Understanding totally asymmetric simple-exclusion-process transport on networks: Generic analysis via effective rates and explicit vertices

Ben Embley, Andrea Parmeggiani, and Norbert Kern

Phys. Rev. E 80, 041128 – Published 26 October 2009

Abstract

In this paper we rationalize relevant features of totally asymmetric simple-exclusion processes on topologies more complex than a single segment. We present a mean-field framework, exploiting the previously introduced notion of effective rates, which we express in terms of the average particle density on explicitly introduced junction sites. It allows us to construct the phase behavior as well as the current-density characteristic from well-known results for a linear totally asymmetric simple-exclusion-process segment in a very systematic and generic way. We validate the approach by studying a fourfold vertex in all variations in the number of entering/exiting segments and compare our predictions to simulation data. Generalizing the notion of particle–hole symmetry to take into account the topology at a junction shows that the average particle density at the junction constitutes a relevant directly observable parameter which gives detailed insight into the transport process. This is illustrated by a complete study of a simple network with figure-of-eight topology. Finally we generalize the approach to handle rate bias at a junction and discuss the surprisingly rich phenomenology of a biased figure-of-eight structure. This example highlights that the proposed framework is generic and readily extends to other topologies.

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Received 15 April 2009

DOI:https://doi.org/10.1103/PhysRevE.80.041128

Authors & Affiliations

Ben Embley^*

School of Chemical Engineering and Analytical Science, The University of Manchester, P.O. Box 88, Sackville Street, Manchester M60 1QD, United Kingdom

Andrea Parmeggiani^†

Laboratoire de Dynamique des Interactions Membranaires Normales et Pathologiques, CNRS-UMR 5235, CC107, Université Montpellier II, Place Eugène Bataillon, 34095 Montpellier Cedex 5, France

Norbert Kern^‡

Laboratoire des Colloïdes, Verres et Nanomatériaux, CNRS-UMR 5587, CC069, Université Montpellier II, Place Eugène Bataillon, 34095 Montpellier Cedex 5, France

^*ben.embley@manchester.ac.uk
^†andrea.parmeggiani@univ-montp2.fr
^‡norbert.kern@univ-montp2.fr

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Vol. 80, Iss. 4 — October 2009

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Figure 1
Examples of the pure (a) LD, (b) MC, and (c) HD phases for TASEP. Particles are flowing from left to right, the boundary values for the density are $α$ at the contact with the entrance reservoir and $1 - β$ at the contact with the exit reservoir. Note that the curvatures of the LD and HD profiles may be convex or concave, depending on the actual values of the entrance/exit rates.Reuse & Permissions
Figure 2
Schematic of a (trivial) twofold vertex V(1:1), i.e., two successive linear TASEP segments coupled via a junction site (labeled as $^{\sim}$ ). $α$ and $β$ are the overall entrance and exit rates, respectively. $β_{A}$ and $α_{B}$ are the effective exit/entrance rates at the points where the segments connect to the junction. The phase diagram of each segment is that of linear TASEP, where the abscissa is the effective entrance rate to a segment, while the ordinate is the effective exit rate from the same segment. The coupling is implemented by expressing the effective rates which link “segments” to the junction site in terms of that site’s density $\tilde{ρ}$ . Current conservation then supplies the matching condition required to determine the phases of the combined system. Although a vertex V(1:1) is trivial, the example serves to illustrate our approach. Phases of the subsystems are labeled HD: for a high-density phase of the incoming segment and :LD for a low-density phase of the outgoing segment, etc.Reuse & Permissions
Figure 3
(Color online) Examples of the current-matching procedure (cf. Sec. 2B) at a junction for the trivial case V(1:1). For this schematic, we plot $J_{A} (α; \tilde{ρ})$ and $J_{B} (\tilde{ρ}; β)$ , the intersection of which (labeled by a circle) corresponds to the physical solution for which the currents match correctly. Segments of the individual curves are labeled to indicate the phase they correspond to, and the nature of the physical solution may therefore be determined by reading off the phases of each segment at the point of intersection. The corresponding density at the junction $\tilde{ρ}$ may also be read off directly. In (a), $α \geq 1 / 2$ and $β < 1 / 2$ ; in (b), $β > α > 1 / 2$ . The sequence (a) to (b) shown therefore illustrates the switching of the solution as the exit rate $β = 0, \dots, 1$ sweeps out its entire range and $α > 1 / 2$ is kept constant. Here we switch from a HD:HD solution [i.e., in (a)] to the rather particular case where the overall solution, MC/HD:MC/LD, involves coexisting phases in both of segments A and B [i.e., in (b)]. This latter case corresponds a junction density of $\tilde{ρ} = 1 / 2$ .Reuse & Permissions
Figure 4
(a) Schematic of a $V (m : n)$ junction, i.e., a junction with $m$ incoming segments and $n$ outgoing segment. The junction is labeled with a tilde. (b) Illustration for notation of topologies.Reuse & Permissions
Figure 5
Schematic representation of the fourfold junction V(2:2) as used for the explicit-vertex analysis. Both incoming branches couple to a reservoir at density $α$ and both outgoing segments couple to a reservoir at density $1 - β$ . Phases are labeled as “2A:2B,” e.g., 2HD:2LD stands for both incoming segments in a high-density phase $(A_{1} = A_{2} = HD)$ and both outgoing segments in a low-density phase $(B_{1} = B_{2} = LD)$ . Note that we have simplified the notation based on symmetry, writing 2A:2B rather than “ $(A_{1} ∥ A_{2}) : (B_{1} ∥ B_{2})$ .”Reuse & Permissions
Figure 6
Mean-field phases for the individual segments of V(2:2): (a) incoming segments $(A_{1}, A_{2})$ , identical to linear TASEP, and (b) for outgoing segments $(B_{1}, B_{2})$ , with a rescaled axis such that only part of the linear TASEP behavior is accessible. The effective rates linking segments to the junction site are expressed in terms of the junction density $\tilde{ρ}$ , which subsequently makes it easier to deduce the phase behavior of the overall system. In order to distinguish incoming and outgoing segments, we label the subsystems HD: for a high-density phase of an incoming segment, :LD for a low-density phase of an outgoing segments, etc.Reuse & Permissions
Figure 7
(Color online) Illustration of the current-matching procedure for one particular case $(α, β > 1 / 2)$ of V(2:2). The current through any individual segment is known in terms of its input/output rates, which are $(α, 1 - \tilde{ρ})$ and $(\tilde{ρ} / 2, β)$ , respectively. For a given pair $(α, β)$ , the currents $J_{A_{1, 2}} (α; \tilde{ρ})$ and $J_{B_{1, 2}} (\tilde{ρ}; β)$ are piecewise functions of $\tilde{ρ}$ [illustrated here for the domain $α, β > 1 / 2$ ; cf. Eqs. (15, 16)]. The current flowing through the segments are compatible at the intersection of the two curves, which fixes the junction density (here $\tilde{ρ} = 2 / 3$ , irrespective of $α$ and $β$ ) and therefore the current $J = 2 \times J_{A_{1}, A_{2}} = 2 \times J_{B_{1}, B_{2}} = 2 \times 2 / 9$ crossing the system. The corresponding phases are determined from the parts of the current relations at which the intersection occurs (here $A_{1}, A_{2} = HD :$ and $B_{1}, B_{2} = : LD$ ). Once the input as well as the output rates exceed 1/2, the system is therefore in a state where the current saturates at a value of $J_{max} = 2 \times 2 / 9$ , irrespective of the actual values of these rates.Reuse & Permissions
Figure 8
Mean-field phase diagram for a fourfold junction V(2:2), where both incoming branches couple to a reservoir at density $α$ and both outgoing segments couple to a reservoir at density $1 - β$ . Phases are labeled as “ $A_{1, 2} : B_{1, 2}$ ,” e.g., 2HD:2LD stands for both incoming segments in a high-density phase $(A_{1, 2} = B_{1, 2} = HD)$ and both outgoing segments in a low-density phase $(A_{1, 2} = B_{1, 2} = LD)$ . The labels along the transition lines recall that these correspond to coexisting phases in the individual segments: for example, the phase boundary line on $β = 1 / 3$ involves segment A being in a HD phase, whereas a LD/HD coexistence is induced on segment B.Reuse & Permissions
Figure 9
(Color online) Current flowing through a fourfold vertex V(2:2). The current is continuous, but, at all phase boundaries, its slope is not. In (a), we plot the (simulation) total current $J$ as a function of both $α$ and $β$ ; a two-dimensional projection is shown in (b). We examine all cases of $(α, β)$ and plot (as a minimum) a $10 \times 10$ grid of points $(α, β)$ , with extra points close to certain transition lines. We also detect the phase combination directly from simulation, which are represented through the data symbols: data detected as 2LD:2LD are plotted as (blue) stars, 2HD:2LD as (turquoise) squares, and 2HD:2HD as (red) crosses; black squares (unfilled) are data for which either LD or HD phases are detected in each branch (i.e., 2LD/HD:2LD/HD coexistence: cf. Fig. 8 and also [18] for a method of domain wall detection). The black lines represent theoretical phase boundaries, based on Eq. (17), which are thus corroborated by the simulation data.Reuse & Permissions
Figure 10
(Color online) Mean-field phase diagram for (a) a fourfold junction V(1:3) where the incoming branch is coupled to a reservoir at density $α$ and all three outgoing segments coupled to a reservoir at density $1 - β$ , as obtained by an effective-rate analysis similar to the one for the symmetric case V(2:2) (cf. Sec. 3B). The value indicated on the $β$ axis, which characterizes the boundary between HD:3HD and MC:3LD, is $β^{*} = \frac{1}{2} - \frac{1}{\sqrt{6}} \approx 0.09$ . The separation line between phases HD:3HD and LD:3LD obeys $α (1 - α) = 3 \times β (1 - β)$ . (b) In this phase diagram for a junction V(3:1) (cf. Sec. 3C), the value indicated on the $α$ axis is $α^{*} \approx 0.09$ , i.e., the same as $β^{*}$ in the previous figure. The separation line between phases 3LD:LD and 3HD:HD obeys $3 \times α (1 - α) = β (1 - β)$ . Labels in italics refer to the phase boundaries only; dotted phase boundaries indicate continuous transitions.Reuse & Permissions
Figure 11
(Color online) Current $J (α, β)$ passing through a fourfold junction V(1:3). In (a), we plot the (simulation) total current $J$ as a function of both $α$ and $β$ . In (b), we use a 2D projection to examine all cases of $(α, β)$ and plot (as a minimum) a $10 \times 10$ grid of points $(α, β)$ , many more close to the curved phase boundaries. Symbols are attributed from simulations, where systems in phase LD:3LD are plotted as (blue) stars, MC:3LD as (turquoise) squares, and HD:3HD as (red) crosses. Based on the black lines, which represent theoretical mean-field phase transitions, the phase and current predictions in Eq. (17) can be judged accurate. This indicates that the analysis of Sec. 3A correctly captures the phase behavior [cf. Fig. 10a].Reuse & Permissions
Figure 12
(a) The figure of eight and corresponding effective rates at the junction. (b) Effective rates, corresponding to the bulk of the figure of eight, required for the corresponding periodic-on-average V(2:2).Reuse & Permissions
Figure 13
Simulation data for a figure-of-eight network constructed both as a periodic-on-average vertex V(2:2) and as a proper closed system, respectively. Simulations use $L = 100$ . Open symbols refer to open but periodic-on-average boundary conditions [i.e., V(2:2)]; closed symbols refer to closed boundary conditions (i.e., figure of eight). (a) shows the density at the junction site $\tilde{ρ}$ as a function of the overall density $ρ$ ; the flat line at $\tilde{ρ} = 2 / 3$ corresponds to the saturated-current plateau. The occurrence of two different slopes reveals the absence of particle–hole symmetry at the junction. (b) shows the total current $J$ as a function of the overall density $ρ$ ; the plateau is $J_{max} = 4 / 9$ .Reuse & Permissions
Figure 14
(Color online) (a) Schematic representation of the figure of eight with a rate bias at the junction. A is the favored branch, which is selected with a probability $σ_{A} > 1 / 2$ ; branch B is selected with probability $σ_{B} = 1 - σ_{A}$ . (b) Representation of the topology in terms of two parallel segments joined at junction sites, which are later to be matched. (c) The mean-field phases for a biased figure of eight are easily identified by considering the phases of two parallel sections A and B. For a general output rate $β$ , two TASEP-like phase diagrams may be superposed in terms of the density at the junction $\tilde{ρ}$ after having expressed the input rates for the favored/unfavored branches A/B in terms of the associated bias $σ_{A}$ (and $σ_{B}$ ). In particular, the bias favoring branch A $(σ_{A} > 1 / 2)$ opens up a zone for the new $HD ∥ LD$ phase, which widens as the bias becomes stronger. Note that the dashed parts of the phase diagram $(α > 1 / 2)$ are in fact inaccessible since the density at a junction cannot exceed unity. Since both branches are in parallel, no matching of currents is required at this stage. To conclude for a figure-of-eight topology, we must furthermore impose the condition of periodicity on average, $β = 1 - \tilde{ρ}$ , which therefore limits the relevant phases to the first antidiagonal (dashed green line).Reuse & Permissions
Figure 15
Theoretical and Monte Carlo simulation (closed system) results for the junction density $\tilde{ρ}$ : (a) an asymmetric split $σ_{A} = 0.75$ , $σ_{B} = 0.25$ ; (b) $σ_{A} = 0.90$ , $σ_{B} = 0.10$ . Solid lines represent analytical predictions, gray grids indicate mean-field values for transition densities $ρ$ and $\tilde{ρ}$ . Theoretical predictions capture the behavior of $\tilde{ρ}$ , although in simulations the transitions are not as sharp and fall below mean-field predictions, just as for the symmetric case [cf. Fig. 13a]. Note that the slopes in the $(LD ∥ LD)$ and $(HD ∥ HD)$ phases are independent of the exact junction asymmetry [cf. Eq. (49)], whereas the slope of the curve between the two plateaus depends on $σ_{A} = 1 - σ_{B}$ .Reuse & Permissions
Figure 16
(Color online) Theoretical and Monte Carlo simulation results for the currents $J_{A}$ and $J_{B}$ passing through individual branches and overall current $J$ passing through a fourfold junction: (a) an asymmetric split $σ_{A} = 0.75$ , $σ_{B} = 0.25$ ; (b) $σ_{A} = 0.90$ , $σ_{B} = 0.10$ . Solid lines represent theoretical predictions. As in the absence of bias ( $σ_{A} = σ_{B} = 1 / 2$ , mean-field predictions for the current tend to underestimate the plateau current in a coexistence region [cf. Figs. 13b, 15].Reuse & Permissions
Figure 17
(Color online) (a) Theoretical mean-field results illustrating the two coexistence regions (dashed) for the biased figure of eight as a function of the bias $σ_{A}$ . The first zone corresponds to the presence of a domain wall in the favored branch A, whereas the second zone corresponds to a domain wall in branch B. For a given bias $σ_{A}$ the current is constant within either of these domains (cf. Figs. 15, 16). (b) Theoretical and Monte Carlo simulation results for the current $J$ passing through a fourfold junction, with various values for the bias. Plotting several overall currents on the same graph allows us to demonstrate the limiting behavior of a biased figure of eight. In the limit $σ_{A} \to 0.5$ , we recover the symmetric V(2:2) and the figure of eight [cf. Fig. 13b]. In the limit $σ_{A} \to 1$ , we recover a standard TASEP parabola with $J = 2 ρ (1 - 2 ρ)$ (i.e., half of the sites become unusable). The underestimation of the overall current $J$ tends to decrease as the system approaches more and more closely a standard linear TASEP: mean-field predictions work least well in the symmetric case.Reuse & Permissions

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