Homogenization of fully overdamped Frenkel–Kontorova models

In this paper, we consider the fully overdamped Frenkel–Kontorova model. This is an infinite system of coupled first-order ODEs. Each ODE represents the microscopic evolution of one particle interacting with its neighbors and submitted to a fixed periodic potential. After a proper rescaling, a macroscopic model describing the evolution of densities of particles is obtained. We get this homogenization result for a general class of Frenkel–Kontorova models. The proof is based on the construction of suitable hull functions in the framework of viscosity solutions.