Full distribution of first exit times in the narrow escape problem

TL;DR

This paper derives the full distribution of first exit times in the narrow escape problem, revealing the variability and geometry dependence of escape times, with implications for biological and technological systems.

Contribution

It provides the first derivation of the complete first reaction time distribution in the NEP, highlighting the impact of geometry and barriers on escape dynamics.

Findings

FRTs can be much shorter than the mean, causing temporal scale defocusing.

Initial distance to the target strongly influences typical escape times.

Barrier effects lead to repeated attempts, affecting overall escape time distribution.

Abstract

In the scenario of the narrow escape problem (NEP) a particle diffuses in a finite container and eventually leaves it through a small "escape window" in the otherwise impermeable boundary, once it arrives to this window and over-passes an entropic barrier at the entrance to it. This generic problem is mathematically identical to that of a diffusion-mediated reaction with a partially-reactive site on the container's boundary. Considerable knowledge is available on the dependence of the mean first-reaction time (FRT) on the pertinent parameters. We here go a distinct step further and derive the full FRT distribution for the NEP. We demonstrate that typical FRTs may be orders of magnitude shorter than the mean one, thus resulting in a strong defocusing of characteristic temporal scales. We unveil the geometry-control of the typical times, emphasising the role of the initial distance to the target as a decisive parameter. A crucial finding is the further FRT defocusing due to the barrier, necessitating repeated escape or reaction attempts interspersed with bulk excursions. These results add new perspectives and offer a broad comprehension of various features of the by-now classical NEP that are relevant for numerous biological and technological systems.