Optimal transport with Coulomb cost and the semiclassical limit of density functional theory
[Transport optimal avec coût coulombien et limite semi-classique de la théorie de la fonctionnelle de la densité]
Journal de l’École polytechnique — Mathématiques, Tome 4 (2017), pp. 909-934.

Nous présentons des progrès récents en vue de la détermination de la limite semi-classique de la fonctionnelle universelle de Levy-Lieb ou Hohenberg-Kohn en théorie de la fonctionnelle de la densité pour des systèmes coulombiens. Nous donnons en particulier une preuve du fait que, pour des systèmes de bosons avec un nombre arbitraire de particules, la limite est le problème de transport optimal multi-marginal à coût coulombien, de même que pour les systèmes de fermions à deux ou trois particules. Nous établissons des comparaisons avec des résultats antérieurs. Nous nous appuyons sur certaines techniques de la théorie du transport optimal.

We present some progress in the direction of determining the semiclassical limit of the Levy-Lieb or Hohenberg-Kohn universal functional in density functional theory for Coulomb systems. In particular we give a proof of the fact that for Bosonic systems with an arbitrary number of particles the limit is the multimarginal optimal transport problem with Coulomb cost and that the same holds for Fermionic systems with two or three particles. Comparisons with previous results are reported. The approach is based on some techniques from the optimal transportation theory.

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DOI : https://doi.org/10.5802/jep.59
Classification : 49J45,  49N15,  49K30
Mots clés : Théorie de la fonctionnelle de la densité, transport optimal multi-marginal, problème de Monge-Kantorovich, théorie de la dualité, coût coulombien
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Bindini, Ugo; De Pascale, Luigi. Optimal transport with Coulomb cost and the semiclassical limit of density functional theory. Journal de l’École polytechnique — Mathématiques, Tome 4 (2017), pp. 909-934. doi : 10.5802/jep.59. http://www.numdam.org/articles/10.5802/jep.59/

[1] Ambrosio, L.; Gigli, N.; Savaré, G. Gradient flows in metric spaces and in the space of probability measures, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2008 | Zbl 1145.35001

[2] Bindini, U. Γ-convergence and optimal transportation in density functional theory (2016) (Master Thesis)

[3] Braides, A. Γ-convergence for beginners, Oxford Lecture Series in Mathematics and its Applications, 22, Oxford University Press, Oxford, 2002 | MR 1968440 | Zbl 1198.49001

[4] Buttazzo, G.; Champion, T.; De Pascale, L. Continuity and estimates for multimarginal optimal transportation problems with singular costs, Appl. Math. Optim. (2017) (on-line first) | Article | Zbl 1400.49051

[5] Buttazzo, G.; De Pascale, L.; Gori-Giorgi, P. Optimal-transport formulation of electronic density-functional theory, Phys. Rev. A, Volume 85 (2012) no. 6 (#062502) | Article

[6] Colombo, M.; De Pascale, L.; Di Marino, S. Multimarginal optimal transport maps for one-dimensional repulsive costs, Canad. J. Math., Volume 67 (2015) no. 2, pp. 350-368 | Article | MR 3314838 | Zbl 1312.49052

[7] Cotar, C.; Friesecke, G.; Klüppelberg, C. Density functional theory and optimal transportation with Coulomb cost, Comm. Pure Appl. Math., Volume 66 (2013) no. 4, pp. 548-599 | Article | MR 3020313 | Zbl 1266.82057

[8] Cotar, C.; Friesecke, G.; Klüppelberg, C. Smoothing of transport plans with fixed marginals and rigorous semiclassical limit of the Hohenberg-Kohn functional (2017) (arXiv:1706.05676) | Zbl 1394.82015

[9] Dal Maso, G. An introduction to Γ-convergence, Progress in Nonlinear Differential Equations and their Applications, 8, Birkhäuser Boston, Inc., Boston, MA, 1993 | MR 1201152 | Zbl 0816.49001

[10] De Pascale, L. Optimal transport with Coulomb cost. Approximation and duality, ESAIM Math. Model. Numer. Anal., Volume 49 (2015) no. 6, pp. 1643-1657 | Article | MR 3423269 | Zbl 1330.49048

[11] Levy, M. Universal variational functionals of electron densities, first-order density matrices, and natural spin-orbitals and solution of the v-representability problem, Proc. Nat. Acad. Sci. U.S.A., Volume 76 (1979) no. 12, pp. 6062-6065 | Article | MR 554891

[12] Lewin, M. Semi-classical limit of the Levy-Lieb functional in density functional theory (2017) (arXiv:1706.02199)

[13] Lieb, E. H. Density functionals for Coulomb systems, Int. J. Quantum Chem., Volume 24 (1983) no. 3, pp. 243-277 | Article

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