Fully inertial active Brownian particle in a harmonic potential

Uhlenbeck and Ornstein presented their theory of a passive Brownian particle in a harmonic potential in 1930. Despite intensive research on active Brownian particles since the 1990s, a complete theory of their inertial motion in a confining force field has yet to be developed. Here, we present rigorous calculations of the steady-state position and velocity autocorrelation functions of an active (self-propelled) Brownian particle with translational and rotational inertia in a two-dimensional harmonic trap. Our results show that the well-known active Ornstein-Uhlenbeck particle inertial model can be used as an approximated treatment of the dynamics of a fully inertial active Brownian particle in a trap, but only within a relatively small range of system parameters. Beyond this range, rotational inertia significantly impacts particle dynamics. In particular, for sufficiently strong confinement, the theory predicts a new mode of active particle dynamics that differs radically from all known dynamical modes of trapped active and passive Brownian particles. In this case, rotational inertia suppresses the oscillatory behavior of the mean-square displacement evolution function, resulting in additional quasiballistic motion. We discuss possible experimental implementations of this dynamical mode.