External forces in the continuum limit of discrete systems with non-convex interaction potentials: Compactness for a $\Gamma$-development
Abstract
This paper is concerned with equilibrium configurations of one-dimensional particle system with non-convex nearest-neighbour and next-to-nearest-neighbour interactions and its passage to the continuum. The goal is to derive compactness results for a -development of the energy with the novelty that external forces are allowed. In particular, the forces may depend on Lagrangian or Eulerian coordinates. Our result is based on a new technique for deriving compactness results which are required for calculating the first-order -limit: instead of comparing a configuration of atoms to a global minimizer of the -limit, we compare the configuration to a minimizer in some subclass of functions which in some sense are "close to" the configuration. This new technique is required due to the additional presence of forces with non-convex potentials. The paper is complemented with the study of the minimizers of the -limit.
Keywords
Cite
@article{arxiv.1811.09857,
title = {External forces in the continuum limit of discrete systems with non-convex interaction potentials: Compactness for a $\Gamma$-development},
author = {Marcello Carioni and Julian Fischer and Anja Schlömerkemper},
journal= {arXiv preprint arXiv:1811.09857},
year = {2019}
}