Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 4 / 35C (Statistical Physics)

A first-order phase transition has a discontinuous first derivative of the equilibrium free energy, such as entropy or volume, and latent heat. A second-order transition has continuous first derivatives but a discontinuous or divergent second derivative.

At fixed temperature, volume, and particle number, spontaneous changes satisfy $\Delta F\le 0$ because $F=E-TS$ accounts for heat exchanged with the reservoir. Equilibrium therefore minimizes the Helmholtz free energy.

Minimization gives $m=0$ for $T>T_c$ and $m=\pm \sqrt{a(T_c-T)/b}$ below. ThusUsing $S=-F_T$ and $C=-T{F}_{TT}$, entropy is continuous but the low-temperature heat capacity has the additional $T{a}^2/(2b)$. The jump is $T_c{a}^2/(2b)$, so the transition is second order.

Writing $x=m^2$, nonzero equilibria satisfy $a(T-T_c)+bx+c{x}^2=0$. For $b<0$, coexistence occurs at $x=-3b/(4c)$ and $T-T_c=3b^2/(16ac)$, so magnetization jumps and the transition is first order.

For $b=0$, $m=0$ above $T_c$ and $m^4=a(T_c-T)/c$ below: this is the tricritical continuous case. Below $T_c$,so the extra heat capacity is $a^{3/2}T/[4\sqrt{c(T_c-T)}]$ and diverges on approach from below; above it is absent.