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

TL;DR

This paper rigorously derives mean-field equations describing the complex dynamics of Ginzburg-Landau vortices under pinning and forcing effects, revealing insights into their collective behavior in large systems.

Contribution

It provides a rigorous derivation of mean-field evolution equations for vortices with impurities and applied currents, accounting for various flow regimes.

Findings

Derivation of fluid-like mean-field equations for vortex dynamics

Analysis of vortex behavior under competing forces

Discussion of homogenization and large-system limits

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.