Extended Fick-Jacobs equation: Variational approach

Abstract

We derive an extended Fick-Jacobs equation for the diffusion of noninteracting particles in a two- and symmetric three-dimensional channels of varying cross section A(x), using a variational approach. The result is a diffusion differential equation of second order in only one space (longitudinal) coordinate. This equation is tested on the task of calculating the stationary flux through a hyperboloidal tube, and its solution is compared with that of other methods.

Transport of Brownian particles confined to a weakly corrugated channel

“…In this case, the description of diffusive transport in the absence of an external force can be simplified by reducing the dimensionality of the problem via the Fick-Jacobs approximation, in which the motion in the cross-section is transformed into an entropic barrier to longitudinal transport. 1,[4][5][6][7] In the presence of an external field a relatively simple extension of the Fick-Jacobs approximation has been used to calculate the average velocity of Brownian particles. 1,8,9 Alternatively, Laachi et al 10 used the standard long-wave asymptotic perturbation analysis and obtained analogous results for the average velocity to leading order in ⑀.…”

Range of applicability of modified Fick-Jacobs equation in two dimensions

“…During the last two decades, the problem of the derivation of the modified FJ equation has attracted attention of many researchers. [3][4][5][6][7][8][9][10][11][12][13][14] The reason is that quasi-one-dimensional systems of varying geometry play an important role in different processes ranging from controlled drug delivery to entropic transport of different substances in soils and biological tissues. Along with the problem of deriving the modified FJ equation, there are also questions of the range of applicability of this approximate one-dimensional description and the accuracy of the expressions for the effective position-dependent diffusivity obtained by different researchers.…”

Communications: Drift and diffusion in a tube of periodically varying diameter. Driving force induced intermittency

“…[1][2][3][4][5] At F = 0 the major approach to the problem is based on the Fick-Jacobs equation 6 generalized by Zwanzig 7 and Reguera and Rubi; 8 a more sophisticated approach has been developed by Kalinay and Percus. 9 The generalized FickJacobs equation is a one-dimensional Smoluchowski equation that describes diffusion in the entropy potential. For onedimensional Brownian motion in a regular periodic potential it has been shown 10 that ͑i͒ the effective mobility monotonically increases with F from eff ͑0͒ Ͻ 0 to eff ͑ϱ͒ = 0 , where 0 is the particle mobility in the absence of the periodic potential; ͑ii͒ dependence D eff ͑F͒ is nonmonotonic: first it increases from D eff ͑0͒ Ͻ D 0 to its maximum value, which is larger than the particle diffusion coefficient in the absence of the periodic potential D 0 , and then decreases approaching D eff ͑ϱ͒ = D 0 from above as F → ϱ.…”