  We derive generalized Fluctuation-Dissipation Relations (FDR) holding for a
general stochastic dynamics that includes as subcases both equilibrium models
for passive colloids and non-equilibrium models used to describe active
particles. The relations reported here differ from previous formulations of the
FDR because of their simplicity: they require only the microscopic knowledge of
the dynamics instead of the whole expression of the steady-state probability
distribution function that, except for linear interactions, is unknown for
systems displaying non-vanishing currents. From the response function, we can
extrapolate generalized versions of the Mesoscopic Virial equation and the
equipartition theorem, which still holds far from equilibrium. Our results are
tested in the case of equilibrium colloids described by underdamped or
overdamped Langevin equations and for models describing the non-equilibrium
behavior of active particles. Both the Active Brownian Particle and the Active
Ornstein-Uhlenbeck particle models are compared in the case of a single
particle confined in an external potential.
