Time-dependent Ginzburg-Landau equation ($\Gamma ({\partial }_t+iq\varphi /\hslash )\Psi =-\delta F/\delta {\Psi }^{\ast }$)

A phenomenological near-transition relaxation equation for a charged complex order parameter, with a positive real relaxation coefficient in its simplest dissipative form. Under $\Psi \mapsto e^{iq\chi /\hslash }\Psi$, $\mathbf{A}\mapsto \mathbf{A}+\nabla \chi$ and $\varphi \mapsto \varphi -{\partial }_t\chi$, the displayed time derivative transforms covariantly, as does the spatial derivative in the Ginzburg-Landau theory of superconductivity. Microscopic applicability depends on regime and scattering processes; this equation is distinct from a universal conservative condensate wave equation.