First-Passage-Time Asymmetry for Biased Run-and-Tumble Processes

TL;DR

This paper investigates the asymmetry in first-passage times for biased run-and-tumble processes, providing analytical results, conditions for duality restoration, and measures of asymmetry, with implications for active matter systems.

Contribution

It offers the first analytical framework for understanding first-passage-time asymmetry in biased run-and-tumble models, including conditions for duality and quantitative measures of asymmetry.

Findings

No equality between first-passage time distributions in opposite bias directions.

Conditions identified for asymptotic duality restoration at large barrier distances.

Quantitative measures of asymmetry depend on hidden tumbling dynamics.

Abstract

We explore first-passage phenomenology for biased active processes with a renewal-type structure, focusing in particular on paradigmatic run-and-tumble models in both discrete and continuous state spaces. In general, we show there is no equality between distributions of conditional first-passage times to symmetric barriers positioned in and against the bias direction. However, we give conditions for such a duality to be restored asymptotically (in the limit of a large barrier distance) and highlight connections to the Gallavotti-Cohen fluctuation relation and the method of images. Our general trajectory arguments of first-passage-time distributions for asymmetric run-and-tumble processes to escape from an interval of arbitrary width are supported by exact analytical results, which we derive extending Montroll's defect technique. Furthermore, we quantify the degree of violation of first-passage duality using Kullback-Leibler divergence and signal-to-noise ratios associated with the first-passage times to the two barriers. We reveal an intriguing dependence of such measures of first-passage asymmetry on the underlying often hidden tumbling dynamics which may inspire inference techniques based on first-passage-time statistics in active systems.