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The mean-squared displacement (MSD) is an averaged quantity widely used toassess anomalous diffusion. In many cases, such as molecular motors with finiteprocessivity, dynamics of the system of interest produce trajectories ofvarying duration. Here we explore the effects of finite processivity ondifferent measures of the MSD. We do so by investigating a deceptively simpledynamical system: a one-dimensional random walk (with equidistant jump lengths,symmetric move probabilities, and constant step duration) with anorigin-directed detachment bias. By tuning the time dependence of thedetachment bias, we find through analytical calculations and trajectorysimulations that the system can exhibit a broad range of anomalous diffusion,extending beyond conventional diffusion to superdiffusion and evensuperballistic motion. We analytically determine that protocols with atime-increasing detachment lead to an ensemble-averaged velocity increasing intime, thereby providing the effective acceleration that is required to push thesystem above the ballistic threshold. MSD analysis of burnt-bridges ratchetssimilarly reveals superballistic behavior. Because superdiffusive MSDs areoften used to infer biased, motor-like dynamics, these findings provide acautionary tale for dynamical interpretation.

Apparent superballistic dynamics in one-dimensional random walks with biased detachment