Expansion of the propagation of chaos for Bird and Nanbu systems

TL;DR

This paper extends the understanding of the propagation of chaos in Bird and Nanbu particle systems, providing error expansions, convergence proofs, and a central limit theorem under broader conditions.

Contribution

It introduces a new expansion of the error in propagation of chaos for generalized Bird and Nanbu systems using coupling and branching process techniques.

Findings

Error expansion in propagation of chaos in terms of particle number

Almost sure convergence of the systems

Central limit theorem for empirical measures

Abstract

The Bird and Nanbu systems are particle systems used to approximate the solution of the mollied Boltzmann equation. In particular, they have the propagation of chaos property. Following [GM94, GM97, GM99], we use coupling techniques and results on branching processes to write an expansion of the error in the propagation of chaos in terms of the number of particles, for slightly more general systems than the ones cited above. This result leads to the proof of the a.s convergence and the centrallimit theorem for these systems. In particular, we have a central-limit theorem for the empirical measure of the system under less assumptions then in [M{\'e}l98]. As in [GM94, GM97, GM99], these results apply to the trajectories of particles on an interval [0; T].