Equilibrium Configurations of Equal Charges on a Sphere 2025, 40 - Conjectures

From simulation results, extrapolate that locally stable equilibrium states of the surface Coulomb problem coincides often with the characteristics of planar magnetic dipole configurations and the eventual 3D spherical Coulomb problem. In the simplest cases, expect regular polyhedra or pologonals with forms independent of the specifics of the forces. Assume that the potential energy of these systems can be derived from the superposition of pair-wise interactions, identical up to irregularities in the polygonality. For spherical charge systems, the force laws become dominant in the transitions at N ≥ 7. Below this, all force laws generate identical equilibrium patterns. The strict geometry experiences decreasing relevance in the specifics of the solution, which breaks down to solutions of a 2D Tammes problem. Optimal configurations tend to be asymmetric. Often solutions only have the identity transformation as their invariant isometry. There is no apparent difference between force laws generating solutions with high regularity and those without. Diversity is taken to be prerequisite for complexity.

Stable configurations are characterized by dipole moment, center of charge, nearest neighbor angle and average number of edges, meeting at the vertices of the associated polyhedra. These are considered soft logarithmic interactions.

Domain structures of planar magnetic dipole arrays are robust under scaling, but not under shifts in the strength of multipolarities. The structural stabilities then experience clear hierarchy. At large N, the expected uniformity in charge distribution is violated by observed distinct charge plane minima, likely as a consequence of some pseudo-random variables. The non-uniformity persists likely even in cases that the local minima are nearly degenerate in energy.