Asymptotic analysis of boundary layers in a repulsive particle system

TL;DR

This paper analyzes the boundary layer behavior in a finite particle system with repulsive interactions, using asymptotic and variational methods to understand equilibrium configurations and decay properties.

Contribution

It introduces a rigorous asymptotic and energy-based framework for boundary layer analysis in repulsive particle systems, covering dislocation models.

Findings

Derived equilibrium equations with boundary layer corrections

Established decay rates into the bulk

Developed a Gamma-convergence based energy framework

Abstract

This paper studies the boundary behaviour at mechanical equilibrium at the ends of a finite interval of a class of systems of interacting particles with monotone decreasing repulsive force. Our setting covers pile-ups of dislocations, dislocation dipoles and dislocation walls. The main challenge is to control the nonlocal nature of the pairwise particle interactions. Using matched asymptotic expansions for the particle positions and rigorous development of an appropriate energy via Gamma-convergence, we obtain the equilibrium equation solved by the boundary layer correction, associate an energy with an appropriate scaling to this correction, and provide decay rates into the bulk.