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1、TheStatisticalEnsembleDistributionFunctionandProbabilityDensity•Giventheformofthehamiltonianforaspecificsystem,theintegrationofHamilton’sequationsallowsoneinprincipletouniquelydetermineallthedynamicalvariablesatanytime,providedthatsufficientdataareavailabletoyiel
2、dtheirvaluesatafixedtimet.0•However,therearenormallyfartoofewdatafurnishedbyactualobservationtosatisfythisrequirement,sothatfurtherinformationcanbebasedonlyuponstatementsofprobability.Ensemble•Consideranyset(x)ofdynamicalvariables(k=k1,2,…,2n),fortheset(p,q)ofmom
3、entaandiicoordinates.•Imagineanarbitrarilylargenumberνofidenticallyreproducedsystems,eachpertainingtoaseparatesetofthesevariables.•Suchamanifoldiscalledanensemble,withtheunderstandingthatanyoneofitsmembersshallberequiredtobeapossiblerepresentativeofindividualsyst
4、emunderinvestigation.•Theensemblecanbecharacterizedbyνdifferentpointsinthephasespacesuchthateachofthemrepresentsaseparatememberwiththecorrespondingset(x)givenbyktheinstantaneousvaluesofthedynamicalvariables.•Considerthepointswithinagivenregion,denotedby∆νtheirnum
5、beratacertintime,onethenhasbydefinition∆ν∆P=νfortheprobabilityoffindingatthattimethedynamicalvariablesofthesystemunderinvestigationwithintherangeofvaluesindicatedbythisregion.•Inthelimitofaninfinitesimalregionwitharangeofvariablesbetweenxandx+dx,ledtothekkkdiffer
6、entialprobabilitydνdP=ν•Bychoosingνsufficientlylarge,therepresentativepointstobesodenselylocated,theirnumberdνwithinanysuchregionasacontinuouslyvaryingquantity•Therefore2nndλ=dx=dpdqΠkΠiik=1i=1•Onemayrewrite,dν=D(x)dλwhereD(x)isdistributionfunction.•Byintegration
7、overthewholeofphasespace,ν=∫D(x)dλ•Hence,probabilityoffindingthesetofvariables(x)withinktheinfinitesimalregionofvolumedλD(x)dλdP=∫D()xdλ•OnecanwritedP=ρ(x)dλD(x)ρ()x=∫D()xdλ•istheprobabilitydensity,therefore,∫ρ(x)dλ=1MeanValues•ProbabilityPoffindingthedynamicalva
8、riablesRwithinarangeoftheirvalues,representedbyagivenregionRofphasespace•FromthedifferentialprobailitiesdP,referringtoanyvolumeelementdλwithinthatregion,P=ρ(x)
