Asymptotic Fixed-Speed Reduced Dynamics for Kinetic Equations in Swarming

TL;DR

This paper analyzes the asymptotic behavior of particle systems in collective dynamics, deriving reduced models with fixed speed by averaging fast dynamics, applicable to models like Cucker-Smale and Vicsek.

Contribution

It introduces a rigorous asymptotic analysis that reduces complex kinetic models to simpler fixed-speed dynamics, incorporating effects like attraction, repulsion, and alignment.

Findings

Reduction of Cucker-Smale to Vicsek model without noise

Development of a formal expansion for diffusion effects

Handling measure solutions in the asymptotic analysis

Abstract

We perform an asymptotic analysis of general particle systems arising in collective behavior in the limit of large self-propulsion and friction forces. These asymptotics impose a fixed speed in the limit, and thus a reduction of the dynamics to a sphere in the velocity variables. The limit models are obtained by averaging with respect to the fast dynamics. We can include all typical effects in the applications: short-range repulsion, long-range attraction, and alignment. For instance, we can rigorously show that the Cucker-Smale model is reduced to the Vicsek model without noise in this asymptotic limit. Finally, a formal expansion based on the reduced dynamics allows us to treat the case of diffusion. This technique follows closely the gyroaverage method used when studying the magnetic confinement of charged particles. The main new mathematical difficulty is to deal with measure solutions in this expansion procedure.