Monte-Carlo and polyhedron-based simulations I: extremal states of the logarithmic N-body problem on a sphere

The problem of N bodies on the surface of the sphere interacting by a logarithmic potential is examined for selected N ranging from $4$ to $40,962$, comparing the energies found by placing points at the vertices of certain polyhedrons to the lowest energies found by a Monte Carlo algorithm. The polyhedron families are generated from simple polyhedrons through two triangular face splitting operations which are used iteratively to increase the number of vertices. The closest energy of these polyhedron vertex configurations to the Monte Carlo-generated minimum energy is identified and the two energies are found to agree well. Finally the energy per particle pair is found to asymptotically approach a mean field theory limit of $- 1/2 (log(2) - 1)$, approximately $0.153426$, for both the polyhedron and the Monte Carlo-generated energies. The deterministic algorithm of generating polyhedrons is shown to be a method able to generate consistently good approximations to the extremal energy configuration for a wide range of numbers of points.