# From the Hartree equation to the Vlasov-Poisson system: strong convergence for a class of mixed states

## TL;DR

This paper proves that, for large numbers of interacting fermions, the quantum Hartree dynamics strongly converges to the classical Vlasov-Poisson system in trace norm, specifically for certain mixed states.

## Contribution

It establishes strong trace norm convergence from the quantum Hartree equation to the classical Vlasov-Poisson system for a class of mixed states in the mean-field limit.

## Key findings

- Strong convergence in trace norm for large N
- Convergence proven for Coulomb and gravitational interactions
- Applicable to a specific class of mixed quasi-free states

## Abstract

We consider the evolution of $N$ fermions interacting through a Coulomb or gravitational potential in the mean-field limit as governed by the nonlinear Hartree equation with Coulomb or gravitational interaction. In the limit of large $N$, we study the convergence in trace norm towards the classical Vlasov-Poisson equation for a special class of mixed quasi-free states.