Narrow escape and leakage of Brownian particles.pdf

a r X i v : 0 8 0 8 .2 2 8 8 v 1 [ m a t h - p h ] 1 7 A u g 2 0 0 8 APS/123-PRE Narrow escape and leakage of Brownian particles A. Singer? Department of Mathematics and PACM, Princeton University, Fine Hall, Washington Road, Princeton NJ 08544-1000 USA Z. Schuss? Department of Mathematics, Tel-Aviv University, Tel-Aviv 69978, Israel D. Holcman? Department of Mathematics, Weizmann Institute of Science, Rehovot 76100, Israel and Visiting address: De?partement de Mathe?matiques et de Biologie, Ecole Normale Supe?rieure, 46 rue d’Ulm 75005 Paris, France. (Dated: August 18, 2008) Abstract Questions of flux regulation in biological cells raise renewed interest in the narrow escape problem. The often inadequate expansions of the narrow escape time are due to a not so well known fact that the boundary singularity of Green’s function for Poisson’s equation with Neumann and mixed Dirichlet-Neumann boundary conditions in three-dimensions contains a logarithmic singularity. Using this fact, we find the second term in the expansion of the narrow escape time and in the expansion of the principal eigenvalue of the Laplace equation with mixed Dirichlet-Neumann boundary conditions, with small Dirichlet and large Neumann parts. We also find the leakage flux of Brownian particles that diffuse from a source to an absorbing target on a reflecting boundary of a domain, if a small perforation is made in the reflecting boundary. PACS numbers: 05.40.-Jc., 87.10.-e ?Electronic address: amits@ ?Electronic address: schuss@post.tau.ac.il ?Electronic address: david.holcman@weizmann.ac.il 1 I. INTRODUCTION The narrow escape problem in diffusion theory, which goes back to Lord Rayleigh [1], is to calculate the mean first passage time, also called the narrow escape time (NET), of a Brownian particle to a small absorbing window on the otherwise reflecting boundary of a bounded domain. The renewed interest in the small hole problem is due to its relevance in molecular biology and biophysics.