Optimal transport in physics

Project ID: 2531ad1584

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UCL Lead department: Statistical Science

Project Summary:

Density functional theory (DFT) is a simplified version of quantum mechanics, widely used in molecular simulations in chemistry, physics and materials science. It was introduced by Hohenberg-Kohn-Sham in the 1960s in two fundamental papers as an approximate computational method for solving the many-electrons Schroedinger equation. The inventor of DFT, Walter Kohn, received the 1998 Nobel Prize in Chemistry.

The mathematical structure of DFT is as follows: Minimise an approximate energy functional F[rho] which depends on the single-electron density rho(x), a function in R3. Catch: exact quantum mechanics energy requires knowledge of electron-pair density rho_2(x,y), a function in R6, which entails correlations. Standard way out: start by assuming statistical independence (known as “mean field” in physics), add semi-empirical corrections to F[rho] accounting for correlations. Note that interactions are known not to be weaker than single-electron terms.

Mathematically, the best DFT models used in practice have no rigorous connection to full quantum mechanics, and an accompanying lack of systematic improvability. Additionally, the mathematical properties of the celebrated Hohenberg-Kohn (HK) functional from DFT are not fully understood and any new insight could influence the models used by practitioners.

We plan to use techniques from optimal transport -a very active area in mathematical analysis - and from the mathematical theory of large points configurations, to advance the DFT problem by rigorous mathematical methods.