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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 todoi:10.1090/s0002-9947-2012-05650-9 fatcat:rfmci7skhbb3nod4anbrl5s4hi