Hartree Corrections in a Mean-field Limit for Fermions with Coulomb Interaction

TL;DR

This paper analyzes the mean-field limit for fermions with Coulomb interaction, demonstrating that free dynamics with phase correction and Hartree dynamics effectively approximate the many-body system, with Hartree providing a more accurate model.

Contribution

It proves that in the mean-field limit, the many-body fermionic dynamics can be approximated by free and Hartree dynamics, establishing the Hartree equations as an effective description.

Findings

Free dynamics with phase factor approximates many-body evolution.

Hartree dynamics offers a better approximation with smaller error.

Derivation of Hartree equations from many-body Coulomb interactions.

Abstract

We consider the many-body dynamics of fermions with Coulomb interaction in a mean-field scaling limit where the kinetic and potential energy are of the same order for large particle numbers. In the considered limit the spatial variation of the mean-field is small. We prove two results about this scaling limit. First, due to the small variation, i.e., small forces, we show that the many-body dynamics can be approximated by the free dynamics with an appropriate phase factor with the conjectured optimal error term. Second, we show that the Hartree dynamics gives a better approximation with a smaller error term. In this sense, assuming that the error term in the first result is optimal, we derive the Hartree equations from the many-body dynamics with Coulomb interaction in a mean-field scaling limit.