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

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

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques, Informatique et Calcul Scientifique

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.

Mots clés: Particle systems; periodic homogenization; Frenkel-Kontorova models; Hamilton-Jacobi equations; hull function

Domaines: Equations aux dérivées partielles [math.AP]

Nicolas Forcadel, Cyril Imbert, Régis Monneau. Homogenization of accelerated Frenkel-Kontorova models with $n$ types of particles. Transactions of the American Mathematical Society, 2012, 364, pp.6187-6227. ⟨10.1090/S0002-9947-2012-05650-9⟩. ⟨hal-00387818v3⟩