Double milling in self-propelled swarms from kinetic theory

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2009, 2(2): 363-378. doi: 10.3934/krm.2009.2.363

Double milling in self-propelled swarms from kinetic theory

José A. Carrillo ^1, , M. R. D’Orsogna ^2, and V. Panferov ^2,

1.	ICREA-Departament de Matemàtiques, Universitat Autònoma de Barcelona, E-08193 Bellaterra, Spain
2.	Department of Mathematics, California State University, Northridge, CA 91330-8313, United States, United States

Received December 2008 Revised March 2009 Published May 2009

Abstract
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We present a kinetic theory for swarming systems of interacting, self-propelled discrete particles. Starting from the Liouville equation for the many-body problem we derive a kinetic equation for the single particle probability distribution function and the related macroscopic hydrodynamic equations. General solutions include flocks of constant density and fixed velocity and other non-trivial morphologies such as compactly supported rotating mills. The kinetic theory approach leads us to the identification of macroscopic structures otherwise not recognized as solutions of the hydrodynamic equations, such as double mills of two superimposed flows. We find the conditions allowing for the existence of such solutions and compare to the case of single mills.

Keywords: kinetic theory, Interacting particle systems, swarming, milling patterns..

Mathematics Subject Classification: 92D50, 82C40, 82C22, 92C1.

Citation: José A. Carrillo, M. R. D’Orsogna, V. Panferov. Double milling in self-propelled swarms from kinetic theory. Kinetic & Related Models, 2009, 2 (2) : 363-378. doi: 10.3934/krm.2009.2.363

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