Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 41 / 1 / ii / Solution

For $u>0$, the quartic Landau free energy at $h=0$ minimizes at $m=0$ for $r>0$ and at $m=\pm \sqrt{-r/u}$ for $r<0$. The order parameter changes continuously at $r=0$, giving a continuous phase transition and a divergent response. The minimized potential is $0$ above the transition and $-r^2/(4u)$ below it, so its first temperature derivative is continuous while its second derivative jumps.

For $u<0$, retain $v>0$. A first-order phase transition occurs when the $m=0$ minimum and a nonzero minimum have equal free energy. Combining stationarity $r+u{m}^2+v{m}^4=0$ with equality of potential values givesThe order parameter jumps at a positive $r$ before the disordered minimum loses local stability. Generic temperature dependence then also produces a jump in entropy and a latent heat. Equality of global minima locates coexistence; the disappearance of a metastable minimum instead defines a spinodal point.