caa a22 4500 467892520 CHVBK 20180406152753.0 cr unu---uuuuu 170328e20060201xx s 000 0 eng 10.1007/s10955-005-8026-6 doi (NATIONALLICENCE)springer-10.1007/s10955-005-8026-6 Narrow Escape, Part I [Elektronische Daten] [A. Singer, Z. Schuss, D. Holcman, R. Eisenberg] 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. Springer Science + Business Media, Inc., 2006 Brownian motion nationallicence Exit problem nationallicence Singular perturbations nationallicence Singer A. Department of Mathematics, Yale University, 10, Hillhouse Ave., PO Box 208283, 06520-8283, New Haven, CT, USA aut Schuss Z. Department of Mathematics, Tel-Aviv University, 69978, Tel-Aviv, Israel aut Holcman D. Department of Mathematics, Weizmann Institute of Science, 76100, Rehovot, Israel aut Eisenberg R. Department of Molecular Biophysics and Physiology, Rush Medical Center, 1750 Harrison St., 60612, Chicago, IL, USA aut Journal of Statistical Physics Kluwer Academic Publishers-Plenum Publishers 122/3(2006-02-01), 437-463 0022-4715 122:3<437 2006 122 10955 https://doi.org/10.1007/s10955-005-8026-6 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s10955-005-8026-6 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Singer A. Department of Mathematics, Yale University, 10, Hillhouse Ave., PO Box 208283, 06520-8283, New Haven, CT, USA aut NATIONALLICENCE 700 1- Schuss Z. Department of Mathematics, Tel-Aviv University, 69978, Tel-Aviv, Israel aut NATIONALLICENCE 700 1- Holcman D. Department of Mathematics, Weizmann Institute of Science, 76100, Rehovot, Israel aut NATIONALLICENCE 700 1- Eisenberg R. Department of Molecular Biophysics and Physiology, Rush Medical Center, 1750 Harrison St., 60612, Chicago, IL, USA aut NATIONALLICENCE 773 0- Journal of Statistical Physics Kluwer Academic Publishers-Plenum Publishers 122/3(2006-02-01), 437-463 0022-4715 122:3<437 2006 122 10955 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer