Article

Journal of Statistical Physics

, Volume 122, Issue 3, pp 437-463

Narrow Escape, Part I

  • A. SingerAffiliated withDepartment of Mathematics, Yale University Email author 
  • , Z. SchussAffiliated withDepartment of Mathematics, Tel-Aviv University
  • , D. HolcmanAffiliated withDepartment of Mathematics, Weizmann Institute of Science
  • , R. S. EisenbergAffiliated withDepartment of Molecular Biophysics and Physiology, Rush Medical Center

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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 \(\partial\Omega_a\). 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 \(|\Omega|^{1/3}\) (\(|\Omega|\) is the volume), and show that the mean escape time is \(E\tau\sim{\frac{|\Omega|}{2\pi Da}} K(e)\), where e is the eccentricity 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 \(E\tau\sim{\frac{|\Omega|}{4aD}}\), which was derived by heuristic considerations. For the special case of a spherical domain, we obtain the asymptotic expansion \(E\tau={\frac{|\Omega|}{4aD}} [1+\frac{a}{R} \log \frac{R}{a} + O(\frac{a}{R})]\). 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 \(\varepsilon=|\partial\Omega_a|_g/|\Omega|_g\ll1\), we show that \(E\tau ={\frac{|\Omega|_g}{D\pi}}[\log{\frac{1}{\varepsilon}}+O(1)]\). This result is applicable to diffusion in membrane surfaces.

Key Words

Brownian motion Exit problem Singular perturbations