Nonlinear Hartree equation as the mean field limit of weakly coupled fermions

TL;DR

This paper proves that the dynamics of weakly interacting fermions can be approximated by a nonlinear Hartree equation within a fixed time interval, with an error of order 1/N, for a broad class of initial data.

Contribution

It establishes a rigorous mean field limit for fermions with analytic pair interactions, connecting many-body quantum dynamics to a nonlinear PDE.

Findings

Error between fermionic system and Hartree solution is of order 1/N.

The result holds for a general class of initial data.

The approximation is valid up to a fixed time T.

Abstract

We consider a system of N weakly interacting fermions with a real analytic pair interaction. We prove that for a general class of initial data there exists a fixed time T such that the difference between the one particle density matrix of this system and the solution of the non-linear Hartree equation is of order 1/N for any time t less or equal T.