A non-linear kinetic model of self-propelled particles with multiple equilibria
File(s)
Accepted version
OA Location
Author(s)
Buttà, Paolo
Flandoli, Franco
Ottobre, Michela
Zegarlinski, Boguslaw
Type
Journal Article
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 ft, 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 the solution of the PDE.
Date Issued
2019-08-01
Date Acceptance
2019-01-07
Citation
Kinetic and Related Models, 2019, 12 (4), pp.791-827
ISSN
1937-5093
Publisher
American Institute of Mathematical Sciences
Start Page
791
End Page
827
Journal / Book Title
Kinetic and Related Models
Volume
12
Issue
4
Copyright Statement
© 2019 American Institute of Mathematical Sciences. All rights reserved.
Subjects
Science & Technology
Physical Sciences
Mathematics, Applied
Mathematics
Nonlinear kinetic PDEs
self-organization
Vicsek model
scaling limit of interacting particle systems
non ergodic McKean-Vlasov process
COLLECTIVE BEHAVIOR
HYDRODYNAMIC LIMIT
PHASE-TRANSITION
EXISTENCE
EQUATIONS
DYNAMICS
PATTERNS
SYSTEM
ORDER
Applied Mathematics
Publication Status
Published