Asymptotic expansion of the mean-field approximation

We established and estimate the full asymptotic expansion in integer powers of 1 N of the [ $\sqrt$ N ] first marginals of N-body evolutions lying in a general paradigm containing Kac models and non-relativistic quantum evolution. We prove that the coefficients of the expansion are, at any time, explicitly computable given the knowledge of the linearization on the one-body meanfield kinetic limit equation. Instead of working directly with the corresponding BBGKY-type hierarchy, we follows a method developed in [22] for the meanfield limit, dealing with error terms analogue to the v-functions used in previous works. As a by-product we get that the rate of convergence to the meanfield limit in 1 N is optimal.