From particle to kinetic and hydrodynamic descriptions of flocking

TL;DR

This paper connects particle, kinetic, and hydrodynamic models of flocking, establishing conditions for flocking behavior and proving it in each modeling framework, including for arbitrary initial data.

Contribution

It derives a kinetic and hydrodynamic model from the Cucker-Smale particle model and proves flocking behavior without moment closure.

Findings

Flocking occurs under strong long-range interactions in particle models.

Kinetic model exhibits time-asymptotic flocking for any compact initial data.

Hydrodynamic model also demonstrates flocking without higher moment closure.

Abstract

We discuss the Cucker-Smale's (C-S) particle model for flocking, deriving precise conditions for flocking to occur when pairwise interactions are sufficiently strong long range. We then derive a Vlasov-type kinetic model for the C-S particle model and prove it exhibits time-asymptotic flocking behavior for arbitrary compactly supported initial data. Finally, we introduce a hydrodynamic description of flocking based on the C-S Vlasov-type kinetic model and prove flocking behavior \emph{without} closure of higher moments.