Optimal-transport formulation of electronic density-functional theory

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The most challenging scenario for Kohn-Sham density functional theory, that is when the electrons move relatively slowly trying to avoid each other as much as possible because of their repulsion (strong-interaction limit), is reformulated here as an optimal transport (or mass transportation theory) problem, a well established field of mathematics and economics. In practice, we show that solving the problem of finding the minimum possible internal repulsion energy for $N$ electrons in a given density $\rho(\rv)$ is equivalent to find the optimal way of transporting $N-1$ times the density $\rho$ into itself, with cost function given by the Coulomb repulsion. We use this link to put the strong-interaction limit of density functional theory on firm grounds and to discuss the potential practical aspects of this reformulation.