Statistical Physics, Second Revised and Enlarged Edition

Ensembles–alarger view 133

withthe partitionfunctionbeing

Z=

∑

j

exp(εεj/kkkBT) (2.24)

the labeljgoingover allone-particle states.

The parallel result for the ensemble involves the energy states of one assembly.

Since an assemblyisitselfalarge andcomplex entity,ithas very many states of

aparticular energy.Infact, these are themicrostatesofChapter 1; an assemblyof

energyEEjhas(EEj,V,N)such states. Since the allowed values ofEare very close

indeed, we usuallyadopt thedensity ofstates approximation anddefineG(E)δEas

the number ofone-assemblystates (= microstates) withenergy betweenEandE+δE.

The assembly partition functionZZAis defined as the sum over all one-assembly states

oftheBoltzmannfactors,i.e.

ZZA=

∑

j

(EEj,V,N)exp(−EEj/kkkBT) (12. 5 a)

or

ZZA=

∫∞

0

∫∫

G(E)exp(−E/kkkBT)dE (12. 5 b)

The distribution is specified by the numberN(E)δEoftheNNAassemblies in the usual

energyrange, or rather more usefully bythe proportionP(E)δE=N(E)δE/NNA.The

Boltzmann resultthenbecomes

P(E)=N(E)/NNA=G(E)exp(−E/kkkBT)/ZZA (12.6)

As before we may use (12. 5 ) and (12.6) to calculate the thermodynamic functions for
the ensemble–andhencefor the average assembly.Wewrite the totalfree energy
asNNAFand the total entropyasNNAS(compare the definitionUabove). Thus the
symbolsU,SandFrefer to the required average values per assembly. Following the
‘royalroute’ ofChapter 2, we obtainfor the average assembly

F=−kkkBTlnZZA (12.7)compare(2.28)

Hence all the thermal properties of the assemblymaybe calculated from the

microstates.

Several comments follow concerningthe new viewpoint as expressed in (12.5),

(12.6)and(12.7).

1 .Temperature. Thevalue ofTwhichappearsinthe equationsisdeterminedfrom

the properties ofthewhole ensemble(just asβinChapter 2 was a propertyof

the whole assembly). Therefore, as remarked above, when we concentrate on the

behaviour ofjust one assemblywearediscussingits properties whenin contact

withalarge (conceptually infinite)heat reservoiratthis temperatureT.