Infinite-body optimal transport with Coulomb Cost

TL;DR

This paper studies infinite-body optimal transport problems with Coulomb cost, showing that the optimal solution is an independent product measure, and connects these results to quantum mechanics and the semiclassical limit.

Contribution

It proves that for certain costs, the optimizer is a product measure, contrasting with finite-body cases, and links the infinite-body problem to quantum chemistry functionals.

Findings

Optimal measure is an independent product measure for certain costs.

Infinite-body OT approximates N-body OT as N grows large.

Coulomb cost case relates to the SCE functional in quantum chemistry.

Abstract

We introduce and analyze symmetric infinite-body optimal transport (OT) problems with cost function of pair potential form. We show that for a natural class of such costs, the optimizer is given by the independent product measure all of whose factors are given by the one-body marginal. This is in striking contrast to standard finite-body OT problems, in which the optimizers are typically highly correlated, as well as to infinite-body OT problems with Gangbo-Swiech cost. Moreover, by adapting a construction from the study of exchangeable processes in probability theory, we prove that the corresponding $N$-body OT problem is well approximated by the infinite-body problem. To our class belongs the Coulomb cost which arises in many-electron quantum mechanics. The optimal cost of the Coulombic N-body OT problem as a function of the one-body marginal density is known in the physics and quantum chemistry literature under the name SCE functional, and arises naturally as the semiclassical limit of the celebrated Hohenberg-Kohn functional. Our results imply that in the inhomogeneous high-density limit (i.e. $N\to\infty$ with arbitrary fixed inhomogeneity profile $\rho/N$), the SCE functional converges to the mean field functional. We also present reformulations of the infinite-body and N-body OT problems as two-body OT problems with representability constraints and give a dual characterization of representable two-body measures which parallels an analogous result by Kummer on quantum representability of two-body density matrices.