 Ginzburg–Landau theory

A long-wavelength order-parameter description extending a local Landau free energy by gradient terms. For a one-component complex field, a common isotropic form is $\mathcal{F}=\int [\alpha |\Psi {|}^2+\beta |\Psi {|}^4/2+K|\nabla \Psi {|}^2]d^{3}r$. A charged field instead uses a covariant gradient. Static minimization determines equilibrium profiles; a dynamical law requires additional input. Ginzburg-Landau theory of superconductivity and the Ginzburg-Landau functional of a dilute Bose gas realize this structure with different microscopic normalizations.

The Ginzburg–Landau theory is a mathematical framework used to describe phase transitions and critical phenomena, particularly in superconductivity and superfluidity. Developed by Vitaly Ginzburg and Lev Landau in the mid-20th century, this theory provides a macroscopic description of these systems using order parameters and a free energy functional.