SLOW MOTION OF PARTICLE SYSTEMS AS A LIMIT OF A REACTION-DIFFUSION EQUATION WITH HALF-LAPLACIAN IN DIMENSION ONE

摘要

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 ε. As ε 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.