Time Variation of the Ginzburg-Landau Order Parameter

We use a nonequilibrium form of the Green's-function formulation of the BCS theory of superconductivity to investigate the circumstances under which differential equations in space and time, i.e., "time-dependent Ginzburg-Landau equations," give a valid description of the space and time variation of the order parameter $\Psi$ in superconductors. We find that if the variations are sufficiently slow, time-dependent Ginzburg-Landau equations exist near absolute zero and near the transition temperature. In the former case the equation has wave-like character, and in the latter case it is of diffusion type, with the restriction that either the characteristic frequency of the time variation of $\Psi$ is greater than the gap frequency or the ratio of the Fermi velocity to the product of the characteristic wavelength and frequency of the space-time variation of $\Psi$ is greater than unity. Under all other circumstances and at general temperatures, there are no differential equations to describe the variations of $\Psi$. We discuss also the influence of slowly varying time-dependent fields and derive the dependence of charge and current densities on the variations of $\Psi$. Local electrodynamics are assumed. The necessary modifications for the case of dilute superconducting alloys are described. Applications are made to the questions of collective modes, nucleation, and London's theory near absolute zero.