Transport phenomena in collective dynamics: from micro to social hydrodynamics

On the mean-field and semi-classical limit of the quantum N-body problem.

François Golse

École Polytechnique
[SLIDES]

The Hartree equation is derived from the N-body Schrödinger equation in the mean-field limit, assuming that N>>1 and that the coupling constant (i.e. the strength of the interaction potential) is of order O(1/N). When the interaction potential is bounded with Lipschitz continuous gradient, one can prove that the convergence rate for the mean-field limit is uniform in the Planck constant. This convergence rate estimate is based on a kind of interpolation argument involving two very different bounds: (a) a convergence rate estimate in trace norm for the Dyson series representing the solution to the BBGKY hierarchy, and (b) an estimate involving a quantum analogue of the quadratic Monge-Kantorovich or Vasershtein distance used in optimal transport. (Work in collaboration with C. Mouhot, T. Paul, M. Pulvirenti).