Bragg-Williams theory of phase transition

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 $$ E = -Jsum_{langle i,j rangle} sigma_isigma_j approx -Jsum_{langle i,j rangle} m^2 = -frac12 JNz m^2, $$ 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.