An external dual charge approach to the optimal transport with Coulomb cost

TL;DR

This paper introduces a dual charge approach to solve multimarginal optimal transport problems with Coulomb cost, revealing the dual potential as an electrostatic potential and enabling discretization in various dimensions.

Contribution

It establishes that the dual Kantorovich potential corresponds to an electrostatic potential from an external charge, providing a novel perspective and discretization method for Coulomb cost problems.

Findings

Dual Kantorovich potential is an electrostatic potential from an external charge.

Discretization methods developed for 1D and 3D cases.

Theoretical insights linking optimal transport and electrostatics.

Abstract

In this paper, we study the multimarginal optimal transport with Coulomb cost, also known in the physics literature as the Strictly-Correlated Electrons (SCE) functional. We prove that the dual Kantorovich potential is an electrostatic potential induced by an external charge density, which we call the dual charge. We study its properties and use it to discretize the potential in one and three space dimensions.