Asymptotic expansion of the mean-field approximation

TL;DR

This paper develops a detailed asymptotic expansion for the mean-field approximation in N-body systems, providing explicit coefficients and optimal convergence rates, applicable to models like Kac and quantum evolutions.

Contribution

It introduces a method to compute the full asymptotic expansion coefficients explicitly, improving understanding of convergence in mean-field limits without relying on BBGKY hierarchies.

Findings

Explicit formulas for expansion coefficients at any time.

Optimal rate of convergence to the meanfield limit.

Applicability to Kac models and quantum evolutions.

Abstract

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.