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We study a stationary state of a single self-propelled, athermal particle inlinear and quadratic external potentials. The self-propulsion is modeled as afluctuating force evolving according to the Ornstein-Uhlenbeck process,independently of the state of the particle. Without an external potential, inthe long time limit, the self-propelled particle moving in a viscous mediumperforms diffusive motion, which allows one to identify an effectivetemperature. We show that in the presence of a linear external potential thestationary state distribution has an exponential form with the sedimentationlength determined by the effective temperature of the free self-propelledparticle. In the presence of a quadratic external potential the stationarystate distribution has a Gaussian form. However, in general, this distributionis not determined by the effective temperature of the free self-propelledparticle.

A self-propelled particle in an external potential: is there an effective temperature?