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, D0 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 × 104 trajectories, with time step 
t = 10-6, D0 = 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.