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1、StatisticalMechanicalEnsemblesDiannanLuDepartmentofChemicalEngineeringMicroscopicAndMacroscopicVariablesCentralquestioninStatisticalMechanics:Ifparticles(atoms,Molecules,electrons,nuclei,orevenlivingcells)obeycertainmicroscopiclawswithspecifiedinterparticleinteractions,whataret
2、heobservablepropertiesofamacroscopicsystemcontainingalargenumberofsuchparticles?Examplesofmicroscopicandmacroscopicvariables:MicroscopicvariablesMacroscopicvariables3×1023positions(x,y,z)n+2independentthermodynamicvariables—3×1023velocities(vx,vy,vz)e.g.forn=1,N,V,TRuleofthumb:
3、Ifanobservableofamanyparticlesystemcanbespecifiedbyasmallnumberofothermacroscopicproperties,weassumethattheobservablecanbedescribedwithstatisticalmechanicsPhasespaceFor1particles,inonedimension(x,px)xphasespacemicrostate↔pointinphasespaceSurfaceinphasespace↔constraintst=0t=t—li
4、keconstantP,T,…ThepropertiesofphasespaceM1ϕmeans=∑ϕitimeaverageMi=1(longenough)pxFormultidimensionalspacedefinedbythemicroscopicvariablesofsystem!N!N!!!!!!!!(r,p)=(r1,r2,r3,"rN;p1,p2,p3,",pN)!!!!risthepositionpisthemomentump=mutotalpotentialenergymassvelocity!!!!!!dri!dui∂U(r1,
5、r2,r3,!,rN)EvolutionofasystemintimeisdescribedbyNewton’slaws:=uimi=−!dtdt∂riEnsemblesAnensembleisacollectionofallmicrostatesofasystem,consistentwiththeconstraintswithwhichwecharacterizeasystemmacroscopically.Example:Acollectionofallpossiblestatesofthe1023moleculesofgasinthecont
6、ainerofvolumeVwithagiventotalenergyU.Q:Whatisthephasespaceofone-dimensionalharmonicoscillatorwithagivenenergy?xpxErgodicHypothesisForsufficientlylongtimes,amacroscopicsystemwillevolvethrough(orwillcomearbitrarilycloseto)allmicroscopicstatesconsistentwiththemacroscopicconstrains
7、weimposeinordertocontrolthesystem.Inotherwords,experimentalmeasurements(performedbytimeaverages)andensembleaveragesareequivalent.Postulate1:Thereexiststableequilibriumstatesfullycharacterizedbyn+2independentmacroscopicvariables.AgeneralpropertyFwillobeythefollowingrule:valueofp
8、ropertyFinmicrostatevF=P×F=F(1)observed∑vvensembleaver
