Optimal and non-optimal lattices for non-completely monotone interaction potentials
Research output: Contribution to journal › Journal article › Research › peer-review
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.
Original language | English |
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Journal | Analysis and Mathematical Physics |
Volume | 9 |
Issue number | 4 |
Pages (from-to) | 2033–2073 |
ISSN | 1664-2368 |
DOIs | |
Publication status | Published - 2019 |
- Completely monotone functions, Laplace transform, Lattice energies, Lennard-Jones potentials, Theta functions, Triangular lattice
Research areas
Links
- https://arxiv.org/pdf/1806.02233.pdf
Accepted author manuscript
ID: 223821982