Yet another look at narrow escape through a tube

TL;DR

This paper derives precise asymptotic formulas for the escape time of a diffusing particle through a narrow tube, resolving conflicting estimates and highlighting the effects of spatially varying diffusivity.

Contribution

It introduces a novel asymptotic analysis combining matched asymptotics and probabilistic methods to accurately determine escape times through a tube.

Findings

New escape time formula consistent with previous estimates in special cases

Importance of spatially varying diffusivity on escape times

Relevance to biological processes like asymmetric cell division

Abstract

The narrow escape problem concerns the time needed for a diffusing particle to exit a confining domain through a small hole in the boundary. While this problem is now well-understood, determining the escape time for a particle that must exit through a narrow tube has proven challenging. Indeed, relying on analogies with electrodynamics, parameter fits to simulations, and heuristics, a variety of conflicting estimates for this escape time have been offered over the last three decades, some of which are counterintuitive and arguably non-physical. In this paper, we combine matched asymptotic analysis and probabilistic methods to determine the exact asymptotics of the narrow escape time through a tube. We obtain a new escape time formula which reduces to the various prior estimates in certain special cases. If the diffusivity in the tube differs from the diffusivity in the rest of the domain, our results reveal the importance of the form of the multiplicative noise inherent to any diffusivity that varies in space. We discuss our results in the context of asymmetric cell division.