Density functional theory and multi-marginal optimal transport: Introduction

Density functional theory (DFT) is one of the workhorses of quantum chemistry and material science. In principle, the joint probability of finding a specific electron configuration in a material is governed by a Schrödinger wave equation. But numerically computing this joint probability is computationally infeasible, due to the complexity scaling exponentially in the number of electrons. DFT aims to circumvent this difficulty by focusing on the marginal probability of one electron. In the last decade, a connection was found between DFT and a multi-marginal optimal transport problem with a repulsive cost. I will give a brief introduction to this topic, including some open problems, and recent progress.