Mean-field limit for the stochastic Vicsek model

TL;DR

This paper rigorously derives the mean-field PDE for the stochastic Vicsek model with noise, demonstrating convergence of particle laws to the PDE solution on a surface, and establishing existence and uniqueness results.

Contribution

It provides the first rigorous derivation of the kinetic mean-field PDE for the Vicsek model with noise on a surface, including convergence and well-posedness analysis.

Findings

Convergence of particle law to the PDE solution as N tends to infinity

Existence and uniqueness of solutions for both the particle system and PDE

Extension of coupling methods to surface-based models

Abstract

We consider the continuous version of the Vicsek model with noise, proposed as a model for collective behavior of individuals with a fixed speed. We rigorously derive the kinetic mean-field partial differential equation satisfied when the number N of particles tends to infinity, quantifying the convergence of the law of one particle to the solution of the PDE. For this we adapt a classical coupling argument to the present case in which both the particle system and the PDE are defined on a surface rather than on the whole space. As part of the study we give existence and uniqueness results for both the particle system and the PDE.