Hydrodynamic equations for self-propelled particles:microscopic derivation and stability analysis

摘要

Considering a gas of self-propelled particles with binary interactions, we derivethe hydrodynamic equations governing the density and velocity fields from themicroscopic dynamics, in the framework of the associated Boltzmann equation.Explicit expressions for the transport coefficients are given, as a function of themicroscopic parameters of the model. We show that the homogeneous statewith zero hydrodynamic velocity is unstable above a critical density (whichdepends on the microscopic parameters), signalling the onset of a collectivemotion. Comparison with numerical simulations on a standard model of self-propelled particles shows that the phase diagram we obtain is robust, in thesense that it depends only slightly on the precise definition of the model. Whilethe homogeneous flow is found to be stable far from the transition line, itbecomes unstable with respect to finite-wavelength perturbations close to thetransition, implying a non-trivial spatio-temporal structure for the resultingflow. We find solitary wave solutions of the hydrodynamic equations, quitesimilar to the stripes reported in direct numerical simulations of self-propelledparticles.