Statistical Mechanics - Physics at Oregon State University

168 CHAPTER 8. MEAN FIELD THEORY: CRITICAL TEMPERATURE.
−NkBT log 2 − NqJ
2T [TC − T ] m 2
(8.46)
Because T < Tc the coefficient in front of the m 2 term is negative, and lowers
the energy. This shows that the solution with m = 0 has the lowest energy. The
problem is also symmetric, for h = 0 there are two possible values for m with
opposite sign.
Note that for T > Tc and hence m = 0 we have
G(T > Tc, h = 0, N) = −NkBT log(2) (8.47)
This result makes sense. The internal energy for zero moment is zero in the
mean field approximation. Also, the magnetization is zero, and hence from
G = U − T S − HM we have S = NkB log(2), which tells us that the number of
states available to the system is equal to 2 N . All states are available, which is
what we would expect.
8.4 Density-matrix approach (Bragg-Williams approximation.
A second way to obtain the solutions for the Ising model is via the maximum
entropy principle. This approach might seem quite tedious at first, and it is
indeed, but it is also much easier to improve upon in a systematic fashion. This
principle states that we have to maximize
S = −kBTr [ρ log(ρ)] (8.48)
over all density matrices ρ consistent with our model. Similar to what we found
in a previous chapter for the canonical ensemble, with constraints Trρ = 1 ,
Trρ(H) = U, and Trρ(M) = M, the density matrix maximizing this expression
for the entropy is given by
ρ =
1
Tr e −β(H−hM)e−β(H−hM)
(8.49)
This gives the exact solution, and the temperature and field have to be set in
such a manner as to give U and M. The trace of the operator in 8.49 is still
hard to calculate, however.
Maximum (and minimum) principles are very powerful. Instead of varying
the entropy over all possible density matrices, we select a subset and maximize
over this subset only. This gives us an approximate density matrix and the
quality of the approximation is determined by our ability to find a good subset.
Many calculations in physics in general are based on minimum or maximum
principles.
The constraints we need are incorporated via Lagrange multipliers, and the
function we need to maximize is