38 research outputs found
Sufficient Condition for a Compact Local Minimality of a Lattice
We give a sufficient condition on a family of radial parametrized long-range
potentials for a compact local minimality of a given -dimensional Bravais
lattice for its total energy of interaction created by each potential. This
work is widely inspired by the paper of F. Theil about two dimensional
crystallization
Optimal lattice configurations for interacting spatially extended particles
We investigate lattice energies for radially symmetric, spatially extended
particles interacting via a radial potential and arranged on the sites of a
two-dimensional Bravais lattice. We show the global minimality of the
triangular lattice among Bravais lattices of fixed density in two cases: In the
first case, the distribution of mass is sufficiently concentrated around the
lattice points, and the mass concentration depends on the density we have
fixed. In the second case, both interacting potential and density of the
distribution of mass are described by completely monotone functions in which
case the optimality holds at any fixed density.Comment: 17 pages. 1 figure. To appear in Letters in Mathematical Physic
On Born's conjecture about optimal distribution of charges for an infinite ionic crystal
We study the problem for the optimal charge distribution on the sites of a
fixed Bravais lattice. In particular, we prove Born's conjecture about the
optimality of the rock-salt alternate distribution of charges on a cubic
lattice (and more generally on a d-dimensional orthorhombic lattice).
Furthermore, we study this problem on the two-dimensional triangular lattice
and we prove the optimality of a two-component honeycomb distribution of
charges. The results holds for a class of completely monotone interaction
potentials which includes Coulomb type interactions. In a more general setting,
we derive a connection between the optimal charge problem and a minimization
problem for the translated lattice theta function.Comment: 32 pages. 3 Figures. To appear in Journal of Nonlinear Science. DOI
:10.1007/s00332-018-9460-
Dimension reduction techniques for the minimization of theta functions on lattices
We consider the minimization of theta functions
amongst
lattices , by reducing the dimension of the
problem, following as a motivation the case , where minimizers are
supposed to be either the BCC or the FCC lattices. A first way to reduce
dimension is by considering layered lattices, and minimize either among
competitors presenting different sequences of repetitions of the layers, or
among competitors presenting different shifts of the layers with respect to
each other. The second case presents the problem of minimizing theta functions
also on translated lattices, namely minimizing . Another way to reduce dimension is by considering
lattices with a product structure or by successively minimizing over concentric
layers. The first direction leads to the question of minimization amongst
orthorhombic lattices, whereas the second is relevant for asymptotics
questions, which we study in detail in two dimensions.Comment: 45 pages. 7 figure
Minimization of energy per particle among Bravais lattices in R^2 : Lennard-Jones and Thomas-Fermi cases
We study the two dimensional Lennard-Jones energy per particle of lattices
and we prove that the minimizer among Bravais lattices with sufficiently large
density is triangular and that is not the case for sufficiently small density.
We give other results about the global minimizer of this energy. Moreover we
study the energy per particle stemming from Thomas-Fermi model in two
dimensions and we prove that the minimizer among Bravais lattices with fixed
density is triangular. We use a result of Montgomery from Number Theory about
the minimization of Theta functions in the plane.Comment: 15 pages, 4 figures. Final Versio
Local variational study of 2d lattice energies and application to Lennard-Jones type interactions
Local variational study of 2d lattice energies and application to Lennard-Jones type interactions
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.Comment: 30 pages. 18 Figures. Published in Nonlinearity, Volume 31, Issue 9,
p. 3973-4005. DOI:10.1088/1361-6544/aac75