Statistical Mechanics - Physics at Oregon State University

8 CHAPTER 1. FOUNDATION OF STATISTICAL MECHANICS.
which gives
This leads to
− 1
2
(N(1 + x) + 1) log(1
2
x = N↑ − N↓
N
= 2N↑ − N
N
(1.15)
N↑ = 1
N(1 + x) (1.16)
2
N↓ = 1
N(1 − x) (1.17)
2
log(g(N, x)) ≈ − 1
2 log(2πN)
(1 + x)) − 1
2
(N(1 − x) + 1) log(1 (1 − x)) (1.18)
2
Next we have to remember that this approximation is only valid for values
of x close to zero. Hence we can use log(1 + x) ≈ x − 1
2 x2 to get
log(g(N, x)) ≈ − 1
2
log(g(N, x)) ≈ − 1
2
log(g(N, x)) ≈ − 1
2
which gives the final result
log(2πN) − 1
2
1
(N(1 + x) + 1)(log(1 ) + x −
2 2 x2 )
− 1
1
(N(1 − x) + 1)(log(1 ) − x −
2 2 2 x2 ) (1.19)
log(2πN) − (N + 1) log(1
2
1
1
) − ((N + 1) + Nx)(x −
2 2 x2 )
− 1
1
((N + 1) − Nx)(−x −
2 2 x2 ) (1.20)
1
log(2πN) + (N + 1) log(2) − (N − 1)x2
2
g(N, x) ≈
2
πN 2N 1 −
e 2 x2 (N−1)
(1.21)
(1.22)
Hence we have shown that the multiplicity function becomes Gaussian for
large values of N and small values of x. The Gaussian form is reduced by a
factor e from its maximum at x = 0 at a value x = x0 with
2
x0 =
(1.23)
N − 1