Theta functions and minimization of interaction energies

Orateur: BETERMIN Laurent
Localisation: Université de Heidelberg, Allemagne
Type: Séminaire problèmes spectraux en physique mathématique
Site: IHP
Salle: 314
Date de début: 11/04/2016 - 14:00
Date de fin: 11/04/2016 - 15:00

The goal of this talk is to explain how to minimize a finite lattice energy of the type $E_f(L) =\sum_{p\in L-0} f(|p|^2)$, among Bravais lattices $L\subset\mathbb R^2$. Our method uses a two-dimensional theta function associated with $L$.
We will show the optimality of the triangular lattice :
1) at any fixed density, if the potential $f$ is completely monotonic ;
2) at fixed high density, for a large class of potentials ;
3) among all Bravais lattices, without density constraint, for some potentials $f$ with a well.
Moreover, we will give some results of non-optimality of the triangular lattice at low density, and some numerical results, if $f$ is the classical Lennard-Jones potential.