Harnack inequalities and discrete - continuum error estimates for a chain of atoms with two - body interactions

Benguria, Rafael; Dolbeault, Jean; Monneau, Régis

Benguria, Rafael; Dolbeault, Jean; Monneau, Régis (2009), Harnack inequalities and discrete - continuum error estimates for a chain of atoms with two - body interactions, Journal of Statistical Physics, 134, 1, p. 27-51. http://dx.doi.org/10.1007/s10955-008-9662-4

Type

Article accepté pour publication ou publié

Lien vers un document non conservé dans cette base

http://hal.archives-ouvertes.fr/hal-00267954/en/

Date

2009

Nom de la revue

Journal of Statistical Physics

Volume

134

Numéro

Éditeur

Springer

Pages

27-51

Résumé (EN)

In the three-dimensional euclidean space, we consider deformations of an infinite linear chain of atoms where each atom interacts with all others through a two-body potential. We compute the effect of an external force applied to the chain. At equilibrium, the positions of the particles satisfy an Euler-Lagrange equation. For large classes of potentials, we prove that every solution is well approximated by the solution of a continuous model. We establish an error estimate between the discrete and the continuous solution based on a Harnack lemma of independent interest. Finally we apply our results to some Lennard-Jones potentials.

Mots-clés

Two-body interactions; thermodynamic limit; Harnack inequality; Cauchy-Born rule; error estimates; discrete-continuum; nonlinear elasticity