I will present a summary of works developed in the last years with Z. Ammari about the bosonic mean field limit with a phase-space point of view, the mean field limit looking formally like a semiclassical limit on some infinite dimensional (in most cases) phase-space. The presence of a small parameter, 1/N where N is the typical number of particles, allows to bypass the well known difficulties of microlocal tools in infinite dimensions. Wigner measures are naturally introduced within this approach and catch all the asymptotic information, in particular the dynamics of the asymptotic BBGKY hierarchy in very general cases. A simple "compactness" criterion on Wigner measures, easy to check in many usual cases and also easy to contradict by constructive examples, elucidates the possible obstructions to the BBGKY interpretation of Wigner measures. Our last result solves completely the mean field dynamics of bosons with a Coulombic (repulsive or attractive in dimension 3) pair interaction potential, for general initial data.
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