  We present a stochastic description of a model of N mutually repelling active
spheres in the presence of external fields and characterize its steady state
behavior. To reproduce the effects of the experimentally observed persistence
of the trajectories of the active particles we consider a Gaussian forcing
having a non vanishing correlation time $\tau$, whose finiteness is a measure
of the activity of the system. With these ingredients we show that it is
possible to develop a statistical mechanical approach similar to the one
employed in the study of equilibrium liquids and to obtain the explicit form of
the many-particle distribution function by means of the multidimensional
unified colored noise approximation. Such a distribution plays a role analogous
to the Gibbs distribution in equilibrium statistical mechanics and provides a
complete information about the microscopic state of the system. From here we
develop a method to determine the one and two-particle distribution functions
in the spirit of the Born-Green-Yvon (BGY) equations of equilibrium statistical
mechanics. The resulting equations which contain extra-correlations induced by
the activity allow to determine the stationary density profiles in the presence
of external fields, the pair correlations and the pressure of active fluids. In
the low density regime we obtain the effective pair potential $\phi(r)$ acting
between two isolated particles separated by a distance, $r$, showing the
existence an effective attraction between them induced by activity. Based on
these results, in the second half of the paper we propose a mean field theory
as an approach simpler than the BGY hierarchy and use it to derive a van der
Waals expression of the equation of state, which can serve as the basis to
understand the phase behavior of active fluids.
