Biased diffusive transport of Brownian particles through irregularly shaped, narrow confining quasi-one-dimensional structures is investigated. The complexity of the higher dimensional diffusive dynamics is reduced by means of the so-called Fick-Jacobs approximation, yielding an effective one-dimensional stochastic dynamics. Accordingly, the elimination of transverse, equilibrated degrees of freedom stemming from geometrical confinements and/or bottlenecks cause entropic potential barriers which the particles have to overcome when moving forward noisily. The applicability and the validity of the reduced kinetic description is tested by comparing the approximation with Brownian dynamics simulations in full configuration space. This non-equilibrium transport in such quasi-one-dimensional irregular structures implies for moderate-to-strong bias a characteristic violation of the Sutherland-Einstein fluctuation-dissipation relation.
The Fick--Jacobs equation has been widely studied, because of its applications in the diffusion and transport of non-interacting particles in narrow channels. It is also known that a modified version of this equation can be used to describe the same system with particles interacting through a hard-core potential. In this work we present a system that can be exactly solved using the Fick--Jacobs equation. The exact results of the particle concentration profile along the channel $n$, the current, $J$, and the mobility, $μ$, of particles as a function of an external force are contrasted with Mont…
We study the translocation of polymers across varying-section channels. Using systematic approximations, we derive a simplified model that reduces the problem of polymer translocation through varying-section channels to that of a point-like particle under the action of an effective potential. Such a model allows us to identify the relevant parameters controlling the polymers dynamics and, in particular, their translocation time. By comparing our analytical results with numerical simulations we show that, under suitable conditions, our model provides reliable predictions of the dynamics of both…
We examine the dispersion of Brownian particles in a symmetric two dimensional channel, this classical problem has been widely studied in the literature using the so called Fick-Jacobs' approximation and its various improvements. Most studies rely on the reduction to an effective one dimensional diffusion equation, here we drive an explicit formula for the diffusion constant which avoids this reduction. Using this formula the effective diffusion constant can be evaluated numerically without resorting to Brownian simulations. In addition a perturbation theory can be developed in $\varepsilon = …
In the last decade the Fick-Jacobs approximation has been exploited to capture the transport across constrictions. Here, we review the derivation of the Fick-Jacobs equation with particular emphasis on its linear response regime. We show that for fore-aft symmetric channels the flux of non-interacting systems is fully captured by its linear response regime. For this case we derive a very simple formula that captures the correct trends and that can be exploited as a simple tool to design experiments or simulations. Finally, we show that higher order corrections in the flux may appear for non-sy…
We show the existence of a resonant behavior of the current of Brownian particles confined in a pulsating channel. The interplay between the periodic oscillations of the shape of the channel and a force applied along its axis leads to an increase of the particle current as a function of the noise level. A regime of current inversion is also observed for particular values of the oscillation frequency and the applied force. The model proposed to obtain these new behaviors of the current is based on the Fick-Jacobs equation in which the entropic barrier and the effective diffusion coefficient dep…
Using our generalized Fick-Jacobs approach [Martens et al., PRL 110, 010601 (2013); Martens et al., Eur. Phys. J. Spec. Topics 222, 2453-2463 (2013)] and extensive Brownian dynamics simulations, we study particle transport through three-dimensional periodic channels of different height. Directed motion is caused by the interplay of constant bias acting along the channel axis and a pressure-driven flow. The tremendous change of the flow profile shape in channel direction with the channel height is reflected in a crucial dependence of the mean particle velocity and the effective diffusion coeffi…
The effective diffusivity of Brownian tracer particles confined in periodic micro-channels is smaller than the microscopic diffusivity due to entropic trapping. Here, we study diffusion in two-dimensional periodic channels whose cross-section presents singular points, such as abrupt changes of radius or the presence of thin walls, with openings, delimiting periodic compartments composing the channel. Dispersion in such systems is analyzed using the Fick-Jacobs' approximation. This approximation assumes a much faster equilibration in the lateral than in the axial direction, along which the disp…
We revisit the classic problem of the effective diffusion constant of a Brownian particle in a square lattice of reflecting impenetrable hard disks. This diffusion constant is also related to the effective conductivity of non-conducting and infinitely conductive disks in the same geometry. We show how a recently derived Green's function for the periodic lattice can be exploited to derive a series expansion of the diffusion constant in terms of the disk's volume fraction $φ$. Secondly we propose a variant of the Fick-Jacobs approximation to study the large volume fraction limit. This combinatio…
We discuss two-dimensional diffusion of a Brownian particle confined to a periodic asymmetric channel with soft walls modeled by a parabolic potential. In the channel, the particle experiences different noise intensities, or temperatures, in the transversal and longitudinal directions. The model is inspired by the famous Feynman's ratchet and pawl. Using ideas of the Fick-Jacobs approximation we derive the mean velocity of the particle, the effective diffusion coefficient, and the stationary probability density in the potential unit cell. The derived results are exact for small channel width. …
In this paper we investigate the diffusion of particles inside a chain of asymmetric cavities. We are considering particles that interact through a hard--core potential and are driven by an external force. We show that the difference in the current when the force is applied to the left and to the right strongly depends on the concentration inside the cavity. We found that, when the concentration is high enough, the hard--core interaction vanishes and inverts the asymmetric effect of the cavity. We also introduce a new equation, a modification to the Fick--Jacobs equation, to describe this syst…
We study the diffusive behavior of biased Brownian particles in a two dimensional confined geometry filled with the freezing obstacles. The transport properties of these particles are investigated for various values of the obstacles density $η$ and the scaling parameter $f$, which is the ratio of work done to the particles to available thermal energy. We show that, when the thermal fluctuations dominate over the external force, i.e., small $f$ regime, particles get trapped in the given environment when the system percolates at the critical obstacles density $η_c \approx 1.2$. However, as $f$ i…
