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 jth order marginals FNⱼ(t), for factorized initial data F(0)⊗ N, are explicitly expressed, modulo N-∞, 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(N12-α) for any α>0. We establish and estimate the full asymptotic expansion in integer powers of 1N of FNⱼ(t), j = O(√ 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 1N is optimal in the sense that the first correction to the mean-field limit does not vanish.