Boundary-layer analysis of repelling particles pushed to an impenetrable barrier

TL;DR

This paper analyzes the equilibrium distribution of particles near an impenetrable barrier, revealing a boundary layer phenomenon where a small subset of particles deviate from the continuum prediction, and establishes a convergence result for this boundary layer.

Contribution

It introduces a first-order onvergence result that characterizes the boundary layer profile of particles near the barrier, bridging discrete and continuum models.

Findings

Discrete boundary layer of particles at the barrier identified

Convergence of boundary particles to a specific profile proven

Continuum limit does not fully capture boundary behavior

Abstract

This paper considers the equilibrium positions of $n$ particles in one dimension. Two forces act on the particles; a nonlocal repulsive particle-interaction force and an external force which pushes them to an impenetrable barrier. While the continuum limit as $n \to \infty$ is known for a certain class of potentials, numerical simulations show that a discrete boundary layer appears at the impenetrable barrier, i.e. the positions of $o(n)$ particles do not fit to the particle density predicted by the continuum limit. In this paper we establish a first-order $\Gamma$-convergence result which guarantees that these $o(n)$ particles converge to a specific continuum boundary-layer profile.