Minimization of energy per particle among Bravais lattices in ℝ²: Lennard–Jones and Thomas–Fermi cases

Abstract

We prove in this paper that the minimizer of Lennard-Jones energy per particle among Bravais lattices is a triangular lattice, i.e. composed of equilateral triangles, in R 2 for large density of points, while it is false for sufficiently small density. We show some characterization results for the global minimizer of this energy and finally we also prove that the minimizer of the Thomas-Fermi energy per particle in R 2 among Bravais lattices with fixed density is triangular.

The crystallization conjecture: a review

“…This weight is non-negative when ρ is large enough. Using this argument, Bétermin and Zhang [22] have proved that, at high density, the optimum of (30) is reached by the hexagonal lattice in 2D, for the Lennard-Jones potential V LJ . Imposing that ρ is large means that particles are close to each other.…”

“…Under this assumption and the usual stability condition (9), the limit in (22) can be shown to exist and to be independent of the sequence Ω N . Potentials that are not integrable are also sometimes considered, but then one should divide the energy by the appropriate power of N .…”

“…In order for the limit (22) to exist, we also need an upper bound. Since the particles are confined to the domain Ω, we cannot send them to infinity anymore, hence we loose most of the properties of Section 1.2.…”

“…If crystallization is assumed, it is possible to determine the most favorable periodic configurations by comparing their energy per particle e ∞ and e(ρ), defined by (8) and (22). In some cases, this question may be related to a problem in analytic number theory, involving special functions.…”

“…as in (22). In order to formalize the crystallization problem at positive temperature, it is convenient to consider the empirical measures (also called k-point correlation functions [88,173]), which are similar to the measure µ N introduced in (15).…”