  We study analytically the emergence of spontaneous collective motion within
large bidimensional groups of self-propelled particles with noisy local
interactions, a schematic model for assemblies of biological organisms. As a
central result, we derive from the individual dynamics the hydrodynamic
equations for the density and velocity fields, thus giving a microscopic
foundation to the phenomenological equations used in previous approaches. A
homogeneous spontaneous motion emerges below a transition line in the
noise-density plane. Yet, this state is shown to be unstable against spatial
perturbations, suggesting that more complicated structures should eventually
appear.
