Statistical Mechanics - Physics at Oregon State University

208 CHAPTER 9. GENERAL METHODS: CRITICAL EXPONENTS.
Z(T, h, N) =< e+|T N |e+ > + < e−|T N |e− > (9.86)
Therefore the partition function in terms of the eigenvalues is
and the magnetic free energy is
Z(T, h, N) = t N + + t N −
(9.87)
G(T, h, N) = −kBT log(t N + + t N − ) (9.88)
Now we use t+ > t− and rewrite this in the form
G(T, h, N) = −kBT log(t N + [1 +
t−
t+
G(T, h, N) = −NkBT log(t+) − kBT log(1 +
N
]) (9.89)
t−
t+
N
) (9.90)
In the thermodynamic limit N → ∞ the second term becomes zero, because
| < 1 and we find
| t−
t+
G(T, h, N) = −NkBT log e βJ
cosh(βh) + e2βJ sinh 2 (βh) + e−2βJ
(9.91)
In the limit h = 0 we recover our previous result. The magnetization per
particle m follows from
m = − 1
∂G
=
N ∂h T,N
or
kBT eβJ β sinh(βh) + 1
with the final result
2 [e2βJ sinh 2 (βh) + e−2βJ 1 − ] 2 [e2βJ 2 sinh(βh) cosh(βh)β]
e βJ cosh(βh) +
e 2βJ sinh 2 (βh) + e −2βJ
e
m =
βJ sinh(βh)
e2βJ sinh 2 (βh) + e−2βJ e 2βJ sinh 2 (βh) + e −2βJ + e βJ cosh(βh)
e βJ cosh(βh) +
e 2βJ sinh 2 (βh) + e −2βJ
sinh(βh)
m =
sinh 2 (βh) + e−4βJ (9.92)
(9.93)
(9.94)