Homogenization of fully overdamped Frenkel-Kontorova models

Nicolas Forcadel (1, 2) , Cyril Imbert (3) , Régis Monneau (1)

1 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques, Informatique et Calcul Scientifique
2 Commands - Control, Optimization, Models, Methods and Applications for Nonlinear Dynamical Systems
3 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

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.

Mots clés: Slepcev formulation; Particle systems; periodic homogenization; Frenkel-Kontorova models; Hamilton-Jacobi equations; hull function; cumulative distribution function; Slepcev formulation.

Domaines: Equations aux dérivées partielles [math.AP]

Nicolas Forcadel, Cyril Imbert, Régis Monneau. Homogenization of fully overdamped Frenkel-Kontorova models. 2008. ⟨hal-00266994v1⟩