Classical Statistical Mechanics

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1、ClassicalStatisticalMechanicsAmacrostatehasNparticlesarrangedamongmvolumes,withNi(i=1:::m)particlesintheithvolume.ThetotalnumberofallowedmicrostateswithdistinguishableparticlesisXmN!W=Qm;lnW=lnN!lnNi!:iNi!iForalargenumberofparticles,useStirling'sformulalnN!=NlnNN:XmlnW=NlnNN

2、(NilnNiNi):iTheoptimumstateisthemacrostatewiththelargestpossiblenumberofmicrostates,whichisfoundbymaximizingW,sub-jecttotheconstraintthatthetotalnumberofparticlesNisxed(N=0).Inaddition,werequirethatthetotalenergybeconserved.Ifwiistheenergyoftheithstate,thisis!XmXmwiNi=wiN

3、i=0:iiWiththeseconstraints,theminimizationis"!#XmXmlnWNiwiNi=0:iiXm[lnNiwi]Ni=0:iNwiwi=kTi=e=e;whichisthefamiliarMaxwell-Boltzmann,orclassical,distri-butionfunction.QuantumStatisticalMechanicsInthequantummechanicalview,onlywithinacertainphasespacevolumeareparticlesindist

4、inguishable.Theminimumphasespaceisoforderh3.Nowdenotethenumberofmi-crostatespercellofphasespaceofvolumeh3asWi.ThenthenumberofmicrostatespermacrostateisYW=Wi:iNotewehavetoconsiderboththeparticlesandthecompart-mentsintowhichtheyareplaced.Iftheithcellhasncom-partments,therearenseq

5、uencesofNi+n1itemstobearranged.Therearen(Ni+n1)!waystoarrangetheparti-clesandcompartments,butwehaveovercountedbecausetherearen!permutationsofcompartmentsinacell,andtheorderinwhichparticlesareaddedtothecellisalsoirrelevant(thefactorNi!wehadintheclassicalcase).ThusYn(Ni+n1)!Y(

6、Ni+n1)!W==:Ni!n!Ni!(n1)!iiOptimizingthis,wendXlnW=[(n+Ni1)ln(n+Ni1)NilnNii(n1)ln(n1)lnNiwiNi]Xn+Ni1=lnlnwiNi=0;Niior1Newi=kT1i=(n1):Infact,thisistherelevantexpressionwhenthereisnolimittothenumberofparticlesthatcanbeputintothecompart-mentofsizeh3,i.e.,for

7、bosons.Further,inthecasewhenbosonsarephotons,theconditionNdoesnotapply,andthefactor1.Forfermions,only2particlescanbeputintoacompart-ment,where2isthespindegeneracy.Thus,phasespaceiscomposedof2nhalf-compartments,eitherfullorempty.Therearenomorethan2nthingstobearrangedandtherefo

8、renomorethan2n!microstates.Butagain,weovercounted.ForN