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July  2009, 23(3): 785-826. doi: 10.3934/dcds.2009.23.785

Homogenization of some particle systems with two-body interactions and of the dislocation dynamics

1. 

CERMICS, Paris Est-ENPC, 6 & 8 avenue Blaise Pascal, Cité Descartes, Champs sur Marne, 77455 Marne la Vallée Cedex 2, France, France

2. 

CEREMADE, UMR CNRS 7534, université Paris-Dauphine, Place de Lattre de Tassigny, 75775 Paris Cedex 16, France

Received  January 2008 Revised  July 2008 Published  November 2008

This paper is concerned with the homogenization of some particle systems with two-body interactions in dimension one and of dislocation dynamics in higher dimensions.
The dynamics of our particle systems are described by some ODEs. We prove that the rescaled "cumulative distribution function'' of the particles converges towards the solution of a Hamilton-Jacobi equation. In the case when the interactions between particles have a slow decay at infinity as $1/x$, we show that this Hamilton-Jacobi equation contains an extra diffusion term which is a half Laplacian. We get the same result in the particular case where the repulsive interactions are exactly $1/x$, which creates some additional difficulties at short distances.
We also study a higher dimensional generalisation of these particle systems which is particularly meaningful to describe the dynamics of dislocations lines. One main result of this paper is the discovery of a satisfactory mathematical formulation of this dynamics, namely a Slepčev formulation. We show in particular that the system of ODEs for particle systems can be naturally imbedded in this Slepčev formulation. Finally, with this formulation in hand, we get homogenization results which contain the particular case of particle systems.
Citation: Nicolas Forcadel, Cyril Imbert, Régis Monneau. Homogenization of some particle systems with two-body interactions and of the dislocation dynamics. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 785-826. doi: 10.3934/dcds.2009.23.785
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