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

Rafael Benguria (1) , Jean Dolbeault (2) , Régis Monneau (3)

1 Departamento de Fisica [UC Santiago]
2 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
3 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques, Informatique et Calcul Scientifique

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.

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

Domains: Analysis of PDEs [math.AP]

Rafael Benguria, Jean Dolbeault, Régis Monneau. Harnack inequalities and discrete - continuum error estimates for a chain of atoms with two - body interactions. Journal of Statistical Physics, 2009, 134, pp.27-51. ⟨hal-00267954⟩