Bragg-Williams theory of phase transition 
Hi I am currently studying Bragg wiliam theory and I dont understand how they derive the equations f(T,m). Actually i dont understand how they got S/N given by equation 8. Can someone explain?
 A: Well, as usual the entropy is the logarithm of the number of microstates. If the total number of spins is $N$ and there are $N_{\rm up}$ "$+$ spins", then this corresponds to a total number of possible configurations given by $\frac{N!}{N_{\rm up}!(N-N_{\rm up})!} = {}^NC_{N_{\rm up}}$ (the number of ways of choosing which of the $N$ spins are "$+$ spins"). This is exactly the first identity in (8).
The second identity in (8) follows from the fact that the magnetization density is
$$
m= \frac{N_{\rm up} - N_{\rm down}}{N} = \frac{2 N_{\rm up} - N}{N} = 2 \frac{N_{\rm up}}{N} - 1,
$$
so that
$$
N_{\rm up} = \frac{(1+m)N}{2}.
$$
Of course, this way of computing the entropy totally ignores the fact that the different configurations do not have the same energy, which is why the Bragg-Williams theory only provides an approximation.
Then, they go on getting another approximation, this time for the energy, writing
$$
E = -J\sum_{\langle i,j \rangle} \sigma_i\sigma_j \approx -J\sum_{\langle i,j \rangle} m^2 = -\frac12 JNz m^2,
$$
since there are $\frac12 N z$ pairs of nearest neighbors.
Once they have the (approximate) entropy and energy, they compute the free energy by the usual thermodynamic relation.
