  We investigate the non-equilibrium character of self-propelled particles
through the study of the linear response of the active Ornstein-Uhlenbeck
particle (AOUP) model. We express the linear response in terms of correlations
computed in the absence of perturbations, proposing a particularly compact and
readable fluctuation-dissipation relation (FDR): such an expression explicitly
separates equilibrium and non-equilibrium contributions due to self-propulsion.
As a case study, we consider non-interacting AOUP confined in single-well and
double-well potentials. In the former case, we also unveil the effect of
dimensionality, studying one, two, and three-dimensional dynamics. We show that
information about the distance from equilibrium can be deduced from the FDR,
putting in evidence the roles of position and velocity variables in the
non-equilibrium relaxation.
