Diffusion as a singular homogenization of the Frenkel-Kontorova model

Nathaël Alibaud (1) , Ariela Briani (2, 3) , Régis Monneau (4)

1 LMB - Laboratoire de Mathématiques de Besançon (UMR 6623)
2 LMPT - Laboratoire de Mathématiques et Physique Théorique
3 OC - Optimisation et commande
4 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques, Informatique et Calcul Scientifique

In this work, we consider a general fully overdamped Frenkel-Kontorova model. This model describes the dynamics of a 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.

Mots clés: particle systems; periodic homogenization; Frenkel-Kontorova models; Hamilton-Jacobi equations; nonlinear diffusion

Domaines: Optimisation et contrôle [math.OC]; Equations aux dérivées partielles [math.AP]

Nathaël Alibaud, Ariela Briani, Régis Monneau. Diffusion as a singular homogenization of the Frenkel-Kontorova model. Journal of Differential Equations, 2011, 251 (4-5), pp.785-815. ⟨10.1016/j.jde.2011.05.020⟩. ⟨hal-00372407⟩