Note on crystallization for alternating particle chains

TL;DR

This paper proves the crystallization of alternating particle chains with mirror symmetric potentials, including Coulomb interactions, showing equidistant configurations are optimal under various conditions.

Contribution

It establishes crystallization results for one-dimensional alternating particle chains with inverse power law and Coulomb interactions, extending previous understanding.

Findings

Crystallization at any scale for neutral and non-neutral systems with inverse power law interactions.

Minimality of equidistant configuration at high density for certain systems.

Necessary condition for crystallization based on Fourier transform positivity.

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.