Kinetic hierarchy and propagation of chaos in biological swarm models

TL;DR

This paper analyzes two biological swarm models, deriving their kinetic hierarchies and examining the infinite particle limit, revealing that the CL process does not satisfy propagation of chaos, with numerical simulations suggesting similar behavior in the BDG process.

Contribution

It establishes the master equations and BBGKY hierarchies for two swarm models and investigates their behavior in the infinite particle limit.

Findings

The CL process hierarchy does not satisfy propagation of chaos.

Numerical simulations show the BDG process behaves similarly to the CL process.

Both models exhibit complex kinetic behaviors at large scales.

Abstract

We consider two models of biological swarm behavior. In these models, pairs of particles interact to adjust their velocities one to each other. In the first process, called 'BDG', they join their average velocity up to some noise. In the second process, called 'CL', one of the two particles tries to join the other one's velocity. This paper establishes the master equations and BBGKY hierarchies of these two processes. It investigates the infinite particle limit of the hierarchies at large time-scale. It shows that the resulting kinetic hierarchy for the CL process does not satisfy propagation of chaos. Numerical simulations indicate that the BDG process has similar behavior to the CL process.