Thomson problem in the disk

TL;DR

This paper studies the classical ground state configurations of many Coulomb charges confined in a disk, introducing algorithms that efficiently handle large systems and reveal complex defect structures as the number of charges increases.

Contribution

The authors developed a new algorithm to efficiently find global energy minima for large Coulomb systems in a disk, accounting for boundary effects and enabling the study of unprecedentedly large configurations.

Findings

Identified the significant role of peripheral charges in energy minimization.

Achieved computational analysis of systems with up to 40,886 charges.

Observed increasingly complex topological defect structures with larger N.

Abstract

We investigate the classical ground state of a large number of charges confined inside a disk and interacting via the Coulomb potential. By realizing the important role that the peripheral charges play in determining the lowest energy solutions, we have successfully implemented an algorithm that allows us to work with configurations with a desired number of border charges. This feature brings a consistent reduction in the computational complexity of the problem, thus simplifying the search of global minima of the energy. Additionally, we have implemented a divide and conquer approach which has allowed us to study configurations of size never reached before (the largest one corresponding to $N=40886$ charges). These last configurations, in particular, are seen to display an increasingly rich structure of topological defects as $N$ gets larger.