  Active matter systems are driven out of thermal equilibrium by a lack of
generalized Stokes-Einstein relation between injection and dissipation of
energy at the microscopic scale. We consider such a system of interacting
particles, propelled by persistent noises, and show that, at small but finite
persistence time, their dynamics still satisfy a time-reversal symmetry. To do
so, we compute perturbatively their steady-state measure and show that, for
short persistent times, the entropy production rate vanishes. This endows such
systems with an effective Fluctuation-Dissipation theorem akin to that of
thermal equilibrium systems. Last we show how interacting particle systems with
viscous drags and correlated noises can be seen as in equilibrium with a
visco-elastic bath but driven out of equilibrium by non-conservative forces,
hence providing an energetic insight on the departure of active systems from
equilibrium.
