Statistical Mechanics - Physics at Oregon State University

1.3. STATES OF A SYSTEM. 7
g(N, M) = 2 N
M
(1.8)
Note that if we can only measure the values of N and M for our Ising system,
all the states counted in g(N,M) are indistinguishable! Hence this multiplicity
function is also a measure of the degeneracy of the states.
Thermodynamic limit again!
What happens to multiplicity functions when N becomes very large? It
is in general a consequence of the law of large numbers that many multiplicity
functions can be approximated by Gaussians in the thermodynamic
limit. In our model Ising system this can be shown as follows. Define the
relative magnetization x by x = M
N . The quantity x can take all values between
−1 and +1, and when N is large x is almost a continuous variable. Since g(N,M)
is maximal for M = 0, we want to find an expression for g(N,M) for x ≪ 1. If
N is large, and x is small, both N↑ and N↓ are large too. Stirling’s formula now
comes in handy:
Leaving out terms of order 1
N
and with
we find
N! ≈ √ 2πNN N 1
1
−N+
e 12N +O(
N2 )
and smaller one finds
(1.9)
log(N!) ≈ N log(N) − N + 1
log(2πN) (1.10)
2
log(g(N, x)) = log(N!) − log(N↑!) − log(N↓!) (1.11)
log(g(N, x) ≈ N log(N) − N↑ log(N↑) − N↓ log(N↓)
+ 1
2 (log(2πN) − log(2πN↑) − log(2πN ↓)) (1.12)
log(g(N, x) ≈ (N↑ + N↓) log(N) − N↑ log(N↑) − N↓ log(N↓)
− 1
1
log(2πN) +
2 2 (2 log(N) − log(N↑) − log(N↓)) (1.13)
log(g(N, x)) ≈ − 1
2 log(2πN) − (N↑ + 1
) log(N↑
2 N ) − (N↓ + 1
) log(N↓ ) (1.14)
2 N
Next we express N↑ and N↓ as a function of x and N via