Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 303 / 1 / a / Solution

Landau theory treats the order parameter as spatially uniform and expands the free-energy density in powers allowed by its symmetries. The Landau-Ginzburg theory promotes it to a field $\varphi (\mathbf{x})$ and adds gradient terms such as $|\nabla \varphi {|}^2$. It therefore describes spatial fluctuations, interfaces, defects, and correlation lengths, while reducing to Landau theory for uniform fields.