Effective One-dimensional Description Research Articles

Dichotomic ratchet in a two-dimensional corrugated channel.

We consider a particle diffusing in a two-dimensional (2D) channel of varying width h(x). It is driven by a force of constant magnitude f but random orientation there or back along the channel. We derive the effective generalized Fick-Jacobs equation for this system, which describes the dynamics of such a particle in the longitudinal coordinate x. Aside from the effective diffusion coefficient D(x), our mapping also generates an additional effective potential -γ(x) added to the entropic potential -log[h(x)]. It acquires an increasing or decreasing component in asymmetric periodic channels, and thus it explains appearance of the ratchet current. We study this effect on a trial example and compare the results of our true 2D theory with a commonly used effective one-dimensional description; the data are verified by the numerical solution of the full 2D problem.

Two-dimensional diffusion biased by a transverse gravitational force in an asymmetric channel: Reduction to an effective one-dimensional description.

We focus on the derivation of a general position-dependent effective diffusion coefficient to describe two-dimensional (2D) diffusion in a narrow and smoothly asymmetric channel of varying width under a transverse gravitational external field, a generalization of the symmetric channel case using the projection method introduced earlier by Kalinay and Percus [P. Kalinay and J. K. Percus, J. Chem. Phys. 122, 204701 (2005)10.1063/1.1899150]. To this end, we project the 2D Smoluchowski equation into an effective one-dimensional generalized Fick-Jacobs equation in the presence of constant force in the transverse direction. The expression for the diffusion coefficient given in Eq.(34) is our main result. This expression is a more general effective diffusion coefficient for narrow 2D channels in the presence of constant transverse force, which contains the well-known previous results for a symmetric channel obtained by Kalinay, as well as the limiting cases when the transverse gravitational external field goes to zero and infinity. Finally, we show that diffusivity can be described by the interpolation formula proposed by Kalinay, D_{0}/[1+(1/4)w^{'2}(x)]^{-η}, where spatial confinement, asymmetry, and the presence of a constant transverse force can be encoded in η, which is a function of channel width (w), channel centerline, and transverse force. The interpolation formula also reduces to well-known previous results, namely, those obtained by Reguera and Rubi [D. Reguera and J. M. Rubi, Phys. Rev. E 64, 061106 (2001)10.1103/PhysRevE.64.061106] and by Kalinay [P. Kalinay, Phys. Rev. E 84, 011118 (2011)10.1103/PhysRevE.84.011118].

Effective diffusivity of a Brownian particle in a two-dimensional periodic channel of abruptly alternating width.

We study diffusion of a Brownian particle in a two-dimensional periodic channel of abruptly alternating width. Our main result is a simple approximate analytical expression for the particle effective diffusivity, which shows how the diffusivity depends on the geometric parameters of the channel: lengths and widths of its wide and narrow segments. The result is obtained in two steps: first, we introduce an approximate one-dimensional description of particle diffusion in the channel, and second, we use this description to derive the expression for the effective diffusivity. While the reduction to the effective one-dimensional description is standard for systems of smoothly varying geometry, such a reduction in the case of abruptly changing geometry requires a new methodology used here, which is based on the boundary homogenization approach to the trapping problem. To test the accuracy of our analytical expression and thus establish the range of its applicability, we compare analytical predictions with the results obtained from Brownian dynamics simulations. The comparison shows excellent agreement between the two, on condition that the length of the wide segment of the channel is equal to or larger than its width.

A Partial Differential Equation for the Mean--Return-Time Phase of Planar Stochastic Oscillators

Stochastic oscillations are ubiquitous in many systems. For deterministic systems, the oscillator's phase has been widely used as an effective one-dimensional description of a higher dimensional dynamics, particularly for driven or coupled systems. Hence, efforts have been made to generalize the phase concept to the stochastic framework. One notion of phase due to Schwabedal and Pikovsky is based on the mean--return-time (MRT) of the oscillator but has so far been described only in terms of a numerical algorithm. Here we develop the boundary condition by which the partial differential equation for the MRT has to be complemented in order to obtain the isochrons (lines of equal phase) of a two-dimensional stochastic oscillator, and rigorously establish the existence and uniqueness of the MRT isochron function (up to an additive constant). We illustrate the method with a number of examples: the stochastic heteroclinic oscillator (which would not oscillate in the absence of noise), the isotropic Stuart--Landau oscillator, the Newby--Schwemmer oscillator, and the Stuart--Landau oscillator with polarized noise. For selected isochrons we confirm by extensive stochastic simulations that the return-time from an isochron to the same isochron (after one complete rotation) is always the mean--first-passage time (irrespective of the initial position on the isochron). Put differently, we verify that Schwabedal and Pikovsky's criterion for an isochron is satisfied. In addition, we discuss how to extend the construction to arbitrary finite dimensions. Our results will enable development of analytical tools to study and compare different notions of phase for stochastic oscillators.

Non-steady-state diffusion in two-dimensional periodic channels

The dynamics of a freely diffusing particle in a two-dimensional channel with cross-sectional area A(x) can be effectively described by a one-dimensional diffusion equation under the action of a potential of mean force U(x)=-k_{B}Tln[A(x)] (where k_{B}T is the thermal energy) in a system with a spatially dependent diffusion coefficient D(x). Several attempts to derive expressions relating D(x) to A(x) and its derivatives have been made, which were based on considering stationary flows in periodic channels. Here we take an alternative approach and consider non-steady-state single-particle diffusion in an open periodic channel. The approach allows us to express D(x) as a series of terms of increasing powers of ε-a parameter associated with the aspect ratio of the channel. When the expansion is truncated at the leading term, we recover the expression suggested by Zwanzig [J. Phys. Chem. 96, 3926 (1992)10.1021/j100189a004] for D(x). Furthermore, comparison of the first few terms in our expansion for D(x) with the one proposed by Kalinay and Percus [Phys. Rev. E 74, 041203 (2006)10.1103/PhysRevE.74.041203] shows that they are consistent with each other. In the limit of long-wavelength channels (ε≪1), the expansion converges rapidly and the leading approximation provides a very accurate description of the two-dimensional dynamics. For short-wavelength channels, the expansion does not converge and the validity of the effective one-dimensional description is questionable.

Effects of curved midline and varying width on the description of the effective diffusivity of Brownian particles

Axial diffusion in channels and tubes of smoothly-varying geometry can be approximately described as one-dimensional diffusion in the entropy potential with a position-dependent effective diffusion coefficient, by means of the modified Fick–Jacobs equation. In this work, we derive analytical expressions for the position-dependent effective diffusivity for two-dimensional asymmetric varying-width channels, and for three-dimensional curved midline tubes, formed by straight walls. To this end, we use a recently developed theoretical framework using the Frenet–Serret moving frame as the coordinate system (2016 J. Chem. Phys. 145 074105). For narrow tubes and channels, an effective one-dimensional description reducing the diffusion equation to a Fick–Jacobs-like equation in general coordinates is used. From this last equation, one can calculate the effective diffusion coefficient applying Neumann boundary conditions.

Engineering bright matter-wave solitons of dipolar condensates

We present a comprehensive analysis of the form and interaction of dipolar bright solitons across the full parameter space afforded by dipolar Bose–Einstein condensates, revealing the rich behavior introduced by the non-local nonlinearity. Working within an effective one-dimensional description, we map out the existence of the soliton solutions and show three collisional regimes: free collisions, bound state formation and soliton fusion. Finally, we examine the solitons in their full three-dimensional form through a variational approach; along with regimes of instability to collapse and runaway expansion, we identify regimes of stability which are accessible to current experiments.

On the description of Brownian particles in confinement on a non-Cartesian coordinates basis.

We developed a theoretical framework to study the diffusion of Brownian point-like particles in bounded geometries in two and three dimensions. We use the Frenet-Serret moving frame as the coordinate system. For narrow tubes and channels, we use an effective one-dimensional description reducing the diffusion equation to a Fick-Jacobs-like equation. From this last equation, we can calculate the effective diffusion coefficient applying Neumann boundary conditions. On one hand, for channels with a straight axis our theoretical approximation for the effective coefficient does coincide with the reported in the literature [D. Reguera and J. M. Rubí, Phys. Rev. E 64, 061106 (2001) and P. Kalinay and J. K. Percus, ibid. 74, 041203 (2006)]. On the other hand, for tubes with a straight axis and circular cross-section our analytical expression does not coincide with the reported by Rubí and Reguera and by Kalinay and Percus, although it is practically identical.

Integral formula for the effective diffusion coefficient in two-dimensional channels.

The effective one-dimensional description of diffusion in two-dimensional channels of varying cross section is revisited. The effective diffusion coefficient D(x), extending Fick-Jacobs equation, depending on the longitudinal coordinate x, is derived here without use of scaling of the transverse coordinates. The result of the presented method is an integral formula for D(x), calculating its value at x as an integral of contributions from the neighboring positions x^{'} depending on h(x^{'}), a function shaping the channel. Unlike the standard formulas based on the scaling, the new proposed formula also describes D(x) correctly near the cusps, or in wider channels.

Smoluchowski Equation Approach in Channel-Facilitated Transport Problems: Counter-Intuitive Analytical Results and Supporting Experiments

When transport occurs in quasi one-dimensional confining structures, it is intuitively appealing to introduce an effective one-dimensional description, where the spatial constraints are accounted for by entropy potentials. In equilibrium, the effect of entropic contribution is not different from that of the bona fide potential of mean force. However, for transport problems the situation can be different. We analyze the problem within the framework of a continuous diffusion model using the Smoluchowski equation (for a recent recapitulation see Berezhkovskii and Bezrukov. On the applicability of entropy potentials in transport problems. European Physical Journal - Special Topics, 2014, 223:3063-3077). Several of our analytical findings are quite counter-intuitive. Three of the examples to be discussed are: (i) “Sticking” to the channel slows down translocation (a particle spends more time in the channel) but increases the particle flux; (ii) “Up-hill” and “downhill” particle translocation times (and their distributions) are identical; (iii) An optimal channel should exhibit the most pronounced binding on the side that is opposite to the side of the oncoming particles. Some supporting experiments and applications of the theory to particular channels will be discussed.

Entropic transport without external force in confined channel with oscillatory boundary

The dynamics of point-like Brownian particles in a periodic confined channel with oscillating boundaries has been studied. Directional transport (DT) behavior, characterized by net displacement along the horizontal direction, is observed even without external force which is necessary for the conventional DT where the boundaries are static. For typical parameter values, the average velocity Vt of DT reaches a maximum with the variation of the noise intensity D, being alike to the phenomenon of stochastic resonance. Interestingly, we find that Vt shows nontrivial dependences on the particle gravity G depending on the noise level. When the noise is large, Vt increases monotonically with G indicating that heavier particle moves faster, while for small noise, Vt shows a bell-shape dependence on G, suggesting that a particle with an intermediate weight may move the fastest. Such results were not observed for DT in a channel with static boundaries. To understand these findings, we have adopted an effective one-dimensional coarsening description, which facilitates us to introduce an effective entropic force along the horizontal direction. The average force is apparently nonzero due to the oscillatory boundary, hence leading to the net transport, and it shows similar dependences as Vt on the noise intensity D and particle gravity G. The dependences of the DT behavior on other parameters describing the oscillatory channel have also been investigated, showing that DT is more pronounced for larger oscillation amplitude and frequency, and asymmetric geometry within a channel period and phase difference between neighboring periods are both necessary for the occurrence of DT.

Order induced by dilution in pyrochlore XY antiferromagnets

XY pyrochlore antiferromagnets are well-known to exhibit order-by-disorder through both quantum and thermal selection. In this paper we consider the effect of substituting non-magnetic ions onto the magnetic sites in a pyrochlore XY model with generally anisotropic exchange tuned by a single parameter $J^{\pm\pm}/J^\pm$. The physics is controlled by two points in this space of parameters $J^{\pm\pm}/J^\pm=\pm 2$ at which there are line modes in the ground state and hence an $O(L^2)$ ground state degeneracy intermediate between that of a conventional magnet and a Coulomb phase. At each of these points, single vacancies seed pairs of line defects. Two line defects carrying incompatible spin configurations from different vacancies can cross leading to an effective one-dimensional description of the resulting spin texture. In the thermodynamic limit at finite density, we find that dilution selects a state "opposite" to the state selected by thermal and quantum disorder which is understood from the single vacancy limit. The latter finding hints at the possibility that Er$_{2-x}$Y$_x$Ti$_2$O$_7$ for small $x$ exhibits a second phase transition within the thermally selected $\psi_2$ state into a $\psi_3$ state selected by the quenched disorder.

Molecular branch of a small highly elongated Fermi gas with an impurity: Full three-dimensional versus effective one-dimensional description

We consider an impurity immersed in a small Fermi gas under highly elongated harmonic confinement. The impurity interacts with the atoms of the Fermi gas through an isotropic short-range potential with three-dimensional free-space $s$-wave scattering length ${a}_{3\text{D}}$. We investigate the energies of the molecular branch, i.e., the energies of the state that corresponds to a gas consisting of a weakly bound diatomic molecule and ``unpaired'' atoms, as a function of the $s$-wave scattering length ${a}_{3\text{D}}$ and the ratio $\ensuremath{\eta}$ between the angular trapping frequencies in the tight and weak confinement directions. The energies obtained from our three-dimensional description that accounts for the dynamics in the weak and tight confinement directions are compared with those obtained within an effective one-dimensional framework, which accounts for the dynamics in the tight confinement direction via a renormalized one-dimensional coupling constant. Our theoretical results are related to recent experimental measurements.

Unbiased Diffusion to Escape Complex Geometries: Is Reduction to Effective One-Dimensional Description Adequate to Assess Narrow Escape Times?

This study is devoted to unbiased diffusion of point Brownian particles inside a tube of varying cross-section (see Figure 1). An expression for the mean survival time, , of the particles inside the tube is obtained in terms of the bulk diffusion constant, D0 and the system’s geometrical parameters, namely, the tube’s axial semi-length, L, the minor radius, , and the slope of the tube’s wall, . Our expression for correctly retrieves the limit behavior of the system under several conditions. We ran Monte Carlo numerical simulations to compute the mean survival time by averaging the survival time of 5 × 104 trajectories, with time step t = 10-6, D0 = 1, and L = 1. The simulations show good agreement with our model. When the geometrical parameters of this system are varied while keeping constant the tube’s enclosed volume, it resembles the problems of Narrow Escape Time (J. Chem. Phys. 116(22), 9574 (2007)). A previous study on the use of the reduction to effective one-dimension technique (J. Mod. Phys. 2, 284 (2011)) in complex geometries has shown excellent agreement between the theoretical model and numerical simulations. However, in this particular system, the general assumptions of the Hill problem are seemingly inapplicable. The expression obtained shows good agreement with our simulations when 0 ≤ ≤ 1, but fails when grows larger. On the other hand, some errors are found when 0, but the expression holds reasonably well for a broad range of values of . These comparisons between simulations and theoretical predictions, and the expressions obtained for , are the main results of this work.

Hydrodynamically Enforced Entropic Trapping of Brownian Particles

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.

Diffusion in two-dimensional conical varying width channels: Comparison of analytical and numerical results

This study is devoted to the unbiased motion of a point-like brownian particle in two-dimensional tilted asymmetric channels of varying width formed by straight walls. An effective one-dimensional description in terms of the generalized Fick-Jacobs equation is used to derive formulas that yield the particle's effective diffusion coefficient as a function of the geometric parameters of the channel. To such end, we use the formulas obtained by Bradley [Phys. Rev. E 80, 061142 (2009)] and by Dagdug and Pineda [J. Chem. Phys. 137, 024107 (2012)] to study two-dimensional diffusion in narrow and smoothly asymmetric channels of varying width. Comparison with brownian dynamics simulation results allows us to establish the domain of applicability of both the one-dimensional description and the effective diffusion coefficient formulas.

Effective one-dimensional description of confined diffusion biased by a transverse gravitational force

Diffusion of pointlike noninteracting particles in a two-dimensional channel of varying cross section is considered. The particles are biased by a constant force in the transverse direction. A recurrence mapping procedure is applied, which enables the derivation of an effective one-dimensional (1D) evolution equation that governs the 1D density of the particles in the channel. In the limit of stationary flow, an extended Fick-Jacobs equation is reached, which is corrected by an effective diffusion coefficient D(x) that depends on the longitudinal coordinate x. The result is an approximate formula for D(x) that also involves the influence of the transverse force. The calculations are verified by the stationary diffusion in a linear cone, which is exactly solvable.

Unbiased Diffusion to Escape through Small Windows: Assessing the Applicability of the Reduction to Effective One-Dimension Description in a Spherical Cavity

This study is devoted to unbiased motion of a point Brownian particle that escapes from a spherical cavity through a round hole. Effective one-dimensional description in terms of the generalized Fick-Jacobs equation is used to derive a formula which gives the mean first-passage time as a function of the geometric parameters for any value of a, where a is the hole’s radius. This is our main result and is given in equation (19). This result is a generalization of the Hill’s formula, which is restricted to small values of a.

Unbiased diffusion in tubes with corrugated walls.

This study is devoted to unbiased motion of a point Brownian particle in a tube with corrugated walls made of conical sections of a varying length. Effective one-dimensional description in terms of the generalized Fick-Jacobs equation is used to derive a formula which gives the effective diffusion coefficient of the particle as a function of the geometric parameters of the tube. Comparison with the results of Brownian dynamics simulations allows us to establish the domain of applicability of both the one-dimensional description and the formula for the effective diffusion coefficient.

Non-equilibrium in soil hydraulic modelling

A notorious problem in modelling water dynamics within the unsaturated zone is preferential flow and temporal non-equilibrium between water content and water potential. These phenomena originate from the ubiquitous spatial heterogeneity of soil hydraulic properties and substantially hampers the prediction of solute transport from the soil surface towards groundwater. The characteristic scale of these processes is the soil profile or ‘pedon scale’ where modelling of water dynamics is typically based on the well established Richards equation which has been augmented by various conceptual approaches to represent preferential flow phenomena. In this paper we investigate through numerical case studies hydraulic non-equilibrium during infiltration of water into relatively dry, heterogeneous soil profiles, which is the most prominent scenario where non-equilibrium conditions affect the overall dynamics of water and solutes. We start from well-defined local heterogeneities in a two-dimensional domain and we analyze the dynamics of water content and water potential at the models full spatial resolution. Based on our numerical results we approach the problem of how to represent the heterogeneous setting by an effective one-dimensional description. For practical purposes of large scale applications, such one-dimensional schemes are highly required. As a prerequisite, soil types need to be represented by a reduced set of effective hydraulic properties. The main challenge is to preserve preferential flow phenomena in such simplified one-dimensional models. We demonstrate that classical approaches of volume averaging do not reflect non-equilibrium effects. From the different concepts for representing hydraulic non-equilibrium we focus on an approach proposed by Ross and Smettem [Ross, P., Smettem, K.R.J., 2000. A simple treatment of physical non-equilibrium water flow in soil. Soil Sci. Soc. Am. J. 64, 1926–1930] to decouple water content and water potential. This concept is corroborated by our numerical results and we discuss the possibility to link non-equilibrium dynamics and the related model parameters to observable structural properties of the material.