Statistical Mechanics - Physics at Oregon State University

176 CHAPTER 8. MEAN FIELD THEORY: CRITICAL TEMPERATURE.
The results of this section are identical to the mean-field results. In this
section we derived the mean-field results via the density matrix and approximated
the density matrix by ignoring all correlations. This is another way of
understanding how mean-field theory ignores correlations.
The form of the function G is very similar to forms we used in our treatment
of Landau theory in thermodynamics. We can show this in more detail by
expanding the function in powers of m for values of m near zero. Using
∞ (−1)
(1+m) log(1+m) = (1+m)
k+1
k
mk ∞ (−1)
=
k+1
k
mk ∞ (−1)
+
k+1
k
k=1
k=1
k=1
mk+1
(8.94)
and the fact that we have to add the same expression with negative m shows
that even exponents are the only ones to survive. Hence
(1+m) log(1+m)+(1−m) log(1−m) = 2
or
or
∞
k=2,even
(1 + m) log(1 + m) + (1 − m) log(1 − m) = 2
(−1) k+1
k
mk +2
∞
k=2,even
This is an even function of m as needed, and we have
∞
k=2,even
1
k(k − 1) mk
(−1) k
k − 1 mk
G(h, T, N; m) = − 1
2 JNqm2
− hNm + NkBT − log(2) + 1
2 m2 + 1
12 m4
(8.95)
(8.96)
(8.97)
1
N G(h, T, N; m) = [−hm] + [−NkBT log(2)] + 1
2 [kBT − Jq]m 2 + 1
4 [kBT
3 ]m4
(8.98)
This is exactly the model for a second order phase transition that we discussed
in Thermodynamics. Hence the Ising model in the mean field approximation
shows a second order phase transition at a temperature Tc given by
kBTc = qJ. The critical exponent β is therefore 1 (Note that β is in statistical
mechanics used for 1
kBT
2
, but also denotes a critical exponent. Do not confuse
these two!). The relation between the macroscopic Landau theory and statistical
mechanics is very important, it shows how the parameters in a thermodynamical
model can be derived from a statistical mechanical theory on a microscopic
level.