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