Dynamics of Non-interacting 1D Inertial Run and Tumble Particles

Abstract

We generalize the well known Run-and-Tumble dynamics to include inertial effects. In contrast to the usual overdamped Run-and-Tumble particle (RTP) in one spatial dimension which `runs' with a constant velocity until random `tumblings' flip the sign of its velocity, the inertial RTP `runs' with a constant acceleration, and random `tumblings' result in a change of the sign of the acceleration. We study the dynamics of one-dimensional inertial RTP in free space. In particular, we focus on two scenarios, where the waiting time distribution between two consecutive tumbling events is exponential and power-law in nature. For the first case, i.e., when the acceleration flips with a constant rate, we show that the mean-squared displacement grows as t^3 at late times whereas it shows a t^4 behaviour at short times. We also compute the short-time and long-time position distributions. At short times, the particle appears to be most likely to be away from the origin, which is a generic feature shown by all active particles. At long times, on the other hand, the distribution approaches a non-diffusive Gaussian limit. For the power-law waiting time distribution, we use a trajectory-based approach to compute the moments and short-time distribution which shows very different features compared to the exponential case. The particles are seen crowding away from the origin at large times in simulations.