Journal of Modern Physics, 2011, 2, 284-288 doi:10.4236/jmp.2011.24037 Published Online April 2011 (http://www.SciRP.org/journal/jmp) Copyright © 2011 SciRes. JMP Unbiased Diffusion to Escape through Small Windows: Assessing the Applicability of the Reduction to Effective One-Dimension Description in a Spherical Cavity Marco-Vinicio Vázquez 1 , Leonardo Dagdug 1,2 1 Departamento de Física, Universidad Autónoma Metropolitana-Iztapalapa, México 2 Mathematical and Statistical Computing Laboratory, Division of Computational Bioscience, Center for Information Technology, National Institutes of Health, Bethesda, USA. E-mail: mvvg@xanum.uam.mx, dll@xanum.uam.mx Received January 21, 2011; revised March 1, 2011; accepted March 6, 2011 Abstract This study is devoted to unbiased motion of a point Brownian particle that escapes from a spherical cavity through a round hole. Effective one-dimensional description in terms of the generalized Fick-Jacobs equation is used to derive a formula which gives the mean first-passage time as a function of the geometric parameters for any value of a, where a is the hole’s radius. This is our main result and is given in Equation (19). This result is a generalization of the Hill’s formula, which is restricted to small values of a. Keywords: Diffusion, Brownian Particle, Fick-Jacobs Equation, Narrow-Escape Time 1. Introduction The first-passage time, namely, the probability that a diffusing particle or a random-walk first reaches a speci- fied site (or set of sites) at a specified time, is known to underlie a wide range of stochastic processes of practical interest [1]. Indeed, chemical and bio-chemical reactions [2,3], animals searching for food [4], the spread of sexu- ally transmitted diseases in a human social network or of viruses through the world wide web [5], and trafficking receptors on biological membranes [6], are often con- trolled by first encounter events [7]. Studying the narrow scape time (NET), the mean time which a Brownian par- ticle spends before to be trapped in an opening window to exit a cavity for the first time, has particular impor- tance. The applications goes from cellular biology to biochemical reactions in cellular micro-domains as den- dritic spines, synapses and micro-vesicles, among others [6,8]. For those cases where the particles first must exit the domain in order to live up to their biological function, the narrow scape time is the limiting quantity and the first step in the modeling of such processes [7]. Experimentally, high-resolution crystallography of bacterial porins and other large channels demonstrates that their pores can be envisaged as tunnels whose cross sections change signiÞcantly along the channel axis. For some of them, variation in cross-section area exceeds an order of magnitude [9,10]. This leads to the so-called entropic walls and barriers in theoretical description of transport through such structures. In addition to biologi- cal systems, diffusion in confined geometries are also important for understanding transport in synthetic nanopores [11-13], transport in zeolites [14], controlled drugs release [15], and nanostructures of complex ge- ometries [16], among others. Theoretically, the transport in systems of varying ge- ometry has been deeply studied in recent years since these systems are ubiquitous in nature and technology [18-23]. Diffusion in two and three dimension, has been formulated as a one dimension problem in terms of the effective one-dimensional concentration of diffusing molecules. If one assumes that the distribution of the solute in any cross section of the tube is uniform as it is at equilibrium, directing the x-axis along the center line of a tube, one can write an approximate one-dimensional effective diffusion equation as           , , = , , cxt cxt DxAx t x x Axt                         (1) where   Dx   = is a position-dependent diffusion coeffi- cient,   2 π A x rx     is the cross section area of the tube of radius   rx , and is the effective   , cxt