Dynamics and escape of active particles in a harmonic trap

The dynamics of active particles is of interest at many levels and is the focus of theoretical and experimental research. There have been many attempts to describe the dynamics of particles affected by random active forces in terms of an effective temperature. This kind of description is tempting due to the similarities (or lack thereof) to systems in or near thermal equilibrium. However, the generality and validity of the effective temperature is not yet fully understood. Here we study the dynamics of trapped particles subjected to both thermal and active forces. The particles are not overdamped. Expressions for the effective temperature due to the potential and kinetic energies are derived, and they differ from each other. A third possible effective temperature can be derived from the escape time of the particle from the trap, using a Kramers-like expression for the mean escape time. We find that over a large fraction of the parameter space, the potential energy effective temperature is in agreement with the escape temperature, while the kinetic effective temperature only agrees with the former two in the overdamped limit. Moreover, we show that the specific implementation of the random active force, and not only its first two moments and the two point autocorrelation function, affects the escape-time distribution.