# Asymptotic Analysis of Boundary Layers in a Repulsive Particle System

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## 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. This setting covers, for instance, pile-ups of dislocations, dislocation dipoles and dislocation walls. The main challenge in characterising the boundary behaviour 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.

## Keywords

Particle system Boundary layer Discrete-to-continuum asymptotics Matched asymptotic expansions \(\Gamma \)-convergence## Mathematics Subject Classification (2010)

74Q05 74G10 41A60## Notes

### Acknowledgements

The authors would like to thank Mark Peletier for valuable discussions, and TU Eindhoven for providing funds to cover research visits by TH and CH. TH and PvM would also like to thank the Hausdorff Research Institute for Mathematics in Bonn for hosting them during the junior workshop ‘Analytic approaches to scaling limits for random systems’ during which work on this project was carried out.

### Conflict of Interest

The authors declare that there is no conflict of interest regarding this work.

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