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 ±1 values with some 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 R1(x)=(∣x∣l)α[γ1 θ(x)+γ2 θ(−x)] and R2(x)=(∣x∣l)α[γ2 θ(x)+γ1 θ(−x)] with α≥0 . For γ1>γ2 , 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 γ1 and γ2 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 γ1>γ2 . On the other hand, for γ1<γ2 , the large t behaviour of P(x,t) is drastically different than γ1=γ2 case where the distribution decays as t−12 . 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 γ1>γ2 case is exponential which we numerically demonstrate....