Run-and-Tumble particle in inhomogeneous media in one dimension

Abstract

We investigate the run and tumble particle (RTP), also known as persistent Brownian motion, in one dimension. A telegraphic noise $\sigma (t)$ drives the particle which changes between $\pm 1$ values with some rates. Denoting the rate of flip from $1$ to $-1$ as $R_1$ and the converse rate as $R_2$, we consider the position and direction dependent rates of the form $R_1(x)={\left(\frac{\mid x\mid }{l}\right)}^{\alpha }\left[{\gamma }_1\theta (x)+{\gamma }_2\theta (-x)\right]$ and $R_2(x)={\left(\frac{\mid x\mid }{l}\right)}^{\alpha }\left[{\gamma }_2\theta (x)+{\gamma }_1\theta (-x)\right]$ with $\alpha \ge 0$. For ${\gamma }_1>{\gamma }_2$, we find that the particle exhibits a steady-state probability distriution even in an infinite line whose exact form depends on $\alpha$. For $\alpha =0$ and $1$, we solve the master equations exactly for arbitrary ${\gamma }_1$ and ${\gamma }_2$ at large $t$. From our explicit expression for time-dependent probability distribution $P(x,t)$ we find that it exponentially relaxes to the steady-state distribution for ${\gamma }_1>{\gamma }_2$. On the other hand, for ${\gamma }_1<{\gamma }_2$, the large $t$ behaviour of $P(x,t)$ is drastically different than ${\gamma }_1={\gamma }_2$ case where the distribution decays as $t^{-\frac{1}{2}}$. Contrary to the latter, detailed balance is not obeyed by the particle even at large $t$ in the former case. For general $\alpha$, we argue that the approach to the steady state in ${\gamma }_1>{\gamma }_2$ case is exponential which we numerically demonstrate....

arXiv comment44 pages, 32 figures

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