Statistical Mechanics - Physics at Oregon State University

72 CHAPTER 4. STATISTICS OF INDEPENDENT PARTICLES.
< no >=
1
e ɛo−µ
k B T − 1
(4.15)
Again, this only depends on the values of the temperature and chemical potential
(as given by the external reservoir) and the energy of the orbital. No other
properties of the orbital play a role. The distribution function for Bosons is
therefore
fBE(ɛ; T, µ) =
1
e ɛ−µ
k B T − 1
(4.16)
This function is called the Bose-Einstein distribution function. The only difference
with the Fermi-Dirac distribution function is the minus sign in the
denominator. This is a small difference, but with large consequences. The
Bose-Einstein distribution function has the following properties:
lim
ɛ→∞ fF D(ɛ; T, µ) = 0 (4.17)
lim fF D(ɛ; T, µ) = ∞ (4.18)
ɛ↓µ
and it is the last infinity that is the cause of all differences. The big difference
between the two different distribution functions is the maximal value they can
attain. For fermions the distribution function never exceeds the value of one,
while for bosons there is no upper limit.
Limit of small occupation numbers.
It is possible to choose T and µ in such a manner that the value of the distribution
function is always much less than one for all orbital energies. In order to
use this argument it is essential that the orbital energy has a lower limit (which
is zero in our case of particles in a box). This requirement is quite natural.
All realistic Hamiltonians are bounded from below, and the energy spectrum
always has a minimum. The only exception is the Dirac Hamiltonian in first
quantization, but in that case the existence of negative energy states is easily
resolved in second quantization, where the negative energy states correspond to
positrons with normal positive energies.
The requirement for small occupation numbers is that
µ(T ) ≪ ɛmin − kBT (4.19)
Note that this does not mean that in the limit T → ∞ this inequality is not
obeyed! The chemical potential also has a temperature dependence which needs
to be taken into account. When the inequality above is valid, we have
e ɛ−µ
k B T ≫ 1 (4.20)