First-passage, transition path, and looping times in conical varying-width channels: Comparison of analytical and numerical results

This paper deals with transitions of diffusing point particles between the two ends of expanding and narrowing two-dimensional conical channels. The particle trajectory starts from the reflecting boundary and ends as soon as the absorbing boundary is reached for the first time. Any such trajectories can be divided into two segments: the looping segment and the transition path segment. The latter is the last part of the trajectory that leaves the reflecting boundary and goes to the absorbing boundary without returning to the reflecting one. The remaining portion of the trajectory is the looping part, where a number of loops that begin and end at the same reflecting boundary are made without touching the absorbing boundary. Because axial diffusion of a smoothly varying channel can be approximately described as one-dimensional diffusion in the presence of an entropy potential with position-dependent effective diffusivity, we approach the problem in terms of the modified Fick–Jacobs equation. This allows us to derive analytical expressions for mean first-passage time, as well as looping and transition path times. Comparison with results from Brownian dynamics simulations allows us to establish the domain of applicability of the one-dimensional description. We also compare our results with those obtained for three-dimensional conical tubes [A. M. Berezhkovskii, L. Dagdug, and S. M. Bezrukov, J. Chem. Phys. 147, 134104 (2017)].

Abstract licence: CC BY

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