A continuum model for alignment of self-propelled particles with anisotropy and density-dependent parameters

Abstract

We consider the macroscopic model derived by Degond and Motsch from a time-continuous version of the Vicsek model, describing the interaction orientation in a large number of self-propelled particles. In this article, we study the influence of a slight modification at the individual level, letting the relaxation parameter depend on the local density and taking in account some anisotropy in the observation kernel (which can model an angle of vision). The main result is a certain robustness of this macroscopic limit and of the methodology used to derive it. With some adaptations to the concept of generalized collisional invariants, we are able to derive the same system of partial differential equations, the only difference being in the definition of the coefficients, which depend on the density. This new feature may lead to the loss of hyperbolicity in some regimes. We provide then a general method which enables us to get asymptotic expansions of these coefficients. These expansions shows, in some effective situations, that the system is not hyperbolic. This asymptotic study is also useful to measure the influence of the angle of vision in the final macroscopic model, when the noise is small.

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Cited by in corpus (8)

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• Phase transitions, hysteresis, and hyperbolicity for self-organized alignment dynamics
• Self-Organized Hydrodynamics with congestion and path formation in crowds
• Global weak solutions for Kolmogorov-Vicsek type equations with orientational interactions
• A Boltzmann model for rod alignment and schooling fish
• Coupled Self-Organized Hydrodynamics and Navier-Stokes models: local well-posedness and the limit from the Self-Organized Kinetic-fluid models
• The microscopic phase density functional approach to the construction of the kinetic and hydrodynamic description for the system of self-propelled particles
• Macroscopic limit of a Fokker-Planck model of swarming rigid bodies