Narrow Escape, Part I

TL;DR

This paper derives an asymptotic approximation for the mean escape time of a Brownian particle through a small elliptical window in a bounded domain, revealing how geometry influences escape dynamics.

Contribution

It provides a new asymptotic formula for mean escape time considering elliptical windows, extending classical results and including effects of domain and window geometry.

Findings

Mean escape time scales with the inverse of the window size.

Elliptical window shape affects escape time through the elliptic integral.

Special cases recover classical formulas for circular windows and spherical domains.

Abstract

A Brownian particle with diffusion coefficient $D$ is confined to a bounded domain of volume $V$ in $\rR^3$ by a reflecting boundary, except for a small absorbing window. The mean time to absorption diverges as the window shrinks, thus rendering the calculation of the mean escape time a singular perturbation problem. We construct an asymptotic approximation for the case of an elliptical window of large semi axis $a\ll V^{1/3}$ and show that the mean escape time is $E\tau\sim\ds{\frac{V}{2\pi Da}} K(e)$, where $e$ is the eccentricity of the ellipse; and $K(\cdot)$ is the complete elliptic integral of the first kind. In the special case of a circular hole the result reduces to Lord Rayleigh's formula $E\tau\sim\ds{\frac{V}{4aD}}$, which was derived by heuristic considerations. For the special case of a spherical domain, we obtain the asymptotic expansion $E\tau=\ds{\frac{V}{4aD}} [1+\frac{a}{R} \log \frac{R}{a} + O(\frac{a}{R}) ]$. This problem is important in understanding the flow of ions in and out of narrow valves that control a wide range of biological and technological function.