The Peierls-Nabarro model as a limit of a Frenkel-Kontorova model

TL;DR

This paper demonstrates that a generalized Frenkel-Kontorova model describing atomic positions in a crystal converges to a Peierls-Nabarro PDE model under suitable rescaling, bridging discrete atomic models and continuum descriptions.

Contribution

It establishes a rigorous connection between a discrete atomistic model and a continuum PDE model using viscosity solutions, extending the understanding of crystal dislocation dynamics.

Findings

Discrete model solutions converge to PDE solutions under rescaling

The limit model is a coupled system of elliptic and evolution PDEs

The convergence is proved within the viscosity solutions framework

Abstract

We study a generalization of the fully overdamped Frenkel-Kontorova model in dimension $n\geq 1.$ This model describes the evolution of the position of each atom in a crystal, and is mathematically given by an infinite system of coupled first order ODEs. We prove that for a suitable rescaling of this model, the solution converges to the solution of a Peierls-Nabarro model, which is a coupled system of two PDEs (typically an elliptic PDE in a domain with an evolution PDE on the boundary of the domain). This passage from the discrete model to a continuous model is done in the framework of viscosity solutions.