Statistical Physics, Second Revised and Enlarged Edition

58 Gases: the distributions

Againthe numeratoristhe product ofnifactors eachapproximatelyequalto (buta
little larger than)gi.Hence for the dilute boson gas

tttBE≈

∏

i

gnii

ni!

Therefore in the dilute limittttFDandtttBEtend to the same value, one from below and

onefrom above. This‘classical’limitiscalledthe Maxwell–Boltzmann (MB) case,

discussed bythese famous scientists before quantum mechanics was conceived. And

we write the answer to our third counting problem as

tttMB=

∏

i

gnii

ni!

(5.6)

5 .4 The three distributions

It now remains toderive thethree equilibriumdistributionsfor thethree counting

methods of section 5 .3. The techniques are precisely those set up in section 1. 5 and

already workedthroughforlocalizedparticlesinChapter 2. Theaimistofindthe most

probabledistribution consistent withthe macrostatebymaximizingthe expression

fort({ni}), just as in section 2.1. 5.

5.4.1 Fermi–Dirac statistics

We require to maximizetttFD, or more convenientlylntttFD,subject to the usual

macrostate conditions

∑

i

ni=N (5.7)

and

∑

i

niεi=U (5.8)

Asbefore, the methodistosimplify lntusingStirling’s approximation (Appendix
B), and then to find the conditional maximum using the Lagrange method. An outline
derivationfollows.
Takinglogarithms of (5.4)gives

lntttFD=

∑

i

{gilngi−nilnni−(gi−ni)ln(gi−ni)}

The Lagrange methodwritesdownfor the maximum condition

d(lnt)+αd(N)+βd(U)=0( 5 .9)