Density functional theory and optimal transportation with Coulomb cost

Fluid Mechanics, Hamiltonian Dynamics, and Numerical Aspects of Optimal Transportation October 14, 2013 - October 18, 2013

October 17, 2013 (03:30 PM PDT - 04:30 PM PDT)

Speaker(s): Gero Friesecke (TU München)
Location: SLMath: Eisenbud Auditorium

Primary Mathematics Subject Classification No Primary AMS MSC

Secondary Mathematics Subject Classification No Secondary AMS MSC

Video

v1190

Abstract

Density functional theory (DFT) is a computationally feasible electronic structure model which simplifies full quantum mechanics and for which Walter Kohn received a Nobel prize in 1998. In the semiclassical limit, DFT reduces to a multi-marginal optimal transport problem [1]. Considerable insight into the limit problem had been built up, prior to our work, by physicists (Seidl, Perdew, Levy, Gori-Giorgi, Savin), who essentially developed a considerable amount of optimal transport theory without knowing they were doing optimal transport. The goal of my talk is three-fold (i) to explain the connection DFT--optimal transport and compare physicist's and OT theory approaches, for instance the Gangbo-McCann formula for the optimal map in terms of the Kantorovich potential is arrived at in an intriguingly simple way by physicists (ii) to discuss what is known rigorously about the limit problem, including -- justification of the formal semiclassical limit [1] -- qualitative theory of OT problems with Coulomb cost, including the question whether ''Kantorovich minimizers'' must be ''Monge minimizers'' (yes for 2 particles, open for N particles, no for infinitely many particles) [1,2] -- exactly soluble cases (N=2 with radial density; N=infinity) [1, 2] (iii) to present a natural hierarchy of further approximations of the limit functional related to representability constraints on the pair density which survive in the classical limit [3], and discuss the important (open) problem of characterizing N-representable pair densities

Supplements

	Friesecke 1.99 MB application/pdf	Download

Video/Audio Files

v1190

H.264 Video	v1190.m4v 331 MB video/mp4 rtsp://videos.msri.org/data/000/018/644/original/v1190.m4v	Download
Quicktime	v1190.mov 464 MB video/quicktime rtsp://videos.msri.org/data/000/018/645/original/v1190.mov	Download

Troubles with video?

Please report video problems to itsupport@slmath.org.

See more of our Streaming videos on our main VMath Videos page.