On the effective dynamics of Bose-Fermi Mixtures

In this talk, I will be discussing recent work with Esteban Cardenas and my advisor Natasa Pavlovic: https://arxiv.org/abs/2309.04638. In this work, we described the dynamics of a Bose-Einstein condensate interacting with a degenerate Fermi gas, at zero temperature. First, we analyzed the mean-field approximation of the many-body Schrödinger dynamics and proved the emergence of a coupled Hartree-type system of equations. We then obtained rigorous error control that yields a non-trivial scaling window in which the approximation is meaningful. Second, starting from this Hartree system, we identified a novel scaling regime in which the fermion distribution behaves semi-clasically, but the boson field remains quantum-mechanical. In that regime, the bosons are much lighter and more numerous than the fermions. We then proved convergence to a coupled Vlasov-Hartee system of equations with an explicit convergence rate.