First passage of a run-and-tumble particle with exponentially-distributed tumble duration in the presence of a drift

TL;DR

The study develops a one-dimensional run-and-tumble particle model with a constant drift and exponentially distributed tumble times on a finite interval with absorbing ends. By formulating forward Fokker–Planck-type equations and applying Laplace transforms, the authors obtain a linear ODE system whose diagonalization yields explicit, exact expressions for exit probabilities and conditional MFPTs for all initial internal velocity states. They systematically solve the problem for negative and positive drift, derive integration-constant fixes from boundary conditions, and analyze half-line limits to connect with established RTP results, including detailed consistency checks. Additionally, they introduce Milne interpolation lengths to characterize where conditional MFPTs vanish and provide closed-form expressions involving the Lambert W function, revealing how the finite tumble duration and drift interplay to modify traditional instantaneous-tumble results. The work extends RTP theory by combining drift and positive tumble times, with clear limiting cases and potential extensions to variance, other boundary conditions, and more general potentials."

Abstract

We consider a run-and-tumble particle on a finite interval $[a,b]$ with two absorbing end points. The particle has an internal velocity state that switches between three values $v,0,-v$ at exponential times, thus incorporating positive tumble times. Moreover, a constant drift is added to the run-and-tumble motion at all times. The combination of these two features constitutes the main novelty of our model. The densities of the first-passage time through $a$ (given the initial position and velocity states) satisfy certain forward Fokker--Planck equations, whose Laplace transforms induce evolution equations for the exit probabilities and mean first-passage times of the particle. We solve these equations explicitly for all possible initial states. We consider the limiting regimes of instantaneous tumble and/or the limit of large $b$ to confirm consistency with existing results in the literature. In particular, in the limit of a half-line (large $b$), the mean first-passage time conditioned on the exit through $a$ is an affine function of the initial position if the drift is positive, as in the case of instantaneous tumble.

Figures (4)

• Figure 1: The probability of exit from the half-line $[a,+\infty[$ for a positive subcritical drift, given an initial state (a) with zero internal velocity (tumbling), (b) with positive internal velocity, (c) with negative internal velocity. The stars correspond to the results of direct numerical simulations with 100,000 particles (the simulation was run until the time reached ten times the value of the mean conditional first-passage time calculated in Section 5).
• Figure 2: The mean first passage-time at $a$ of a particle starting in the half-line $[a,+\infty[$ (in the presence of a negative drift), given an initial state (a) with zero internal velocity (tumbling), (b) with positive internal velocity, (c) with negative internal velocity. The stars correspond to the results of direct numerical simulations with 100,000 particles (the simulation was run until the last particle left the system through $a$).
• Figure 3: The conditional mean first passage-time at $a$ of a particle starting in the half-line $[a,+\infty[$ (in the presence of a positive drift), given an initial state (a) with zero internal velocity (tumbling), (b) with positive internal velocity, (c) with negative internal velocity. The stars correspond to the results of direct numerical simulations with 100,000 particles (the simulation was run until the time reached ten times the predicted value of the mean conditional first-passage time).
• Figure 4: The Milne length for a particle starting its motion in a positive internal velocity state, with a negative drift.