# An integral-representation result for continuum limits of discrete energies with multi-body interactions

## TL;DR

This paper establishes a mathematical framework for understanding the continuum limits of discrete lattice energies with complex multi-body interactions, ensuring the limit can be represented as an integral on Sobolev spaces.

## Contribution

It provides a new compactness and integral representation theorem for atomistic lattice energies with multi-body interactions, including homogenization and applications to Jacobian determinants and Lennard-Jones energies.

## Key findings

- Proved a compactness and integral representation theorem for lattice energies.
- Derived conditions for the limit to be an integral on Sobolev spaces.
- Applied results to multibody interactions like discrete Jacobian determinants and Lennard-Jones energies.

## Abstract

We prove a compactness and integral-representation theorem for sequences of families of lattice energies describing atomistic interactions defined on lattices with vanishing lattice spacing. The densities of these energies may depend on interactions between all points of the corresponding lattice contained in a reference set. We give conditions that ensure that the limit is an integral defined on a Sobolev space. A homogenization theorem is also proved. The result is applied to multibody interactions corresponding to discrete Jacobian determinants and to linearizations of Lennard-Jones energies with mixtures of convex and concave quadratic pair-potentials.