[Transport optimal avec coût coulombien et limite semi-classique de la théorie de la fonctionnelle de la densité]
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.
Accepté le :
Publié le :
DOI : 10.5802/jep.59
Keywords: Density functional theory, multimarginal optimal transportation, Monge-Kantorovich problem, duality theory, Coulomb cost
Mot clés : Théorie de la fonctionnelle de la densité, transport optimal multi-marginal, problème de Monge-Kantorovich, théorie de la dualité, coût coulombien
@article{JEP_2017__4__909_0,
author = {Bindini, Ugo and De Pascale, Luigi},
title = {Optimal transport with {Coulomb} cost and the semiclassical limit of density functional theory},
journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
pages = {909--934},
publisher = {Ecole polytechnique},
volume = {4},
year = {2017},
doi = {10.5802/jep.59},
mrnumber = {3714366},
language = {en},
url = {http://www.numdam.org/articles/10.5802/jep.59/}
}
TY - JOUR AU - Bindini, Ugo AU - De Pascale, Luigi TI - Optimal transport with Coulomb cost and the semiclassical limit of density functional theory JO - Journal de l’École polytechnique — Mathématiques PY - 2017 SP - 909 EP - 934 VL - 4 PB - Ecole polytechnique UR - http://www.numdam.org/articles/10.5802/jep.59/ DO - 10.5802/jep.59 LA - en ID - JEP_2017__4__909_0 ER -
%0 Journal Article %A Bindini, Ugo %A De Pascale, Luigi %T Optimal transport with Coulomb cost and the semiclassical limit of density functional theory %J Journal de l’École polytechnique — Mathématiques %D 2017 %P 909-934 %V 4 %I Ecole polytechnique %U http://www.numdam.org/articles/10.5802/jep.59/ %R 10.5802/jep.59 %G en %F JEP_2017__4__909_0
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/
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