Statistical Mechanics - Physics at Oregon State University

70 CHAPTER 4. STATISTICS OF INDEPENDENT PARTICLES.
tial µ are specified. The volume V of the gas determines the orbital energies
ɛo(V ). We will now divide this gas in a large number of subsystems, each of
which contains exactly one orbital. If all these subsystems are in thermal and
diffusive equilibrium with the reservoir, they are in equilibrium with each other.
Therefore, the properties of the total system can be obtained by adding the
corresponding properties of all subsystems and the total system is really the
sum of the subsystems! The properties of each subsystem, or each orbital o,
follow from
Zo(T, µ, V ) =
?
n=0
e n(µ−ɛo)
k B T (4.6)
because the energy of a state of the subsystem with n particles in the orbital
o is simple nɛo. Quantum effects are still allowed to play a role here, and they
determine the upper limit of the summation. The Pauli exclusion principle for
fermions tells us that we can have at most one particle in each orbital and hence
for fermions there are only two terms in the sum. For bosons, on the other hand,
there is no such a restriction, and the upper limit is ∞.
Fermions.
We will first consider the case of Fermions. The grand partition function for
an orbital is
Z(T, µ, V ) = 1 + e µ−ɛo
k B T (4.7)
Once we have the partition function for each orbital, we can calculate the average
number of particles < no > in that orbital. In terms of the probabilities Pn of
finding n particles in this orbital we have < no >= 0P0 + 1P1. Hence we find
< n o >=
e µ−ɛo
k B T
1 + e µ−ɛo
k B T
=
1
e ɛo−µ
k B T + 1
(4.8)
Hence the average number of particles in a subsystem depends on T and µ and
ɛo(V ). The only quantity of the orbital we need to know to determine this
average is the energy! No other aspects of the orbital play a role. In general,
the function of the energy ɛ which yields the average number of particles in an
orbital with energy ɛ is called a distribution function and we have in the case
of Fermions:
fF D(ɛ; T, µ) =
1
e ɛ−µ
k B T + 1
(4.9)
This function is called the Fermi-Dirac distribution function. The shape of this
function is well known, and is given in figure (4.1).
The Fermi-Dirac distribution function has the following general properties: