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We present a stochastic description of a model of N mutually interacting active particles in the presence of external fields and characterize its steady state behavior in the absence of currents. 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 τ, 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 ϕ acting between two isolated
particles separated by a distance, r, showing the existence an effective attraction.
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