Dynamics And Kinetic Limit For A System Of Noiseless D-Dimensional Vicsek-Type Particles

TL;DR

This paper studies the dynamics of a $d$-dimensional system of self-propelled particles inspired by Vicsek's model, proving stability of invariant manifolds and deriving a kinetic equation with decreasing entropy.

Contribution

It introduces a new analysis of Vicsek-type particles, establishing stability results and deriving a nonlinear kinetic equation in the kinetic limit.

Findings

Existence of an invariant manifold in phase space.

Exponential asymptotic stability of the invariant manifold.

Derivation of a Vlasov-type kinetic equation with decreasing entropy.

Abstract

We analyze the continuous time evolution of a $d$-dimensional system of $N$ self propelled particles with a kinematic constraint on the velocities inspired by the original Vicsek's one \cite{VCB-JCS}. Interactions among particles are specified by a pairwise potential in such a way that the velocity of any given particle is updated to the weighted average velocity of all those particles interacting with it. The weights are given in terms of the interaction rate function. When the size of the system is fixed, we show the existence of an invariant manifold in the phase space and prove its exponential asymptotic stability. In the kinetic limit we show that the particle density satisfies a nonlinear kinetic equation of Vlasov type, under suitable conditions on the interaction. We study the qualitative behaviour of the solution and we show that the Boltzmann-Vlasov entropy is strictly decreasing in time.