Asymptotic expansion of the mean-field approximation

We consider the $ N $-body quantum evolution of a particle system in the mean-field approximation. We show that the $ j $th order marginals $ F^N_j(t) $, for factorized initial data $ F(0)^{\otimes N} $, are explicitly expressed, modulo $ N^{-\infty} $, out of the solution $ F(t) $ of the corresponding non-linear mean-field equation and the solution of its linearization around $ F(t) $. The result is valid for all times $ t $, uniformly in $ j = O(N^{\frac12-\alpha}) $ for any $ \alpha>0 $. We establish and estimate the full asymptotic expansion in integer powers of $ \frac1N $ of $ F^N_j(t) $, $ j = O(\sqrt N) $, whose computation at order $ n $ involves a finite number of operations depending on $ j $ and $ n $ but not on $ N $. Our results are also valid for more general models including Kac models. As a by-product we get that the rate of convergence to the mean-field limit in $ \frac1N $ is optimal in the sense that the first correction to the mean-field limit does not vanish.