Steady state, relaxation and first-passage properties of a run-and-tumble particle in one-dimension

TL;DR

This paper provides exact analytical results for the probability distribution, steady state, relaxation, and first-passage properties of a one-dimensional run-and-tumble particle, revealing behaviors distinct from Brownian motion.

Contribution

It derives exact distributions and first-passage properties of a 1D run-and-tumble particle, including steady states and multi-modal distributions, extending understanding beyond classical Brownian models.

Findings

Probability distribution approaches Gaussian at long times

Distribution exhibits multi-modal forms at intermediate times

Steady state peaks at boundaries in finite domains

Abstract

We investigate the motion of a run-and-tumble particle (RTP) in one dimension. We find the exact probability distribution of the particle with and without diffusion on the infinite line, as well as in a finite interval. In the infinite domain, this probability distribution approaches a Gaussian form in the long-time limit, as in the case of a regular Brownian particle. At intermediate times, this distribution exhibits unexpected multi-modal forms. In a finite domain, the probability distribution reaches a steady state form with peaks at the boundaries, in contrast to a Brownian particle. We also study the relaxation to the steady state analytically. Finally we compute the survival probability of the RTP in a semi-infinite domain. In the finite interval, we compute the exit probability and the associated exit times. We provide numerical verifications of our analytical results.