MIT粒子统计学课程_Lec4.pdfVIP

I.IStabilityConditions

TheconditionsderivedinsectionI.Garesimilartothewellknownrequirementsfor

mechanicalstability.AparticlemovinginanexternalpotentialUsettlestoastable

equilibriumataminimumvalueofU.Inadditiontothevanishingoftheforce−U′,this

isaconsequenceofthelossofenergytofrictionalprocesses.Stableequilibriumoccursat

aminimumofthepotentialenergy.Forathermodynamicsystem,equilibriumoccursat

theextremumoftheappropriatepotential,forexampleatthemaximumvalueofentropy

foranisolatedsystem.Therequirementthatspontaneouschangesshouldalwayslead

toanincreasedentropy,placesimportantconstraintsonequilibriumresponsefunctions,

discussedinthissection.

Considerahomogeneoussystematequilibrium,characterizedbyintensivestatefunc­

tions(T,J,),andextensivevariables(E,x,N).Nowimaginethatthesystemisarbitrar­

ilydividedintotwoequalparts,andthatonepartspontaneouslytransferssomeenergy

totheotherpartintheformofworkorheat.Thetwosubsystems,AandB,initially

havethesamevaluesfortheintensivevariables,whiletheirextensivecoordinatessatisfy

EA+EB=E,xA+xB=x,andNA+NB=N.Aftertheexchangeofenergybetween

thetwosubsystems,thecoordinatesofAchangeto

(EA+δE,xA+δx,NA+δN),and(TA+δTA,JA+δJA,A+δA),(I.60)

andthoseofBto

(EB−δE,xB−δx,NB−δN),and(TB+δTB,JB+δJB,B+δB).(I.61)

NotethattheoverallsystemismaintainedatconstantE,x,andN.Sincet