Do real electrons solve the Thomson Problem? 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.
 A: If the sphere has no resistance, the initially distributed electrons will move in a complicated way and will never reach the local or global minima to stay - they will "oscillate". 
In order to reach some minimum to stay, there should be losses of excess of energy. 
If the resistance is high, the electrons may find local minima and stay there (the extreme case - an insulator). 
If the resistance is weak enough, then reaching the global minimum is possible.
A: The question is a bit ambiguous.
On a real conductor (like a copper ball) there are lots of movable charges. All the movable charges – the original and the added ones – will move to a position minimizing potential energy. If the conductor was made of metal, it would not be possible to distinguish between the original and the added charges, so this would not give a solution to the Thomson problem. But even if you could mark the added charges in some way, their charge would be shielded by the movable charges of the conductor and they could move freely. So, this would not solve Thomson problem either.
What you might have thought of would be a sphere, in which the added charges can move freely, but in which other charges do not exist. To make the experiment complete, the sphere would never the less have to be constructed in a way, that the electrons can not leave it. If such a sphere could be constructed – might be for some other charged particles but electrons – these charges would tend to find local optima of the minimization problem.
Quantum effects would be relevant only, if the sphere is sufficiently small and the charges are sufficiently lightweight. Another practical problem using electrons or similar particles would be cooling. If you do not remove energy form the system, the particles will not move to the minimum anyway or not stay there. For larger particles like charged table tennis balls the problem might be solved however. Never the less this will be more difficult than solving the problem numerically and again it will find local optima only.
A: If the sphere is a good conductor then the electrons will be all over the sphere. If it is an insulator then they will remain where you put them. 
