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 $σ(t)$ 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 $R_1(x)=\left(\frac{\mid x \mid}{l}\right) ^α\left[γ_1~θ(x)+γ_2 ~θ(-x)\right]$ and $R_2(x)=\left(\frac{\mid x \mid}{l}\right) ^α\left[γ_2~θ(x)+γ_1 ~θ(-x)\right]$ with $α\geq 0$. For $γ_1 >γ_2$, we find that the particle exhibits a steady-state probability distriution even in an infinite line whose exact form depends on $α$. For $α=0$ and $1$, we solve the master equations exactly for arbitrary $γ_1$ and $γ_2$ 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 $γ_1 > γ_2$. On the other hand, for $γ_1<γ_2$, the large $t$ behaviour of $P(x,t)$ is drastically different than $γ_1=γ_2$ 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 $α$, we argue that the approach to the steady state in $γ_1>γ_2$ case is exponential which we numerically demonstrate....