Statistical Mechanics - Physics at Oregon State University

7.2. CLASSICAL FORMULATION OF STATISTICAL MECHANICAL PROPERTIES.143
where the normalization constant C follows from Trρ = 1, or
C −1 1
= D(U, V, N) =
N!h3N
d Xδ(U − H( X)) (7.15)
Note that in classical mechanics there are no problems with discrete levels, the
set of possible values for H( X) is continuous. If the particles are not identical,
the factor N! should be omitted. The function D(U) is called the density of
states. This density of states also relates the micro-canonical and the canonical
ensemble via
Z =
1
N!h3N
which leads to
d Xe −βH( X) =
1
N!h3N
d
X
Z(T, V, N) = duD(u, V, N)e −βu
duδ(u − H( X))e −βu
(7.16)
(7.17)
The partition function is the Laplace transform of the density of states. In
many cases it is easier to evaluate Z and then this relation can be inverted
to obtain the density of states D. One important technical detail is that the
energy integral is bounded at the lower end. There is a ground state energy,
and a minimum energy level. We can always shift our energy scale so that the
minimum energy is zero. In that case we can write
D(U, V, N) = 1
2πı
Grand partition function.
c+ı∞
c−ı∞
dte st Z(t, V, N) (7.18)
The grand canonical ensemble follows from the density matrix defined in the
much larger space of states with all possible numbers of particles. The grand
partition function now involves a sum over all possible combinations of particles
and is equal to
Z(T, µ, V ) = 1
N!h3N
d Xe −β(H( X,N)−µN)
(7.19)
This leads to
N
Z(T, µ, V ) =
N
e µN
k B T Z(T, N, V ) (7.20)
which is exactly the same relation as we derived in before. This shows that
our classical formalism is consistent with the quantum statistical approach. In
deriving classical statistical mechanics from its quantum analogue, the factors
N! and h 3N come in naturally. It is also possible to postulate classical statistical
mechanics directly, but then these factors have to be introduced in some ad-hoc
fashion.