| Hydrodynamic equations governing the density and velocity fields have been derived for a gas of self-propelled particles with binary interactions, starting from the microscopic dynamics. The homogeneous state with zero hydrodynamic velocity is unstable above a critical density, signaling the onset of a collective motion. While the homogeneous flow is found to be stable far from the transition line, it becomes unstable with respect to finite-wavelength perturbations close to the transition, implying a non trivial spatio-temporal structure for the resulting flow. Solitary wave solutions of the hydrodynamic equations, quite similar to the stripes reported in direct numerical simulations of self-propelled particles, are found and analyzed in some details. Finally, a few perspectives on statistical approaches to simple models of active matter are also discussed, with emphasis on the issue of density fluctuations. |
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