Past exam of the mathematics course of the University of Cambridge / 2015 / ii / Paper 4 / 32C (Statistical Physics)

For the canonical partition function $Z={\sum }_{\{s\}}{e}^{-\beta E}$, differentiation at fixed temperature gives ${\partial }_B\log Z=\beta ⟨{\sum }_i{s}_i⟩$. Thus the average spin per particle, or magnetization in these units, is $\boxed{m=(N\beta {)}^{-1}{\partial }_B\log Z}$.

Write $s_i=m+{\delta }_i$ and neglect the fluctuation product ${\delta }_i{\delta }_j$ in the mean-field approximation. Then $s_i{s}_j\simeq m(s_i+s_j)-m^2$. There are $ND$ neighbour pairs, and each spin enters $2D$ pairs. The mean-field energy isThe spins now factorize, givingThe constant corrects the double counting in the interaction; omitting it would give an incorrect mean-field free energy.

The single-spin average in the effective field is $\tanh (\beta B_{\mathrm{eff}})$. Requiring it to equal the assumed average gives the mean-field self-consistency equationAt $B=0$, the slope of its right side at the origin is $2DJ/(k_{B}T)$. Thus $\boxed{T_c=2DJ/k_B}$. For $T>T_c$, $m=0$ is the only solution and is the stable minimum: thermal fluctuations dominate. For $T<T_c$, two stable nonzero solutions $\pm m(T)$ represent spontaneous magnetization, while the zero solution is unstable. At $T_c$ they merge continuously. Expanding $\tanh x=x-x^3/3+\cdots$ near the transition yields $m^2\sim 3(T_c-T)/T_c$, so the mean-field critical exponent is one-half. These are predictions of the approximation, not a proof of the exact Ising transition in every dimension.

The proposed Landau free energy has stationary equation $m[a(T-T_c)+b{m}^2]=0$. For $T>T_c$ its minimum is $m=0$. For $T<T_c$ the minima are $\boxed{m=\pm \sqrt{a(T_c-T)/b}}$, while zero becomes a maximum; at $T_c$ the positive quartic term keeps the free energy bounded below. Thus the two symmetry-related states, continuous transition and square-root onset agree with the mean-field approximation.

More explicitly the mean-field entropy expansion gives $F=F_0+N[k_B(T-T_c)m^2/2+k_{B}T{m}^4/12+O(m^6)]$ at zero field. Near $T_c$, replacing $T$ by $T_c$ in the quartic coefficient gives the positive constants $a=N{k}_B$ and $b=N{k}_B{T}_c/3$, and reproduces the preceding square-root amplitude. The polynomial is a local expansion near the transition, not a global expression at every temperature.