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

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 $p\ge 1$. 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 $p\ge 1$, 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 ${\gamma }_{\mathrm{opt}}$ that minimizes the MFPT and we characterize in detail its dependence on $x_0$. We find that ${\gamma }_{\mathrm{opt}}\propto 1/x_0$ as $x_0\to 0$, while ${\gamma }_{\mathrm{opt}}$ converges to a nontrivial constant as $x_0\to \infty$. In contrast, for $p=1$, there is no finite optimum and ${\gamma }_{\mathrm{opt}}\to \infty$ in this case. These analytical results are confirmed by our numerical simulations.

Main text: 7+eps pages, 5 figures. Supp. Mat.: 12 pages (revised version)

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