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