A non-linear kinetic model of self-propelled particles with multiple equilibria

TL;DR

This paper introduces a non-linear kinetic PDE model for self-propelled particles on a one-dimensional torus, analyzing existence, uniqueness, multiple equilibria, and mean-field convergence of particle systems.

Contribution

It presents a novel continuum PDE model with multiple equilibria for self-propelled particles and proves well-posedness and convergence results.

Findings

Existence and uniqueness of PDE solutions established.

Multiple stationary states identified for specific interaction functions.

Mean-field convergence of particle systems to PDE solutions proved.

Abstract

We introduce and analyse a continuum model for an interacting particle system of Vicsek type. The model is given by a non-linear kinetic partial differential equation (PDE) describing the time-evolution of the density $f_t$, in the single particle phase-space, of a collection of interacting particles confined to move on the one-dimensional torus. The corresponding stochastic differential equation for the position and velocity of the particles is a conditional McKean-Vlasov type of evolution (conditional in the sense that the process depends on its own law through its own conditional expectation). In this paper, we study existence and uniqueness of the solution of the PDE in consideration. Challenges arise from the fact that the PDE is neither elliptic (the linear part is only {\em hypoelliptic}) nor in gradient form. Moreover, for some specific choices of the interaction function and for the simplified case in which the density profile does not depend on the spatial variable, we show that the model exhibits multiple stationary states (corresponding to the particles forming a coordinated clockwise/anticlockwise rotational motion) and we study convergence to such states as well. Finally, we prove mean-field convergence of an appropriate $N$-particles system to the solution of our PDE: more precisely, we show that the empirical measures of such a particle system converge weakly, as $N \rightarrow \infty$, to the solution of the PDE.