On the size of chaos in the mean field dynamics

TL;DR

This paper estimates the error in approximating N-particle dynamics by a mean field kinetic equation, showing convergence rates and applying to stochastic, Boltzmann, Povzner, and quantum systems.

Contribution

It introduces a new approach based on the correlation error evolution to analyze mean field convergence, extending applicability to various models.

Findings

Convergence rate of O(j^2/N) for the chaos size.

Applicable to stochastic jump processes and quantum systems.

Provides a unified framework for different mean field models.

Abstract

We consider the error arising from the approximation of an N-particle dynamics with its description in terms of a one-particle kinetic equation. We estimate the distance between the j-marginal of the system and the factorized state, obtained in a mean field limit as N $\rightarrow$ $\infty$. Our analysis relies on the evolution equation for the "correlation error" rather than on the usual BBGKY hierarchy. The rate of convergence is shown to be O(j 2 N) in any bounded interval of time (size of chaos), as expected from heuristic arguments. Our formalism applies to an abstract hierarchical mean field model with bounded collision operator and a large class of initial data, covering (a) stochastic jump processes converging to the homogeneous Boltzmann and the Povzner equation and (b) quantum systems giving rise to the Hartree equation.