Statistical Mechanics - Physics at Oregon State University

134 CHAPTER 6. DENSITY MATRIX FORMALISM.
For large values of N we can replace the exponent by √ π
N
or
1
s(u, N) = kB
N log
Nɛ N
dx g(x)
ω
1
s(u, N) = kB
N log
Nɛ ω
N
The geometrical factor g(x) is easily calculated
g(x) =
x
0
dx2
x−x2
0
− x) and we get
√
π
N δ(u
− x) (6.89)
ɛ
δ( u
ɛ
g( u
ɛ )
√
π
N
x−···−xN−1
1
dx3 · · ·
dxN =
0
(N − 1)! xN−1
Hence the entropy is
1
s(u, N) = kB
N log
Nɛ N 1
u
√
N−1 π
ω (N − 1)! ɛ N
1
s(u, N) = kB
N log
N N 1
ω N! uN ɛ √ π 1
u
(6.90)
(6.91)
(6.92)
(6.93)
We can now ignore the factor ɛ √ π 1
u in the logarithm, since that leads to a term
proportional to 1 , which disappears in the thermodynamic limit.
N
For large values of N the value of N! is about N N e−N and hence we find
1
s(u, N) = kB
N log ω −N e N u N
= kB log( u
) + 1 (6.94)
ω
The first conclusion for our model system is that the entropy is equal to
NkB log( U
Nω ) in the thermodynamic limit. Hence the temperature and energy
are related by U = NkBT . The second conclusion is that in the thermodynamic
limit the value of S does not depend on ɛ. Hence we can now take the limit of
ɛ → 0 very easily, and use delta functions. This is what we tried to show. For
a finite system the use of delta functions is not justified, such a procedure only
makes sense in the thermodynamic limit.
6.4 Equivalence of entropy definitions for canonical
ensemble.
In the previous section we used the information-theoretical definition of the entropy
to discuss the microcanonical ensemble. The resulting density matrix was
exactly what we expected from a discussion in chapter 2. Only those states with
energy U are available, and all have the same probability. Since the canonical
and grand-canonical ensemble are based on the microcanonical ensemble, the