Statistical Mechanics - Physics at Oregon State University

82 CHAPTER 4. STATISTICS OF INDEPENDENT PARTICLES.
we need absolute convergence, that is we need to be able to find a minimum value
of N for which the partial sum is a good approximation for all temperatures.
Hence we need for every ɛ > 0 we a value Nɛ such that
N > Nɛ ⇒ |SN(T ) − S(T )| < ɛ (4.58)
In that case we can interchange summation and limits, and can integrate the
sum of the series by integrating each term and sum the resulting integrals.
In our case, for large values of the quantum numbers the energy is very large
and the distribution function can be approximated by
lim
ɛ→∞ fF D(ɛ) ≈ e µ−ɛ
kB T (4.59)
Hence the terms in the series for N decay very rapidly. If we consider a temperature
interval [0, Tmax] we always have that e −ɛ −ɛ
kB T k e B Tmax . Therefore we
can always use the value at Tmax to condition the convergence of the series.
Hence we find that the series converges uniformly on the interval [0, Tmax], with
endpoints included. We may interchange the limit T → 0 and the sum and get
N =
Θ(ɛF − ɛo) (4.60)
orb
If we use free particle energies, which are always positive, we find immediately
that ɛF > 0 because N > 0. Note that the only problem is when we take
T → ∞. That limit needs to be analyzed separately, since we now have an
infinite number of infinitesimally small terms to sum. We will see later how
to deal with that limit. We already know the answer, though, from a previous
section. At large temperatures we can replace the sum by an integral, if the
orbital energies are free particle energies. This is always true for large quantum
numbers, due to Bohr’s correspondence principle.
Grand partition function.
The grand partition function for a Fermi gas involves a sum over states.
If we enumerate the orbitals for the independent particles, a state of the total
system can be specified by the occupation of each orbital, or by a set of numbers
{n1, n2, . . .}. Here ni denotes the number of particles in orbital i. Since we are
dealing with fermions ni is zero or one only. Examples of states are {0, 0, 0, . . .}
for a state with no particles or {1, 1, 0, . . .} for a state with one particle in orbit
1 and in orbit 2.
The grand partition function is
Z(T, µ, V ) =
{n1,n2,...}
The total number of particles is easy to find:
e 1
k B T (µN({n1,n2,...})−E({n1,n2,...}))
(4.61)