Statistical Mechanics - Physics at Oregon State University

86 CHAPTER 4. STATISTICS OF INDEPENDENT PARTICLES.
Z(T, µ, V ) =
∞
orb
n=0
e n(µ−ɛo)
kB T
=
Zo(T, µ, V ) (4.81)
The only difference is the limit of the summation now going to infinity. The
partition function for one orbital is in this case
Zo(T, µ, V ) =
and is always positive since µ < min(ɛ).
The grand potential follows from
1
orb
1 − e µ−ɛo
k B T
(4.82)
Ω(T, µ, V ) = −kBT
log(Zo(T, µ, V )) (4.83)
in the same way we found for fermions. The total number of particles is equal
to minus the derivative of Ω with respect to µ, and is, of course:
∂Ω
N = − =
∂µ
fBE(ɛo) (4.84)
because
∂ log Zo
∂µ T,V
T,V
orb
orb
= 1
kBT Z−1 o
e µ−ɛo
k B T
(1 − e µ−ɛo
k B T ) 2
(4.85)
which is equal to the distribution function divided by kBT .
A very useful formula for the entropy, again relating the entropy to probabilities,
is
∂Ω
S = − =
∂T V,µ
−kB (fBE(ɛo) log(fBE(ɛo)) − (1 + fBE(ɛo)) log(1 + fBE(ɛo))) (4.86)
orb
The second term always gives a positive contribution, but the first term is
negative if fBE(ɛo) > 1. But since we can combine terms according to
1 + fBE(ɛo)
S = kB fBE(ɛo) log( ) + log(1 + fBE(ɛo))
fBE(ɛo)
orb
(4.87)
we see that each negative term in equation (4.86) is cancelled by a larger positive
term, because the first term in this expansion is always positive. The second
term in equation (4.86) does not have a simple physical interpretation, but is
directly related to the physical phenomenon of stimulated emission.