The narrow escape problem revisited

TL;DR

This paper provides explicit formulas for the mean narrow escape time in various geometries, applicable to different stochastic processes including anomalous diffusion and external forces, aiding in understanding biological reaction rates.

Contribution

It offers a generalized analytical approach to estimate the mean NET considering domain volume and starting point, applicable to diverse stochastic processes.

Findings

Explicit scaling dependence of mean NET on domain volume and starting point.

Analytical method applicable to anomalous diffusion and external force fields.

Broad applicability to biological and physical systems.

Abstract

The time needed for a particle to exit a confining domain through a small window, called the narrow escape time (NET), is a limiting factor of various processes, such as some biochemical reactions in cells. Obtaining an estimate of the mean NET for a given geometric environment is therefore a requisite step to quantify the reaction rate constant of such processes, which has raised a growing interest in the last few years. In this Letter, we determine explicitly the scaling dependence of the mean NET on both the volume of the confining domain and the starting point to aperture distance. We show that this analytical approach is applicable to a very wide range of stochastic processes, including anomalous diffusion or diffusion in the presence of an external force field, which cover situations of biological relevance.