2018/06/05 by Bétermin, Laurent, Petrache, Mircea
#FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Optimization and Control (math.OC)
paper · doi:10.48550/arxiv.1806.02233
We investigate the minimization of the energy per point E_f among d-dimensional Bravais lattices, depending on the choice of pairwise potential equal to a radially symmetric function f(|x|2). We formulate criteria for minimality and non-minimality of some lattices for E_f at fixed scale based on the sign of the inverse Laplace transform of f when f is a superposition of exponentials, beyond the class of completely monotone functions. We also construct a family of non-completely monotone functions having the triangular lattice as the unique minimizer of E_f at any scale. For Lennard-Jones type potentials, we reduce the minimization problem among all Bravais lattices to a minimization over the smaller space of unit-density lattices and we establish a link to the maximum kissing problem. New numerical evidence for the optimality of particular lattices for all the exponents are also given. We finally design one-well potentials f such that the square lattice has lower energy E_f than the triangular one. Many open questions are also presented.