Attempt to describe phase slips by means of an adiabatic approximation

In the description of non equilibrium situations in a superconductor at temperatures far below its critical temperature, the Keldysh-Usadel technique (KUT) is required. However, the non-stationary KUT has not been applied to realistic circuits. Moreover, the stationary KUT has been applied in situations where time independence is not guaranteed. As a plausibility check for this procedure, we resort to a toy model (the Ginzburg-Landau model), in situations that involve very slow evolution. We find that, even in these situations, neglecting the explicit influence of time variation leads to inaccurate or even qualitatively wrong description of phase slips.