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Kinetic theory for systems of self-propelled particles with metric-free interactions

Yen-Liang Chou¹, Rylan Wolfe^1,2, and Thomas Ihle^1,3

¹Department of Physics, North Dakota State University, Fargo, North Dakota 58108-6050, USA
²Department of Chemistry and Biochemistry, North Dakota State University, Fargo, North Dakota 58105-5516, USA
³Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Straße 38, 01187 Dresden, Germany

Phys. Rev. E 86, 021120 – Published 17 August, 2012

DOI: https://doi.org/10.1103/PhysRevE.86.021120

Abstract

A model of self-driven particles similar to the Vicsek model [Phys. Rev. Lett. 75, 1226 (1995)] but with metric-free interactions is studied by means of a novel Enskog-type kinetic theory. In this model, $N$ particles of constant speed $v_{0}$ try to align their travel directions with the average direction of a fixed number of closest neighbors. At strong alignment a global flocking state forms. The alignment is defined by a stochastic rule, not by a Hamiltonian. The corresponding interactions are of genuine multibody nature. The theory is based on a Master equation in $3 N$ -dimensional phase space, which is made tractable by means of the molecular chaos approximation. The phase diagram for the transition to collective motion is calculated and compared to direct numerical simulations. A linear stability analysis of a homogeneous ordered state is performed using the kinetic but not the hydrodynamic equations in order to achieve high accuracy. In contrast to the regular metric Vicsek-model no instabilities occur. This confirms previous direct simulations that, for Vicsek-like models with metric-free interactions, there is no formation of density bands and that the flocking transition is continuous.

Article Text

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