Motivated by biological models of solvation, this dissertation consists of analysis of models of electrostatic free energy of charged systems that incorporate both continuum and discrete idealizations of charges. Discrete models of charge can yield vastly divergent results than the corresponding continuum models for systems with a small number of particles, but will

be shown in Chapter 2 to be asymptotically equivalent in the large particle number limit.

In Section 2.1 the energy of a given continuum charge distribution is shown to be representable as a limit of approximating discrete charge distributions by way of properties of harmonic functions and Riemann sum approximations. In Section 2.2 the problem is reversed in that a given sequence of collections of point charges is shown to have a limiting continuum charge density with the same limiting electrostatic energy.

Motivated by application to a minimization problem common in molecular

modelling, potential theory, and fluid mechanics, Chapter 3 details a delicate multiscale construction to generalize the results of Chapter 2 to more general measures, requiring the further development of the theory of Radon measures and their Fourier

transforms, facilitated by a gradient flow evolution of the domain.

The analysis of Chapter 4 concerns a hybrid model of solvation that incorporates statistical mechanics and the classical Coulomb energy of a system, allowing for both continuous and discrete distributions of charge whose equilibrium configuration is described by the Poisson–Boltzmann Equation. Motivated by application to optimization, a modified free energy functional is constructed by way of a Legendre transform and is shown to be equivalent.

Despite the long history of competition between these models, a precise treatment of the question has never been addressed, to this author’s knowledge. This work is significant in bridging the gap between scales, and furthermore has application to

a wide variety of physical and biological modelling problems.