- #1

CAF123

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## Homework Statement

Consider a D dimensional Ising model with N sites, defined by the Hamiltonian $$\mathcal H = -J \sum_{\langle i j \rangle} \sigma_i \sigma_j - h \sum_i \sigma_i$$ where the sum extends over nearest neighbours and each spin variable ##\sigma_i = \pm 1##. For a given spin configuration we denote the number of spins up and spins down by ##N_+## and ##N_-## respectively. The magnetisation is defined by ##m=(N_+ - N_-)/N##

a) Using Stirling's approximation, show that the entropy of the system can be written as $$S/N = \log 2 - \frac{1}{2} (1+m) \log(1+m) - \frac{1}{2} (1-m) \log(1-m)$$ and that in the mean field approximation $$f(T,m) = -\frac{1}{2} Jzm^2 + \frac{1}{2} T((1+m) \log(1+m) - (1-m) \log(1-m)) - T\log 2$$

## Homework Equations

$$S = \ln \Omega,$$ in units of ##k_B=1##

## The Attempt at a Solution

There are N sites and on each site, the spin can be up or down. So isn't the total number of spin configurations (or number of microstates) just ##2^N##? This would give ##S = N \log 2##, the first term in that expansion but I don't see how the other terms come about.

Thanks!