Narrow escape, part I

A Brownian particle with diffusion coefficient D is confined to a bounded domain Omega by a reflecting boundary, except for a small absorbing window. partial derivative 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 vertical bar Omega vertical bar(1/3) (vertical bar Omega vertical bar is the volume), and show that the mean escape is time is E tau similar to vertical bar Omega vertical bar/2 pi 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 E tau similar to vertical bar Omega vertical bar/4aD, which was derived by heuristic considerations. For the special case of a shperical domain, we obtain the asymptotic expansion E tau = vertical bar Omega vertical bar/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 Omega is a two-dimensional bounded Riemannian manifold with metric g and e = vertical bar partial derivative Omega(a)vertical bar(g)/vertical bar Omega vertical bar(g) << 1, we show that E tau = vertical bar Omega vertical bar g/D pi[log 1/epsilon + O(1)]. This result is applicable to diffusion in membrane surfaces.