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 → ∞. 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(j2N) 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.
On the Size of Chaos in the Mean Field Dynamics / Paul, T; Pulvirenti, M; Simonella, S. - In: ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS. - ISSN 0003-9527. - (2019).
On the Size of Chaos in the Mean Field Dynamics
Simonella S
2019
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 → ∞. 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(j2N) 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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.