Propagation of chaos for topological interactions

TL;DR

This paper rigorously proves that a kinetic equation describing topological interactions among particles emerges as the mean-field limit of a finite particle system, confirming propagation of chaos under initial independence assumptions.

Contribution

It provides a rigorous derivation of the kinetic equation from a particle model with topological interactions, establishing propagation of chaos in the mean-field limit.

Findings

Kinetic equation derived from particle system

Propagation of chaos proven in the limit

Mean-field limit confirmed for topological interactions

Abstract

We consider a $N$-particle model describing an alignment mechanism due to a topological interaction among the agents. We show that the kinetic equation, expected to hold in the mean-field limit $N \to \infty$, as following from the previous analysis in [A. Blanchet, P. Degond, Topological interactions in a Boltzmann-type framework, J. Stat. Phys., 163 (2016), pp. 41-60.] can be rigorously derived. This means that the statistical independence (propagation of chaos) is indeed recovered in the limit, provided it is assumed at time zero.