Mean-field dynamics for Ginzburg-Landau vortices with pinning and applied force

Abstract

We consider the time-dependent Ginzburg-Landau equation in the whole plane with terms modeling pinning and applied forces. The Ginzburg-Landau vortices are then subjected to three forces: their mutual repulsive interaction, a constant applied force pushing them in a fixed direction, and the pinning force attracting them towards the local minima of the pinning potential. The competition between the three is expected to lead to possible glassy effects. We first rigorously study the limit in which the number of vortices \(N_\epsilon\) blows up as the inverse Ginzburg-Landau parameter \(\epsilon\) goes to \(0\), and we derive via a modulated energy method the limiting fluid-like mean-field evolution equations. These results hold in the case of parabolic, conservative, and mixed-flow dynamics in appropriate regimes of \(N_\epsilon\to\infty\). We next consider the problem of homogenization of the limiting mean-field equations when the pinning potential oscillates rapidly: we formulate a number of questions and heuristics on the appropriate limiting stick-slip equations, as well as some rigorous results on the simpler regimes.