Homogenization of accelerated Frenkel-Kontorova models with $n$ types of particles

TL;DR

This paper studies a system of particles with different types interacting under a periodic potential, showing that their microscopic dynamics converge to a macroscopic Hamilton-Jacobi equation after rescaling.

Contribution

It extends homogenization results to systems with multiple particle types and damping, deriving a macroscopic limit for their dynamics.

Findings

Convergence of particle systems to a homogenized Hamilton-Jacobi equation.

Inclusion of multiple particle types in the homogenization framework.

Establishment of the macroscopic limit under damping and periodic interactions.

Abstract

We consider systems of ODEs that describe the dynamics of particles. Each particle satisfies a Newton law (including the acceleration term) where the force is created by the interactions with the other particles and with a periodic potential. The presence of a damping term allows the system to be monotone. Our study takes into account the fact that the particles can be different. After a proper hyperbolic rescaling, we show that the solutions to this system of ODEs converge to the solution of a macroscopic homogenized Hamilton-Jacobi equation.