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

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 ±1 values at certain rates. Denoting the rate of flip from 1 to -1 as R₁ and the converse rate as R₂, we consider the position- and direction-dependent rates of the form ${R}_{1}\left(x\right)={\left(\frac{\vert x\vert }{l}\right)}^{\alpha }\left[{\gamma }_{1}\enspace \theta \left(x\right)+{\gamma }_{2}\enspace \theta \left(-x\right)\right]$ and ${R}_{2}\left(x\right)={\left(\frac{\vert x\vert }{l}\right)}^{\alpha }\left[{\gamma }_{2}\enspace \theta \left(x\right)+{\gamma }_{1}\enspace \theta \left(-x\right)\right]$ with α ⩾ 0. For α = 0 and 1, we solve the master equations exactly for arbitrary γ₁ and γ₂ at large t. From our analytical expression for the time-dependent probability distribution P(x, t) we find that for γ₁ > γ₂ the distribution relaxes to a steady state exponentially, whereas for γ₁ ⩽ γ₂ the distribution does not reach a steady state and can be described by a non-trivial scaling form. We interestingly find that these features of the probability distribution P(x, t) in the two regimes γ₁ > γ₂ and γ₁ ⩽ γ₂ also remain valid for general α > 0. In particular, for general α, we argue and numerically demonstrate that the approach to the steady state in γ₁ > γ₂ case is exponential. On the other hand, for γ₁ ⩽ γ₂, the distribution P(x, t) remains time dependent and possesses certain scaling behavior. For γ₁ = γ₂ we derive the scaling behavior as well as the scaling function rigorously, whereas for γ₁ < γ₂ we provide heuristic arguments to obtain the scaling behavior and the corresponding scaling functions. We also study the dynamics on a semi-infinite line with an absorbing barrier at the origin. For α = 0 and 1, we analytically compute the survival probabilities and the corresponding first-passage time distributions. For general α ⩾ 0, we provide approximate calculations to compute the behavior of the survival probability for t → ∞ in which limit it approaches a finite value for γ₁ < γ₂ but goes to zero for γ ⩾ γ₂. We also study the approach to the large t value in both cases. Finally, we consider RTP in a finite interval [0, M] and compute the associated exit probabilities from that interval for all α. All our analytic results are verified with a numerical simulation.