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

Abstract

A model of self-driven particles similar to the Vicsek model [Phys. Rev. Lett. 75 (1995) 1226] but with metric-free interactions is studied by means of a novel Enskog-type kinetic theory. In this model, N particles of constant speed v0 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 multi-body nature. The theory is based on a Master equation in 3N-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.

21 pages, 17 figures

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• Numerical treatment of the Boltzmann equation for self-propelled particle systems
• Active matter beyond mean-field: Ring-kinetic theory for self-propelled particles
• Fluctuation-Induced First Order Transition to Collective Motion
• Flocking in One Dimension: Asters and Reversals
• Comment on Ihle, "Towards a quantitative kinetic theory of polar active matter"
• Discussion on Peshkov et al., "Boltzmann-Ginzburg-Landau approach for continuous descriptions of generic Vicsek-like models"
• Hydrodynamics of a Granular Gas in a Heterogeneous Environment
• Reentrant phase behavior in binary topological flocks with nonreciprocal alignment
• Gaussian theory for spatially distributed self-propelled particles
• Reduced description method in the kinetic theory of Brownian motion with active fluctuations