We derive equations of motion for the mean-squared displacement (MSD) of an active Brownian particle (ABP) in a crowded environment modeled by a dense system of passive Brownian particles, and of a passive tracer particle in a dense active-Brownian particle system, using a projection-operator scheme. The interaction of the tracer particle with the dense host environment gives rise to strong memory effects. Evaluating these approximately in the framework of a recently developed mode-coupling theory for the glass transition in active Brownian particles (ABP-MCT), we discuss the various regimes of activity-induced super-diffusive motion and density-induced sub-diffusive motion. The predictions of the theory are shown to be in good agreement with results from an event-driven Brownian dynamics simulation scheme for the dynamics of two-dimensional active Brownian hard disks.
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