Local time for run and tumble particle

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 $\alpha \ge 0$. 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 $\alpha (\ge 0)$ with coefficients appearing from appropriate exit problems. All our analytical findings are supported with the numerical simulations.

18 pages, 9 figures

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