AOUP in the presence of Brownian noise: a perturbative approach

By working in the small persistence time limit, we determine the steady-state distribution of an active Ornstein Uhlenbeck particle (AOUP) experiencing, in addition to self-propulsion, a Gaussian white noise modeling a bath at temperature T. This allows us to derive analytical formulas for three quantities: the spatial density of a confined particle, the current induced by an asymmetric periodic potential and the entropy production rate. These formulas disentangle the respective roles of the passive and active noises on the steady state of AOUPs, showing that signatures of non-equilibrium can display surprising behaviors as the temperature is varied. Indeed, depending on the potential in which the particle evolves, both the current and the entropy production rate can be non-monotonic functions of T. The latter can even diverge at high temperature for steep enough confining potentials. Thus, depending on context, switching on translational diffusion may drive the particle closer to or further away from equilibrium. We then probe the range of validity of our quantitative derivations by numerical simulations. Finally, we explain how the method presented here to tackle perturbatively an Ornstein Uhlenbeck noise could be further generalized beyond the Brownian case.