Note on Crystallization for Alternating Particle Chains

Abstract

We investigate one-dimensional periodic chains of alternate type of particles interacting through mirror symmetric potentials. The optimality of the equidistant configuration at fixed density—also called crystallization—is shown in various settings. In particular, we prove the crystallization at any scale for neutral and non-neutral systems with inverse power laws interactions, including the three-dimensional Coulomb potential. We also show the minimality of the equidistant configuration at high density for systems involving inverse power laws and repulsion at the origin. Furthermore, we derive a necessary condition for crystallization at high density based on the positivity of the Fourier transform of the interaction potentials sum.

Maximal Theta Functions -- Universal Optimality of the Hexagonal Lattice for Madelung-Like Lattice Energies

“…This conjecture is supported by Corollary 1.2 as well as the recent work of the first author and Knüpfer [10] where the alternate configuration of charges {±1} has been shown to minimize ϕ → E f [Λ, ϕ] in the simple lattice case and for a large set of lattices Λ with charges satisfying some simple assumptions. Furthermore, the same conjecture has been stated in [8] for the one-dimensional case and solved in the same paper for a small class of potentials, including the inverse power laws.…”

Emergence of Rigid Polycrystals from Atomistic Systems with Heitmann–Radin Sticky Disk Energy

“…For illustration purposes, we suppose that w = 0 in (7.34) Here, the addend C in the brackets is due to the fact that there may be x ∈ X T ∩ Q ν S (x j ) with more neighbors in X j than in X T . This, however, can only occur for atoms in x ∈ Q ν S (x j ) such that x ∈ (∂ Q ν S (x j )) 6 ∩ ({y : y − x j , ν = 0}) 6 . Since E(X T ) < +∞, we can apply Lemma 3.1(v) and get that their cardinality is controlled by some universal constant C.…”

“…Recent results for positive temperature including an analysis of boundary layers are obtained in [34,35]. For results on dimers we refer to [6,29].) The first rigorous results for a two-dimensional system were achieved in [32,33,43]; see also the recent paper [18].…”

Complete minimal logarithmic energy asymptotics for points in a compact interval: a consequence of the discriminant of Jacobi polynomials

“…Point sets characterized by means of minimizing a suitably defined potential energy function have applications in a surprising number of problems in various fields of science and engineering ranging from physics over chemistry to geodesy and mathematics. We refer the reader to [4,5,7,8,9,11,13,16,18,19,23,24,28,33,34,42,43,44,46,47,48,49,51,55] and the book [15]. A fundamental question concerns the asymptotic expansion of the minimal energy as the number of points tend to infinity.…”