Equilibrium Configurations of Equal Charges on a Sphere 2025, 37 - Describing the Problem

Reading a Paper this time, which is going to be mostly math, if I'm guessing correctly. The topic has been a recurring question with each iteration while working my thesis, and since I've been floundering a little bit about choosing new math topics, especially if I don't want them to overwhelm me. This publication is 122 pages long, so I would think that this qualifies fine for my self-imposed study material. It's certainly less of a stretch than the string theory I did some time back.

"Complex Systems: Equilibrium Configurations of N Equal Charges on a Sphere" (FERMILAB-PUB-95-075-T) will consist to some amount of computer simulation, as these things tend to do for numbers that are larger than convenient. It's not exactly the topic I was often faced with, but it's somewhat related, populating the particles onto the surface of a sphere, without considering the possibility of suddenly splitting the configuration into layers. It turns out that several different people have proposed different solutions to the problem, usually covering different ranges of particle numbers. Specifically, the angular diversity ratio D_a between the vectors between two particles for larger numbers (here noted as larger than 50), is written as The number of distinct angle divided by N(N - 1)/2.

At large values of D_a, which is typical for N > 50, one can expect irregular configurations, not found in the standard convex polyhedra. A quantitative measure of the difference between random and geometrically irregular distribution is given by the dipole moment. The dipole moments of all the equilibrium Coulomb states typically fall in between 10^-5 and 10^-3 in magnitude. The physical effects are best described as (pairwise) sums of terms, which increases the complexity of each system. The energy sharing is completely symmetric for the equilibrium states of the surface Coulomb problem in small systems. The emergence of unequal distances between particles, leads to energy splitting. As the energy becomes increasingly asymmetrical, the center of the charge shifts through the sphere distribution.

Determining stationary solutions for these problems becomes analytically difficult. They become effectively optimization problems, finding a maximum density of a covered area for the charge distribution.

The surface Coulomb problem is described analytically by a number of N unit vectors describing the polar positions of N point charges into a dimensionless Coulomb energy E(N)=∑_i=1^N∑_j>i^N |r_i-r_j|^-1, which can be translated directly to an expression of Forces, which classically should sum to 0. For equal partial energies, the dipole moments are expected to vanish. At equilibrium, the energy distributions are also expected to be independent of the angle coordinates. The solutions are rotated into one of a standard set of orientations in order to compare the geometric properties. ordering the particles by their partial energies is adequate as orientations are unique for irregular configurations, and any ambivalence are unimportant for symmetric configurations.