Relaxation times for atom dislocations in crystals

TL;DR

This paper analyzes the relaxation times of atom dislocations in crystals modeled by a parabolic PDE, showing exponential decay to steady state after collisions, with implications for understanding dislocation dynamics.

Contribution

It provides the first detailed estimates of relaxation times post-collision for models including long-range interactions and classical dislocation models.

Findings

Dislocation functions relax exponentially fast after collision.

Exponential decay is unusual in nonlocal problems due to polynomial spatial decay.

Coupling between evolution and periodic potential induces exponential relaxation.

Abstract

We study the relaxation times for a parabolic differential equation whose solution represents the atom dislocation in a crystal. The equation that we consider comprises the classical Peierls-Nabarro model as a particular case, and it allows also long range interactions. It is known that the dislocation function of such a model has the tendency to concentrate at single points, which evolve in time according to the external stress and a singular, long range potential. Depending on the orientation of the dislocation function at these points, the potential may be either attractive or repulsive, hence collisions may occur in the latter case and, at the collision time, the dislocation function does not disappear. The goal of this paper is to provide accurate estimates on the relaxation times of the system after collision. More precisely, we take into account the case of two and three colliding points, and we show that, after a small transition time subsequent to the collision, the dislocation function relaxes exponentially fast to a steady state. We stress that the exponential decay is somehow exceptional in nonlocal problems (for instance, the spatial decay in this case is polynomial). The exponential time decay is due to the coupling (in a suitable space/time scale) between the evolution term and the potential induced by the periodicity of the crystal.