Past exam of the condensed matter physics major option course of the University of Oxford / 2016 / 2 / c

Differentiating the Landau theory free energy givesIts stationary values arewhen the discriminant is nonnegative. At $P_0$, the curvature is $a(T-T_c)$: a local minimum above $T_c$ and a local maximum below it. Exactly at $T_c$, $F^{\prime \prime \prime }(0)=-2b\ne 0$, so the origin is a stationary inflection point, not a stable minimum.

At a nonzero stationary point, eliminating $T-T_c$ yields $F^{\prime \prime }=P(2cP-b)$. Between $T_c$ and the upper spinodal, $P_{+}$ is a minimum and $P_{-}$ a maximum. Below $T_c$, $P_{-}<0$ becomes another local minimum, while $P_{+}>0$ is the lower minimum. Thus loss of local stability at $T_c$ is distinct from the equilibrium transition temperature.