Statistical Physics, Second Revised and Enlarged Edition

The three distributions 59

Substitutinglnt=lntttFDfrom above together with (5.7) and (5.8) forNandUgives

afterdifferentiation

∑

i

{ln[(gi−ni)/ni]+α+βεi}dni= 0

The Lagrange method enables one to remove the summation sign for the specific, but

as yet undetermined, values of the multipliersαandβto obtain for the most probable

distribution

ln[(gi−n∗i)n∗i]+α+βεi= 0

Rearranging this expression we find

n∗i=gi/[exp(−α−βεi+ 1 ]

As anticipatedthegroupingofthe stateshas no explicitinfluence on the equilibrium
occupation per state, and the result may be written in terms of a distribution function

fffi=n∗i/gi= 1 /[exp(−α−βεi+ 1 ] (5.10a)

The distribution function contains no detail about the states except their energies, and

soin concordwiththewholedensity ofstates approximationit canbethoughtofas

effectivelya continuousfunction ofthe one-particle energyεior simplyε.Hence a

useful form of(5.10a)is

fffFD(ε)= 1 /[exp(−α−βε)+ 1 ] (5.10b)

Thisisthe Fermi–Diracdistributionfunction.

5.4.2 Bose–Einstein statistics

The derivation of the Bose–Einstein distribution for a boson gas follows analogous

lines.

Takinglogarithms of (5.5) and usingStirling’s approximationgives

lntttBE=

∑

i

{(ni+gi)ln(ni+gi)−nilnni−gilngi}

Substituting this into (5.9) together with the restrictions (5.7) and (5.8) gives

∑

i

{ln[(ni+gi)/ni]+α+βεi}dni= 0

The form of the equilibrium distribution is again obtained by setting each term of the

sum equalto zero

ln[(n∗i+gi)/n∗i]+α+βεi= 0