Statistical Physics, Second Revised and Enlarged Edition

Idealgases in thegrand ensemble 143

Aparticular microstateisdefinedbythe number ofparticlesnnjina statej,for every

one-particle state. And in (13.10a), this corresponds to pickingout the termγγγjnjfrom

thejth bracket. This is not yet the simplest expression forZZZG,however, since the

infinite sumin eachbracket can readily be performed.Wehave alreadymet sums of

this sort in our discussion of an assemblyof harmonic oscillators in section 3.2.1. The

summation is( 1 +γ+γ^2 +γ^3 ...)=( 1 −γ)−^1 .Hence we obtain the final result

ZZZG(BE)=

∏

j

( 1 −γγγj)−^1 (13.10b)

Maxwell–Boltzmann limit. Having said that there are two cases, we now follow

precedent to considerathird!Thepointisthat we can obtainanduse an even simpler

expression forZZZGin the MB limit of either form of statistics, FD or BE. We already

know that the two statistics tend to the same limit for a dilute gas, i.e. one in which the

occupation ofthe statesis verysparse. As usual,theFDandBE cases tendto thelimit

from opposite sides. In the dilute limit, one expects the exclusionprinciple to become

anirrelevance, since the probability ofdouble occupationisalways negligible. The

appropriate expressionisobtainedfrom a compromisebetween (13.9) and(13.10a)

when all theγγγjare small,i.e. 1. It is to replace either equation by

ZZZG(MB)=

∏

j

exp(γγγj) (13.11)

The compromiseinvolvedis seen when one recallsthe power series, appropriatefor

smallγ

expγ= 1 +γ+γ^2 / 2 !+γ^3 / 3 !...

13.3.2 Derivation of the distributions

The distribution is defined as a filling factor for a one-particle state. It tells us the

number ofparticles per state on averageinthermalequilibrium. Earlier wedefined

thedistributionfffias the fractional occupation of states with energyεi.The definition

is even more straightforwardinthe grandassembly, since we now consider the states

individually.Thus thefractionaloccupation ofstatejis preciselythethermalaverage

ofnnj. We denote this average by the symbolnnj,which has exactly the same interpre-

tation as the earlierfffi.As postulatedat theendofsection 13.2.1, thethermalaverage

isobtainedusingtherelative weights ofeachterminthe expressionforZZZG.

Fermi–Dirac. The expressionforZZZGis (13.9), where the product goes over allone-

particle statesj.Nowlet us workout the average occupation ofone ofthose states,

which at the risk of confusing the reader we shall label asi.(There are not enough

lettersinthealphabet;however this notationhas the meritthat theoldfffishould

lookidenticalto the newly derivedni.) Eachone ofthe many(infinite!) microstates