In the mean-field regime, we prove convergence (with explicit bounds) of the many-body von Neumann dynamics with bounded interactions to the Hartree-von Neumann dynamics.
We develop a quantum relative entropy method for the mean-field limit of quantum many-body systems. For closed systems governed by the von Neumann equation, we prove a quantitative stability estimate between the N-body density matrix and the tensorized solution of the Hartree equation. The argument is based on an entropy production identity, a cancellation mechanism for the centered two-body fluctuation, and a combinatorial estimate controlling the remaining mixed moments. As a consequence, we obtain propagation of chaos in trace norm for fixed marginals. We further combine the entropy estimate with known semiclassical Wasserstein bounds to derive a convergence estimate that is uniform in the Planck constant in an appropriate joint mean-field and semiclassical regime. Finally, we extend the method to finite-dimensional open quantum systems governed by Lindblad dynamics. In this setting, we establish an analogous relative entropy estimate for general bounded two-body interactions, where the mean-field potential is defined through partial trace. This shows that the entropy method does not rely on any special tensor-product decomposition of the interaction.
In this paper, we investigate the dynamics of a system of N weakly interacting bosons with singular three-body interactions in three dimensions. By assuming factorized initial data ΨN,0=φ0⊗N and triple collisions, we prove that in the many-particle limit, its mean-field approximation converges to quintic Hartree dynamics. Moreover, we prove that the rate of convergence towards the mean-field quintic Hartree evolution is of O(N−(1+4a)/(3+2a)) for φ0∈H(3/2)+a(R3) where 0≤a<1/2 and O(N−1) for a>1. Our proof is based on and extends the Fock space approach.
We review recent results concerning the evolution of fermionic systems. We are interested in the mean field regime, where particles experience many weak collisions. For fermions, the mean field regime is naturally linked with a semiclassical limit. Assuming some regularity of the interaction potential we show that the many body evolution of initial states close to Slater determinants exhibiting the appropriate semiclassical structure can be approximated by the Hartree-Fock equation. Our method provides precise bounds on the rate of the convergence.