\sect{Phase transition of Ising cluster (based on Exam2002)} Consider a cluster of ${N}$ spins ${s_{i}=\pm 1}$. The interaction between {\em any} two spins is ${-\epsilon s_{i} s_{j}}$, with ${\epsilon>0}$. The interaction of each spin with the external magnetic field is ${-\mathsf{H} s_{i}}$. The total magnetization is defined as ${\mathsf{m}=\sum s_{i}}$. The inverse temperature is ${\beta}$. \begin {itemize} \item[(a)] Show that the partition function can be written as ${Z\left(\beta,\mathsf{H}\right)=\sum_{\mathsf{m}}g\left(\mathsf{m}\right)\exp\left(\half \mathsf{B} m^{2} + h \mathsf{m}\right)}$. Express ${g\left(\mathsf{m}\right)}$ and ${\mathsf{B}}$ and $h$ using ${\left(N, \epsilon, \beta, \mathsf{H} \right)}$. \item[(b)] Assume that ${\mathsf{B}=b/{N}}$, and define the magnetization as ${M={\mathsf{m}}/{N}}$. Write the partition function as ${Z\left(b,h\right)=\sum_{M} \exp\left(-N \mathcal{A}\left(M\right)\right)}$. Write the expressions for ${\mathcal{A}\left(M\right)}$ and for its derivatives ${\mathcal{A}'\left(M\right)}$ and ${\mathcal{A}''\left(M\right)}$. \item[(c)] Determine the critical temperature ${T_{c}}$, and write an equation for the mean field value of ${M}$. Make a qualitative plot of ${\mathcal{A}\left(M\right)}$ below and above the critical temperature. \item[(d)] Write an approximation for ${\mathcal{A}\left(M\right)}$ up to order ${M^{4}}$. On the basis of this expression determine the temperature range where mean filed theory cannot be trusted. Hint: you have to estimate the variance ${\langle M^{2} \rangle}$ in the Gaussian approximation. What happens with this condition in the thermodynamic limit (${N\rightarrow\infty}$)? \item[(e)] Find an expression for the heat capacity in the mean field and in the Gaussian approximations. \end {itemize}