Statistical Physics, Second Revised and Enlarged Edition

22 Distinguishable particles

withthepartitionfunctionZdefined as

Z=

∑

j

exp(−εεj/kkkBT) (2.24)

(From now on we shall for simplicity omit the∗s fromnnj,since all the discussion
willrelate to thermalequilibrium.) Itis worthmakingtwo points about the partition
function. The first is that its common symbolZis used from the German word for sum-
over-states,for thatisallthe partitionfunctionis: the sum over allone-particle states of
the ‘Boltzmannfactors’exp(−εεj/kkkBT).The secondpoint concernsits Englishname.
It is called the partition function because (in thermal equilibrium at temperatureT)nnj
is proportionalto the corresponding terminthe sum. In other wordstheNparticles
are partitionedinto their possible states (labelledbyj)injust the same ratios asZis
split up into the Boltzmann factor terms. This is clear when we rewrite (2.23) as

nnj/N=exp(−εεj/kkkBT)/Z (2.25)

or equivalently

nnj/nk=exp[−(εεj−εk)/kkkBT] (2.2 6 )

In fact the way of writing the Boltzmann distribution given in (2.26) is a very straight-
forwardwayofrememberingit. Andexpressions ofthetype exp(−ε/kkkBT)turn up
in all sorts of different physical situations.

2 .5 Calculation of thermodynamic functions

Tofinishthischapter wediscuss afew practicalities abouthow theBoltzmanndis-
tribution maybe used to calculate thermodynamic functions from first principles. In
the next chapter we apply these ideas to two particular physical cases.
We start withour system atgiven (T,V,N)asdiscussedinthe previous section.
There are then(at least)three useful routes for calculation. The best one to use will
depend on which thermodynamic functions are to be calculated – and there is no
substitutefor experienceindeciding!

Method 1: UseS=kkkBln This methodisoften theshortest to useifonlythe
entropy is required. The point is that one can substitute the Boltzmann distribution
numbers, (2.23),backinto (2.3)inorder to givet∗andhence(equation (2.4)) and
henceS.ThusSisobtainedfrom aknowledgeoftheεεjs(whichdependonV), ofN
andofT(as it enters the Boltzmann distribution).

Method 2: Use the definition ofZ There is a direct shortcut from the partition
function toU.Thisis particularly usefulifonlyUandperhapsdU/dT(=CV,the
heat capacityat constant volume) are wanted.InfactUcanbe workedout at once