### Abstract

A Brownian particle with diffusion coefficient D is confined to a bounded domain Ω by a reflecting boundary, except for a small absorbing window ∂Ω_{a}. The mean time to absorption diverges as the window shrinks, thus rendering the calculation of the mean escape time a singular perturbation problem. In the three-dimensional case, we construct an asymptotic approximation when the window is an ellipse, assuming the large semi axis a is much smaller than |Ω|^{1/3} (|Ω| is the volume), and show that the mean escape time is Eτ ∼ |Ω|/2π Da K(e), where e is the eccentricity and K(·) 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 ∼ |Ω|/4aD, which was derived by heuristic considerations. For the special case of a spherical domain, we obtain the asymptotic expansion Eτ = |Ω|/4aD[1 + a/R log R/a + O(a/R)]. This result is important in understanding the flow of ions in and out of narrow valves that control a wide range of biological and technological function. If Ω is a two-dimensional bounded Riemannian manifold with metric g and ε = |∂Ω_{a}|_{g} ≪ 1, we show that Eτ = |Ω|_{g}/Dπ[log 1/ε + O(1)]. This result is applicable to diffusion in membrane surfaces.

Original language | English (US) |
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Pages (from-to) | 437-463 |

Number of pages | 27 |

Journal | Journal of Statistical Physics |

Volume | 122 |

Issue number | 3 |

DOIs | |

State | Published - Feb 1 2006 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics

### Keywords

- Brownian motion
- Exit problem
- Singular perturbations

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## Cite this

*Journal of Statistical Physics*,

*122*(3), 437-463. https://doi.org/10.1007/s10955-005-8026-6