First-Passage Duality

TL;DR

This paper reveals a duality in first-passage times for diffusing particles hitting an absorber, showing independence from flow direction under certain conditions, with implications for understanding diffusion and transience.

Contribution

It introduces a novel duality principle in first-passage time distributions for diffusing particles under flow, extending to multiple dimensions and linking to spatial dimensionality and transience.

Findings

First-passage time distribution is independent of flow direction when conditioned on reaching the target.

In one dimension, average hitting time is proportional to distance over flow speed, regardless of flow sign.

The duality extends to all moments of the hitting time and applies in two-dimensional radial flow scenarios.

Abstract

We show that the distribution of times for a diffusing particle to first hit an absorber is \emph{independent} of the direction of an external flow field, when we condition on the event that the particle reaches the target for flow away from the target. Thus, in one dimension, the average time for a particle to travel to an absorber a distance $\ell$ away is $\ell/|v|$, independent of the sign of $v$. This duality extends to all moments of the hitting time. In two dimensions, the distribution of first-passage times to an absorbing circle in the radial velocity field $v(r)=Q/(2\pi r)$ again exhibits duality. Our approach also gives a new perspective on how varying the radial velocity is equivalent to changing the spatial dimension, as well as the transition between transience and strong transience in diffusion.