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 $\alpha \geq 0$. 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 $\gamma_1 \gamma_2$ case is exponential which we numerically demonstrate....

中文翻译：

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一维非均质介质中的滚动粒子

我们在一维中研究奔跑和翻滚粒子 (RTP)，也称为持久布朗运动。电报噪声 $\sigma(t)$ 驱动粒子以某些速率在 $\pm 1$ 值之间变化。将 $1$ 到 $-1$ 的翻转率表示为 $R_1$，将相反的速率表示为 $R_2$，我们考虑 $R_1(x)=\left(\frac{\ mid x \mid}{l}\right) ^{\alpha}\left[\gamma_1~\theta(x)+\gamma_2 ~\theta (-x)\right]$ 和 $R_2(x)=\left (\frac{\mid x \mid}{l}\right) ^{\alpha}\left[\gamma_2~\theta(x)+\gamma_1 ~\theta (-x)\right]$ 与 $\alpha \geq 0$。对于$\gamma_1 >\gamma_2$，我们发现即使在精确形式取决于$\alpha$ 的无限直线中，粒子也表现出稳态概率分布。对于 $\alpha =0$ 和 $1$，我们精确地求解任意 $\gamma_1$ 和 $\gamma_2$ 在大 $t$ 的主方程。从我们对时间相关概率分布 $P(x,t)$ 的显式表达式中，我们发现它以指数方式松弛到 $\gamma_1 > \gamma_2$ 的稳态分布。另一方面，对于 $\gamma_1 \gamma_2$ 的情况是指数的，我们用数字证明了......