Abstract
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.
Highlights
Many transport processes in nature involve distinct motility states
The run-and-tumble dynamics is beneficial for bacteria as it allows them to react to the environmental changes by adjusting their average run time or speed [3], change their direction of motion, perform an efficient search [4,5,6,7], or optimize their navigation [8, 9]
By deriving an analytical expression for the time evolution of the mean square displacement, we show how the interplay between the run and tumble velocities, the transition probabilities, and the initial conditions leads to various anomalous transport regimes on short and intermediate time scales
Summary
Many transport processes in nature involve distinct motility states. Of particular interest is the run-and-tumble process, which consists of alternating phases of fast active and slow passive motion. The question arises how the transient short time dynamics, the crossover time to asymptotic diffusion, and Dasymp depend on the run and tumble velocities and the switching probabilities between the two states.
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