TDGL

The time-dependent Ginzburg-Landau (TDGL) equations give the evolution in time of the steady-state equations of the Ginzburg-Landau theory (GL). Although phenomenological, these equations can be very useful in making qualitative predictions about the time evolution of superconductors, particularly in the mixed state where Abrikosov vortices or Pearl vortices may appear.

Because of the phenomenological nature of GL theory, there are a number of different ways to expand its time dependence including different corrections and approximations. For example, in their seminal paper using TDGL to describe the time scale of fluctuations in one-dimensional superconducting wires, McCumber and Halperin adopt the following form (note units are CGS):

$\tau (T)({\frac {\partial }{\partial t}}+i{\frac {2eV}{\hbar }})\psi =(1-|\psi |^{2})\psi +\xi (T)({\frac {\partial }{\partial x}}-i{\frac {2e}{\hbar c}}A_{x})^{2}\psi$

With $\psi$ the order parameter describing the degree of superconducting order; $\tau$ the temperature-dependent GL relaxation time of the order parameter; $V$ the electrochemical potential; $A_{x}$ the magnetic vector potential; and $\xi$ the superconducting coherence length. However, other forms exist. Sometimes the electrochemical potential is dropped for convenience, even though it increases the quantitative accuracy of the TDGL equations, and sometimes other correction terms are added.