Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 4 / 34D (Statistical Physics)

Landau theory represents the appropriate thermodynamic potential as an analytic function of a small order parameter, retaining every low-order term allowed by the system's symmetry. At fixed control parameters, the equilibrium phase is the global minimum of the Landau free energy; local minima describe metastable phases. A continuous phase transition occurs when a minimum changes continuously and loses stability, whereas a first-order phase transition occurs when distinct minima exchange global stability.

Forthe stationary points satisfyIf $\epsilon >0$, $\varphi =0$ is the unique minimum. If $\epsilon <0$, $\varphi =0$ is a maximum and the two degenerate minima areThus the reflection symmetry $\varphi \mapsto -\varphi$ is spontaneously broken below $\epsilon =0$, and

For the sextic even Landau potentialput $q={\varphi }^2$. The stationary-point equation isso the nonzero stationary points haveAt such a point, $\epsilon =-q^2-gq$, and henceA first-order transition occurs when a nonzero minimum has the same free energy as the disordered minimum $G(0)=0$. ThereforeAt $g=0,\epsilon =0$, the first-order line meets the continuous line and forms the tricritical point. For $g>0$, the transition at $\epsilon =0$ is continuous.

For $g<0$, the disordered state $\varphi =0$ is a local minimum when $\epsilon >0$ and loses stability at the disordered spinodal point $\epsilon =0$. The ordered minima exist whileand merge with the intervening maxima at the ordered spinodal $\epsilon =g^2/4$. Consequently the two spinodals enclose a region of metastability. If increasing $\epsilon$ corresponds to heating, the ordered phase can persist above the coexistence curve throughoutwhile the disordered phase can persist below it throughout