Cotar, C;
Friesecke, G;
Pass, B;
(2015)
Infinite-body optimal transport with Coulomb Cost.
Calculus of Variations and Partial Differential Equations
, 54
(1)
pp. 717-742.
10.1007/s00526-014-0803-0.
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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.
| Type: | Article |
|---|---|
| Title: | Infinite-body optimal transport with Coulomb Cost |
| Open access status: | An open access version is available from UCL Discovery |
| DOI: | 10.1007/s00526-014-0803-0 |
| Publisher version: | http://dx.doi.org/10.1007/s00526-014-0803-0 |
| Language: | English |
| Additional information: | This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited. |
| Keywords: | N-representability; Density functional theory; Hohenberg-Kohn functional; N-body optimal transport; Infinite-body optimal transport; Coulomb cost; Exchangecorrelation functional; de Finetti’s Theorem; Finite exchangeability; N-extendability |
| UCL classification: | UCL UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Statistical Science |
| URI: | https://discovery.ucl.ac.uk/id/eprint/1429039 |
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