Discrete-to-continuum limits of multi-body systems with bulk and surface long-range interactions

TL;DR

This paper establishes a rigorous mathematical framework for deriving continuum models from atomistic descriptions of crystalline materials, accounting for complex long-range and multi-body interactions through Gamma-convergence.

Contribution

It provides new conditions for the atomistic-to-continuum limit, including compactness, integral representation, and homogenization results for systems with long-range, multi-body interactions.

Findings

Derived conditions for Gamma-convergence of energy functionals

Proved decoupling of bulk and surface energies in the continuum limit

Applied results to weak-membrane energies with long-range interactions

Abstract

We study the atomistic-to-continuum limit of a class of energy functionals for crystalline materials via Gamma-convergence. We consider energy densities that may depend on interactions between all points of the lattice and we give conditions that ensure compactness and integral-representation of the continuum limit on the space of special functions of bounded variation. This abstract result is complemented by a homogenization theorem, where we provide sufficient conditions on the energy densities under which bulk- and surface contributions decouple in the limit. The results are applied to long-range and multi-body interactions in the setting of weak-membrane energies.