Diffusion with two resetting points.

We study the problem of a target search by a Brownian particle subject to stochastic resetting to a pair of sites. The mean search time is minimized by an optimal resetting rate which does not vary smoothly, in contrast with the well-known single site case, but exhibits a discontinuous transition as the position of one resetting site is varied while keeping the initial position of the particle fixed, or vice versa. The discontinuity vanishes at a "liquid-gas" critical point in position space. This critical point exists provided that the relative weight m of the further site is comprised in the interval [2.9028...,8.5603...]. When the initial position is a random variable that follows the resetting point distribution, a discontinuous transition also exists for the optimal rate as the distance between the resetting points is varied, provided that m exceeds the critical value m_{c}=6.6008.... This setup can be mapped onto an intermittent search problem with switching diffusion coefficients and represents a minimal model for the study of distributed resetting.

Diffusion Resistance of Segmented Channels.

The geometry of ion and metabolite channels in membranes of biological cells and organelles is usually far from that of a regular right cylinder. Rather, the channels have complex shapes that are characterized by the so-called vestibules and constriction zones which play roles of molecular filters determining the channel selectivity. In the present paper we discuss several channel structures with varying radius that approximate most of the cases found in nature, specifically, channels of smoothly varying radius and channels composed of multiple cylindrical sections of different lengths and radii including channels containing very thin circular constrictions. We consider diffusive transport of electrically neutral molecules driven by the concentration gradient and derive analytical expressions for the diffusion resistance - the integral parameter that describes steady-state transport properties of membrane channels.

Permeability and diffusion resistance of porous membranes: Analytical theory and its numerical test.

This study is devoted to the transport of neutral solutes through porous flat membranes, driven by the solute concentration difference in the reservoirs separated by the membrane. Transport occurs through membrane channels, which are assumed to be non-overlapping, identical, straight cylindrical pores connecting the reservoirs. The key quantities characterizing transport are membrane permeability and its diffusion resistance. Such transport problems arising in very different contexts, ranging from plant physiology and cell biology to chemical engineering, have been studied for more than a century. Nevertheless, an expression giving the permeability for a membrane of arbitrary thickness at arbitrary surface densities of the channel openings is still unknown. Here, we fill in the gap and derive such an expression. Since this expression is approximate, we compare its predictions with the permeability obtained from Brownian dynamics simulations and find good agreement between the two.

Fick-Jacobs description and first passage dynamics for diffusion in a channel under stochastic resetting.

The transport of particles through channels is of paramount importance in physics, chemistry, and surface science due to its broad real world applications. Much insight can be gained by observing the transition paths of a particle through a channel and collecting statistics on the lifetimes in the channel or the escape probabilities from the channel. In this paper, we consider the diffusive transport through a narrow conical channel of a Brownian particle subject to intermittent dynamics, namely, stochastic resetting. As such, resetting brings the particle back to a desired location from where it resumes its diffusive phase. To this end, we extend the Fick-Jacobs theory of channel-facilitated diffusive transport to resetting-induced transport. Exact expressions for the conditional mean first passage times, escape probabilities, and the total average lifetime in the channel are obtained, and their behavior as a function of the resetting rate is highlighted. It is shown that resetting can expedite the transport through the channel-rigorous constraints for such conditions are then illustrated. Furthermore, we observe that a carefully chosen resetting rate can render the average lifetime of the particle inside the channel minimal. Interestingly, the optimal rate undergoes continuous and discontinuous transitions as some relevant system parameters are varied. The validity of our one-dimensional analysis and the corresponding theoretical predictions is supported by three-dimensional Brownian dynamics simulations. We thus believe that resetting can be useful to facilitate particle transport across biological membranes-a phenomenon that can spearhead further theoretical and experimental studies.

Trapping of single diffusing particles by a circular disk on a reflecting flat surface. Absorbing hemisphere approximation.

Recent progress in biophysics (for example, in studies of chemical sensing and spatiotemporal cell-signaling) poses new challenges to statistical theory of trapping of single diffusing particles. Here we deal with one of them, namely, trapping kinetics of single particles diffusing in a half-space bounded by a reflecting flat surface containing an absorbing circular disk. This trapping problem is essentially two-dimensional and the question of the angular dependence of the kinetics on the particle starting point is highly nontrivial. We propose an approximate approach to the problem that replaces the absorbing disk by an absorbing hemisphere of a properly chosen radius. This replacement makes the problem angular-independent and essentially one-dimensional. After the replacement one can find an exact solution for the particle propagator (Green's function) that allows one to completely characterize the kinetics. Extensive testing of the theoretical predictions based on the absorbing hemisphere approximation against three-dimensional Brownian dynamics simulations shows excellent agreement between the analytical and simulation results when the particle starts sufficiently far away from the disk. Our approach fails and the angular dependence of the kinetics is important when the distance of the particle starting point from the disk center is comparable with the disk radius. However, even when the initial distance is only two disk radii, the maximum relative error of the theoretical predictions is about 10%. The relative error rapidly decreases as the initial distance increases.

Blocker Effect on Diffusion Resistance of a Membrane Channel: Dependence on the Blocker Geometry.

Being motivated by recent progress in nanopore sensing, we develop a theory of the effect of large analytes, or blockers, trapped within the nanopore confines, on diffusion flow of small solutes. The focus is on the nanopore diffusion resistance which is the ratio of the solute concentration difference in the reservoirs connected by the nanopore to the solute flux driven by this difference. Analytical expressions for the diffusion resistance are derived for a cylindrically symmetric blocker whose axis coincides with the axis of a cylindrical nanopore in two limiting cases where the blocker radius changes either smoothly or abruptly. Comparison of our theoretical predictions with the results obtained from Brownian dynamics simulations shows good agreement between the two.

First-passage times in conical varying-width channels biased by a transverse gravitational force: Comparison of analytical and numerical results.

We study the crossing time statistic of diffusing point particles between the two ends of expanding and narrowing two-dimensional conical channels under a transverse external gravitational field. The theoretical expression for the mean first-passage time for such a system is derived under the assumption that the axial diffusion in a two-dimensional channel of smoothly varying geometry can be approximately described as a one-dimensional diffusion in an entropic potential with position-dependent effective diffusivity in terms of the modified Fick-Jacobs equation. We analyze the channel crossing dynamics in terms of the mean first-passage time, combining our analytical results with extensive two-dimensional Brownian dynamics simulations, allowing us to find the range of applicability of the one-dimensional approximation. We find that the effective particle diffusivity decreases with increasing amplitude of the external potential. Remarkably, the mean first-passage time for crossing the channel is shown to assume a minimum at finite values of the potential amplitude.

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.

Trapping of particles diffusing in two dimensions by a hidden binding site.

We study trapping of particles diffusing in a two-dimensional rectangular chamber by a binding site located at the end of a rectangular sleeve. To reach the site a particle first has to enter the sleeve. After that it has two options: to come back to the chamber or to diffuse to the site where it is trapped. The main result of the present work is a simple expression for the mean particle lifetime as a function of its starting position and geometric parameters of the system. This expression is obtained by an approximate reduction of the initial two-dimensional problem to the effective one-dimensional one which can be solved with relative ease. Our analytical predictions are tested against the results obtained from Brownian dynamics simulations. The test shows excellent agreement between the two for a wide range of the geometric parameters of the system.

Evaluating diffusion resistance of a constriction in a membrane channel by the method of boundary homogenization.

In this paper we analyze diffusive transport of noninteracting electrically uncharged solute molecules through a cylindrical membrane channel with a constriction located in the middle of the channel. The constriction is modeled by an infinitely thin partition with a circular hole in its center. The focus is on how the presence of the partition slows down the transport governed by the difference in the solute concentrations in the two reservoirs separated by the membrane. It is assumed that the solutions in both reservoirs are well stirred. To quantify the effect of the constriction we use the notion of diffusion resistance defined as the ratio of the concentration difference to the steady-state flux. We show that when the channel length exceeds its radius, the diffusion resistance is the sum of the diffusion resistance of the cylindrical channel without a partition and an additional diffusion resistance due to the presence of the partition. We derive an expression for the additional diffusion resistance as a function of the tube radius and that of the hole in the partition. The derivation involves the replacement of the nonpermeable partition with the hole by an effective uniform semipermeable partition with a properly chosen permeability. Such a replacement makes it possible to reduce the initial three-dimensional diffusion problem to a one-dimensional one that can be easily solved. To determine the permeability of the effective partition, we take advantage of the results found earlier for trapping of diffusing particles by inhomogeneous surfaces, which were obtained with the method of boundary homogenization. Brownian dynamics simulations are used to corroborate our approximate analytical results and to establish the range of their applicability.

Peculiarities of the Mean Transition Path Time Dependence on the Barrier Height in Entropy Potentials.

A transition path is a part of a one-dimensional trajectory of a diffusing particle, which starts from point a and is terminated as soon as it comes to point b for the first time. It is the trajectory's final segment that leaves point a and goes to point b without returning to point a. The duration of this segment is called transition path time or, alternatively, direct transit time. We study the mean transition path time in monotonically increasing entropy potentials of the narrowing cones in spaces of different dimensions. We find that this time, normalized to its value in the absence of the potential, monotonically increases with the barrier height for the entropy potential of a narrowing two-dimensional cone, is independent of the barrier height for a narrowing three-dimensional cone, and monotonically decreases with the barrier height for narrowing cones in spaces of higher dimensions. Moreover, we show that as the barrier height tends to infinity, the normalized mean transition path time approaches its universal limiting value n/3, where n = 2, 3, 4, ... is the space dimension. This is in sharp contrast to the asymptotic behavior of this quantity in the case of a linear potential of mean force, for which it approaches zero in this limit.

Biased diffusion in periodic potentials: Three types of force dependence of effective diffusivity and generalized Lifson-Jackson formula.

Diffusive transport of particles in a biased periodic potential is characterized by the effective drift velocity and diffusivity, which are functions of the biasing force. We derive a simple exact expression for the effective diffusivity and use it to show that the force dependence of this quantity may be a nonmonotonic function with a maximum [as shown in the work of Reimann et al. Phys. Rev. Lett. 87, 010602 (2001) for periodic sinusoidal potential] or with a minimum, or a monotonic function. The shape of the dependence is determined by the shape of the periodic potential.

Steady-state flux of diffusing particles to a rough boundary formed by absorbing spikes periodically protruding from a reflecting base.

We study steady-state flux of particles diffusing on a flat surface and trapped by absorbing spikes of arbitrary length periodically protruding from a reflecting base. It is assumed that the particle concentration, far from this comblike boundary, is kept constant. To find the flux, we use a boundary regularization approach that replaces the initial highly rough and heterogeneous boundary by an effective boundary which is smooth and uniform. After such a replacement, the two-dimensional diffusion problem becomes essentially one-dimensional, and the steady-state flux can be readily found. Our main results are simple analytical expressions determining the position of the smooth effective boundary and its uniform trapping rate as functions of the spike length and interspike distance. It is shown that the steady-state flux to the effective boundary is identical to its counterpart to the initial boundary at large distances from this boundary. Our analytical results are corroborated by Brownian dynamics simulations.

Exact Solutions for Distributions of First-Passage, Direct-Transit, and Looping Times in Symmetric Cusp Potential Barriers and Wells.

For a particle diffusing in one dimension, the distribution of its first-passage time from point a to point b is determined by the durations of the particle trajectories that start from point a and are terminated as soon as they touch point b for the first time. Any such trajectory consists of looping and direct-transit segments. The latter is the final part of the trajectory that leaves point a and goes to point b without returning to point a. The rest of the trajectory is the looping segment that makes numerous loops which begin and end at the same point a without touching point b. In this article we discuss general relations between the first-passage time distribution and those for the durations of the two segments. These general relations allow us to find exact solutions for the Laplace transforms of the distributions of the first-passage, direct-transit, and looping times for transitions between two points separated by a symmetric cusp potential barrier or well of arbitrary height and depth, respectively. The obtained Laplace transforms are inverted numerically, leading to nontrivial dependences of the resulting distributions on the barrier height and the well depth.

Trapping of diffusing particles by small absorbers localized in a spherical region.

We study trapping of particles diffusing in a spherical cavity with an absorbing wall containing small static spherical absorbers localized in a spherical region in the center of the cavity. The focus is on the competition between the absorbers and the cavity wall for diffusing particles. Assuming that the absorbers and, initially, the particles are uniformly distributed in the central region, we derive an expression for the particle trapping probability by the cavity wall. The expression gives this probability as a function of two dimensionless parameters: the transparency parameter, characterizing the efficiency of the particle trapping by the absorbers, and the ratio of the absorber-containing region radius to that of the cavity. This work is a generalization of a recent study by Krapivsky and Redner [J. Chem. Phys. 147, 214903 (2017)] who considered the case where the absorber-containing region occupies the entire cavity. The expression for the particle trapping probability is derived in the framework of a steady-state approach which, in our opinion, is much simpler than the time-dependent approach used in the above-mentioned study.

Trapping of diffusing particles by short absorbing spikes periodically protruding from reflecting base.

We study trapping of diffusing particles by a periodic non-uniform boundary formed by absorbing spikes protruding from a reflecting flat base. It is argued that such a boundary can be replaced by a flat uniform partially absorbing boundary with a properly chosen effective trapping rate. Assuming that the spikes are short compared to the inter-spike distance, we propose an approximate expression which gives the trapping rate in terms of geometric parameters of the boundary and the particle diffusivity. To validate this result, we compare some theoretical predictions based on the expression for the effective trapping rate with corresponding quantities obtained from Brownian dynamics simulations.

Unbiased diffusion of Brownian particles in a helical tube.

A theoretical framework based on using the Frenet-Serret moving frame as the coordinate system to study the diffusion of bounded Brownian point-like particles has been recently developed [L. Dagdug et al., J. Chem. Phys. 145, 074105 (2016)]. Here, this formalism is extended to a variable cross section tube with a helix with constant torsion and curvature as a mid-curve. For the sake of clarity, we will divide this study into two parts: one for a helical tube with a constant cross section and another for a helical tube with a variable cross section. For helical tubes with a constant cross section, two regimes need to be considered for systematic calculations. On the one hand, in the limit when the curvature is smaller than the inverse of the helical tube radius R, the resulting coefficient is that obtained by Ogawa. On the other hand, we also considered the limit when torsion is small compared to R, and to the best of our knowledge, the expression thus obtained has not been previously reported in the literature. In the more general case of helical tubes with a variable cross section, we also had to limit ourselves to small variations of R. In this case, we obtained one of the main contributions of this work, which is an expression for the diffusivity dependent on R', torsion, and curvature that consistently reduces to the well-known expressions within the corresponding limits.