Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 50 / 1 / c

A first-order phase transition exchanges distinct global minima of the Landau free energy; the order parameter jumps and a thermal crossing generically has latent heat. A continuous phase transition has minima merging continuously and a diverging correlation length. A local loss of stability at a spinodal point is not the equilibrium coexistence condition.

For the sextic even Landau potential, write $y=M^2$. At a nonzero zero-field stationary point, $r+uy+v{y}^2=0$. To coexist with $M=0$, also require $ry/2+u{y}^2/4+v{y}^3/6=0$. Eliminate $r$ to obtain $y=-3u/(4v)$ and $r=3u^2/(16v)$, which requires $u<0$. On this locusproving that all three equal minima are globally stable. Their curvatures are $3u^2/(16v)$ at zero and $3u^2/(4v)$ at the nonzero minima, so the correlation length is finite when $u<0$ is fixed. For $u>0$, by contrast, the nonzero minimum approaches zero as $r\uparrow 0$, giving the ordinary continuous transition. At fixed $u>0$ and $r<0$, crossing $h=0$ exchanges the two ordered minima discontinuously even though the zero-field thermal transition at $r=0$ is continuous.