Transient anomalous diffusion in run-and-tumble dynamics

We study the stochastic dynamics of a particle with two distinct motility states. Each one is characterized by two parameters: one represents the average speed and the other represents the persistence quantifying the tendency to maintain the current direction of motion. We consider a run-and-tumble process, which is a combination of an active fast motility mode (persistent motion) and a passive slow mode (diffusion). Assuming stochastic transitions between the two motility states, we derive an analytical expression for the time evolution of the mean square displacement. The interplay of the key parameters and the initial conditions as for instance the probability of initially starting in the run or tumble state leads to a variety of transient regimes of anomalous transport on different time scales before approaching the asymptotic diffusive dynamics. We estimate the crossover time to the long-term diffusive regime and prove that the asymptotic diffusion constant is independent of initially starting in the run or tumble state.