Unbiased Diffusion to Escape Complex Geometries: Is Reduction to Effective One-Dimensional Description Adequate to Assess Narrow Escape Times?

Abstract

This study is devoted to unbiased diffusion of point Brownian particles inside a tube of varying cross-section (see Figure 1). An expression for the mean survival time, , of the particles inside the tube is obtained in terms of the bulk diffusion constant, D 0 and the system’s geometrical parameters, namely, the tube’s axial semi-length, L , the minor radius, , and the slope of the tube’s wall, . Our expression for correctly retrieves the limit behavior of the system under several conditions. We ran Monte Carlo numerical simulations to compute the mean survival time by averaging the survival time of 5 × 10 4 trajectories, with time step t = 10 - 6 , D 0 = 1, and L = 1. The simulations show good agreement with our model. When the geometrical parameters of this system are varied while keeping constant the tube’s enclosed volume, it resembles the problems of Narrow Escape Time ( J. Chem. Phys . 116(22), 9574 (2007)). A previous study on the use of the reduction to effective one-dimension technique ( J. Mod. Phys . 2, 284 (2011)) in complex geometries has shown excellent agreement between the theoretical model and numerical simulations. However, in this particular system, the general assumptions of the Hill problem are seemingly inapplicable. The expression obtained shows good agreement with our simulations when 0 ≤ ≤ 1, but fails when grows larger. On the other hand, some errors are found when 0, but the expression holds reasonably well for a broad range of values of . These comparisons between simulations and theoretical predictions, and the expressions obtained for , are the main results of this work.