Equilibrium Configurations of Equal Charges on a Sphere 2025, 38 - Stable States on the Surface

Energy minimization identifies a unique terminal energy for N in the range of 2 to 14. At N = 15, the energy is still unique, but the associated charge configurations are split into enantiometric states, whereas the previous ones are unique as well. The paper identifies energy levels and configurations through a minimizing algorithm, using separate passes and occasionally identifying several minima. Consistency is considered up to chiral transformation. At N = 16, three distinct configurations are identified. The multiplicities of states M(N) can be gathered into a metric M_C(N) = ∑^N_j=2M(j), though M(N) can be approximated with M(N) = Ae^νN, meaning M_C(N) = A(e^νN - e^ν)/(1 - e^-ν) with A = 0.382, ν = 0.0497. The energy distribution is derived from the electrostatic energy of the N-particle Coulomb problem. The electrostatic energy corresponds to a surface in 2N+1 dim space with some peaks, the median of which is identified and corresponds to stable configurations of the problem. The average energy of the set of random states is written ⟨E^Ran(N)⟩ = N²/2. which coincides with the Coulomb energy of a continuous uniform spherical surface charge distribution with total charge N. It's an approximation accurate to 6%. The energy states of an N-body surface Coulomb problem E₁(N) is fit by 0.5N² - 0.5513N^3/2, but is not analytically rigorous. The energy of a distribution of N can be expressed by subtracting the self-energies of the uniformly charged spherical caps from the electrostatic energy on a unit sphere. After correcting for the geometry, the self-energy correction is at about 0.4244N^3/2. Metastable states have closely bunched energies just about the minimum energy states. Their number of course increases with the particle numbers. Generally, ⟨ΔE(N)⟩ = [E_n(N) - E₁(N)]/(n-1) where n is the number of distinct energy levels. State density distribution is not uniform (to be expected for this sort of problem anyway), featuring notably a dip in the energy states just above the ground state.

For the individual charges in the system, the variation of their charge energies in a config is much larger than the variation of the total energy between configs. The charge energy is the sum of the inverse distances to all the other charges, and the variation of the individual energies is a measure of the geometric regularity. Between metastable states, the difference is much finer than this variation. The partial energies in a system are usually unique, up to geometry.