# American Institute of Mathematical Sciences

April  2019, 39(4): 1891-1921. doi: 10.3934/dcds.2019080

## Asymptotic expansion of the mean-field approximation

 1 CMLS, Ecole polytechnique, CNRS, Université Paris-Saclay, 91128 Palaiseau Cedex, France 2 International Research Center on the Mathematics and Mechanics of Complex Systems, MeMoCS, University of L'Aquila, Italy

Received  January 2018 Revised  October 2018 Published  January 2019

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.

Citation: Thierry Paul, Mario Pulvirenti. Asymptotic expansion of the mean-field approximation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 1891-1921. doi: 10.3934/dcds.2019080
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