Local time for run and tumble particle

TL;DR

This paper analyzes the local time statistics of a run and tumble particle in an inhomogeneous medium, deriving exact distributions and scaling behaviors for different inhomogeneity parameters, supported by numerical simulations.

Contribution

It provides exact and scaling analytical results for local time distributions of run and tumble particles in inhomogeneous environments, extending understanding beyond homogeneous cases.

Findings

Exact distribution of local time for $oldsymbol{ ext{alpha}=0}$ case.

Scaling behavior of local time fluctuations for general $oldsymbol{ ext{alpha}}$.

Series sum representation of local time distribution till first passage for all $oldsymbol{ ext{alpha}} extgreater 0$.

Abstract

We investigate the local time $(T_{loc})$ statistics for a run and tumble particle in an one dimensional inhomogeneous medium. The inhomogeneity is introduced by considering the position dependent rate of the form $R(x) = \gamma \frac{|x|^{\alpha}}{l^{\alpha}}$ with $\alpha \geq 0$. For $\alpha =0$, we derive the probability distribution of $T_{loc}$ exactly which is expressed as a series of $\delta$-functions in which the coefficients can be interpreted as the probability of multiple revisits of the particle to the origin starting from the origin. For general $\alpha$, we show that the typical fluctuations of $T_{loc}$ scale with time as $T_{loc} \sim t^{\frac{1+\alpha}{2+\alpha}}$ for large $t$ and their probability distribution possesses a scaling behaviour described by a scaling function which we have computed analytically. In the second part, we study the statistics of $T_{loc}$ till the RTP makes a first passage to $x=M~(>0)$. In this case also, we show that the probability distribution can be expressed as a series sum of $\delta$-functions for all values of $\alpha~(\geq 0)$ with coefficients appearing from appropriate exit problems. All our analytical findings are supported with the numerical simulations.