On the size of chaos in the mean field dynamics

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 $\to$ $\infty$. 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.

References in corpus (1)

• On the derivation of the Hartree equation in the mean field limit: Uniformity in the Planck constant

Cited by in corpus (9)

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• Dynamical large deviations for plasmas below the Debye length and the Landau equation
• Sharp local propagation of chaos for mean field particles with $W^{-1,\infty }$ kernels
• Correspondence between open bosonic systems and stochastic differential equations
• Size of chaos for Gibbs measures of mean field interacting diffusions