Run-and-tumble particle in one-dimensional confining potentials: Steady-state, relaxation, and first-passage properties

Abstract

We study the dynamics of a one-dimensional run-and-tumble particle subjected to confining potentials of the type V (x) = α |x| p, with p > 0. The noise that drives the particle dynamics is telegraphic and alternates between ±1 values. We show that the stationary probability density P(x) has a rich behavior in the (p, α) plane. For p > 1, the distribution has a finite support in [x−, x+] and there is a critical line αc (p) that separates an activelike phase for α>αc (p) where P(x) diverges at x±, from a passivelike phase for α<αc (p) where P(x) vanishes at x±. For p < 1, the stationary density P(x) collapses to a delta function at the origin, P(x) = δ(x). In the marginal case p = 1, we show that, for α<αc, the stationary density P(x) is a symmetric exponential, while for α>αc, it again is a delta function P(x) = δ(x). For the harmonic case p = 2, we obtain exactly the full time-dependent distribution P(x,t), which allows us to study how the system relaxes to its stationary state. In addition, for this p = 2 case, we also study analytically the full distribution of the first-passage time to the origin. Numerical simulations are in complete agreement with our analytical predictions.