Talk page
Title:
Density functional theory and optimal transportation with Coulomb cost
Speaker:
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
Link:
Workshop: