# Effective Poisson equation of density functional theory at positive temperature

Density functional theory (DFT) has been a very successful effective theory of many-body quantum mechanics. In particular, the Kohn-Sham (KS) equations of DFT serve as an accurate model for the electron densities. The KS equations are a case of the Schrodinger-Poisson equations whose electron-electron effective interaction potential only depends on the density of electrons. When the number of electrons are limited, the KS equation can be solved quickly by numerical method at temperature T = 0. Since physically interesting settings are at T > 0, we study the KS equations at positive temperature and give an iterative scheme to construct solutions.

One important class of electronic structures described by the KS equations is a crystalline lattice. At positive temperature, we show that a local perturbation to a crystalline structure induces an electric field governed by the Poisson equation. The latter equation emerges as an effective equation of the KS equations. This is a joint work with Israel M. Sigal.