The question of how $N$ electrons (seen as point charges) on a conducting sphere will arrange themselves in the electrostatic final state was first posed by J.J. Thomson in 1904--hence, aka the Thomson Problem. If these abstract point charge electrons are initially placed randomly, they will migrate along potential gradients to a state with a locally minimum potential energy. The Thomson Problem is seen as finding the geometrical arrangement of the N charges with the global minimum potential energy. Unfortunately there are usually a great number of local minima, e.g. on the order of $10^6$ for $N$ of several hundred, so numerical techniques don't necessarily produce the global minimum and analytic techniques to date have only solved the problem for some small values of $N$.
Notwithstanding the applications of the problem to many other practical phenomena, I have some general questions about the specific real case of electrons on a conducting sphere. If the sphere initially has a random distribution of excess charge ($N$ electrons), will they in fact somehow end up in the global minimum potential energy state, or will they as in a numerical simulation just find a local minimum and be stuck there? Is there any way to know that? If so, how do they do it?
Another question: is it valid to think of the electrons as ultimately stationary points on the vertices of some geometric arrangement on the sphere in the first place? I.e., given quantum effects, statistical considerations, etc.