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.

We study the transport of Brownian particles through a corrugated channel caused by a force field containing curl-free (scalar potential) and divergence-free (vector potential) parts. We develop a generalized Fick-Jacobs approach leading to an effective one-dimensional description involving the potential of mean force. As an application, the interplay of a pressure-driven flow and an oppositely oriented constant bias is considered. We show that for certain parameters, the particle diffusion is significantly suppressed via the property of hydrodynamically enforced entropic particle trapping.

Biased Brownian motion of point-size particles in a three-dimensional tube with smoothly varying crosssection is investigated. In the fashion of our recent work 1 we employ an asymptotic analysis to the stationary probability density in a geometric parameter of the tube geometry. We demonstrate that the leading order term is equivalent to the Fick-Jacobs approximation. Expression for the higher order corrections to the probability density are derived. Using this expansion orders we obtain that in the diffusion dominated regime the average particle current equals the zeroth-order Fick-Jacobs result corrected by a factor including the corrugation of the tube geometry. In particular we demonstrate that this estimate is more accurate for extreme corrugated geometries compared to the common applied method using the spatially dependent diffusion coefficient D(x, f ). The analytic findings are corroborated with the finite element calculation of a sinusoidal-shaped tube. Particle transport in micro-and nanostructured channel structures exhibits peculiar characteristics which differs from other transport phenomena occurring for energetic systems. The theoretical modelling involves Fokker-Planck type dynamics in three dimensions which cannot be solved for arbitrary boundary conditions imposed by the geometrical restrictions. Recently, much effort is drawn on a reduction of the complexity of the problem resulting in the so-called FickJacobs approximation in which (infinitely) fast equilibration in certain spatial directions is assumed. Within the present manuscript we derive a reduction method which (i) corresponds in zeroth order in the expansion parameter, which describes the corrugation of the tube wall, to the celebrated Fick-Jacobs result and (ii) extends the validity of the Fick-Jacobs approximation towards extreme corrugated tube structures.

Abstract. This work deals with the position control of selected patterns in reactiondiffusion systems. Exemplarily, the Schlögl and FitzHugh-Nagumo model are discussed using three different approaches. First, an analytical solution is proposed. Second, the standard optimal control procedure is applied. The third approach extends standard optimal control to so-called sparse optimal control that results in very localized control signals and allows the analysis of second order optimality conditions.

Using the analytical Fick-Jacobs approximation formalism and extensive Brownian dynamics simulations we study particle transport through two-dimensional periodic channels with triangularly shaped walls. Directed motion is caused by the interplay of constant bias acting along the channel axis and a pressure-driven flow. In particular, we analyze the particle mobility and the effective diffusion coefficient. The mechanisms of entropic rectification is revealed in channels with a broken spatial reflection symmetry in presence of hydrodynamically enforced entropic trapping. Due to the combined action of the forcing and the pressure-driven flow field, efficient rectification with a drastically reduced diffusivity is achieved.

We numerically investigate the transport of a suspended overdamped Brownian particle which is driven through a two-dimensional rectangular array of circular obstacles with finite radius. Two limiting cases are considered in detail, namely, when the constant drive is parallel to the principal or the diagonal array axes. This corresponds to studying the Brownian transport in periodic channels with reflecting walls of different topologies. The mobility and diffusivity of the transported particles in such channels are determined as functions of the drive and the array geometric parameters. Prominent transport features, like negative differential mobilities, excess diffusion peaks, and unconventional asymptotic behaviors, are explained in terms of two distinct lengths, the size of single obstacles (trapping length), and the lattice constant of the array (local correlation length). Local correlation effects are further analyzed by continuously rotating the drive between the two limiting orientations.

The problem of front propagation in a three-dimensional channel with spatially varying crosssection is reduced to an equivalent reaction-diffusion-advection equation with boundary-induced advection term. Treating the advection term as a weak perturbation, an equation of motion for the front position is derived. We analyze channels whose cross-sections vary periodically with L along the propagation direction of the front. Taking the Schlögl model as representative example, we calculate analytically the nonlinear dependence of the front velocity on the ratio L/l where l denotes the intrinsic front width. Our analytical results agree well with the results obtained by numerical simulations. In particular, the peculiarity of boundary-induced propagation failure for a finite range of L/l values is predicted by analytical calculations. Lastly, we demonstrate that the front velocity is determined by the suppressed diffusivity of the reactants for L l.

We consider the impact of inertia on biased Brownian motion of point-size particles in a two-dimensional channel with sinusoidally varying width. If the time scales of the problem separate, the adiabatic elimination of the transverse degrees of freedom leads to an effective description for the motion along the channel given by the potential of mean force. The possibility of such description is intimately connected with equipartition. Numerical simulations show that in the presence of external bias the equipartition may break down leading to non-monotonic dependence of mobility on external force and several other interesting effects.