Transient anomalous diffusion in run-and-tumble dynamics

TL;DR

This paper analyzes a run-and-tumble particle model with stochastic state transitions, revealing transient anomalous diffusion behaviors and deriving analytical expressions for mean square displacement, with implications for understanding complex transport phenomena.

Contribution

It provides an analytical framework for transient anomalous diffusion in a two-state run-and-tumble model, highlighting the effects of initial conditions and parameter interplay.

Findings

Transient anomalous transport regimes are identified.

Crossover time to diffusive behavior is estimated.

Asymptotic diffusion constant is independent of initial state.

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.