Minimum energy configurations of classical charges: Large N asymptotics

We study the minimum energy configurations of N particles in of charge -1 ("electrons") in the potential of M particles of charges Zα > 0 ("atomic nuclei"). In a suitable large-N limit, we determine the asymptotic electron distribution explicitly, showing in particular that the number of electrons surrounding each nucleus is asymptotic to the nuclear charge ("screening"). The proof proceeds by establishing, via Gamma-convergence, a coarse-grained variational principle for the limit distribution, which can be solved explicitly.