  We review and extend recent developments on the statistical properties of
Active Ornstein Uhlenbeck particles (AOUPs). In this simplest of models, the
Gaussian white noise of overdamped Brownian colloids is replaced by a Gaussian
colored noise. This suffice to grant this system the hallmark properties of
active matter, while still allowing for analytical progress. We first detail
the perturbative derivation of the steady state of AOUPs in the small
persistence time limit. We show the existence of an effective equilibrium
regime in which detailed-balance is obeyed with respect to a non-Boltzmann
distribution and detail the corresponding fluctuation-dissipation theorem. We
then characterize the departure from equilibrium by computing several relevant
observables (entropy production, ratchet current). At the collective level, we
show AOUPs to experience motility-induced phase separation both in the presence
of pairwise forces or due to quorum-sensing interactions. The latter can be
accounted for by considering the steady-state of AOUPs with spatially varying
propulsion speed or persistence time. Finally, we discuss how the emerging
properties of AOUPs can be characterized from the dynamics of their collective
modes, which we construct explicitly.
