Optimal mean first-passage time of a run-and-tumble particle in a class of one-dimensional confining potentials

TL;DR

This paper derives explicit formulas for the mean first-passage time of a run-and-tumble particle in one-dimensional confining potentials, revealing an optimal tumbling rate that minimizes this time for various potential shapes.

Contribution

It provides the first comprehensive analytical solution for MFPT of RTPs in a class of confining potentials and identifies how the optimal tumbling rate depends on initial position and potential shape.

Findings

Explicit MFPT formulas for all p ≥ 1

Existence of an optimal tumbling rate γ_opt for p > 1

γ_opt scales as 1/x_0 for small x_0

Abstract

We consider a run-and-tumble particle (RTP) in one dimension, subjected to a telegraphic noise with a constant rate $\gamma$, and in the presence of an external confining potential $V(x) = \alpha |x|^p$ with $p \geq 1$. We compute the mean first-passage time (MFPT) at the origin $\tau_\gamma(x_0)$ for an RTP starting at $x_0$. We obtain a closed form expression for $\tau_\gamma(x_0)$ for all $p \geq 1$, which becomes fully explicit in the case $p=1$, $p=2$ and in the limit $p \to \infty$. For generic $p>1$ we find that there exists an optimal rate $\gamma_{\rm opt}$ that minimizes the MFPT and we characterize in detail its dependence on $x_0$. We find that $\gamma_{\rm opt} \propto 1/x_0$ as $x_0 \to 0$, while $\gamma_{\rm opt}$ converges to a nontrivial constant as $x_0 \to \infty$. In contrast, for $p=1$, there is no finite optimum and $\gamma_{\rm opt} \to \infty$ in this case. These analytical results are confirmed by our numerical simulations.