Solution
= Solution
Minimization gives $m=0$ for $T>T_c$ and $m=±\sqrt{a(T_c-T)/b}$ below. Thus
$$
F_{eq}=F_0\quad(T\geq T_c),\qquad F_{eq}=F_0-{a^2(T_c-T)^2\over4b}\quad(T<T_c).
$$
Using $S=-F_T$ and $C=-TF_{TT}$, entropy is continuous but the low-temperature heat capacity has the additional $Ta^2/(2b)$. The jump is $T_ca^2/(2b)$, so the transition is second order.