Optimal and non-optimal lattices for non-completely monotone interaction potentials
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Optimal and non-optimal lattices for non-completely monotone interaction potentials. / Bétermin, Laurent; Petrache, Mircea.
In: Analysis and Mathematical Physics, Vol. 9, No. 4, 2019, p. 2033–2073.Research output: Contribution to journal › Journal article › Research › peer-review
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TY - JOUR
T1 - Optimal and non-optimal lattices for non-completely monotone interaction potentials
AU - Bétermin, Laurent
AU - Petrache, Mircea
PY - 2019
Y1 - 2019
N2 - 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.
AB - 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.
KW - Completely monotone functions
KW - Laplace transform
KW - Lattice energies
KW - Lennard-Jones potentials
KW - Theta functions
KW - Triangular lattice
UR - http://www.scopus.com/inward/record.url?scp=85064686267&partnerID=8YFLogxK
U2 - 10.1007/s13324-019-00299-6
DO - 10.1007/s13324-019-00299-6
M3 - Journal article
AN - SCOPUS:85064686267
VL - 9
SP - 2033
EP - 2073
JO - Analysis and Mathematical Physics
JF - Analysis and Mathematical Physics
SN - 1664-2368
IS - 4
ER -
ID: 223821982