# Derivation of the Hartree equation for compound Bose gases in the mean field limit

## TL;DR

This paper proves that mixtures of different Bose gases can be effectively described by coupled Hartree equations in the mean field limit, providing quantitative convergence bounds for reduced density matrices.

## Contribution

It extends the derivation of Hartree equations to multi-species Bose gases, including magnetic and semi-relativistic cases, with explicit convergence rates.

## Key findings

- Coupled Hartree equations accurately describe multi-species Bose gases.
- Quantitative bounds on convergence rates of reduced density matrices.
- Applicable to both non-relativistic and semi-relativistic regimes.

## Abstract

We consider mixtures of Bose gases of different species. We prove that in the mean field limit and under suitable conditions on the initial condition a system composed of two Bose species can be effectively described by a system of coupled Hartree equations. Moreover, we derive quantitative bounds on the rates of convergence of the reduced density matrices in Sobolev trace norms. We treat both the non-relativistic case in the presence of an external magnetic field $A\in L^2_{\text{loc}}(\R^3;\R^3)$ and the semi-relativistic case.