Journal of Chemistry
Volume 2015 (2015), Article ID 986402, 10 pages
http://dx.doi.org/10.1155/2015/986402

Research Article

Unbiased Diffusion through a Linear Porous Media with Periodic Entropy Barriers: A Tube Formed by Contacting Ellipses

Yoshua Chávez, Marco-Vinicio Vázquez, and Leonardo Dagdug

Universidad Autónoma Metropolitana Iztapalapa, Avenue San Rafael Atlixco 186, 09340 México, DF, Mexico

Received 18 February 2015; Accepted 26 May 2015

Academic Editor: Demeter Tzeli

Copyright © 2015 Yoshua Chávez et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

This work is devoted to the study of unbiased diffusion of point-like Brownian particles through channels with radial symmetry of varying cross-section and elliptic shape. The effective one-dimensional reduction is used with distinct forms of a position-dependent diffusion coefficient, , found in literature, to obtain expressions for (I) narrow escape times from a single open-ended tube, (II) its correspondent effective diffusion coefficient, both as functions of the eccentricity of the tube, ε, where ε = 0 returns the system to a spherical vesicle with two open opposite sides, and (III) finally, Lifson-Jackson formula that is used to compute expressions to assess the mean effective diffusion coefficient for a periodic elliptic channel formed by contacting ellipses, also as a function of the eccentricity. Mathematical expressions are presented and contrasted against computational simulations to validate them.

1. Introduction

Among many systems, well spread in nature and technological applications, the unbiased diffusion of point-like particles confined within quasi-one-dimensional systems, such as pores and channels, is ubiquitous and has been subjected to increasing attention in recent years [1–6]. Diffusive transport in confined environments arises from different contexts of practical and theoretical interest. It is relevant to study in fields as nanotechnology, chemistry, and biology, having direct applications to channels such as pores in zeolites [7], carbon nanotubes [8], synthetic nanopores [9–11], artificial pores produced in solid thin films [12], channels in biological systems [13], and single-nanopore sensors, designed to detect, quantify, and characterize many different types of molecules, for example, single- and double-stranded DNA chains. Experimental techniques, such as high-resolution crystallography of bacterial porins and other large channels, have demonstrated that these can be envisaged as tubes with significantly varying cross-sections along their principal axis. In some of these channels, variations in cross-section areas exceed one order of magnitude [14, 15]. This leads to the so-called entropic-like walls and barriers in the theoretical description of transport through such structures.

When diffusion occurs in quasi-one-dimensional structures, it becomes intuitively appealing to introduce an effective one-dimensional description. In a three-dimensional tube of varying radius, , the -axis directed along the centerline of the tube, the one-dimensional concentration of point-like particles is related to their three-dimensional concentration by the expression

which averages the spatial concentration over the channel’s position-dependent cross-section, . Given the condition of uniform concentration in any cross-section, satisfies the Fick-Jacobs equation

where is the particle’s bulk diffusion coefficient. Later, this result was generalized by Zwanzig [16], showing that the diffusion entering in (2) becomes position-dependent, , by means of introducing fluctuations in the concentration of particles in any cross-section, provided is a slowly varying function of ; that is, . Thus the generalized Fick-Jacobs equation takes the form

Zwanzig also proposed an explicit form to :

which has been extensively used throughout literature but in many cases poorly adjusted to simulated data. Later, Reguera and Rubí generalized Zwanzig’s result, and based on heuristic arguments they suggested that entering into (3) is given by [17]

That has proved to be more reliable given several geometrical conditions [18–22], when used along with (3). Surprisingly, in the same literature, this approach has also shown to be valid, even for a less restrictive condition upon the radius’ rate of change: , thus giving the generalized Fick-Jacobs equation a considerably extended range of applicability. In the same spirit Kalinay and Percus [23, 24] developed a more general theory of reduction to the effective one-dimensional description for radially symmetrical two- (2D) and three-dimensional (3D) tubes, and Dagdug and Pineda [25] later extended these results to nonsymmetrical 2D systems.

A wide range of stochastic processes of practical interest underlies first-passage events, such as the first-passage time, namely, the probability that a diffusing particle or a random walk will first reach a specified site (or set of sites) at a specified time [26]. Indeed, chemical and biochemical reactions [27, 28], animals searching for food [29], and trafficking receptors on biological membranes [30] are often controlled by first-encounter events [31]. Among first-passage events, the narrow escape time (NET)—the mean time a Brownian particle spends before being trapped by an opening window in exiting the cavity for the first time—is a quantity is of particular importance. Its applications range from cellular biology to biochemical reactions in cellular microdomains, such as dendritic spines, synapses, and micro vesicles, among others [30, 32]. The NET is the limiting quantity and the first step in the mathematical modeling of such processes where particles must first exit their domains in order to live up to their biological function [31].

To determine which form of to enter in (3) for a given set of geometries and boundary conditions, we took advantage of the fact that the Mean First-Passage Time (MFPT)—defined as the time it takes a random walker to reach a specified place for the first time, averaged over all the trajectories or realizations of the random walk—, τ, is a quantity often obtained by means of computer simulations. The MFPT is found to satisfy a backward equation:

where , is the Boltzmann constant, and is the absolute temperature, and is the initial position where any particle begins its random walk. The potential is defined as follows:

Equation (7), called entropic-like potential, accounts for the change in cross-section area along the axial length of the tube, taken to be zero at (a reference position, placed somewhere in the system). Then (6) is solved for the appropriate boundary conditions, to obtain an algebraic expression that relates with and the system’s geometrical parameters.

Complex geometries, such as those found in nature [13] and manmade structures at the small scale [33, 34], hardly resemble the simple shapes used so far to model them. Nonetheless, particle transport through these very complex geometries still can be tackled as a problem of diffusion in quasi-one-dimensional structures [5]. Again, Zwanzig had worked, back in 1983, a solution to the study of diffusion of Brownian particles in two-dimensional channels made of periodic boundaries [35], using a sophisticated mathematical result, known as the Lifson-Jackson theorem [36]. More recently, the correspondent Lifson-Jackson formula

has been successfully used in several works on 2D and 3D systems [19, 20, 22, 37, 38], to obtain a mathematical expression for the effective one-dimensional diffusion coefficient averaged for periodic channels, which represents a good assessment of the effective diffusion coefficient behavior in real long, narrow channels (i.e., their length is significantly greater than their radii), with internal structure (also known as corrugated channels).

The present work is divided in three main parts:

(I)Following the methodology developed in previous studies [20, 22], we treat the system shown in Figure 1, a single tube of uniform cross-section and elliptic shape (which we call elliptic tube hereafter) of length , where is given by , is the semilength of the minimum diameter (minor axis), gamma is the semi length of the maximum diameter (major axis) given by the expression , and epsilon is the eccentricity. The openings placed at opposite sides of the tube are of radius . In this system we solve (6) entering (7) and several expressions for the position dependent diffusion coefficient : (the Fick-Jacobs approximation), , and , given by (4) and (5), respectively, and subject to the appropriate boundary conditions (see Figure 3), thus obtaining algebraic expressions of as a function of geometrical parameters of the system (, , ). The resulting theoretical curves will be compared with data obtained by computer simulations.(II) We assess the effective diffusion coefficient for this tube, using the exact solution of (6) for a cylindrical tube (given , and ) to establish the relation , where we have assumed that has the same structure than the bulk diffusion . Then we compare the theoretical curves of with values obtained by computer simulations.(III) Finally, we use the Lifson-Jackson formula (8), along with different forms of —the same set of expressions as before—to obtain mathematical expressions of the mean effective diffusion coefficient, , as a function of geometrical parameters of the system. This last system is a periodic channel, made of juxtaposed elliptic tubes, as seen in Figure 2 (from now on called periodic elliptic tube). The various analytical expressions of , , elliptic tube, and , periodic elliptic tube, obtained along with the graphical comparisons between theory and computer simulated data, constitute the main results of this paper.

(a)

(b)

(c)

Figure 1: Elliptic tube. (a) A tube of circular cross-section where the position-dependent radius is given by , an ellipse in the longitudinal plane, as shown in (b), where is the square of its major radius. is the size of the minor radius of the elliptic shape; the tube’s length is , where . The radius of the openings, at opposing sides of the tube, is , else , where . Panel (c) shows a representation of the entropic-like potential associated with the change of shape in this tube.

(a)

(b)

Figure 2: Periodic elliptic tube. (a) A tube made of juxtaposed elliptic sections (each an elliptic tube as depicted in Figure 1) is called a periodic elliptic tube. Its shape is schematically shown in (b).

Figure 3: Region used to set the limits for integration in this work. The boundary conditions required to solve (6) are explained in text.

2. Theory and Computational Details

2.1. Elliptic Tube: Obtaining Analytical Expressions for the MFPT

Solving (6), given the entropic-like potential (7), requires an explicit form for and second-order boundary conditions. As can be seen in Figure 1, the elliptic tube has cross-sections of uniform circular shape, and position dependent radius, , given by

which, plotted against the transversal plane along the -axis, resembles the characteristic shape of an ellipse, whose center—for convenience—we chose to place at the origin of coordinates. In this way we will have the open ends of the tube at the planes and then model them as absorbing discs. Thus the cross-section, , takes the form

Taking this expression to (7), we have the explicit form of the entropic-like potential , which we need to enter in (6)

where we have already chosen , as the reference position. Regarding the form of (11), it is advisable to introduce the notation , and , which in turn gives , in order to simplify the forthcoming algebra. Notice that is the major radius of the ellipse in Figure 1. In this fashion, we may rewrite (11) to read

Thus, using this result in (6), we obtain

which, upon integration, yields

Then, in order to fix the value of , a boundary condition is needed.

Taking advantage of the symmetry of the system, we can cope with the elliptic semitube, shown in Figure 3, instead of the whole, to carry on the integration process. Thus we place a reflective surface at the middle plane in :

while the tube’s opening at can be treated as an absorbing surface, where

Given these boundary conditions, we can use (15) with (14) to find . Therefore, we can write

Further treatment of (17) requires us to supply an adequate form of . In this work we will use three expressions.

(I) Fick-Jacobs Approximation. Taking to expression (17) yields

which, along with the boundary condition given in (16), yields :

Then, introducing (19) in (18), we obtain

Now, we will use the relation (which can be verified from the definition of ; see Figure 1)

to rewrite (20) as

or in a more useful form:

where we have used the definition . Setting and the initial position of all random walks in , we will have an expression for which is parametrically dependent solely on the eccentricity , and the opening-size at the ends of the elliptic tube is as follows:

As soon as we have this result, it is noticeable that (24) reduces to the exact result for a spherical vesicle with two opposite openings, as , in perfect agreement with the previous work by Vazquez and Dagdug [21].

(II) Zwanzig Approximation. Taking the expression of given by (4) into (17) to carry on the second integration leads us first to write (4) in an explicit form. Here we use to find

So far, we have (4) written down as follows:

In (26) we have defined . Now, with this result in (17), we may write

or, in a simplified form,

where, upon integration, we obtain

Then, to fix the value of , we use (16):

which, after some algebra, leads to

Now we may use (21), and our prior definitions of and , to put (31) in a more useful form:

Again, by choosing and , we obtain, after simplification, an expression of , as a function of geometrical parameters:

(III) Reguera-Rubí Approximation. Following the procedure, previously outlined, we took (5), now written, according to (25), in the form

and then we use this expression in (17),

to carry on the second integration, thus yielding

Using (16), to fix , we have

And taking the last result to (36), after some algebra,

Substituting some previous definitions, , , , we may rewrite (38) in a more useful form:

Then, by setting and in (39), we obtain a simpler relation for the MFPT in terms of geometrical parameters:

2.2. Elliptic Tube: Computing Effective Diffusion Coefficients for the Elliptic Tube

Let us assume that the effective diffusion coefficient for the elliptic tube (of length and open ends) has the same structure than the bulk diffusion; then

Thus, using in (41) the various expressions already obtained for the MFPT, (24), (33), and (40), we can write expressions for the in terms of geometrical parameters of the elliptic tube, given .

(I) Fick-Jacobs

(II) Zwanzig

(III) Reguera-Rubí

where is given by

2.3. Periodic Elliptic Channel: Computing Averaged Diffusion Coefficients

The effective diffusion coefficient averaged for a tube made of identical sections (the elliptic tube of previous section) is given by (8), the Lifson-Jackson formula, where the expected value of a function over an interval of values accessible to (see Figure 3) is defined as

Given the definition of in (7), we have

Then, we can write the integrals in (8) to read as

Now we can use appropriate expressions of to carry out the integration process.

(I) Fick-Jacobs. Taking the Fick-Jacobs approximation, in (48), we can rewrite (8) to read

and carrying out the integrals, we obtain

Finally, combining these two results, we have

(II) Zwanzig. Now, taking (26) in (48) allows us to rewrite (8) in the form

which, in turn, leads to

where is given by

(III) Reguera-Rubí. Finally, taking from (34) to (48), we may rewrite (8) in the form

which leads to the result

2.4. Computational Details

Computational simulations were carried out to compute the mean first-passage time, τ (MFPT), defined above. The particle’s initial position, , was distributed uniformly within the middle plane at . The MFPT is denoted as . When running simulations, we took , , and the time step , so that. The actual particle’s position, , is given by  , where   is the previous position and is a vector of pseudo-random numbers generated with a Gaussian distribution (, ). Each MFPT was obtained by averaging the first-passage times of 5 × 10⁴ trajectories.

In the elliptic tube, the effective diffusion coefficients were computed using the formula given by Berezhkovskii and Weiss [39]:

which requires the recording of several quantities from computer simulations: the global first and second moments of the MFPT, and , the MFPT for particles exiting the system to the left and right openings, and , respectively, and the correspondent fractions of exiting particles, and (where ).

In the periodic elliptic tube, the effective diffusion coefficients were computed from the Einstein’s relation:

where the mean-square displacement, , was recorded from computer simulations and tabulated for a total of 5 × 10⁴ realizations of a random walk.

3. Results and Discussions

Comparisons between theoretical curves and data obtained from simulations (circles) are shown in Figures 4 to 6. In Figure 4, from (24) (black dash and dot line), (33) (blue dashed line), and (40) (orange continuous line) are plotted against the eccentricity of the elliptic tube (, ); see Figure 1. The physical limit corresponds to ; thus, the three expressions show the right behavior , but only (40) shows a good agreement with simulations.

Figure 4: Comparison of from (24), black dash and dot line, (33), blue dashed line, and (40), orange continuous line, with data from computer simulations (circles), are plotted against the eccentricity of the elliptic tube (given , ).

Figure 5 shows theoretical curves of from (42) (black dash and dot line), (43) (blue dashed line), and (44) (orange continuous line) as a function of the eccentricity of the elliptic tube. This time we have a quotient of two quantities that, individually, approach as ; thus remains finite. As grows faster than , we observe a tendency to increase while approaching the limit . The best fit with simulation data comes for the Reguera-Rubí approximation, in (43).

Figure 5: from (42), black dash and dot line, (43), blue dashed line, and (44), orange continuous line, as a function of the eccentricity of the elliptic tube (given , ), compared with Data from computer simulations (circles).

Figure 6: for the system in Figure 2, from (52), black dash and dot line, (54), blue dashed line, and (57), orange continuous line, as a function of the eccentricity of each link in the periodic elliptic tube (given , ), compared with Data from computer simulations (circles).

Figure 6 shows theoretical curves of for the system in Figure 2, from (52) (black dash and dot line), (54) (blue dashed line), and (57) (orange continuous line), as a function of the eccentricity of each link in the periodic elliptic tube (, ). The inset graphic shows the slight difference between (54) and (57), which is practically negligible. Despite the apparent difference shown in relation to simulation data, the mean error remains below . Expression (54) has a clear advantage over (57); that is, the first is analytic, while the latter has an integral, which, in turn, has to be solved numerically. When choosing error grows faster and above 10–30%, suggesting that the narrow-tube approximation is no longer applicable to this system.

Final expressions shown in Figures 4 to 6, discussed above, are the main result of this paper.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was carried out with the support of the CONACyT (Consejo Nacional de Ciencia y Tecnología-National Science and Technology Council) Project Grant no. 176452. Y. Chávez thanks CONACyT (Fellowship no. 269180).

References

P. S. Burada, Entropic Transport in Confined Media, Universität Augsburg, Augsburg, Germany, 2008.

P. S. Burada, P. Hänggi, F. Marchesoni, G. Schmid, and P. Talkner, “Diffusion in confined geometries,” ChemPhysChem, vol. 10, no. 1, pp. 45–54, 2009.

P. S. Burada, G. Schmid, and P. Hänggi, “Entropic transport: a test bed for the Fick-Jacobs approximation,” Philosophical Transactions of the Royal Society of London Series A: Mathematical, Physical & Engineering Sciences, vol. 367, no. 1901, pp. 3157–3171, 2009.

P. S. Burada, G. Schmid, P. Talkner, P. Hänggi, D. Reguera, and J. M. Rubí, “Entropic particle transport in periodic channels,” BioSystems, vol. 93, no. 1-2, pp. 16–22, 2008.

P. Malgaretti, I. Pagonabarraga, and J. M. Rubi, “Entropic transport in confined media: a challenge for computational studies in biological and soft-matter systems,” Frontiers in Physics, vol. 1, article 23, 2013.

L. Karwacki, M. H. F. Kox, D. A. M. de Winter et al., “Morphology-dependent zeolite intergrowth structures leading to distinct internal and outer-surface molecular diffusion barriers,” Nature Materials, vol. 8, no. 12, pp. 959–965, 2009.

R. Haul, “J. Kärger, D. M. Ruthven: Diffusion in Zeolites and other Microporous Solids, J. Wiley & Sons INC, New York 1992. ISBN 0-471-50907-8. 605 Seiten, Preis: £ 117,” Berichte der Bunsengesellschaft für Physikalische Chemie, vol. 97, no. 1, pp. 146–147, 1993.

A. Berezhkovskii and G. Hummer, “Single-file transport of water molecules through a carbon nanotube,” Physical Review Letters, vol. 89, no. 6, Article ID 064503, 2002.

M. Gershow and J. A. Golovchenko, “Recapturing and trapping single molecules with a solid-state nanopore,” Nature Nanotechnology, vol. 2, no. 12, pp. 775–779, 2007.

I. D. Kosińska, I. Goychuk, M. Kostur, G. Schmid, and P. Hänggi, “Rectification in synthetic conical nanopores: a one-dimensional Poisson-Nernst-Planck model,” Physical Review E—Statistical, Nonlinear, and Soft Matter Physics, vol. 77, no. 3, Article ID 031131, 2008.

L. T. Sexton, L. P. Horne, S. A. Sherrill, G. W. Bishop, L. A. Baker, and C. R. Martin, “Resistive-pulse studies of proteins and protein/antibody complexes using a conical nanotube sensor,” Journal of the American Chemical Society, vol. 129, no. 43, pp. 13144–13152, 2007.

S. Howorka and Z. Siwy, “Nanopore analytics: sensing of single molecules,” Chemical Society Reviews, vol. 38, no. 8, pp. 2360–2384, 2009.

B. Hille, Ion Channels of Excitable Membranes, Sinauer Associates Inc., Sunderland, Mass, USA, 3rd edition, 2001.

S. W. Cowan, T. Schirmer, G. Rummel et al., “Crystal structures explain functional properties of two E. coli porins,” Nature, vol. 358, no. 6389, pp. 727–733, 1992.

L. Song, M. R. Hobaugh, C. Shustak, S. Cheley, H. Bayley, and J. E. Gouaux, “Structure of staphylococcal α-hemolysin, a heptameric transmembrane pore,” Science, vol. 274, no. 5294, pp. 1859–1865, 1996.

R. Zwanzig, “Diffusion past an entropy barrier,” Journal of Physical Chemistry, vol. 96, no. 10, pp. 3926–3930, 1992.

D. Reguera and J. M. Rubí, “Kinetic equations for diffusion in the presence of entropic barriers,” Physical Review E, vol. 64, no. 6, Article ID 061106, 2001.

A. M. Berezhkovskii, M. A. Pustovoit, and S. M. Bezrukov, “Diffusion in a tube of varying cross section: numerical study of reduction to effective one-dimensional description,” The Journal of Chemical Physics, vol. 126, no. 13, Article ID 134706, 2007.

M. V. Vázquez, A. M. Berezhkovskii, and L. Dagdug, “Diffusion in linear porous media with periodic entropy barriers: a tube formed by contacting spheres,” Journal of Chemical Physics, vol. 129, no. 4, Article ID 046101, 2008.

M.-V. Vázquez and L. Dagdug, “Numerical study assessing the applicability of the reduction to effective one-dimensional description of diffusion in a hemispherical shaped tube,” Journal of Non-Newtonian Fluid Mechanics, vol. 165, no. 17-18, pp. 987–991, 2010.

M.-V. Vazquez and L. Dagdug, “Unbiased diffusion to escape through small windows: assessing the applicability of the reduction to effective one-dimension description in a spherical cavity,” Journal of Modern Physics, vol. 2, no. 4, pp. 284–288, 2011.

Y. Chávez, G. Chacón-Acosta, M. Vázquez, and L. Dagdug, “Unbiased diffusion to escape complex geometries: is reduction to effective one-dimensional description adequate to assess narrow escape times?” Applied Mathematics, vol. 5, no. 8, pp. 1218–1225, 2014.

P. Kalinay and J. K. Percus, “Projection of two-dimensional diffusion in a narrow channel onto the longitudinal dimension,” Journal of Chemical Physics, vol. 122, no. 20, Article ID 204701, 2005.

P. Kalinay and J. K. Percus, “Corrections to the Fick-Jacobs equation,” Physical Review E, vol. 74, no. 4, Article ID 041203, 6 pages, 2006.

L. Dagdug and I. Pineda, “Projection of two-dimensional diffusion in a curved midline and narrow varying width channel onto the longitudinal dimension,” Journal of Chemical Physics, vol. 137, no. 2, Article ID 024107, 2012.

S. Redner, A Guide to First-Passage Processes, Cambridge University Press, Cambridge, UK, 2001.

M. Coppey, O. Bénichou, R. Voituriez, and M. Moreau, “Kinetics of target site localization of a protein on DNA: a stochastic approach,” Biophysical Journal, vol. 87, no. 3, pp. 1640–1649, 2004.

P. Hanggi, P. Talkner, and M. Borkovec, “Reaction-rate theory: fifty years after Kramers,” Reviews of Modern Physics, vol. 62, no. 2, pp. 251–341, 1990.

O. Bénichou, M. Coppey, M. Moreau, P.-H. Suet, and R. Voituriez, “Optimal search strategies for hidden targets,” Physical Review Letters, vol. 94, no. 19, Article ID 198101, 2005.

D. Holcman and Z. Schuss, “Escape through a small opening: receptor trafficking in a synaptic membrane,” Journal of Statistical Physics, vol. 117, no. 5-6, pp. 975–1014, 2004.

O. Bénichou and R. Voituriez, “Narrow-escape time problem: time needed for a particle to exit a confining domain through a small window,” Physical Review Letters, vol. 100, no. 16, Article ID 168105, 2008.

Z. Schuss, A. Singer, and D. Holcman, “The narrow escape problem for diffusion in cellular microdomains,” Proceedings of the National Academy of Sciences of the United States of America, vol. 104, no. 41, pp. 16098–16103, 2007.

E. Lippmaa, M. Mägi, A. Samoson, M. Tarmak, and G. Engelhardt, “Investigation of the structure of zeolites by solid-state high-resolution 29Si NMR spectroscopy,” Journal of the American Chemical Society, vol. 103, no. 17, pp. 4992–4996, 1981.

P. Chen, J. Gu, E. Brandin, Y.-R. Kim, Q. Wang, and D. Branton, “Probing single DNA molecule transport using fabricated nanopores,” Nano Letters, vol. 4, no. 11, pp. 2293–2298, 2004.

R. Zwanzig, “Effective diffusion coefficient for a Brownian particle in a two-dimensional periodic channel,” Physica A: Statistical Mechanics and its Applications, vol. 117, no. 1, pp. 277–280, 1983.

S. Lifson and J. L. Jackson, “On the self-diffusion of ions in a polyelectrolyte solution,” The Journal of Chemical Physics, vol. 36, no. 9, pp. 2410–2414, 1962.

L. Dagdug, M.-V. Vazquez, A. M. Berezhkovskii, and S. M. Bezrukov, “Unbiased diffusion in tubes with corrugated walls,” Journal of Chemical Physics, vol. 133, no. 3, Article ID 034707, 2010.

I. Pineda, M.-V. Vazquez, A. M. Berezhkovskii, and L. Dagdug, “Diffusion in periodic two-dimensional channels formed by overlapping circles: comparison of analytical and numerical results,” Journal of Chemical Physics, vol. 135, no. 22, Article ID 224101, 2011.

A. M. Berezhkovskii and G. H. Weiss, “Propagators and related descriptors for non-Markovian asymmetric random walks with and without boundaries,” The Journal of Chemical Physics, vol. 128, Article ID 044914, 2008.