Statistical Mechanics - Physics at Oregon State University

150 CHAPTER 7. CLASSICAL STATISTICAL MECHANICS.
Similar to what we derived before, it is possible to show that the last, exponential,
factor in the logarithm does not give a contribution to the integral. We
expand the logarithm to get:
1
S(U, V, N) = −kB
N!h3N
d XC 1
ɛ √ (U−H)2
e− ɛ
π 2
log(C) − log(ɛ √ (U − H)2
π) −
ɛ2
(7.34)
The third term in the integral is very small when |U − H| ≫ ɛ due to the
exponential. It is also very small when |U − H| ≪ ɛ because of the quadratic
term. Hence it only contributes when |U − H| ≈ ɛ. In that case the integrant
is proportional to 1
ɛ . But the volume in phase space that corresponds to this
value is proportional to ɛ6N−1 and hence the total contribution vanishes when
ɛ goes to zero.
The second factor in the logarithm gives a contribution kB log(ɛ √ π) to the
entropy, and this factor vanishes in the thermodynamic limit. Hence we find
with
S(U, V, N) = −kB log(C) = kB log(D(U, V, N)) (7.35)
D(U, V, N) = C −1 =
1
N!h3N ɛ √
π
d (U−H)2
−
Xe ɛ2 (7.36)
Here we can again take the limit ɛ → 0 to recover the expression we derived
before. But note that we have used the thermodynamic limit, and we did take
it before the limit ɛ → 0. That is technically not correct. This was a topic of
much discussion in the early days of the development of statistical mechanics.
For an ideal gas the density of states D follows from
D(U, V, N) =
1
N!h3N
dp 3 1 · · · dp 3 N δ(U −
N
i=1
p 2 i
2m )
dr 3 1 · · · dr 3 N
(7.37)
The spatial integral is easy and gives a factor V N . The integral over the momenta
is equal to the surface area of a 3N-dimensional hyper-sphere of radius
√ 2mU. This leads to
If N is even, Γ( 3N
2
D(U, V, N) =
) = ( 3N
2
N V
N!h3N 2π 3N
√2mU3N−1 2
Γ( 3N
2 )
(7.38)
3N
−1)! and if N is odd, Γ( 2 ) = √ 3N−1
− π2 2 (3N −2)!!
In both cases we can approximate log(Γ( 3N 3N−1
2 )) by ( 2 ) log( 3N 3N
2 ) − 2 .
This assumes that N is very large (thermodynamic limit) and that we can
replace N −1 by N. The entropy in this limit is (again ignoring terms not linear
in N)