Homogenization of Frenkel-Kontorova models and dislocation dynamics

In this talk I will present Frenkel-Kontorova models which can be seen as an infinite system of coupled ODEs describing the motion of particles moving in a periodic landscape and in interaction with their nearest neighbors. The macroscopic dynamics can be seen as a conservation law (describing the conservation of particle densities).

The effective macroscopic dynamics is obtained by homogenization in the framework of viscosity solutions for Hamilton-Jacobi equations. I will present various results for overdamped or accelerated systems of particles and also for dislocations dynamics.