On the effective dynamics of Bose-Fermi mixtures

TL;DR

This paper rigorously derives effective equations describing the coupled dynamics of Bose-Einstein condensates and degenerate Fermi gases at zero temperature, revealing a novel semi-classical regime with explicit convergence rates.

Contribution

It introduces a new scaling regime where fermions are semi-classical and bosons quantum, providing rigorous derivation and error estimates for the coupled system.

Findings

Derivation of a coupled Hartree system from many-body Schrödinger dynamics.

Identification of a semi-classical fermion regime with quantum bosons.

Proof of convergence to a coupled Vlasov-Hartree system with explicit rates.

Abstract

In this work, we describe the dynamics of a Bose-Einstein condensate interacting with a degenerate Fermi gas, at zero temperature. First, we analyze the mean-field approximation of the many-body Schr\"odinger dynamics and prove emergence of a coupled Hartree-type system of equations. We obtain rigorous error control that yields a non-trivial scaling window in which the approximation is meaningful. Second, starting from this Hartree system, we identify a novel scaling regime in which the fermion distribution behaves semi-clasically, but the boson field remains quantum-mechanical; this is one of the main contributions of the present article. In this regime, the bosons are much lighter and more numerous than the fermions. We then prove convergence to a coupled Vlasov-Hartee system of equations with an explicit convergence rate.