Kinetic limits for pair-interaction driven master equations and biological swarm models

TL;DR

This paper introduces a class of master equations for binary interaction systems, proves a propagation of chaos result generalizing Kac's theorem, and studies kinetic limits in biological swarm models, revealing potential loss of chaos over time.

Contribution

It generalizes propagation of chaos results to pair-interaction driven master equations and applies this to biological swarm models, highlighting conditions for chaos loss.

Findings

Propagation of chaos holds for the class of master equations.

Kinetic limits can lead to loss of chaos at large times.

Invariant densities may not be chaotic in some models.

Abstract

We consider a class of stochastic processes modeling binary interactions in an N-particle system. Examples of such systems can be found in the modeling of biological swarms. They lead to the definition of a class of master equations that we call pair interaction driven master equations. We prove a propagation of chaos result for this class of master equations which generalizes Mark Kac's well know result for the Kac model in kinetic theory. We use this result to study kinetic limits for two biological swarm models. We show that propagation of chaos may be lost at large times and we exhibit an example where the invariant density is not chaotic.