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 ∈ L 2 loc (R 3 ; R 3 ) and the semi-relativistic case.

The derivation of the Hartree equation from many-body systems of Bosons in the mean field limit has been very intensively studied in the last couple of years. However, very few results exist showing convergence of the k-th marginal of the N-body density matrix to the projection to the k-fold tensor product of the solution of the Hartree equation in stronger trace norms like the energy trace norm, see [MS], [Lu]. This issue is from a physical view point very important. The reason is that one can then approximate expectation values of certain observables of the N-body system by means of the Hartree equation, with relaxation of the very restrictive assumption that the observables are bounded operators. Here we consider the non-relativistic case. We prove, assuming only H 1 -regularity of the initial data, convergence in the energy trace norm without rates, and convergence in any other weaker Sobolev trace norm with rates. Our proof is simple and uses the functional a N introduced by Pickl in [Pi].

In this work, we show convergence of the [Formula: see text]-particle bosonic Schrödinger equation towards the Hartree equation. Hereby, we extend the results of [I. Anapolitanos and M. Hott, A simple proof of convergence to the Hartree dynamics in Sobolev trace norms, J. Math. Phys. 57(12) (2016) 122108; I. Anapolitanos, M. Hott and D. Hundertmark, Derivation of the Hartree equation for compound Bose gases in the mean field limit, Rev. Math. Phys. 29(07) (2017) 1750022]. We first consider the semi-relativistic Hartree equation in the defocusing and the focusing cases. We show that Pickl’s projection method [P. Pickl, Derivation of the time dependent Gross–Pitaevskii equation without positivity condition on the interaction, J. Statist. Phys. 140(1) (2010) 76–89; P. Pickl, A simple derivation of mean field limits for quantum systems, Lett. Math. Phys. 97(2) (2011) 151–164; P. Pickl, Derivation of the time dependent Gross–Pitaevskii equation with external fields, Rev. Math. Phys. 27(1) (2015) 1550003], can be adapted to this problem. Next, we extend this result to the case of fractional Hartree equations with potentials that are more singular than the Coulomb potential. Finally, in the non-relativistic case, we derive the Hartree equation assuming only [Formula: see text] initial data for potentials with a quantitative bound on the convergence rate.

In this paper, we study the quantum fluctuation dynamics in a Bose gas on a torus Λ Ă R 3 that exhibits Bose-Einstein condensation, beyond the leading order Hartree-Fock-Bogoliubov (HFB) fluctuations. Given a mean-field Hamiltonian and Bose-Einstein condensate (BEC) with density N , we extract a quantum Boltzmann type dynamics from a second-order Duhamel expansion upon subtracting both the HFB dynamics and the BEC dynamics. Using a Fock-space approach, we provide explicit error bounds. Given an approximately quasi-free initial state, we determine the time evolution of the centered correlation functions xay, xaay ´xay 2 , xa `ay ´|xay| 2 for mesoscopic time scales. For large but finite N , we consider both the case of fixed system size |Λ| " 1, and the case |Λ| " N 6 53 ´. In the case |Λ| " 1, we show that the Boltzmann collision operator contains subleading terms that can become dominant, depending on time-dependent coefficients assuming particular values in Q; this phenomenon is reminiscent of the Talbot effect. For the case´, we prove that the collision operator is well approximated by the expression predicted in the literature. THOMAS CHEN AND MICHAEL HOTT 7.1. Discrete case 78 7.2. Approximation by continuous limit 81 Appendix A. Introductory observations 83 Appendix B. Proof of convergence to mean field equations 85 B.1. Error bounds due to HFB evolution 89 B.2. Various error bounds 90 References 95