPrice,ChuckRogers,MichaelDubson,Leo Radzihovsky,VictorGurarie,MichaelHermele,RahulNandkishore,DanDessau,Dmitry Reznik,MinhyeaLee,MatthewGlaser,JosephMacLennan,KyleMcElroy,MurrayHolland, HeatherLewandowski,JohnCumalat,ShanthadeAlwis,AlexConley,JamieNagle,Paul Romatschke,NoahFinkelstein,KathyPerkins,JohnBlanco,KevinStenson,LorenHough, MeredithBetterton,IvanSmalyukh,ColinWest,EleanorHodby,andEricCornell.Inadditiontothose,specialthanksarealsoduetoothercolleagueswhohavereadsectionsof thethirdorfourtheditionmanuscript,orofferedvaluablesuggestions:EdmondMeyer, MatthewGrau,AndrewSisler,MichaelFoss-Feig,PeterJoot,JeffJustice,StephenH.White, andHarveyLeff.
P.D.B.wouldliketoexpresshisspecialgratitudetoRajKumarPathriaforthehonorof beingaskedtojoinhimascoauthoratthetimeofpublicationofthethirdeditionofhis highlyregardedtextbook.HeandhiswifeErikatreasurethefriendshipstheyhavedevelopedwithRajandhislovelywifeRajKumariPathria.
P.D.B.dedicatesthiseditiontoErika,foreverything.
Prefacetothethirdedition Thesecondeditionof Statisticalmechanics waspublishedin1996.Thenewmaterial addedatthattimefocusedonphasetransitions,criticalphenomena,andtherenormalizationgroup–topicsthathadundergonevasttransformationsduringtheyearsfollowing thepublicationofthefirsteditionin 1972.In2009,R.K.Pathria(R.K.P.)andthepublishersagreeditwastimeforathirdeditiontoincorporatetheimportantchangesthathad occurredinthefieldsincethepublicationofthesecondeditionandinvitedPaulD.Beale (P.D.B.)tojoinascoauthor.Thetwoauthorsagreedonthescopeoftheadditionsand changesandP.D.B.wrotethefirstdraftofthenewsectionsexceptforAppendixF,which waswrittenbyR.K.P.Bothauthorsworkedverycloselytogethereditingthedraftsandfinalizingthisthirdedition.
Thenewtopicsaddedtothiseditionare:
• Bose–EinsteincondensationanddegenerateFermigasbehaviorinultracoldatomic gases: Sections7.2,8.4,11.2.A,and11.9.ThecreationofBose–Einsteincondensatesin ultracoldgasesduringthe1990sandindegenerateFermigasesduringthe2000sled toarevolutioninatomic,molecular,andopticalphysics,andprovidedavaluablelink tothequantumbehaviorofcondensedmattersystems.SeveralofP.D.B.’sfriendsand colleaguesinphysicsandJILAattheUniversityofColoradohavebeenleadersinthis excitingnewfield.
• Finite-sizescalingbehaviorofBose–Einsteincondensates: AppendixF.Wedevelopan analyticaltheoryforthebehaviorofBose–Einsteincondensatesinafinitesystem, whichprovidesarigorousjustificationforsinglingoutthegroundstateinthecalculationofthepropertiesoftheBose–Einsteincondensate.
• Thermodynamicsoftheearlyuniverse: Chapter9.Thesequenceofthermodynamic transitionsthattheuniversewentthroughshortlyaftertheBigBangleftbehindmilepoststhatastrophysicistshaveexploitedtolookbackintotheuniverse’searliestmoments.Majoradvancesinspace-basedastronomybeginningaround1990haveprovidedavastbodyofobservationaldataabouttheearlyevolutionoftheuniverse.These includetheHubbleSpaceTelescope’sdeepspacemeasurementsoftheexpansionof theuniverse,theCosmicBackgroundExplorer’sprecisemeasurementsofthetemperatureofthecosmicmicrowavebackground,andtheWilkinsonMicrowaveAnisotropy Probe’smappingoftheangularvariationsinthecosmicmicrowavebackground.These datasetshaveledtoprecisedeterminationsoftheageoftheuniverseanditscompositionandearlyevolution.Coincidentally,P.D.B.’sfacultyofficeislocatedinthetower namedafterGeorgeGamow,amemberofthefacultyattheUniversityofColoradoin the1950sand1960sandaleaderinthetheoryofnucleosynthesisintheearlyuniverse.
• Chemicalequilibrium: Section6.6.Chemicalpotentialsdeterminetheconditionsnecessaryforchemicalequilibrium.Thisisanimportanttopicinitsownright,butalso playsacriticalroleinourdiscussionofthethermodynamicsoftheearlyuniversein Chapter9.
• MonteCarloandmoleculardynamicssimulations: Chapter16.Computersimulations havebecomeanimportanttoolinmodernstatisticalmechanics.Weprovideherea briefintroductiontoMonteCarloandmoleculardynamicstechniquesandalgorithms.
• Correlationfunctionsandscattering: Section10.7.Correlationfunctionsarecentralto theunderstandingofthermodynamicphases,phasetransitions,andcriticalphenomena.Thedifferencesbetweenthermodynamicphasesareoftenmostconspicuousin thebehaviorofcorrelationfunctionsandthecloselyrelatedstaticstructurefactors.We havecollecteddiscussionsfromthesecondeditionintooneplaceandaddednewmaterial.
• Thefluctuation–dissipationtheoremandthedynamicalstructurefactor: Sections15.3.A, 15.6.A,and15.6.B.Thefluctuation–dissipationtheoremdescribestherelationbetweennaturalequilibriumthermodynamicfluctuationsinasystemandtheresponse ofthesystemtosmalldisturbancesfromequilibrium,anditisoneofthecornerstonesofnonequilibriumstatisticalmechanics.Wehaveexpandedthediscussionof thefluctuation–dissipationtheoremtoincludeaderivationofthekeyresultsfromlinearresponsetheory,adiscussionofthedynamicalstructurefactor,andanalysisofthe Brownianmotionofharmonicoscillatorsthatprovidesusefulpracticalexamples.
• PhaseequilibriumandtheClausius–Clapeyronequation: Sections4.6and4.7.Muchof thetextisdevotedtousingstatisticalmechanicsmethodstodeterminetheproperties ofthermodynamicphasesandphasetransitions.Thisbriefoverviewofphaseequilibriumandthestructureofphasediagramslaysthegroundworkforlaterdiscussions.
• Exactsolutionsofone-dimensionalfluidmodels: Section13.1.One-dimensionalfluid modelswithshort-rangeinteractionsdonotexhibitphasetransitionsbuttheydodisplayshort-rangecorrelationsandotherbehaviorstypicalofdensefluids.
• Exactsolutionofthetwo-dimensionalIsingmodelonafinitelattice: Section13.4.A.This solutionentailsanexactcountingofthemicrostatesofthemicrocanonicalensemble andprovidesanalyticalresultsfortheenergydistribution,internalenergy,andheat capacityofthesystem.Thissolutionalsodescribesthefinite-sizescalingbehaviorof theIsingmodelnearthetransitionpointandprovidesanexactframeworkthatcanbe usedtotestMonteCarlomethods.
• Summaryofthermodynamicassembliesandassociatedstatisticalensembles: AppendixH.Weprovideasummaryofthermodynamicrelationsandtheirconnectionsto statisticalmechanicalensembles.Mostofthisinformationcanbefoundelsewherein thetext,butwethoughtitwouldbehelpfultoprovidearundownoftheseimportant connectionsinoneplace.
• Pseudorandomnumbergenerators: AppendixI.Pseudorandomnumbergeneratorsare indispensableincomputersimulations.Weprovidesimplealgorithmsforgenerating uniformandGaussianpseudorandomnumbersanddiscusstheirproperties.
• Dozensofnewhomeworkproblems.
Theremainderofthetextislargelyunchanged.
Thecompletionofthistaskhasleftusindebtedtomanyafriendandcolleague.R.K.P. hasalreadyexpressedhisindebtednesstoagoodnumberofpeopleontwopreviousoccasions–in1972andin1996–so,atthistime,hewillsimplyreiteratethemanywordsof gratitudehehasalreadywritten.Inadditionthough,hewouldliketothankPaulBealefor hiswillingnesstobeapartnerinthisprojectandforhisdiligenceincarryingoutthetask athandbotharduouslyandmeticulously.
Onhispart,P.D.B.wouldliketothankhisfriendsattheUniversityofColoradoBoulderforthemanyconversationshehashadwiththemovertheyearsaboutresearchand pedagogyofstatisticalmechanics,especiallyNoelClark,TomDeGrand,JohnPrice,Chuck Rogers,MikeDubson,andLeoRadzihovsky.Hewouldalsoliketothankthefacultyofthe DepartmentofPhysicsforaccordinghimthehonorofservingasthechairofthisoutstandingdepartment.
Specialthanksarealsoduetomanyfriendsandcolleagueswhohavereadsectionsof themanuscriptandhaveofferedmanyvaluablesuggestionsandcorrections,especially TomDeGrand,MichaelShull,DavidNesbitt,JamieNagle,MattGlaser,MurrayHolland, LeoRadzihovsky,VictorGurarie,EdmondMeyer,MatthewGrau,AndrewSisler,Michael Foss-Feig,AllanFranklin,ShanthadeAlwis,DmitriReznik,andEricCornell.
P.D.B.wouldliketotakethisopportunitytoextendhisthanksandbestwishesto ProfessorMichaelE.Fisher,whosegraduatestatisticalmechanicscourseatCornellintroducedhimtothiselegantfield.HewouldalsoliketoexpresshisgratitudetoRajPathriafor invitinghimtobepartofthisproject,andforthefunandengagingdiscussionstheyhave hadduringthepreparationofthisnewedition.Raj’sthoughtfulcounselalwaysprovedto bevaluableinimprovingthetext.
P.D.B.’sgreatestthanksgotoMatthew,Melanie,andErikafortheirloveandsupport. R.K.P.
Prefacetothesecondedition Thefirsteditionofthisbookwaspreparedovertheyears1966to1970,whenthesubjectof phasetransitionswasundergoingacompleteoverhaul.Theconceptsofscalinganduniversalityhadjusttakenroot,buttherenormalizationgroupapproach,whichconverted theseconceptsintoacalculationaltool,wasstillobscure.Notsurprisingly,mytextofthat timecouldnotdojusticetotheseemergingdevelopments.OvertheinterveningyearsI havefeltincreasinglyconsciousofthisratherseriousdeficiencyinthetext;sowhenthe timecametoprepareanewedition,mymajoreffortwenttowardcorrectingthatdeficiency.
Despitetheaforementionedshortcoming,thefirsteditionofthisbookhascontinued tobepopularsinceitsoriginalpublicationin1972.I,therefore,decidednottotinkerwith itunnecessarily.Nevertheless,tomakeroomforthenewmaterial,Ihadtoremovesome sectionsfromthepresenttext,which,Ifelt,werenotbeingusedbythereadersasmuchas therestofthebookwas.Thismayturnouttobeadisappointmenttosomeindividualsbut Itrusttheywillunderstandthelogicbehinditand,ifneedbe,Iwillgobacktoacopyofthe firsteditionforreference.I,onmypart,hopethatagoodmajorityoftheuserswillnotbe inconveniencedbythesedeletions.Asforthematerialretained,Ihaveconfinedmyselfto makingonlyeditorialchanges.Thesubjectofphasetransitionsandcriticalphenomena, whichhasbeenmymainfocusofrevision,hasbeentreatedinthreenewchaptersthat providearespectablecoverageofthesubjectandessentiallybringthebookuptodate. Thesechapters,alongwithacollectionofmorethan60homeworkproblems,will,Ibelieve, enhancetheusefulnessofthebookforbothstudentsandinstructors.
Thecompletionofthistaskhasleftmeindebtedtomany.Firstofall,asmentionedin theprefacetothefirstedition,Ioweaconsiderabledebttothosewhohavewrittenonthis subjectbeforeandfromwhosewritingsIhavebenefitedgreatly.Itisdifficulttothankthem allindividually;thebibliographyattheendofthebookisanobvioustributetothem.Asfor definitivehelp,IammostgratefultoDr.SurjitSingh,whoadvisedmeexpertlyandassisted megenerouslyintheselectionofthematerialthatcomprisesChapters11to13ofthenew text;withouthishelp,thefinalproductmightnothavebeenascoherentasitnowappears tobe.Onthetechnicalside,IamverythankfultoMrs.DebbieGuenther,whotypedthe manuscriptwithexceptionalskillandproofreaditwithextremecare;hertaskwasclearly anarduousonebutsheperformeditwithgoodcheer–forwhichIadmirehergreatly.
Finally,Iwishtoexpressmyheartfeltappreciationformywife,wholetmedevote myselffullytothistaskoveraratherlongperiodoftimeandwaitedforitscompletion ungrudgingly.
Prefacetothefirstedition ThisbookhasarisenoutofthenotesoflecturesthatIgavetothegraduatestudentsat McMasterUniversity(1964–1965),theUniversityofAlberta(1965–1967),theUniversityof Waterloo(1969–1971),andtheUniversityofWindsor(1970–1971).Whilethesubjectmatter,initsfinerdetails,haschangedconsiderablyduringthepreparationofthemanuscript, thestyleofpresentationremainsthesameasfollowedintheselectures.
Statisticalmechanicsisanindispensabletoolforstudyingphysicalpropertiesof matter“inbulk”onthebasisofthedynamicalbehaviorofits“microscopic”constituents.
Foundedonthewell-laidprinciplesof mathematicalstatistics ontheonehandand Hamiltonianmechanics ontheother,theformalismofstatisticalmechanicshasprovedtobeof immensevaluetothephysicsofthelast100years.Inviewoftheuniversalityofitsappeal, abasicknowledgeofthissubjectisconsideredessentialforeverystudentofphysics,irrespectiveofthearea(s)inwhichhe/shemaybeplanningtospecialize.Toprovidethis knowledge,inamannerthatbringsouttheessenceofthesubjectwithduerigorbutwithoutunduepain,isthemainpurposeofthiswork.
Thefactthat thedynamicsofaphysicalsystemisrepresentedbyasetofquantumstates andtheassertionthat thethermodynamicsofthesystemisdeterminedbythemultiplicity ofthesestates constitutethebasisofourtreatment.Thefundamentalconnectionbetween themicroscopicandthemacroscopicdescriptionsofasystemisuncoveredbyinvestigatingtheconditionsforequilibriumbetweentwophysicalsystemsinthermodynamic contact.Thisisbestaccomplishedbyworkinginthespiritofthequantumtheoryright fromthebeginning;theentropyandotherthermodynamicvariablesofthesystemthen followinamostnaturalmanner.Aftertheformalismisdeveloped,onemay(ifthesituationpermits)goovertothelimitoftheclassicalstatistics.Thismessagemaynotbenew, buthereIhavetriedtofollowitasfarasisreasonablypossibleinatextbook.Indoingso,an attempthasbeenmadetokeepthelevelofpresentationfairlyuniformsothatthereader doesnotencounterfluctuationsoftoowildacharacter.
Thistextisconfinedtothestudyofthe equilibriumstates ofphysicalsystemsandis intendedtobeusedfora graduatecourse instatisticalmechanics.Withinthesebounds, thecoverageisfairlywideandprovidesenoughmaterialfortailoringagoodtwo-semester course.Thefinalchoicealwaysrestswiththeindividualinstructor;I,forone,regardChapters1to9(minus afewsectionsfromthesechapters plus afewsectionsfromChapter13) asthe“essentialpart”ofsuchacourse.ThecontentsofChapters10to12arerelatively advanced(notnecessarilydifficult);thechoiceofmaterialoutofthesechapterswilldependentirelyonthetasteoftheinstructor.Tofacilitatetheunderstandingofthesubject, thetexthasbeenillustratedwithalargenumberofgraphs;toassesstheunderstanding,a largenumberofproblemshavebeenincluded.Ihopethesefeaturesarefounduseful. xxi
Ifeelthatoneofthemostessentialaspectsofteachingistoarousethecuriosityofthe studentsintheirsubject,andoneofthemosteffectivewaysofdoingthisistodiscusswith them(inareasonablemeasure,ofcourse)thecircumstancesthatledtotheemergence ofthesubject.Onewould,therefore,liketostopoccasionallytoreflectuponthemanner inwhichthevariousdevelopmentsreallycameabout;atthesametime,onemaynotlike theflowofthetexttobehamperedbythediscontinuitiesarisingfromanintermittent additionofhistoricalmaterial.Accordingly,Idecidedtoincludeinthisaccountahistorical introductiontothesubjectwhichstandsseparatefromthemaintext.Itrustthereaders, especiallytheinstructors,willfinditofinterest.
Forthosewhowishtocontinuetheirstudyofstatisticalmechanicsbeyondtheconfines ofthisbook,afairlyextensivebibliographyisincluded.Itcontainsavarietyofreferences –oldaswellasnew,experimentalaswellastheoretical,technicalaswellaspedagogical.I hopethatthiswillmakethebookusefulforawiderreadership.
Thecompletionofthistaskhasleftmeindebtedtomany.Likemostauthors,Ioweconsiderabledebttothosewhohavewrittenonthesubjectbefore.Thebibliographyatthe endofthebookisthemostobvioustributetothem;nevertheless,Iwouldliketomention, inparticular,theworksoftheEhrenfests,Fowler,Guggenheim,Schrödinger,Rushbrooke, terHaar,Hill,LandauandLifshitz,Huang,andKubo,whohavebeenmyconstantreferenceforseveralyearsandhaveinfluencedmyunderstandingofthesubjectinavarietyof ways.Asforthepreparationofthetext,IamindebtedtoRobertTeshima,whodrewmost ofthegraphsandcheckedmostoftheproblems,toRavindarBansal,VishwaMittar,and SurjitSingh,whowentthroughtheentiremanuscriptandmadeseveralsuggestionsthat helpedmeunkinktheexpositionatanumberofpoints,toMaryAnnetts,whotypedthe manuscriptwithexceptionalpatience,diligence,andcare,andtoFredHetzel,JimBriante, andLarryKry,whoprovidedtechnicalhelpduringthepreparationofthefinalversion.
AsthisworkprogressedIfeltincreasinglygratifiedtowardProfessorsF.C.Auluckand D.S.KotharioftheUniversityofDelhi,withwhomIstartedmycareerandwhoinitiated meintothestudyofthissubject,andtowardProfessorR.C.Majumdar,whotookkeen interestinmyworkonthisandeveryotherprojectthatIhaveundertakenfromtimeto time.IamgratefultoDr.D.terHaar,oftheUniversityofOxford,who,asthegeneraleditor ofthisseries,gavevaluableadviceonvariousaspectsofthepreparationofthemanuscript andmadeseveralusefulsuggestionstowardtheimprovementofthetext.Iamthankfulto ProfessorsJ.W.Leech,J.Grindlay,andA.D.SinghNagioftheUniversityofWaterloofor theirinterestandhospitalitythatwentalongwayinmakingthistaskapleasantone.
Thefinaltributemustgotomywife,whosecooperationandunderstanding,atall stagesofthisprojectandagainstallodds,havebeensimplyoverwhelming.
R.K.P.
Historicalintroduction Statisticalmechanicsisaformalismthataimsatexplainingthephysicalpropertiesofmatter inbulk onthebasisofthedynamicalbehaviorofits microscopic constituents.The scopeoftheformalismisalmostasunlimitedastheveryrangeofthenaturalphenomena, forinprincipleitisapplicabletomatterinanystatewhatsoever.Ithas,infact,beenapplied,withconsiderablesuccess,tothestudyofmatterinthesolidstate,theliquidstate,or thegaseousstate,mattercomposedofseveralphasesand/orseveralcomponents,matter underextremeconditionsofdensityandtemperature,matterinequilibriumwithradiation(as,forexample,inastrophysics),matterintheformofabiologicalspecimen,and soon.Furthermore,theformalismofstatisticalmechanicsenablesustoinvestigatethe nonequilibrium statesofmatteraswellasthe equilibrium states;indeed,theseinvestigationshelpustounderstandthemannerinwhichaphysicalsystemthathappenstobe“out ofequilibrium”atagiventime t approachesa“stateofequilibrium”astimepasses.
Incontrastwiththepresentstatusofitsdevelopment,thesuccessofitsapplications, andthebreadthofitsscope,thebeginningsofstatisticalmechanicswererathermodest. Barringcertainprimitivereferences,suchasthoseofGassendi,Hooke,andsoon,thereal workonthissubjectstartedwiththecontemplationsofBernoulli(1738),Herapath(1821), andJoule(1851),who,intheirownindividualways,attemptedtolayafoundationforthe so-called kinetictheoryofgases –adisciplinethatfinallyturnedouttobetheforerunner ofstatisticalmechanics.Thepioneeringworkoftheseinvestigatorsestablishedthefact thatthepressureofagasarosefromthemotionofitsmoleculesandcould,therefore,be computedbyconsideringthedynamicalinfluenceofthemolecularbombardmentonthe wallsofthecontainer.Thus,BernoulliandHerapathcouldshowthat,ifthetemperature remainedconstant,thepressure P ofanordinarygaswasinverselyproportionaltothevolume V ofthecontainer(Boyle’slaw),andthatitwasessentiallyindependentoftheshape ofthecontainer.This,ofcourse,involvedtheexplicitassumptionthat, atagiventemperature T ,the(mean)speedofthemoleculeswasindependentofbothpressureandvolume. Bernoullievenattemptedtodeterminethe(first-order)correctiontothislaw,arisingfrom the finite sizeofthemolecules,andshowedthatthevolume V appearinginthestatement ofthelawshouldbereplacedby (V b),where b isthe“actual”volumeofthemolecules.1
Joulewasthefirsttoshowthatthepressure P wasdirectlyproportionaltothesquare ofthemolecularspeed c ,whichhehadinitiallyassumedtobethesameforallmolecules. Krönig(1856)wentastepfurther.Introducingthe“quasistatistical”assumptionthat, at anytime t ,one-sixthofthemoleculescouldbeassumedtobeflyingineachofthesix
1 Asiswellknown,this“correction”wascorrectlyevaluated,muchlater,byvanderWaals(1873),whoshowed that,forlarge V,b is fourtimes the“actual”volumeofthemolecules;seeProblem1.4.
“independent”directions,namely, +x, x, +y, y, +z,and z,hederivedtheequation
where n isthenumberdensityofthemoleculesand m themolecularmass.Krönig,too, assumedthemolecularspeed c tobethesameforallmolecules;sofrom(1),heinferred thatthekineticenergyofthemoleculesshouldbedirectlyproportionaltotheabsolute temperatureofthegas.
Krönigjustifiedhismethodinthesewords:“Thepathofeachmoleculemustbeso irregularthatitwilldefyallattemptsatcalculation.However,accordingtothelawsof probability,onecouldassumeacompletelyregularmotioninplaceofacompletelyirregularone!”Itmust,however,benotedthatitisonlybecauseofthespecialformofthe summationsappearinginthecalculationofthepressurethatKrönig’sargumentleadsto thesameresultastheonefollowingfrommorerefinedmodels.Inotherproblems,suchas theonesinvolvingdiffusion,viscosity,orheatconduction,thisisnolongerthecase.
ItwasatthisstagethatClausiusenteredthefield.Firstofall,in 1857,hederivedthe ideal-gaslawunderassumptionsfarlessstringentthanKrönig’s.HediscardedbothleadingassumptionsofKrönigandshowedthatequation(1)wasstilltrue;ofcourse, c 2 now becamethe mean-squarespeed ofthemolecules.Inalaterpaper(1859),Clausiusintroducedtheconceptofthe meanfreepath andthusbecamethefirsttoanalyzetransport phenomena.Itwasinthesestudiesthatheintroducedthefamous“Stosszahlansatz”–the hypothesisonthenumberofcollisions(amongthemolecules)–which,lateron,playeda prominentroleinthemonumentalworkofBoltzmann.2 WithClausius,theintroduction ofthemicroscopicandstatisticalpointsofviewintothephysicaltheorywasdefinitive, ratherthanspeculative.Accordingly,Maxwell,inapopulararticleentitled“Molecules,” writtenforthe EncyclopediaBritannica,referredtoClausiusasthe“principalfounderof thekinetictheoryofgases,”whileGibbs,inhisClausiusobituarynotice,calledhimthe “fatherofstatisticalmechanics.”3
TheworkofClausiusattractedMaxwelltothefield.Hemadehisfirstappearancewith thememoir“Illustrationsinthedynamicaltheoryofgases”(1860),inwhichhewentmuch furtherthanhispredecessorsbyderivinghisfamouslawofthe“distributionofmolecular speeds.”Maxwell’sderivationwasbasedonelementaryprinciplesofprobabilityandwas clearlyinspiredbytheGaussianlawof“distributionofrandomerrors.”Aderivationbased ontherequirementthat“theequilibriumdistributionofmolecularspeeds,onceacquired, shouldremaininvariantundermolecularcollisions”appearedin 1867.ThisledMaxwell
2 Foranexcellentreviewofthisandrelatedtopics,seeEhrenfestandEhrenfest(1912).
3 Forfurtherdetails,refertoMontroll(1963),whereanaccountisalsogivenofthepioneeringworkofWaterston(1846, 1892).
toestablishwhatisknownas Maxwell ’stransportequation,which,ifskillfullyused,leads tothesameresultsasonegetsfromthemorefundamentalequationduetoBoltzmann.4
Maxwell’scontributionstothesubjectdiminishedconsiderablyafterhisappointment, in1871,astheCavendishProfessoratCambridge.BythattimeBoltzmannhadalready madehisfirststrides.InBoltzmann(1868, 1871)hegeneralizedMaxwell’sdistribution lawtopolyatomicgases,alsotakingintoaccountthepresenceofexternalforces,ifany; thisgaverisetothefamous Boltzmannfactor exp( βε),where ε denotesthe total energyofamolecule.Theseinvestigationsalsoledtothe equipartitiontheorem.Boltzmann furthershowedthat,justliketheoriginaldistributionofMaxwell,thegeneralizeddistribution(whichwenowcallthe Maxwell–Boltzmanndistribution )isstationarywithrespectto molecularcollisions.
In1872camethecelebrated H-theorem,whichprovidedamolecularbasisforthenaturaltendencyofphysicalsystemstoapproach,andstayin,astateofequilibrium.Thisestablishedaconnectionbetweenthemicroscopicapproach(whichcharacterizesstatistical mechanics)andthephenomenologicalapproach(whichcharacterizedthermodynamics) muchmoretransparentlythaneverbefore;italsoprovidedadirectmethodforcomputingtheentropyofagivenphysicalsystemfrompurelymicroscopicconsiderations.As acorollarytothe H -theorem,BoltzmannshowedthattheMaxwell–Boltzmanndistributionisthe only distributionthatstaysinvariantundermolecularcollisionsandthatany otherdistribution,undertheinfluenceofmolecularcollisions,willultimatelygoovertoa Maxwell–Boltzmanndistribution;seeBoltzmann(1872, 1875).In 1876 Boltzmannderived hisfamoustransportequation(seeBoltzmann(1876, 1877, 1879, 1884)),which,inthe handsofChapmanandEnskog(Chapman(1916, 1917);Enskog(1917)),provedtobean extremelypowerfultoolforinvestigatingmacroscopicpropertiesofsystemsinnonequilibriumstates.
Things,however,provedquiteharshforBoltzmann.His H -theorem,andtheconsequent irreversible behaviorofphysicalsystems,cameunderheavyattack,mainlyfrom Loschmidt(1876, 1877)andZermelo(1896).WhileLoschmidtwonderedhowtheconsequencesofthistheoremcouldbereconciledwiththereversiblecharacterofthebasic equationsofmotionofthemolecules,Zermelowonderedhowtheseconsequencescould bemadetofitwiththe quasiperiodic behaviorofclosedsystems(whicharoseinviewof theso-calledPoincarécycles).Boltzmanndefendedhimselfagainsttheseattackswithall hismightbut,unfortunately,couldnotconvincehisopponentsofthecorrectnessofhis viewpoint.Atthesametime,theenergeticists,ledbyMachandOstwald,werecriticiz-
4 ThisequivalencehasbeendemonstratedinGuggenheim(1960),wherethecoefficientsofviscosity,thermal conductivity,anddiffusionofagasofhardsphereshavebeencalculatedonthebasisofMaxwell’stransport equation.
ingthevery(molecular)basisofthekinetictheory,5 whileKelvinwasemphasizingthe “nineteenth-centurycloudshoveringoverthedynamicaltheoryoflightandheat.”6
AllthisleftBoltzmanninastateofdespairandinducedinhimapersecutioncomplex.7 Hewroteintheintroductiontothesecondvolumeofhistreatise VorlesungenüberGastheorie (1898):8
Iamconvincedthattheattacks(onthekinetictheory)restonmisunderstandingsand thattheroleofthekinetictheoryisnotyetplayedout.Inmyopinionitwouldbea blowtoscienceifcontemporaryoppositionweretocausekinetictheorytosinkinto theoblivionwhichwasthefatesufferedbythewavetheoryoflightthroughtheauthorityofNewton.Iamawareoftheweaknessofoneindividualagainsttheprevailing currentsofopinion.Inordertoinsurethatnottoomuchwillhavetoberediscovered whenpeoplereturntothestudyofkinetictheoryIwillpresentthemostdifficultand misunderstoodpartsofthesubjectinasclearamannerasIcan.
Weshallnotdwellanyfurtheronthekinetictheory;wewouldrathermoveontothedevelopmentofthemoresophisticatedapproachknownasthe ensembletheory,whichmay infactberegardedasthestatisticalmechanicsproper.9 Inthisapproach,thedynamical stateofagivensystem,ascharacterizedbythegeneralizedcoordinates qi andthegeneralizedmomenta pi ,isrepresentedbya phasepoint G(qi ,pi ) ina phasespace ofappropriate dimensionality.Theevolutionofthedynamicalstateintimeisdepictedbythe trajectory ofthe G-pointinthephasespace,the“geometry”ofthetrajectorybeinggovernedbythe equationsofmotionofthesystemandbythenatureofthephysicalconstraintsimposed onit.Todevelopanappropriateformalism,oneconsidersthegivensystemalongwith aninfinitelylargenumberof“mentalcopies”thereof;thatis,an ensemble ofsimilarsystemsunderidenticalphysicalconstraints(though,atanytime t ,thevarioussystemsin theensemblewoulddifferwidelyinrespectoftheirdynamicalstates).Inthephasespace, then,onehasaswarmofinfinitelymany G-points(which,atanytime t ,arewidelydispersedand,withtime,movealongtheirrespectivetrajectories).Thefictionofahostof infinitelymany,identicalbutindependent,systemsallowsonetoreplacecertaindubious assumptionsofthekinetictheoryofgasesbyreadilyacceptablestatementsofstatisticalmechanics.TheexplicitformulationofthesestatementswasfirstgivenbyMaxwell (1879),whoonthisoccasionusedtheword“statistico-mechanical”todescribethestudy
5 ThesecriticsweresilencedbyEinstein,whoseworkontheBrownianmotion(1905b)establishedatomic theory onceandforall
6 Thefirstofthesecloudswasconcernedwiththemysteriesofthe“aether,”andwasdispelledbythetheoryof relativity.Thesecondwasconcernedwiththeinadequacyofthe“equipartitiontheorem,”andwasdispelledby thequantumtheory.
7 SomepeopleattributeBoltzmann’ssuicideonSeptember5,1906,tothiscause.
8 QuotationfromMontroll(1963).
9 Forareviewofthehistoricaldevelopmentofkinetictheoryleadingtostatisticalmechanics,seeBrush(1957, 1958, 1961a,b, 1965–1966).
ofensembles(ofgaseoussystems)–though,8yearsearlier,Boltzmann(1871)hadalready workedwithessentiallythesamekindofensembles.
Themostimportantquantityintheensembletheoryisthe densityfunction, ρ(qi ,pi ; t), ofthe G-pointsinthephasespace;astationarydistribution (∂ρ/∂t = 0) characterizesa stationaryensemble,whichinturnrepresentsasystem inequilibrium.MaxwellandBoltzmannconfinedtheirstudytoensemblesforwhichthefunction ρ dependedsolelyonthe energy E ofthesystem.Thisincludedthespecialcaseof ergodic systems,whichwereso definedthat“theundisturbedmotionofsuchasystem,ifpursuedforanunlimitedtime, wouldultimatelytraverse(theneighborhoodof)eachandeveryphasepointcompatible withthe fixed value E oftheenergy.”Consequently,the ensembleaverage, f ,ofaphysicalquantity f ,takenat any giventime t ,wouldbethesameasthe long-timeaverage, f , pertainingto any givenmemberoftheensemble.Now, f isthevalueweexpecttoobtain forthequantityinquestionwhenwemakeanappropriatemeasurementonthesystem; theresultofthismeasurementshould,therefore,agreewiththetheoreticalestimate f . Wethusacquirearecipetobringaboutadirectcontactbetweentheoryandexperiment. Atthesametime,welaydownarationalbasisforamicroscopictheoryofmatterasan alternativetotheempiricalapproachofthermodynamics!
AsignificantadvanceinthisdirectionwasmadebyGibbs,who,withhis Elementary PrinciplesofStatisticalMechanics (1902),turnedensembletheoryintoamostefficient toolforthetheorist.Heemphasizedtheuseof“generalized”ensemblesanddeveloped schemeswhich,inprinciple,enabledonetocomputeacompletesetofthermodynamic quantitiesofagivensystemfrompurelymechanicalpropertiesofitsmicroscopicconstituents.10 Initsmethodsandresults,theworkofGibbsturnedouttobemuchmore generalthananyprecedingtreatmentofthesubject;itappliedtoanyphysicalsystem thatmetthesimple-mindedrequirementsthat(i)itwasmechanicalinstructureand(ii)it obeyedLagrange’sandHamilton’sequationsofmotion.Inthisrespect,Gibbs’sworkmay beconsideredtohaveaccomplishedforthermodynamicsasmuchasMaxwellhadaccomplishedforelectrodynamics.
ThesedevelopmentsalmostcoincidedwiththegreatrevolutionthatPlanck’sworkof 1900 broughtintophysics.Asiswellknown,Planck’s quantumhypothesis successfully resolvedtheessentialmysteriesoftheblackbodyradiation–asubjectwherethethree best-establisheddisciplinesofthe19thcentury,namely,mechanics,electrodynamics,and thermodynamics,wereallfocused.Atthesametime,ituncoveredboththestrengthsand theweaknessesofthesedisciplines.Itwouldhavebeensurprisingifstatisticalmechanics, whichlinkedthermodynamicswithmechanics,couldhaveescapedtherepercussionsof thisrevolution.
ThesubsequentworkofEinstein(1905a)onthephotoelectriceffectandofCompton (1923a,b)onthescatteringofx-raysestablished,sotosay,the“existence”ofthe quan-
10 InmuchthesamewayasGibbs,butquiteindependentlyofhim,Einstein(1902, 1903)alsodevelopedthe theoryofensembles.
tumofradiation,orthe photon aswenowcallit.11 Itwasthennaturalforsomeoneto derivePlanck’sradiationformulabytreatingblackbodyradiationasa gasofphotons inthe samewayasMaxwellhadderivedhislawofdistributionofmolecularspeedsforagasof conventionalmolecules.But,then,doesagasofphotonsdiffersoradicallyfromagasof conventionalmoleculesthatthetwolawsofdistributionshouldbesodifferentfromone another?
TheanswertothisquestionwasprovidedbythemannerinwhichPlanck’sformula wasderivedbyBose.Inhishistoricpaperof 1924,Bosetreatedblackbodyradiationasa gasofphotons;however,insteadofconsideringtheallocationofthe“individual”photons tothevariousenergystatesofthesystem,hefixedhisattentiononthenumberofstates thatcontained“aparticularnumber”ofphotons.Einstein,whoseemstohavetranslated Bose’spaperintoGermanfromanEnglishmanuscriptsenttohimbytheauthor,atonce recognizedtheimportanceofthisapproachandaddedthefollowingnotetohistranslation:“Bose’sderivationofPlanck’sformulaisinmyopinionanimportantstepforward. Themethodemployedherewouldalsoyieldthequantumtheoryofanidealgas,whichI proposetodemonstrateelsewhere.”
ImplicitinBose’sapproachwasthefactthatinthecaseofphotonswhatreallymattered was“thesetofnumbersofphotonsinvariousenergystatesofthesystem”andnotthe specificationasto“whichphotonwasinwhichstate”;inotherwords,photonswere mutuallyindistinguishable.EinsteinarguedthatwhatBosehadimpliedforphotonsshouldbe trueformaterialparticlesaswell(forthepropertyofindistinguishabilityaroseessentially fromthewavecharacteroftheseentitiesand,accordingtodeBroglie,materialparticles alsopossessedthatcharacter).12 Intwopapers,whichappearedsoonafter,Einstein(1924, 1925)appliedBose’smethodtothestudyofanidealgasandtherebydevelopedwhatwe nowcall Bose–Einsteinstatistics.Inthesecondofthesepapers,thefundamentaldifference betweenthenewstatisticsandtheclassical Maxwell–Boltzmann statisticscomesoutso transparentlyintermsoftheindistinguishabilityofthemolecules.13 Inthesamepaper, Einsteindiscoveredthephenomenonof Bose–Einsteincondensation,which,13yearslater, wasadoptedbyLondon(1938a,b)asthebasisforamicroscopicunderstandingofthecuriouspropertiesofliquid 4 Heatlowtemperatures.
FollowingtheenunciationofPauli’sexclusionprinciple(1925),Fermi(1926)showed thatcertainphysicalsystemswouldobeyadifferentkindofstatistics,namely,the Fermi–
11 Strictlyspeaking,itmightbesomewhatmisleadingtociteEinstein’sworkonthephotoelectriceffectasa proofoftheexistenceofphotons.Infact,manyoftheeffects(includingthephotoelectriceffect),forwhichit seemsnecessarytoinvokephotons,canbeexplainedawayonthebasisofawavetheoryofradiation.Theonly phenomenaforwhichphotonsseemindispensablearetheonesinvolving fluctuations,suchastheHanbury Brown–TwisseffectortheLambshift.Fortherelevanceoffluctuationstotheproblemofradiation,seeterHaar (1967, 1968).
12 Ofcourse,inthecaseofmaterialparticles,thetotalnumber N (oftheparticles)willalsohavetobeconserved; thishadnottobedoneinthecaseofphotons.Fordetails,seeSection6.1.
13 Itisherethatoneencountersthe correct methodofcounting“thenumberofdistinctwaysinwhich gi energy statescanaccommodate ni particles,”dependingonwhethertheparticlesare(i)distinguishableor(ii)indistinguishable.Theoccupancyoftheindividualstateswas,ineachcase, unrestricted,thatis, ni = 0, 1, 2,....
Diracstatistics,inwhichnotmorethanoneparticlecouldoccupythesameenergystate (ni = 0, 1).ItseemsimportanttomentionherethatBose’smethodof 1924 leadstothe Fermi–Diracdistributionaswell,providedthatonelimitstheoccupancyofanenergystate to atmost oneparticle.14
Soonafteritsappearance,theFermi–DiracstatisticswereappliedbyFowler(1926)to discusstheequilibriumstatesofwhitedwarfstarsandbyPauli(1927)toexplaintheweak, temperature-independentparamagnetismofalkalimetals;ineachcase,onehadtodeal witha“highlydegenerate”gasofelectronsthatobeyFermi–Diracstatistics.Inthewake ofthis,Sommerfeldproducedhismonumentalworkof 1928 thatnotonlyputtheelectrontheoryofmetalsonaphysicallysecurefoundation,butalsogaveitafreshstartinthe rightdirection.Thus,Sommerfeldcouldexplainpracticallyallthemajorpropertiesofmetalsthatarosefromconductionelectronsand,ineachcase,obtainedresultsthatshowed muchbetteragreementwithexperimentthantheonesfollowingfromtheclassicaltheoriesofRiecke(1898, 1900),Drude(1900),andLorentz(1904–1905).Aroundthesametime, Thomas(1927)andFermi(1928)investigatedtheelectrondistributioninheavieratoms andobtainedtheoreticalestimatesfortherelevantbindingenergies;theseinvestigations ledtothedevelopmentoftheso-called Thomas–Fermimodel oftheatom,whichwaslater extendedsothatitcouldbeappliedtomolecules,solids,andnucleiaswell.15
Thus,thewholestructureofstatisticalmechanicswasoverhauledbytheintroduction oftheconceptofindistinguishabilityof(identical)particles.16 Thestatisticalaspectofthe problem,whichwasalreadythereinviewofthelargenumberofparticlespresent,was nowaugmentedbyanotherstatisticalaspectthatarosefromtheprobabilisticnatureof thewave-mechanicaldescription.Onehad,therefore,tocarryouta two-fold averagingof thedynamicalvariablesoverthestatesofthegivensysteminordertoobtaintherelevant expectationvalues.Thatsortofasituationwasboundtonecessitateareformulationof theensembletheoryitself,whichwascarriedoutstepbystep.First,Landau(1927)and vonNeumann(1927)introducedtheso-called densitymatrix,whichwasthequantummechanicalanalogofthe densityfunction oftheclassicalphasespace;thiswaselaborated,bothfromstatisticalandquantum-mechanicalpointsofview,byDirac(1929–1931). Guidedbytheclassicalensembletheory,theseauthorsconsideredboth microcanonical and canonical ensembles;theintroductionof grandcanonical ensemblesinquantum statisticswasmadebyPauli(1927).17
TheimportantquestionastowhichparticleswouldobeyBose–Einsteinstatisticsand whichFermi–DiracremainedtheoreticallyunsettleduntilBelinfante(1939),Pauli(1940), andBelinfanteandPauli(1940)discoveredthevitalconnectionbetweenspinandstatis-
14 Dirac,whowasthefirsttoinvestigatetheconnectionbetweenstatisticsandwavemechanics,showed,in 1926,thatthewavefunctionsdescribingasystemofidenticalparticlesobeyingBose–Einstein(orFermi–Dirac) statisticsmustbesymmetric(orantisymmetric)withrespecttoaninterchangeoftwoparticles.
15 Foranexcellentreviewofthismodel,seeMarch(1957).
16 Ofcourse,inmanyasituationwherethewavenatureoftheparticlesisnotsoimportant,classicalstatistics continuetoapply.
17 AdetailedtreatmentofthisdevelopmenthasbeengivenbyKramers(1938).
