MATS4400 Density Functional Theory for strong correlated systems and Optimal Transport, 5 op [kotisivu]

The Strictly-Correlated-Electrons (SCE) density functional theory (SCE DFT) approach, originally proposed by Michael Seidl [Phys. Rev. A 60, 4387 (1999)], could pave the way to a formulation of density functional theory, alternatively to widely used Kohn-Sham DFT, especially aimed at the study of strongly-correlated systems. The course aim to present that theory focusing on some mathematical aspects of SCE DFT relying to optimal transportation theory with Coulomb costs. We will present both the physical intuition of the problem and rigorous mathematical results (guided by physical intuition) on the interface between Density Functional Theory and Optimal Transport.

Optimal Transport Theory is a very active field of mathematics with many applications in geometry, PDEs, meterology, probability, statistics, immaging (see the books written by the Fields medalists Cédric Villani). In this course, we will revisited that theory by considering Coulomb cost functions motivated by SCE DFT.

1st part (M. Seidl, ∼ 3 weeks): Ground state problem for Many-body Schödinger Equation; A brief introduction of Density Functional Theory: Hohenberg-Kohn functional, Kohm Sham Equations; Density Functional Theory for strongly correlated systems (adiabatic limit). Co-motion functions for spherically symmetric systems.

2nd part (A. Gerolin, ∼ 5 weeks): Duality between the space of finite measures and continuous bounded functions; Monge and Monge-Kantorovich problems, existence of optimal plans, Kantorovich duality and existence of Kantorovich potentials. Monge problem (two marginal case) and Wasserstein distances. Multi-marginal Optimal Transport for the attractive harmonic case (existence of Monge minimizers). Study the two electrons (marginals) case for Coulomb costs and the N electrons (multi-marginal) case for radially symmetric densities.

Depending of the interests of the students the following topics (not limited of) can be covered as well: Regularity of Kantorovich potentials for Coulomb costs. Entropic Transport. Optimal Transport for Repulsive harmonic costs. Semi-classical limit of the Hohenberg-Kohn functional.

From the mathematical point of view, we expect at the end of the courses the students will be familiarized with and combine techniques on Calculus of Variations, Functional Analysis, Convex Analysis and Measure Theory. The course has a good environment to introduce master students to applied analysis. In particular, in putting in mathematical grounds a problem that comes from physics. Finally, we hope to present some numerical aspects and point out open problems in the field.

This is a master level course in mathematics. No background in physics will be necessary to follow this course.

There will be around three homework assignments and a final exam. Grades will be based on the homeworks (1/3) and the final exam (2/3). Collaboration between student are encouraged during homework (but not during the exams). If you collaborate with other students during a homework you must specify it. Moreover, each student must submit a homework solution individually.

8 weeks, starting in January 2018.

Lecture notes will be posted online or send by e-mail every week. Other references will be provided
during the lectures.

Teachers:

Michael Seidl (Physics, VU-Amsterdam) and Augusto Gerolin (Mathematics, University of Jyväskylä).
E-mail: michael.seidl@physik.uni-regensburg.de / augusto.gerolin@jyu.fi.