Statistical Mechanics - Physics at Oregon State University

4.6. BOSON GAS. 85
S = −kB (fF D log(fF D) + (1 − fF D) log(1 − fF D)) (4.77)
orb
This is very similar to the expression we had before in terms of probabilities.
The big difference is again that this time we sum
over orbitals. When we
summed over many body states we had S = −kB s Ps log(Ps). But now
we sum over single particle orbitals, which is a much simpler sum. We can
remember the formula above by the following analogy. For each orbital there
are two states, it is either occupied or empty and the first term in the expression
for S is related to the probability of an orbital being occupied, the second to
the probability of the orbital being empty.
4.6 Boson gas.
Only a minus sign different!
The treatment of a gas of identical, independent bosons is almost identical to
that for fermions. There is one important exception, though. The distribution
function for bosons is
fBE(ɛ) =
1
e ɛ−µ
k B T − 1
(4.78)
for which we need that µ < min(ɛ), otherwise the number of particles in a
given orbital with energy below µ would be negative! The fermion distribution
function is always less than one, but for bosons the distribution function can take
any positive value depending on how close µ is to ɛ in units of kBT . Obviously,
since f(ɛ) is a monotonically decreasing function of ɛ the orbital with the lowest
energy will have the largest population and this orbital will cause problems in
the limit µ → min(ɛ).
Grand partition function.
The grand partition function is calculated in a similar way as for fermions,
with the important difference that the number of particles in each orbital can
be between zero and infinity
Z(T, µ, V ) =
∞
Z(T, µ, V ) =
∞
n=0 n2=0 ni
∞
e 1
k B T (µ o no− o noɛo)
∞
n=0 n2=0 ni orb
e no
k B T (µ−ɛo)
(4.79)
(4.80)