Active Brownian and inertial particles in disordered environments: Short-time expansion of the mean-square displacement

We consider an active Brownian particle moving in a disordered two-dimensional energy or motility landscape. The averaged mean-square displacement (MSD) of the particle is calculated analytically within a systematic short-time expansion. As a result, for overdamped particles, both an external random force field and disorder in the self-propulsion speed induce ballistic behavior adding to the ballistic regime of an active particle with sharp self-propulsion speed. Spatial correlations in the force and motility landscape contribute only to the cubic and higher-order powers in time for the MSD. Finally, for inertial particles two superballistic regimes are found where the scaling exponent of the MSD with time is alpha = 3 and alpha = 4. We confirm our theoretical predictions by computer simulations. Moreover, they are verifiable in experiments on self-propelled colloids in random environments.