Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 64 / 1 / a

An order parameter distinguishes thermodynamic phases. For a ferromagnet it is the magnetization; for a fluid near its liquid-gas critical point it can be the density relative to its critical value. In a system with a scalar order parameter, introduce a slowly varying real scalar field $\varphi (x)$ whose spatial average represents that order parameter. The Landau-Ginzburg theory organizes the effective free energy as a local expansion consistent with the symmetry:Here $K>0$ is the stiffness in the gradient energy, and $H$ is the field conjugate to an order parameter. The even powers assume a symmetry $\varphi \mapsto -\varphi$ at $H=0$. Without that symmetry further terms, including a cubic term, may be allowed. The coefficients are smooth functions of temperature and other controls; near a mean-field transition one usually writes $r=a(T-T_0)$ with $a>0$. A positive highest retained even coefficient makes the local potential bounded below.

The partition function is obtained by weighting configurations with $\exp [-\mathcal{F}/(k_{B}T)]$. The Landau approximation replaces this functional average by minimization, taking a uniform $\varphi =M$ in a homogeneous phase. In a symmetric finite system the zero-field average can vanish even below the transition because it averages both signs of $M$. The spontaneous order parameter selects a pure thermodynamic phase by taking the thermodynamic limit before removing the conjugate field:Thus the order parameter measures the ordering of a selected thermodynamic phase, rather than the average over all symmetry-related phases.