Slow motion of particle systems as a limit of a reaction-diffusion equation with half-Laplacian in dimension one

TL;DR

This paper studies the slow motion of particle systems derived from a reaction-diffusion equation with a half-Laplacian, showing that as a small parameter tends to zero, the dynamics converge to a system of ODEs with specific interaction forces.

Contribution

It introduces a rescaling approach for the reaction-diffusion equation with half-Laplacian and rigorously derives the particle system limit with explicit interaction forces.

Findings

Limit dynamics described by ODEs with 1/x interaction forces

Convergence of rescaled reaction-diffusion solutions to particle motion

Connection to dislocation models in materials science

Abstract

We consider a reaction-diffusion equation with a half-Laplacian. In the case where the solution is independent on time, the model reduces to the Peierls-Nabarro model describing dislocations as transition layers in a phase field setting. We introduce a suitable rescaling of the evolution equation, using a small parameter $\varepsilon$. As $\varepsilon$ goes to zero, we show that the limit dynamics is characterized by a system of ODEs describing the motion of particles with two-body interactions. The interaction forces are in $1/x$ and correspond to the well-known interaction between dislocations.