Mean-field limit for the stochastic Vicsek model

François Bolley; José A. Cañizo; José A. Carrillo

doi:10.1016/j.aml.2011.09.011

Mean-field limit for the stochastic Vicsek model

François Bolley, José A. Cañizo, José A. Carrillo

Department of Mathematics

Research output: Contribution to journal › Article › Research › peer-review

62 Citations (Scopus)

Abstract

We consider the continuous version of the Vicsek model with noise, proposed as a model for collective behaviour 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 Rd. As part of the study we give existence and uniqueness results for both the particle system and the PDE. © 2011 Elsevier Ltd. All rights reserved.

Original language	English
Pages (from-to)	339-343
Journal	Applied Mathematics Letters
Volume	25
Issue number	3
DOIs	https://doi.org/10.1016/j.aml.2011.09.011
Publication status	Published - 1 Jan 2012

Keywords

Collective behaviour
Interacting particle systems
Mean-field limit
Vicsek model

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AB - We consider the continuous version of the Vicsek model with noise, proposed as a model for collective behaviour 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 Rd. As part of the study we give existence and uniqueness results for both the particle system and the PDE. © 2011 Elsevier Ltd. All rights reserved.

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