Statistical Physics, Second Revised and Enlarged Edition

The Boltzmann distribution and the partitionfunction 21

Theargumentin outlineisasfollows. Startbyconsideringhowachangeinln
can be brought about

d(ln)=d(lnt∗) for alargesystem

=−

∑

j

lnnn∗jdnnj asin (2.7)

=−

∑

j

(α+βεεj)dnnj using the Boltzmann distribution

=−β

∑

j

εεjdnnj sinceNisfixed

=−β(dU)no work first term of(2.22)

=−β(TdS) first term of(2.21)

Thisidentificationis consistentwithS =kkkBlntogether withβ =− 1 /kkkBT.
It shows clearlythat the two statistical definitions are linked, i.e. that (2.20)
validates (1. 5 ) or vice versa. Part of this consistency is to note that it must
bethe same constant (kkkB,Boltzmann’s constant) whichappearsinbothdefinitions.

2 .4 Theboltzmanndistribution andthepartitionfunction

We now return to discuss the Boltzmann distribution. We have seen that this dis-
tributionisthe appropriate one todescribethethermalequilibrium properties ofan
assemblyofNidentical localized (distinguishable) weaklyinteractingparticles. We
have derived it for an isolated assembly having a fixed volumeVand a fixed internal
energyU.Animportant partinthe resultisplayedbythe parameterβwhichisa
function of the macrostate(U,V,N). However, the upshot of theprevious section
is to note that the Boltzmann distribution is most easily written and understood in
terms of(T,V,N)rather than(U,V,N).Thisisnoinconvenience, sinceitfrequently
happens inpractice that it isTratherthanUthat is known. And it is no embarrass-
mentfrom afundamentalpoint ofview solong as we aredealing withalarge enough
system thatfluctuations are unimportant. Therefore, althoughour methodlogically
determinesT(and other thermodynamic quantities) as a function of (U,V,N)for
anisolatedsystem, we shallusually use the results todescribethebehaviour ofU
(andother thermodynamic quantities) as afunction of(T,V,N). (Thissubject will
be reopened in Chapters 10–12.)
Therefore, we now write theBoltzmanndistribution as

nnj=(N/Z)exp(−εεj/kkkBT) (2.23)