Mean first-passage time at the origin of a run-and-tumble particle with periodic forces

TL;DR

This paper analyzes the mean first-passage time of a run-and-tumble particle under periodic forces, deriving conditions for finite hitting probabilities and explicit formulas for the average time to reach the origin.

Contribution

It provides a novel analysis of first-passage times for run-and-tumble particles in periodic force fields, including explicit formulas and conditions for finite survival probability.

Findings

Derived integral condition for non-zero survival probability.

Obtained closed-form expression for the average first-passage time.

Showed that in the short-period limit, the system behaves like a constant drift case.

Abstract

We consider a run-and-tumble particle on a half-line with an absorbing target at the origin. The particle has an internal velocity state that switches between two opposite values at Poisson-distributed times. The position of the particle evolves according to an overdamped Langevin dynamics with a spatially-periodic force field such that every point in a given period interval is accessible to the particle. The survival probability of the particle satisfies a backward Fokker--Planck equation, whose Laplace transform yields systems of equations for the moments of the first-passage time of the particle at the origin. The mean first-passage time has already been calculated assuming that the particle exits the system almost surely. We calculate the probability that the particle reaches the origin in a finite time, given its initial position and velocity. We obtain an integral condition on the force, under which the particle has a non-zero survival probability. The conditional average of the first-passage time at the origin (over the trajectories that reach the origin) is obtained in closed form. As an application, we consider a piecewise-constant force field that alternates periodically between two opposite values. In the limit where the period is short compared to the mean free path of the particle, the mean first-return time to the origin coincides with the value obtained in the case of an effective constant drift, which we calculate explicitly.