In the mean-field limit the dynamics of a quantum Bose gas is described by a Hartree equation. We present a simple method for proving the convergence of the microscopic quantum dynamics to the Hartree dynamics when the number of particles becomes large and the strength of the two-body potential tends to 0 like the inverse of the particle number. Our method is applicable for a class of singular interaction potentials including the Coulomb potential. We prove and state our main result for the Heisenberg-picture dynamics of "observables", thus avoiding the use of coherent states. Our formulation shows that the mean-field limit is a "semi-classical" limit.Comment: Corrected typos and included an elementary proof of the Kato smoothing estimate (Lemma 6.1

We present a new proof of the convergence of the N −particle Schrödinger dynamics for bosons towards the dynamics generated by the Hartree equation in the mean-field limit. For a restricted class of two-body interactions, we obtain convergence estimates uniform in , up to an exponentially small remainder. For = 0, the classical dynamics in the mean-field limit is given by the Vlasov equation.

In this note we describe some results concerning non-relativistic quantum systems at positive temperature and density confined to macroscopically large regions, Λ, of physical space R 3 which are under the influence of some local, time-dependent external forces. We are interested in asymptotic properties of such systems, as Λ increases to all of R 3 . It might thus appear natural *

No abstract