  We derive the hydrodynamic equations of motion for a fluid of active
particles described by under- damped Langevin equations that reduce to the
Active-Brownian-Particle model, in the overdamped limit. The contraction into
the hydrodynamic description is performed by locally averaging the par- ticle
dynamics with the non-equilibrium many-particle probability density, whose
formal expression is found in the physically relevant limit of high-friction
through a multiple-time-scale analysis. This approach permits to identify the
conditions under which self-propulsion can be subsumed into the fluid stress
tensor and thus to define systematically and unambiguously the local pressure
of the active fluid.
