Quick Relaxation in Collective Motion

TL;DR

This paper provides new theoretical conditions ensuring rapid convergence to equilibrium in flocking models, with polynomial bounds on relaxation time depending on flock size and system geometry.

Contribution

It introduces novel bounds on relaxation time for the Vicsek-Cucker-Smale model using convex geometry and s-energy analysis, extending to robotic pattern-formation systems.

Findings

Convergence time is polynomial in the number of birds for bounded flocks.

New bounds on s-energy improve understanding of agreement systems.

Techniques apply to robotic systems, broadening their relevance.

Abstract

We establish sufficient conditions for the quick relaxation to kinetic equilibrium in the classic Vicsek-Cucker-Smale model of bird flocking. The convergence time is polynomial in the number of birds as long as the number of flocks remains bounded. This new result relies on two key ingredients: exploiting the convex geometry of embedded averaging systems; and deriving new bounds on the s-energy of disconnected agreement systems. We also apply our techniques to bound the relaxation time of certain pattern-formation robotic systems investigated by Sugihara and Suzuki.