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

Paolo Buttà; Franco Flandoli; Michela Ottobre; Boguslaw Zegarlinski

doi:10.3934/krm.2019031

Article Contents

2019, Volume 12, Issue 4: 791-827. Doi: 10.3934/krm.2019031

This issue Previous Article Effect of abrupt change of the wall temperature in the kinetic theory Next Article Well-posedness of Cauchy problem for Landau equation in critical Besov space

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

1.
Dipartimento di Matematica, Sapienza Università di Roma, P.le Aldo Moro 5, 00185 Roma, Italy
2.
Scuola Normale Superiore of Pisa, Piazza dei Cavalieri, 7 56126 Pisa, Italy
3.
Heriot Watt University, Mathematics Department, Edinburgh, EH14 4AS, UK
4.
Imperial College London, South Kensington Campus, London SW7 2AZ, UK

^* Corresponding author: M. Ottobre, m.ottobre@hw.ac.uk
^* Corresponding author: M. Ottobre, m.ottobre@hw.ac.uk

Received: June 30, 2018

Published: May 2019

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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 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 \to \infty $, to the solution of the PDE.

Keywords:

Nonlinear kinetic PDEs,
self-organization,
Vicsek model,
scaling limit of interacting particle systems,
non ergodic McKean-Vlasov process.

Mathematics Subject Classification: Primary: 35A01, 35B40, 82C22; Secondary: 82C40, 92D25.

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