Energy Minimization, Periodic Sets and Spherical Designs

Coulangeon, Renaud; Schürmann, Achill

doi:10.1093/imrn/rnr048

Cited by 27 publications

(49 citation statements)

References 5 publications

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“…It is for instance not clear if f ε defined by (1.8) could satisfy this property. However, we conjecture that the local minimality of Λ 1 for E fε should hold among periodic configurations of fixed density by a small modification of [27,Cor. 4.5].…”

Section: Introduction and Main Resultsmentioning

confidence: 82%

“…• (cristallization at fixed density) The minimization for f (r 2 ) = e −ar 2 being a Gaussian, and amongst lattices of fixed density has been treated in d = 2 (in which case E f is the so-called lattice theta function (1.5)), was studied in the fundamental work [49] (see also higher-dimensional results [27]), which prove that at fixed density the triangular lattice (defined by (1.2)) is the unique minimizer, for all choices of the variance a > 0 . By a change of variable, this means also that for any Gaussian kernel and amongst lattices of any fixed density, the triangular lattice is the unique minimizer.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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Optimal and non-optimal lattices for non-completely monotone interaction potentials

Bétermin

Petrache

2019

Anal.Math.Phys.

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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.

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Section: Introduction and Main Resultsmentioning

confidence: 82%

Section: Introduction and Main Resultsmentioning

confidence: 99%

Optimal and non-optimal lattices for non-completely monotone interaction potentials

Bétermin

Petrache

2019

Anal.Math.Phys.

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“…minimizing the interaction energy among all the possible point configurations -we choose to study the minimization of the energy per point among periodic lattices where the points are interacting via a Morse potential. This point of view has been taken in several previous works in Number Theory [22,25,26,28,31,32,37,49,57,59], optimal point configurations problems [24] and Mathematical Physics [1,13,15,23,52,58]. It is a natural first step for keeping or rejecting periodic structures which could be good candidates for the Crystal Problem associated to the interaction potential.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Minimizing lattice structures for Morse potential energy in two and three dimensions

Bétermin

2019

Journal of Mathematical Physics

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We investigate the local and global optimality of the triangular, square, simple cubic, facecentred-cubic (FCC), body-centred-cubic (BCC) lattices and the hexagonal-close-packing (HCP) structure for a potential energy per point generated by a Morse potential with parameters (α, r 0 ). In dimension 2 and for α large enough, the optimality of the triangular lattice is shown at fixed densities belonging to an explicit interval, using a method based on lattice theta function properties. Furthermore, this energy per point is numerically studied among all twodimensional Bravais lattices with respect to their density. The behaviour of the minimizer, when the density varies, matches with the one that has been already observed for the Lennard-Jones potential, confirming a conjecture we have previously stated for differences of completely monotone functions. Furthermore, in dimension 3, the local minimality of the cubic, FCC and BCC lattices are checked, showing several interesting similarities with the Lennard-Jones potential case. We also show that the square, triangular, cubic, FCC and BCC lattices are the only Bravais lattices in dimensions 2 and 3 being critical points of a large class of lattice energies (including the one studied in this paper) in some open intervals of densities, as we observe for the Lennard-Jones and the Morse potential lattice energies. More surprisingly, in the Morse potential case, we numerically found a transition of the global minimizer from BCC, FCC to HCP, as α increases, that we partially and heuristically explain from the lattice theta functions properties. Thus, it allows us to state a conjecture about the global minimizer of the Morse lattice energy with respect to the value of α. Finally, we compare the values of α found experimentally for metals and rare-gas crystals with the expected lattice ground-state structure given by our numerical investigation/conjecture. Only in a few cases does the known ground-state crystal structure match the minimizer we find for the expected value of α. Our conclusion is that the pairwise interaction model with Morse potential and fixed α is not adapted to describe metals and rare-gas crystals if we want to take into consideration that the lattice structure we find in nature is the ground-state of the associated potential energy.

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“…In dimension d ≥ 3, some authors have studied the critical points and the (local or global) minima of the Jacobi theta function and the Epstein zeta function. In a famous article [218], Sarnak and Strömbergsson determined special local minima in dimensions 4, 8 and 24 (see also [62,58,63]). In dimension 3, Ennola has proved that the face centered cubic (FCC) lattice is a non-degenerate local minimum of ζ 3 (S, s) for all s > 0 [80].…”

Section: Optimal Lattices and Special Functionsmentioning

confidence: 99%

The crystallization conjecture: a review

Blanc¹,

Lewin²

2015

EMS Surv. Math. Sci.

134

191

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In this article we describe the crystallization conjecture. It states that, in appropriate physical conditions, interacting particles always place themselves into periodic configurations, breaking thereby the natural translation-invariance of the system. This famous problem is still largely open. Mathematically, it amounts to studying the minima of a real-valued function defined on R 3N where N is the number of particles, which tends to infinity. We review the existing literature and mention several related open problems, of which many have not been thoroughly studied.

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Energy Minimization, Periodic Sets and Spherical Designs

Cited by 27 publications

References 5 publications

Optimal and non-optimal lattices for non-completely monotone interaction potentials

Optimal and non-optimal lattices for non-completely monotone interaction potentials

Minimizing lattice structures for Morse potential energy in two and three dimensions

The crystallization conjecture: a review

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