Homogenization of some particle systems with two-body interactions and of the dislocation dynamics
Nicolas Forcadel, Cyril Imbert, Régis Monneau · Discrete and Continuous Dynamical Systems · 2008
This paper is concerned with the homogenization of some particlesystems with two-body interactions in dimension one and ofdislocation dynamics in higher dimensions.The dynamics of our particle systems are described bysome ODEs. We prove that the rescaled 'cumulative distributionfunction'' of the particles converges towards the solution of aHamilton-Jacobi equation. In the case when the interactions betweenparticles have a slow decay at infinity as $1/x$, we show that thisHamilton-Jacobi equation contains an extra diffusion term which is ahalf Laplacian. We get the same result in the particular case wherethe repulsive interactions are exactly $1/x$, which creates someadditional difficulties at short distances. We also study a higher dimensional generalisation ofthese particle systems which is particularly meaningful to describethe dynamics of dislocations lines. One main result of this paper isthe discovery of a satisfactory mathematical formulation of thisdynamics, namely a Slepčev formulation. We show in particularthat the system of ODEs for particle systems can be naturallyimbedded in this Slepčev formulation. Finally, with thisformulation in hand, we get homogenization results which contain theparticular case of particle systems.