Narrow Escape, Part I
| dc.creator | Singer, A. | |
| dc.creator | Schuss, Z. | |
| dc.creator | Holcman, D. | |
| dc.creator | Eisenberg, R. S. | |
| dc.date | 2004-12-15 | |
| dc.date.accessioned | 2026-07-07T04:31:44Z | |
| dc.date.available | 2026-07-07T04:31:44Z | |
| dc.description | 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 $Eτ\sim\ds{\frac{V}{2πDa}} K(e)$, 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 $Eτ\sim\ds{\frac{V}{4aD}}$, which was derived by heuristic considerations. For the special case of a spherical domain, we obtain the asymptotic expansion $Eτ=\ds{\frac{V}{4aD}} [1+\frac{a}{R} \log \frac{R}{a} + O(\frac{a}{R}) ]$. 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. | |
| dc.description | This is the first in a series of three papers | |
| dc.identifier | https://arxiv.org/abs/math-ph/0412048 | |
| dc.identifier | http://arxiv.org/abs/math-ph/0412048 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/57928 | |
| dc.subject | Mathematical Physics | |
| dc.subject | Probability | |
| dc.subject | 60J65; 58J05; 58J32; 58J37 | |
| dc.title | Narrow Escape, Part I | |
| dc.type | text |