Abstract

We investigate a new multi-marginal optimal transport problem arising from a dissociation model in the Strong Interaction Limit of Density Functional Theory. In this short note, we introduce such dissociation model, the corresponding optimal transport problem as well as show preliminary results on the existence and uniqueness of Monge solutions assuming absolute continuity of at least two of the marginals. Finally, we show that such marginal regularity conditions are necessary for the existence of an unique Monge solution.

Author information

Contact details are reproduced from the original publication and may be historical.

Augusto Gerolin

Dept. of Mathematics and Statistics, and: Dept. of Chemistry and Biomolecular Sciences, and: Nexus for Quantum Technologies, University of Ottawa, Canada

agerolin@uottawa.ca

Mircea Petrache

Dept. of Mathematics, and: Inst. for Mathematical and Computational Engineering, Pontificia Universidad Católica, Santiago, Chile

mpetrache@mat.puc.cl

Adolfo Vargas-Jiménez

Department of Mathematics and Statistics, University of Ottawa, Canada

Keywords

  • Multi-marginal optimal transport
  • density functional theory
  • dissociation energy

Mathematics Subject Classification

49-XX, 35-XX

References

  1. U. Bindini, L. De Pascale: Optimal transport with Coulomb cost and the semiclassical limit of density functional theory, J. Écol. Polytech. Math. 4 (2017) 909–934.https://doi.org/10.5802/jep.59Google Scholar
  2. G. Bouchitté, G. Buttazzo, T. Champion, L. De Pascale: Dissociating limit in density functional theory with Coulomb optimal transport cost, Ann. Sc. Norm. Sup. Cl. Sci. (5) 22/3 (2021) 1429–1471.https://doi.org/10.2422/2036-2145.201903_014Google Scholar
  3. G. Bouchitté, G. Buttazzo, T. Champion, L. De Pascale: Relaxed multi-marginal costs and quantization effects, Ann. Inst. H. Poincaré, Anal. Non Lin. 38/1 (2021) 61–90.https://doi.org/10.1016/j.anihpc.2020.06.004Google Scholar
  4. Y. Brenier: The least action principle and the related concept of generalized flows for incompressible perfect fluids, J. Amer. Math. Soc. 2 (1989) 225–255.https://doi.org/10.1090/S0894-0347-1989-0969419-8Google Scholar
  5. G. Buttazzo, L. De Pascale, P. Gori-Giorgi: Optimal-transport formulation of electronic density-functional theory, Phys. Rev. A 85/6 (2012), art. no. 062502.https://doi.org/10.1103/PhysRevA.85.062502Google Scholar
  6. G. Carlier: On a class of multidimensional optimal transportation problems, J. Convex Analysis 10/2 (2010) 517–529.https://doi.org/10.68381/jca1032Google Scholar
  7. T. Champion, L. De Pascale: On the twist condition and c-monotone transport plans, Discrete Contin. Dyn. Systems 34/4 (2014) 1339–1353.https://doi.org/10.3934/dcds.2014.34.1339Google Scholar
  8. C. Cotar, G. Friesecke, C. Klüppelberg: Density functional theory and optimal transportation with Coulomb cost, Comm. Pure Appl. Math. 66 (2013) 548–599.https://doi.org/10.1002/cpa.21437Google Scholar
  9. C. Cotar, G. Friesecke, C. Klüppelberg: Smoothing of transport plans with fixed marginals and rigorous semiclassical limit of the Hohenberg-Kohn functional, Arch. Ration. Mech. Anal. 228/3 (2018) 891–922.https://doi.org/10.1007/s00205-017-1208-yGoogle Scholar
  10. L. De Pascale: Optimal transport with Coulomb cost. Approximation and duality, ESAIM, Math. Model. Numer. Anal. 49/6 (2015) 1643–1657.https://doi.org/10.1051/m2an/2015035Google Scholar
  11. S. Di Marino, A. Gerolin, L. Nenna: Optimal transport for repulsive costs, in: Topological Optimization and Optimal Transport in the Applied Sciences, M. Bergounioux et al. (eds.), Radon Series on Computational and Applied Mathematics 17, De Gruyter, Berlin (2017) 204–256.https://doi.org/10.1515/9783110430417-010Google Scholar
  12. A. Fathi, A. Figalli: Optimal transportation on non-compact manifolds, Israel J. Math. 175 (2010) 1–59.https://doi.org/10.1007/s11856-010-0001-5Google Scholar
  13. G. Friesecke, A. Gerolin, P. Gori-Giorgi: The strong-interaction limit of density functional theory, arXiv: 2202.09760 (2022).https://doi.org/10.1007/978-3-031-22340-2_4Google Scholar
  14. W. Gangbo, A. Swiech: Optimal maps for the multidimensional Monge-Kantorovich problem, Commun. Pure Appl. Math. 51 (1998) 23–45.https://doi.org/10.1002/(SICI)1097-0312(199801)51:1<23::AID-CPA2>3.0.CO;2-HGoogle Scholar
  15. A. Gerolin: Multi-Marginal Optimal Transport and Potential Optimization Problems for Schrödinger Operators, PhD thesis, Università degli studi di Pisa (2016).Google Scholar
  16. A. Gerolin, A. Kausamo, T. Rajala: Duality theory for multi-marginal optimal transport with repulsive costs in metric spaces, ESAIM, Control, Optim. Calc. Var. 25 (2019), art. no. 62, 21 p.https://doi.org/10.1051/cocv/2018062Google Scholar
  17. A. Gerolin, A. Kausamo, T. Rajala: Non-existence of optimal transport maps for the multi-marginal repulsive harmonic cost, SIAM J. Math. Analysis 51/3 (2019), art. no. 10.1137/18M1186514.https://doi.org/10.1137/18M1186514Google Scholar
  18. A. Gerolin, A. Kausamo, T. Rajala: Multi-marginal entropy-transport with repulsive cost, Calc. Var. Part. Diff. Equations 59/3 (2020), art. no. 90, 20 p.https://doi.org/10.1007/s00526-020-01735-3Google Scholar
  19. H. Heinich: Problème de Monge pour n probabilités, C. R. Math. Acad. Sci. Paris 334/9 (2002) 793–795.https://doi.org/10.1016/S1631-073X(02)02341-5Google Scholar
  20. P. Hohenberg, W. Kohn: Inhomogeneous electron gas, Phys. Rev. 136 (1964), art. no. B 864.https://doi.org/10.1103/PhysRev.136.B864Google Scholar
  21. H. G. Kellerer: Duality theorems for marginal problems, Zeitschrift Wahrsch. verw. Gebiete 67/4 (1984) 399–432.https://doi.org/10.1007/BF00532047Google Scholar
  22. Y.-H. Kim, B. Pass: A general condition for Monge solutions in the multi-marginal optimal transport problem, SIAM J. Math. Analysis 46 (2014) 1538–1550.https://doi.org/10.1137/130930443Google Scholar
  23. Y.-H. Kim, B. Pass: Multi-marginal optimal transport on Riemannian manifolds, Amer. J. Math. 137 (2015) 1045–1060.https://doi.org/10.1353/ajm.2015.0024Google Scholar
  24. D. P. Kooi, P. Gori-Giorgi: A variational approach to London dispersion interactions without density distortion, J. Phys. Chem. Letters 10/7 (2019) 1537–1541.https://doi.org/10.1021/acs.jpclett.9b00469Google Scholar
  25. M. Levy: Electron densities in search of Hamiltonians, Phys. Rev. A 26/3 (1982) 1200–1208.https://doi.org/10.1103/PhysRevA.26.1200Google Scholar
  26. M. Lewin: Semi-classical limit of the Levy-Lieb functional in density functional theory, C. R. Math. Acad. Sci. Paris 356/4 (2018) 449–455.https://doi.org/10.1016/j.crma.2018.03.002Google Scholar
  27. E. H. Lieb: Density functionals for Coulomb systems, Int. J. Quantum Chem. 24/3 (1983) 243–277.https://doi.org/10.1002/qua.560240302Google Scholar
  28. R. McCann: Polar factorization of maps on Riemannian manifolds, Geom. Funct. Analysis 11 (2001) 589–608.https://doi.org/10.1007/PL00001679Google Scholar
  29. B. Pass: Multi-marginal optimal transport and multi-agent matching problems: uniqueness and structure of solutions, Discrete Contin. Dyn. Systems 34 (2014) 1623–1639.https://doi.org/10.3934/dcds.2014.34.1623Google Scholar
  30. B. Pass: Uniqueness and Monge solutions in the multi-marginal optimal transportation problem, SIAM J. Math. Analysis 43 (2015) 2758–2775.https://doi.org/10.1137/100804917Google Scholar
  31. B. Pass, A. Vargas-Jimenez: Multi-marginal optimal transportation problem for cyclic costs, SIAM J. Math. Analysis 53 (2021) 4386–4400.https://doi.org/10.1137/19M130889XGoogle Scholar
  32. B. Pass, A. Vargas-Jimenez: Monge solutions and uniqueness in multi-marginal optimal transport: weaker conditions on the cost, stronger conditions on the marginals, arXiv: 2202.06783 (2022).Google Scholar
  33. B. Pass, A. Vargas-Jiménez: Monge solutions and uniqueness in multi-marginal optimal transport via graph theory, Advances Math. 428 (2023), art. no. 109101, 43 p.https://doi.org/10.1016/j.aim.2023.109101Google Scholar
  34. F. Santambrogio: Optimal Transport for Applied Mathematicians, Progress in Nonlinear Differential Equations and Their Applications, Birkhäuser, Basel (2015).https://doi.org/10.1007/978-3-319-20828-2Google Scholar
  35. M. Seidl, P. Gori-Giorgi, A. Savin: Strictly correlated electrons in density-functional theory: a general formulation with applications to spherical densities, Phys. Rev. A 75/4 (2007), art. no. 042511/12.https://doi.org/10.1103/PhysRevA.75.042511Google Scholar
  36. C. Villani: Topics in Optimal Transportation, Graduate Studies in Mathematics 58, American Mathematical Society, Providence (2003),https://doi.org/10.1090/gsm/058Google Scholar
  37. S. Vuckovic, A. Gerolin, T. J. Daas, H. Bahmann, G. Friesecke, P. Gori-Giorgi: Density functionals based on the mathematical structure of the strong-interaction limit of DFT, Wiley Interdisciplinary Reviews: Computational Molecular Science 13/2 (2023), art. no. e1634.https://doi.org/10.1002/wcms.1634Google Scholar

Suggested citation

A. Gerolin, M. Petrache, A. Vargas-Jiménez. “Existence and Uniqueness of Monge Minimizers for a Multi-Marginal Optimal Transport Problem with Intermolecular Interactions Cost.” Journal of Convex Analysis 31 (2024), No. 2, 603–618. https://doi.org/10.68381/jca31030

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