Local variational study of 2d lattice energies and application to Lennard-Jones type interactions

TL;DR

This paper analyzes the local minimality of 2D lattice energies, especially Lennard-Jones type, using variational methods, and characterizes how the minimal configurations depend on lattice density and shape.

Contribution

It provides a variational framework for analyzing 2D lattice energies, deriving conditions for local minimality and applying these to Lennard-Jones potentials with new insights into optimal lattice shapes.

Findings

Hessian at square and triangular lattices is diagonal.

Conditions for local minimality of these lattices are established.

Triangular lattice minimizes Lennard-Jones energy for certain densities.

Abstract

In this paper, we focus on finite Bravais lattice energies per point in two dimensions. We compute the first and second derivatives of these energies. We prove that the Hessian at the square and the triangular lattice are diagonal and we give simple sufficient conditions for the local minimality of these lattices. Furthermore, we apply our result to Lennard--Jones type interacting potentials that appear to be accurate in many physical and biological models. The goal of this investigation is to understand how the minimum of the Lennard--Jones lattice energy varies with respect to the density of the points. Considering the lattices of fixed area A, we find the maximal open set to which A must belong so that the triangular lattice is a minimizer (resp. a maximizer) among lattices of area A. Similarly, we find the maximal open set to which A must belong so that the square lattice is a minimizer (resp. a saddle point). Finally, we present a complete conjecture, based on numerical investigations and rigorous results among rhombic and rectangular lattices, for the minimality of the classical Lennard--Jones energy per point with respect to its area. In particular, we prove that the minimizer is a rectangular lattice if the area is sufficiently large.