Passing From Discrete To Continuum Models Of Electrostatic Energy

We analyze the passage from discrete (i.e., point) charges and the corresponding discrete electrostatic energies to a continuum charge density and the corresponding continuum electrostatic energy in the limit of a large number of point charges. Given a continuous function on a bounded region that represents a continuum charge density, we construct a sequence of point charges and prove that the corresponding discrete electrostatic energies converge to the continuum counterpart. In a more general setting, we consider a given, compactly supported, signed Radon measure in the three-dimensional space representing the distribution of charges. We construct a sequence of point charges that converge to the given signed Radon measure and show that the corresponding discrete energies converge to the continuum one defined by the signed Radon measure. Conversely, for any sequence of point charges that satisfy certain reasonable assumptions on local geometry and excluded volumes, we prove that there exists a subsequence converging to a signed Radon measure and that the corresponding discrete energies converge to the continuum one defined by the limiting signed Radon measure. Tools used in our analysis include the explicit constructions of point charges from a given signed Radon measure as well as approximation properties of signed Radon measures. Finally, we apply our discrete-to-continuum analysis to the minimization of electrostatic energy related to the classical balayage problem in the potential theory. Such minimization can be potentially applied to the modeling of charged molecular systems with heterogeneously distributed charges embedded in a continuum solvent.