Past exam of the thermal and statistical physics course of the University of Cambridge / 2014 / Paper 1 / 2 / b

Landau theory expands the free energy in an order parameter. For a scalar with sign-reversal symmetry and zero conjugate field, a useful sextic even Landau potential isEquilibrium selects the global minimum; other local minima can be metastable.

For $b>0$, the minimum changes continuously from $m=0$ to $m\simeq \pm \sqrt{-r/b}$ as $r$ passes below zero. This is a second-order transition at $T_0$: the order parameter is continuous, there is no latent heat, and thermodynamic derivatives such as heat capacity can be discontinuous.

For $b<0$, the sixth-order term stabilizes the free energy. Equality of the ordered and disordered minima, together with $f^{\prime }(m)=0$, givesThe order parameter jumps at $T_t=T_0+r_t/a$. The entropy generally jumps too, giving latent heat; metastable branches and barriers permit hysteresis, though equilibrium coexistence itself does not require hysteresis. The special case $b=0$ is tricritical. Mean-field exponents can fail where fluctuations dominate.