Diffusion as a singular homogenization of the Frenkel–Kontorova model

In this work, we consider a general fully overdamped Frenkel–Kontorova model. This model describes the dynamics of an infinite chain of particles, moving in a periodic landscape. Our aim is to describe the macroscopic behavior of this system. We study a singular limit corresponding to a high density of particles moving in a vanishing periodic landscape. We identify the limit equation which is a nonlinear diffusion equation. Our homogenization approach is done in the framework of viscosity solutions.