Large density expansion of a hydrodynamic theory for self-propelled particles

Abstract

Recently, an Enskog-type kinetic theory for Vicsek-type models for self-propelled particles has been proposed [T. Ihle, Phys. Rev. E 83, 030901 (2011)]. This theory is based on an exact equation for a Markov chain in phase space and is not limited to small density. Previously, the hydrodynamic equations were derived from this theory and its transport coefficients were given in terms of infinite series. Here, I show that the transport coefficients take a simple form in the large density limit. This allows me to analytically evaluate the well-known density instability of the polarly ordered phase near the flocking threshold at moderate and large densities. The growth rate of a longitudinal perturbation is calculated and several scaling regimes, including three different power laws, are identified. It is shown that at large densities, the restabilization of the ordered phase at smaller noise is analytically accessible within the range of validity of the hydrodynamic theory. Analytical predictions for the width of the unstable band, the maximum growth rate and for the wave number below which the instability occurs are given. In particular, the system size below which spatial perturbations of the homogeneous ordered state are stable is predicted to scale with $\sqrt{M}$ where $M$ is the average number of collision partners. The typical time scale until the instability becomes visible is calculated and is proportional to M.

References in corpus (14)

• Novel type of phase transition in a system of self-driven particles
• Interaction Ruling Animal Collective Behaviour Depends on Topological rather than Metric Distance: Evidence from a Field Study
• Collective motion of self-propelled particles interacting without cohesion
• Hydrodynamic equations for self-propelled particles: microscopic derivation and stability analysis
• Spontaneously ordered motion of self-propelled particles
• Pattern formation of microtubules and motors: inelastic interaction of polar rods
• New aspects of the continuous phase transition in the scalar noise model (SNM) of collective motion
• Boltzmann-Ginzburg-Landau approach for continuous descriptions of generic Vicsek-like models
• Kinetic theory for systems of self-propelled particles with metric-free interactions
• Towards a quantitative kinetic theory of polar active matter
• Active matter beyond mean-field: Ring-kinetic theory for self-propelled particles
• 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"
• Reply to comment on "Towards a quantitative kinetic theory of polar active matter" by Bertin et al

Cited by in corpus (7)

• Mesoscale pattern formation of self-propelled rods with velocity reversal
• Giant Kovacs-Like Memory Effect for Active Particles
• Chapman-Enskog expansion for the Vicsek model of self-propelled particles
• Contrarian compulsions produce time dependent flocking of active particles
• Crossover between parabolic and hyperbolic scaling, oscillatory modes and resonances near flocking
• Gaussian theory for spatially distributed self-propelled particles
• Critical assessment of von Mises distribution and an infinite series ansatz for self-propelled particles