Legendre transforms for electrostatic energies

TL;DR

This paper reviews how Legendre transforms can be used to reformulate electrostatic energy functionals, making numerical minimization more straightforward and connecting modern field-theoretic approaches to classical formulations.

Contribution

It introduces a method to convexify electrostatic functionals using Legendre transforms, simplifying numerical minimization and linking different theoretical approaches.

Findings

Convexification of electrostatic functionals via Legendre transforms.

Simplified numerical minimization of electrostatic energies.

Demonstrated equivalence between modern and classical dielectric functionals.

Abstract

We review the use of Legendre transforms in the formulation of electrostatic energies in condensed matter. We show how to render standard functionals expressed in terms of the electrostatic potential, Phi, convex - at the cost of expressing them in terms of the vector field D. This leads to great simplification in the formulation of numerical minimisation of electrostatic energies coupled to other physical degrees of freedom. We also demonstrate the equivalence of recent functionals for dielectrics derived using field theory methods to classical formulations in terms of the electric polarisation.