Entropic particle transport: Higher-order corrections to the Fick-Jacobs diffusion equation

Abstract

Transport of point-size Brownian particles under the influence of a constant and uniform force field through a planar three-dimensional channel with smoothly varying, axis-symmetric periodic side walls is investigated. Here we employ an asymptotic analysis in the ratio between the difference of the widest and the most narrow constriction divided through the period length of the channel geometry. We demonstrate that the leading-order term is equivalent to the Fick-Jacobs approximation. By use of the higher-order corrections to the probability density we show that in the diffusion-dominated regime the average transport velocity is obtained as the product of the zeroth-order Fick-Jacobs result and the expectation value of the spatially dependent diffusion coefficient D(x), which substitutes the constant diffusion coefficient in the common Fick-Jacobs equation. The analytic findings are corroborated with the precise numerical results of a finite element calculation of the Smoluchowski diffusive particle dynamics occurring in a reflection symmetric sinusoidal-shaped channel.

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

“…An alternative derivation of the second formula is given by Martens et al 10 Recently, Dagdug and co-authors proposed a new approach to the reduction of axial diffusion of point particles in two-dimensional channels to the effective one-dimensional description, which allows them to treat channels of arbitrary shapes. 13 The key idea of the approach is to perform the reduction in a curvilinear coordinate system chosen so that the channel boundaries are straight lines.…”

“…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.…”

Non-monotonous polymer translocation time across corrugated channels: Comparison between Fick-Jacobs approximation and numerical simulations

“…The staring point of our model is the Fick-Jacobs approximation [24][25][26] that has already been characterized [27][28][29][30][31][32][33] and exploited for diverse systems ranging from particle splitters [34,35], cooperative rectification [36][37][38] diffusion through porous media [39,40], electro-osmotic systems [41][42][43] and entropic stochastic resonance [44,45] just to mention a few cases among others. The Fick-Jacobs approximation allows us to project the convection-diffusion equation of a noninteracting particle, confined in a two-dimensional (2D) or three-dimensional (3D) corrugated channel, onto a one-dimensional (1D) equation in which the particle dynamics is controlled by an effective potential.…”

Wave propagation in spatially modulated tubes

“…The key assumption is that the modulation of the tube's radius is a small quantity compared to the period length L of the modulation and hence we introduce the dimensionless parameter 40,41 . The latter characterizes the deviation of a modulated tube Ω(x) from a tube with constant diameter, i.e.…”