Numerical study assessing the applicability of the reduction to effective one-dimensional description of diffusion in a hemispherical shaped tube

Computational simulations of a Brownian particle diffusing in a hemispherical shaped tube were run to assess the range of applicability of the reduction of the three-dimensional diffusion to an effective one-dimensional description. On a previous work, for a conical tube, Berezhkovskii et al. [18] found that the one-dimensional description using the Fick-Jacobs' equation with a position dependent diffusion coefficients, D(x) (one suggested by Zwanzig [11], and another by Reguera-Rubí [12]), have a restricted range of applicability. Unexpectedly, our results show that applying the Zwanzig's formula one can predict variation of τ in the whole range of a/R in n→w direction, while the Reguera-Rubí's formula fits numerical data in the w→n direction. This is a remarkable result since it is known that Reguera-Rubí's predicts better the mean first-passage time's behavior without regard of direction in other geometries, and this is the main result of this paper.