In a recent work, Bindini and De Pascale have introduced a regularization of N-particle symmetric probabilities that preserves their one-particle marginals. In this short note, we extend their construction to mixed quantum fermionic states. This enables us to prove the convergence of the Levy–Lieb functional in Density Functional Theory, to the corresponding multi-marginal optimal transport in the semi-classical limit. Our result holds for mixed states of any particle number N, with or without spin.
Dans un travail récent, Bindini et de Pascale ont introduit une régularisation des probabilités symétriques décrivant N particules indiscernables, qui préserve la densité à une particule. Nous étendons ici leur construction aux états quantiques mixtes de fermions. Ceci nous permet de démontrer la convergence de la fonctionnelle de Levy–Lieb, objet central de la théorie de la fonctionnelle de densité (DFT), vers le transport optimal multi-marges associé, à la limite semi-classique. Notre résultat est valable pour les états mixtes de n'importe quel nombre de particules N, avec ou sans spin.
Accepted:
Published online:
Mathieu Lewin 1
@article{CRMATH_2018__356_4_449_0, author = {Mathieu Lewin}, title = {Semi-classical limit of the {Levy{\textendash}Lieb} functional in {Density} {Functional} {Theory}}, journal = {Comptes Rendus. Math\'ematique}, pages = {449--455}, publisher = {Elsevier}, volume = {356}, number = {4}, year = {2018}, doi = {10.1016/j.crma.2018.03.002}, language = {en}, }
Mathieu Lewin. Semi-classical limit of the Levy–Lieb functional in Density Functional Theory. Comptes Rendus. Mathématique, Volume 356 (2018) no. 4, pp. 449-455. doi : 10.1016/j.crma.2018.03.002. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2018.03.002/
[1] U. Bindini, L. De Pascale, Optimal transport with Coulomb cost and the semiclassical limit of Density Functional Theory, ArXiv e-prints, 2017.
[2] Continuity and estimates for multimarginal optimal transportation problems with singular costs, Appl. Math. Optim. (2017) | DOI
[3] Optimal-transport formulation of electronic density-functional theory, Phys. Rev. A, Volume 85 (2012)
[4] (Handbook of Numerical Analysis), Volume vol. X, North-Holland, Amsterdam (2003), pp. 3-270
[5] Equality between Monge and Kantorovich multimarginal problems with Coulomb cost, Ann. Mat. Pura Appl. (4), Volume 194 (2015), pp. 307-320
[6] Density functional theory and optimal transportation with Coulomb cost, Commun. Pure Appl. Math., Volume 66 (2013), pp. 548-599
[7] Infinite-body optimal transport with Coulomb cost, Calc. Var. Partial Differ. Equ., Volume 54 (2015), pp. 717-742
[8] S. Di Marino, A. Gerolin, L. Nenna, Optimal Transportation Theory with Repulsive Costs, ArXiv e-prints, 2015.
[9] N-density representability and the optimal transport limit of the Hohenberg–Kohn functional, J. Chem. Phys., Volume 139 (2013)
[10] Electronic zero-point oscillations in the strong-interaction limit of density functional theory, J. Chem. Theory Comput., Volume 5 (2009), pp. 743-753
[11] Orthonormal orbitals for the representation of an arbitrary density, Phys. Rev. A, Volume 24 (1981), pp. 680-682
[12] Inhomogeneous electron gas, Phys. Rev., Volume 136 (1964), p. B864-B871
[13] Universal variational functionals of electron densities, first-order density matrices, and natural spin-orbitals and solution of the v-representability problem, Proc. Natl. Acad. Sci. USA, Volume 76 (1979), pp. 6062-6065
[14] Density functionals for Coulomb systems, Int. J. Quant. Chem., Volume 24 (1983), pp. 243-277
[15] M. Seidl, S. Di Marino, A. Gerolin, L. Nenna, K.J.H. Giesbertz, P. Gori-Giorgi, The strictly-correlated electron functional for spherically symmetric systems revisited, ArXiv e-prints, 2017.
[16] Optimal Transport. Old and New, Grundlehren der Mathematischen Wissenschaften, Fundamental Principles of Mathematical Sciences, vol. 338, Springer-Verlag, Berlin, 2009
Cited by Sources:
Comments - Policy