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

Forcadel, N.; Imbert, C.; Monneau, R.

doi:10.1090/S0002-9947-2012-05650-9

Homogenization of accelerated Frenkel-Kontorova models with $n$ types of particles
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by N. Forcadel, C. Imbert and R. Monneau PDF

Trans. Amer. Math. Soc. 364 (2012), 6187-6227 Request permission

Abstract:

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.

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