Narrow Escape, Part I

A Brownian particle with diffusion coefficient D is confined to a bounded domain by a reflecting boundary, except for a small absorbing window ∂ 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 | |1/3 (| | is the volume), and show that the mean escape time is Eτ ∼ | | 2πDa 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 ∼ | | 4aD , which was derived by heuristic considerations. For the special case of a spherical domain, we obtain the asymptotic expansion Eτ = | | 4aD [1 + a R log 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 is a two-dimensional bounded Riemannian manifold with metric g and ε = |∂ a |g/| |g 1, we show that Eτ = | |g Dπ [log ε + O(1)]. This result is applicable to diffusion in membrane surfaces.