Mean-field dynamics for Ginzburg-Landau vortices with pinning and forcing

Abstract: We consider the time-dependent 2D Ginzburg-Landau equation in the whole plane with terms modeling impurities and applied currents. The Ginzburg-Landau vortices are then subjected to three forces: their mutual repulsive Coulomb-like interaction, the applied current pushing them in a fixed direction, and the pinning force attracting them towards the impurities. The competition between the three is expected to lead to complicated glassy effects. We 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 fluid-like mean-field evolution equations. These results hold for parabolic, conservative, and mixed-flow dynamics in appropriate regimes of $N_\epsilon\uparrow\infty$. Finally, we briefly discuss some natural homogenization questions raised by this study.