Universal Properties of a Run-and-Tumble Particle in Arbitrary Dimension

TL;DR

This paper derives universal properties of a run-and-tumble particle in any dimension, showing that certain statistics are independent of the dimension and the distribution of speeds, confirmed by simulations.

Contribution

It provides exact, universal results for the sign change probability, maximum time distribution, and record statistics of RTPs across different models and dimensions.

Findings

Sign change probability is independent of dimension.

Distribution of the time of maximum is universal.

Results hold even with random speeds from arbitrary distributions.

Abstract

We consider an active run-and-tumble particle (RTP) in $d$ dimensions, starting from the origin and evolving over a time interval $[0,t]$. We examine three different models for the dynamics of the RTP: the standard RTP model with instantaneous tumblings, a variant with instantaneous runs and a general model in which both the tumblings and the runs are non-instantaneous. For each of these models, we use the Sparre Andersen theorem for discrete-time random walks to compute exactly the probability that the $x$ component does not change sign up to time $t$, showing that it does not depend on $d$. As a consequence of this result, we compute exactly other $x$-component properties, namely the distribution of the time of the maximum and the record statistics, showing that they are universal, i.e. they do not depend on $d$. Moreover, we show that these universal results hold also if the speed $v$ of the particle after each tumbling is random, drawn from a generic probability distribution. Our findings are confirmed by numerical simulations. Some of these results have been announced in a recent Letter [Phys. Rev. Lett. 124, 090603 (2020)].