The Canonical Ensemble and the Central Limit Theor

TheCanonicalEnsembleandtheCentralLimitTheorem.D.SandsandJ.Dunning-Davies,DepartmentofPhysics,UniversityofHull,Hull,England.email:[e-mail][e-mail]ultsofmathematicalstatisticsarebecomingofincreasingimportanceinstatisticalmechanics.Heretheuseofthecentrallimittheoreminconjunctionwiththecanonicalensembleisshowntoleadtoaninterestingandimportantnewinsightintoresultsassociatedwiththecanonicalensemble.Thistheoreticalworkisillustratednumericallyanditisshownhowthisnumericalworkcanformthebasisofanundergraduatelaboratoryexperimentwhichshouldhelptoimplantideasofstatisticalmechanicsinstudents’minds.2Introduction.Itisbecomingmoreandmoreapparentthatthedevelopmentoftheareaofphysicsknownasstatisticalmechanicsindependentlyofmathematicalstatisticsitselfwasatotallyunfortunateoccurrence.Morerecently,withtheworkofsuchasJaynes[1],thisdivisionintotwoapparentlyseparatesubjectshasbeenblurredsomewhat.However,theseparateapproachesanddifferentterminologystilltendtoimposeanunrealisticandartificialbarrier,whichcanprovearealhindrancetostudentsandnewworkersinthefieldalike.AnattempttorectifythisunfortunatesituationisprovidedbythebookonstatisticalphysicsbyLavenda[2].Evenhere,though,itsometimesprovesdifficultfornewcomerstothefieldtoappreciatethepowerfultechniquesthatactualmathematicalstatisticscanbringtobearonproblemsinstatisticalmechanics.Nowhereisthismoreapparentthaninthelackofknowledgeof,andlackofappreciationof,theimportantlimittheoremsofmathematicalstatistics.-particularlytheso-called‘lawsoflargenumbers’andthe‘centrallimittheorems’.Thecentrallimittheorem.Aclassicexampleoftheuseofwell-knownresultsfrommathematicalstatisticsisprovidedbythecanonicalensemble,forwhichtheproperprobabilitydensityisgivenby,(1)where,)()(1mECEmmΓ=Ω-Cbeingaconstantdependingonthesystemunderconsiderationand2mbeingthenumberofdegreesoffreedom.Herealso,Z(β)isgivenby(2)Itfollowsthatthemeanvalueoftheenergyisgivenby(3)and(4)(5)(6)whichisthevarianceofthedistribution.)()();(EZeEfEΩ=-βββ.)()(0∫∞-Ω=dEEeZEββ∫∞-=∂∂-=0)()(1lndEEEeZZEEβββ222211ln∂∂-∂∂=∂∂-∂∂-=∂∂-βββββZZZZZE()202)(1EdEEeEZE-Ω=∫∞-β()()(),2222EEEEEΔ=-=-=3Alltheaboveisquitewell-knownandmanyoftheresultswillbefamiliar.However,withintherealmsofthetheoryofstatisticsitself,specificallyinprobabilitytheory,someofthemostimportantresultsareso-calledlimittheorems.Amongthese,possiblythemostusefularegroupedtogetheraseither‘lawsoflargenumbers’or‘centrallimittheorems’.Theoremsareoftentermed‘lawsoflargenumbers’iftheyareconcernedwithgivingconditionsunderwhichtheaverageofasequenceofrandomvariablesconvergesinsomesensetotheexpectedaverage.Thecentrallimittheorems,ontheotherhand,areconcernedwithfindingtheconditionsunderwhichthesumofalargenumberofrandomvariableshasaprobabilitydistributionwhichapproximatestothenormaldistribution.Withinstatistics,thepropertiesoftheso-called‘normal’distributionhavebeenstudiedingreatdepth.Thisisdueinparttothefactthat,inpractice,manyvariablesaredistributedinaccordancewiththisdistributionorverynearlyaccordingtoit.Itisfoundthatthesamplemeansapproximatelyfollownormaldistributionevenwhentheunderlyingdistributionisnotnormal.Thisresultprovidesthesubjectmatterforacentrallimittheorem,initiallyintroducedbydeMoivreandlaterimprovedbyLaplace,althoughthefirsttrulyrigorousproofwaspresentedbyLindebergtowardsthebeginningofthelastcentury.ThispowerfultheoremstatesthatIftherearex1,x2,….,xnidenticallydistributedrandomvariables,eachwithmeanμandfinitevarianceσ2,thenthevariablex,givenby(7)willhaveadistribution(8)whichapproachesthestandardnormaldistributionwithmean0andvariance1asnbecomesindefinitelylargeTheproofofthistheoremmaybefoundinmoststandardstatisticstexts,forexamplethosebyRényi[3]andCramer[4].However,fromapurelypracticalviewpoint,theimportanceofthistheoremresidesinthefactthatitallowstheuseofresultsassociatedwiththenormaldistributionevenwhenthebasicvariable,x,hasadistributionwhichdiffersmarkedlyfromnormality.Itmightbenotedthatthemorethebasicdistributiondiffersfromnormality,thelargernmustbecometoapproximatenormalityfor.xAlmosttheonlyrestrictionimposedontheunderlyingdistributionisthatthevariancebefinite,Itisencouragingtorealisethatthevastmajorityofphysicsproblemsobeythisrestrictionquitenaturally.Inthepresentcontext,itmaybenoted,therefore,thatthedistributionassociatedwiththecanonicalensemblemaybeapproximatedby(9)∑==niixnx11,/nxzσμ-=()()[][].22exp2/1222EEEEΔΔ--π4Thissomewhatsurprisingresult,showingthatthecanonicaldistributioniseffectivelynormal,isrelativelyunknowninstatisticalmechanics,althoughitisstatedquiteclearlyinthebookbyLavenda[2].Intheorthodoxviewnormallygiveninundergraduatetexts[5-7],thecanonicaldistributionisexpressedas(10)wherepiisinterpretedastheprobabilitythatthesystemisinanenergystateEi.Gibbscalledthis,“thecanonicaldistributionisphase”[8].Hederiveditbyconstructinganensembleofsystems,eachcontainingNparticles,butwithadifferentdistributionofvelocities.Eachsystemintheensembleisrepresentedbyapointin6N-dimensionalspace.TheLiouvilletheoremshows