Optimal-transport formulation of electronic density-functional theory

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 to solve the problem of finding the minimum possible internal repulsion energy for $N$ electrons in a given density $\rho \left(\mathbf{r}\right)$ is equivalent to find the optimal way of transporting $N-1$ times the density $\rho$ into itself, with the cost function given by the Coulomb repulsion. We use this link to set the strong-interaction limit of density-functional theory on firm ground and to discuss the potential practical aspects of this reformulation.

Figure 1

The optimal map $f$ for the density of Eq. (28) with $a=2$.

Figure 2

The optimal map $f$ for the density of Eq. (29) in the case $a=1$.

Figure 3

The optimal map $f$ for the Gaussian density of Eq. (30).

Figure 4

The optimal maps $f_1$ and $f_2=f_1^2$ for $N=3$ and $\rho =dx$ on $[0,1]$.