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Discrete and Continuous Dynamical Systems A 23, 3 (2009) pp 785--826

Versions disponibles :

v1 (06-04-2007)

v2 (15-05-2007)

v3 (27-12-2007)

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

Nicolas Forcadel 1, Cyril Imbert 2, Régis Monneau 1

(2009)

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 Slepcev formulation. We show in particular that the system of ODEs for particle systems can be naturally imbedded in this Slepcev formulation. Finally, with this formulation in hand, we get homogenization results which contain the particular case of particle systems.

1 :	Centre d'Enseignement et de Recherche en Mathématiques, Informatique et Calcul Scientifique (CERMICS)
	INRIA – Ecole des Ponts ParisTech
2 :	CEntre de REcherches en MAthématiques de la DEcision (CEREMADE)
	CNRS : UMR7534 – Université Paris IX - Paris Dauphine

Domaine

:

Mathématiques/Equations aux dérivées partielles

Mots Clés : periodic homogenization – Hamilton-Jacobi equations – moving fronts – two-body interactions – integro-differential operators – Lévy operator – dislocation dynamics – Slepcev formulation – Frenkel-Kontorova model – particle systems.

hal-00140545, version 3

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Contributeur : Nicolas Forcadel <>

Soumis le : Mercredi 26 Décembre 2007, 20:17:31

Dernière modification le : Mercredi 11 Février 2009, 13:55:05