Ground state of a large number of particles on a frozen topography

TL;DR

This paper introduces a universal geometric approach to determine the ground state configurations of a large number of interacting particles constrained on various fixed geometries, providing explicit solutions for spheres and tori.

Contribution

It presents a general, geometry-based method for finding ground states of many particles on fixed topographies, extending solutions beyond small particle numbers.

Findings

Explicit solutions for particles on sphere and torus

Universal applicability of the geometric ansatz

Predictions for further theoretical and numerical validation

Abstract

Problems consisting in finding the ground state of particles interacting with a given potential constrained to move on a particular geometry are surprisingly difficult. Explicit solutions have been found for small numbers of particles by the use of numerical methods in some particular cases such as particles on a sphere and to a much lesser extent on a torus. In this paper we propose a general solution to the problem in the opposite limit of a very large number of particles M by expressing the energy as an expansion in M whose coefficients can be minimized by a geometrical ansatz. The solution is remarkably universal with respect to the geometry and the interaction potential. Explicit solutions for the sphere and the torus are provided. The paper concludes with several predictions that could be verified by further theoretical or numerical work.