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

We consider systems of ODEs that describe the dynamics of particles. Each particle satisfies a Newton law (including a damping term and an acceleration term) where the force is created by the interactions with 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 solutions of these systems of ODEs converge to solutions of some macroscopic homogenized Hamilton-Jacobi equations. AMS Classification: 35B27, 35F20, 45K05, 47G20, 49L25, 35B10.