Weakly coupled, antiparallel, totally asymmetric simple exclusion processes

Róbert Juhász*Research Institute for Solid State Physics and Optics, P.O. Box 49, H-1525 Budapest, HungaryReceived 24 April 2007; revised manuscript received 3 July 2007; published 21 August 2007

We study a system composed of two parallel totally asymmetric simple exclusion processes with openboundaries, where the particles move in the two lanes in opposite directions and are allowed to jump to theother lane with rates inversely proportional to the length of the system. Stationary density profiles are deter-mined and the phase diagram of the model is constructed in the hydrodynamic limit, by solving the differentialequations describing the steady state of the system, analytically for vanishing total current and numerically fornonzero total current. The system possesses phases with a localized shock in the density profile in one of thelanes, similarly to exclusion processes endowed with nonconserving kinetics in the bulk. Besides, the systemundergoes a discontinuous phase transition, where coherently moving delocalized shocks emerge in both lanesand the fluctuation of the global density is described by an unbiased random walk. This phenomenon isanalogous to the phase coexistence observed at the coexistence line of the totally asymmetric simple exclusionprocess, however, as a consequence of the interaction between lanes, the density profiles are deformed and inthe case of asymmetric lane change, the motion of the shocks is confined to a limited domain.

DOI: 10.1103/PhysRevE.76.021117 PACS numbers: 02.50.Ey, 47.11.Qr, 05.60.k

I. INTRODUCTION

The investigation of interacting stochastic driven diffusivesystems plays an important role in the understanding of non-equilibrium steady states 1,2. As opposed to equilibriumstatistical mechanics, phase transitions may occur in thesesystems even in one spatial dimension 3. The paradigmaticmodel of driven lattice gases is the one-dimensional totallyasymmetric simple exclusion process TASEP 4,5, whichexhibits boundary induced phase transitions 6 and thesteady state of which is exactly known 7,8. Beside theoret-ical interest, this model and its numerous variants have founda wide range of applications, such as the description of ve-hicular traffic 9 or modeling of transport processes in bio-logical systems 10. Inspired by the traffic of cytoskeletalmotors 11, such models were introduced where a totallyasymmetric exclusion process is coupled to a finite compart-ment where the motion of particles is diffusive 12–14. Re-cently, the attention has turned to exclusion processes en-dowed with various types of reactions which violate theconservation of particles in the bulk 15–28. The simplestone among these models is the TASEP with “Langmuir ki-netics,” where particles are created and annihilated also atthe bulk sites of the system 16—a process, which mayserve as a simplified model for the cooperative motion ofmolecular motors along a filament from which motors candetach and attach to it again. For these types of systems, thetime scale of nonconserving processes compared to that ofdirected motion and the processes at the boundaries is cru-cial. If the nonconserving reactions occur with rates of largerorder than the inverse of the system size L, then in the largeL limit, they dominate the stationary state. On the contrary,when they are of smaller order than O1/L, they are irrel-evant and the stationary state is identical to that of the un-derlying driven diffusive system. However, in the marginal

case when the rates of nonconserving processes are of orderO1/L, the interplay between them and the boundary pro-cesses may result in intriguing phenomena, such as ergodic-ity breaking 19,24 or the appearance of a localized shock inthe density profile 16, which is in contrast to the delocal-ized shock dynamics at the coexistence line of the TASEP8,29. The formation of domain walls can be observed alsoexperimentally in the transport of kinesin motors in accor-dance with theoretical predictions 30–32.

Other systems which have an intermediate complexitycompared to exclusion processes with bulk reactions andthose coupled to a compartment are the two-channel or mul-tichannel systems. In these models, particles are either con-served by the dynamics in each lane and interaction is real-ized by the dependence of the hop rates on the configurationof the parallel lanes 33–38, or particles can jump betweenlanes 39–42. We study in this work a two-lane exclusionprocess where particles move in the two lanes in oppositedirections. Particles are allowed to change lanes and we re-strict ourselves to the case of weak lane change rates, i.e.they are inversely proportional to the system size. Thismeans that the probability that a marked particle changeslanes during the time it resides in the system is O1. Ifparticles in one of the lanes are regarded as holes, and viceversa, this model can also be interpreted as a two-channeldriven system where particles move in the same direction inthe channels and are created and annihilated in pairs. In thehydrodynamic limit of the model, we shall construct thesteady-state phase diagram by means of analyzing the differ-ential equations describing the model on the macroscopicscale. At the coexistence line, where coherently moving de-localized shocks develop in both lanes, which is reminiscentof the delocalized shock dynamics at the coexistence line ofthe TASEP, the density profiles are studied in the frameworkof a phenomenological domain wall picture based on thehydrodynamic description. Recently, a two-lane exclusionprocess has been investigated with weak, symmetric lanechange, where particles move in the lanes in the same direc-*juhasz@szfki.hu

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tion 42. In this model, the formation of delocalized shocksin both lanes has been found, as well. In our model, even thecase of asymmetric lane change can be treated analytically inthe hydrodynamic limit if the total current is zero, whichholds also at the coexistence line.

The paper is organized as follows. In Sec. II, the model isintroduced and the hydrodynamic description is discussed. InSec. III, the case of symmetric lane change is investigated,while Sec. IV is devoted to the asymmetric case. The resultsare discussed in Sec. V and some of the calculations arepresented in two Appendixes.

II. DESCRIPTION OF THE MODEL

The model we focus on consists of two parallel one-dimensional lattices with L sites, denoted by A and B, thesites of which are either empty or occupied by a particle. Thestate of the system is specified by the set of occupation num-bers ni

A,B which are zero one for empty occupied sites. Weconsider in this system a continuous-time stochastic processwhere the occupations of pairs of adjacent sites change inde-pendently and randomly after an exponentially distributedwaiting time. The possible transitions and the correspondingrates, i.e., the inverses of the mean waiting times, are thefollowing Fig. 1. On chain A, particles attempt to jump tothe adjacent site on their right-hand side, whereas on chain Bto the adjacent site on their left-hand side, with a rate whichis set to unity, and attempts are successful when the targetsite is empty. On the first site of chain A and on the Lth siteof chain B particles are injected with rate , provided thesesites are empty, whereas on the Lth site of chain A and on thefirst site of chain B they are removed with rate . So thesystem may be regarded to be in contact with virtual particlereservoirs with densities and 1− at the entrance and exitsites, respectively. The process described so far is composedof two independent totally asymmetric simple exclusion pro-cesses. The interaction between them is realized by allowinga particle residing at site i of chain AB to hop to site i ofchain BA with rate A B, provided the target site isempty.

As in the case of the TASEP with Langmuir kinetics, onemust distinguish here between three cases, concerning theorder of magnitude of the lane change rates in the large Llimit. If the rates A and B are of larger order than O1/L,then in the limit L→, the interchain processes are domi-nant compared to the effects of the boundary reservoirs and

the horizontal motion of the particles. The densities and in lane A and B, respectively, are expected to be constant farfrom the boundaries and to fulfill the relation A1−=B1−, which is forced by the lane change kinetics.When the interchain hop rates are smaller than O1/L, thenapart from some possible special parameter combinations20 they are irrelevant in the L→ limit and the stationarystate is that of two independent exclusion processes. An in-teresting situation arises if the rates A and B are propor-tional to 1/L. In this case the effects of boundary reservoirsand those of lane change kinetics are comparable and thecompetition between them results in nontrivial density pro-files. We focus on this case in the present work, and param-etrize the lane change rates as A=A /L and B=B /L,with the constants A and B. Setting the lattice constant ato a=1/L and rescaling the time t as = t /L, we are inter-ested in the properties of the system in the continuum limitL→, where the state of the system is characterized by thelocal densities x , and x , on chain A and B, respec-tively, which are functions of the continuous space variablex 0,1 and time . Turning our attention to the subsystemcontaining lane AB alone, we see that the interchain hop-pings can be interpreted as bulk nonconserving processes forthe TASEP in lane AB. The bulk reservoir which theTASEP is connected to is, however, not homogeneous but itis characterized by the position and time-dependent densityx ,(x ,). Generally, driven diffusive systems whichare combined with a weak i.e., O1/L bulk nonconservingprocess are described on the macroscopic scale specifiedabove by the partial differential equation

x,

+J„x,…

x= S„x,… , 1

where S(x , t) is the source term related to the nonconserv-ing process and J is the current as a function of the den-sity in the steady state of the corresponding translation in-variant infinite system without nonconserving processes i.e.,S(x ,)0 17. Under these circumstances, the TASEPhas a product measure stationary state and the current densityrelationship is simply J=1−, hence the currentsin the two lanes are given as JA(x ,)=x ,1−x ,and JB(x ,)=−x ,1−x , in terms of the localdensities. For the source terms in the two lanes, wemay write SA(x , ,x ,)=−SB(x , ,x ,)=B1−x ,x ,−Ax ,1−x , since the lane changeevents at a given site are infinitely rare in the limit L→.Setting these expressions into Eq. 1 we obtain that in thesteady state, where x ,=x ,=0, the density pro-files x and x satisfy the coupled differential equations

2 − 1x + B1 − − A1 − = 0,

2 − 1x + B1 − − A1 − = 0. 2

We mention that one arrives at the same differential equa-tions when in the master equation of the process the expec-tation values of pairs of occupation numbers ninj are re-placed by the products ninj, and afterwards it is turned to

ωA ωBωB ωAβ 1

1α

A

B

1 L

β

α

FIG. 1. Transitions and the corresponding rates in the modelunder study.

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a continuum description with retaining only the first deriva-tives of the densities and neglecting the higher derivativeswhich are at most of the order O1/L almost everywhere.

For the stationary density profiles the boundary conditions0=1= and 1=0=1− are imposed. In fact,we shall keep these boundary conditions only for ,1/2; otherwise, we modify them for practical purposes atthe level of the hydrodynamic description. The reason forthis is the following. In the domain ,1/2 of the TASEP,the so-called maximum current phase, the current is limitedby the maximal carrying capacity in the bulk, J=1/4, whichis realized at the bulk density =1/2 7,8. In this phase,boundary layers form in the stationary density profile at bothends, where the density drops to the bulk value =1/2. Simi-larly, in the case of the TASEP with Langmuir kinetics, if theentrance rate exceeds the value 1/2, then in the densityprofile dictated by the reservoir at the entrance site, a bound-ary layer develops, where the density drops to 1/2. Thewidth of the boundary layer is growing sublinearly with L20, such that in the hydrodynamic limit, it shrinks to x=0and limx→0 x=1/2 holds, independently of , which in-fluences only the shape of the microscopic boundary layer.These considerations apply also to the present model at bothends and for both lanes. Therefore, in order to simplify thetreatment of the problem at the level of the hydrodynamicdescription, we use the effective boundary conditions

0 = 1 = a, 1 = 0 = 1 − b , 3

where amin ,1 /2 and bmin ,1 /2. However, westress that, although the profile propagating from, e.g., theleft-hand boundary lx is continuous at x=0 according tothe effective boundary conditions 3 for 0, a boundarylayer forms on the microscopic scale.

In addition to the boundary layers related to the maximalcarrying capacity in the bulk, the stationary density profilesmay in general contain another type of boundary layer offinite width or a localized shock in the bulk, where the den-sity has a finite variation within a region the width of whichis growing sublinearly with L 16,18. This leads to the ap-pearance of discontinuities in x and x in the hydrody-namic limit, either in the bulk 0x1 in the case of a shockor at x=0,1 in the case of a boundary layer. This is in ac-cordance with the fact that, in general, there does not exist acontinuous solution to the two first-order differential equa-tions, which fulfills all four boundary conditions. Apart fromsome special parameter combinations, there is one disconti-nuity in each lane, which is either in the bulk a shock or atx=0,1 a boundary layer. The location of the discontinuityis determined by the requirement that the currents in bothlanes JA(x) and JB(x) must be continuous functions ofx in the bulk 0x1 17,18. This follows from that thewidth of the shock region is proportional to L, thus the rateof a lane change event is vanishing there in the limit L→.This condition permits only such a shock which separatescomplementary densities on its two sides, i.e., and 1− inlane A or and 1− in lane B. The position of the shock xs,e.g., in lane A is thus given implicitly by the equationlxs=1−rxs, where lx and rx are the solutions on

the two sides of the shock. For the detailed rules on thestability of the discontinuity at x=0,1 see Ref. 17.

Subtracting the two differential equations yields the obvi-ous result that the total current

J x1 − x − x1 − x 4

is a position independent constant. This relation makes itpossible to eliminate one of the functions, say, x and toreduce the problem to the integration of a single differentialequation

d

dx= A

− 1

2± −

1

2 2

+ JK1 − +

2 − 1, 5

where we have introduced the ratio of lane change rates KB /A and the signs in front of the square root are relatedto the two solutions +x1/2 and −x1/2 of the qua-dratic equation 4. Disregarding the simple case K=1, thereare two difficulties about this equation. First, the solutiondepends on the current J as a parameter, which itself dependson the density profiles and is a priori not known. Fortunately,apart from two phases in the phase diagram, x and xsimultaneously fit to the boundary conditions either at x=0or x=1, consequently, the current is exclusively determinedby the entrance and exit rates as J=a1−a−b1−b. In theremaining two phases, the functions x and x meet theboundary conditions at the opposite ends of the system.Here, one may solve Eq. 5 iteratively until self-consistencyis attained. Second, even in the case when J is known, Eq.5 cannot be analytically integrated in general, except forthe case when the current is zero. This is realized in threecases, two of which are related to the symmetries of thesystem. We discuss these possibilities in the rest of the sec-tion.

As the two entrance and exit rates were chosen to beidentical, the obvious relation holds when the rates A andB are interchanged,

x;,,A,B = 1 − x;,,B,A , 6

where the dependence of the profiles on the four parameters, , A, and B is explicitly indicated. This relation, to-gether with Eq. 4, implies that the current changes sign ifA and B are interchanged. Thus A=B implies J=0, thatholds apparently since none of the chains is singled out inthis case.

As a consequence of the particle-hole symmetry of themodel, we have the relation

x;,,A,B = 1 − x;,,A,B . 7

Using Eq. 4, it follows that the current changes sign when and are interchanged, so it must be zero for =. Al-ternatively, this can be seen by interchanging particles andholes in one of the chains, which results in a two-channelsystem where particles move in the channels in the samedirection and particles are created and annihilated in pairs atneighboring sites of the two chains with rates A and B,respectively. Particles are injected and removed in the chan-nels with the same rate, hence the channels are equivalent.

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Since the currents of particles and holes are equal, the totalcurrent must be zero.

The third parameter regime where the current is zero isthe domain , 1/2. Here, as aforesaid, the density pro-files and the current are independent of and in the hy-drodynamic limit. Since the current is zero for = it fol-lows that J=0 in the whole domain , 1/2.

III. SYMMETRIC LANE CHANGE

We start the investigation of the model with the simplecase of equal lane change rates A=B, where thesolutions to the hydrodynamic equations are analyticallyfound and some general features of the model can be under-stood. Since the total current is zero, either x=x orx=1−x must hold. Substituting the former relationinto Eq. 2 yields

ex = ex = const, 8

whereas the latter gives

cx = 1 − cx = x + const. 9

Thus, the profiles are piecewise linear and consist of constantsegments with equal densities in the two lanes and segmentsof slope − in lane AB with complementary densities.Switching off the interchain particle exchange =0, weget two identical TASEPs, which have, apart from the coex-istence line =1/2, constant density profiles in the bulk.In the high-density phase min ,1 /2, the density, be-ing 1−, is controlled by the exit rate and the profile isdiscontinuous at x=0. In the low-density phase min ,1 /2, the density is and a discontinuity appearsat x=1 7,8. In the maximum current phase ,1/2, aswe have already mentioned, the bulk density is 1 /2 andboundary layers appear at both ends. On the other hand, theeffect of symmetric lane change processes is to diminish thedifference between the local densities in the two lanes. Sincethe densities are already equal without the interaction, thissituation is obviously not altered when switching on the ver-tical hopping processes. Consequently, the density profiles inthe bulk are identical to that of the TASEP in these phases.

This is, however, not the case at the coexistence line =1/2. In the TASEP, a sharp domain wall emerges herein the density profile, which separates a low- and a high-density phase with constant densities far from the domainwall and 1−, respectively. The stochastic motion of thedomain wall is described by an unbiased random walk withreflective boundaries 8,29, such that the average stationarydensity profile connects linearly the boundary densities and 1−. Returning to our model, we consider first theclosed system, i.e., ==0. The profiles which fulfill therequirement about the continuity of the currents are depictedin Fig. 2 for various global particle densities limL→

N2L ,

where N is the number of particles in the system. Here, thedensity profiles consist of three segments in general. In themiddle part of the system an equal-density segment is foundFigs. 2a, 2b, and 2d. This region is connected with theboundaries by complementary-density segments on its left-

hand side and on its right-hand side, which are continuous atx=0 and x=1, respectively. In both lanes, the density profileis continuous at one end of the equal-density segment and ashock is located at the other one, such that the two shocks areat opposite ends. The density in the equal-density region andat the same time the location of the shocks depend on theglobal particle density. At =1/2, the equal-density segmentis lacking if 1 Fig. 2c and the profiles are linear if=1.

If particles are allowed to enter and exit from the systemat the boundaries, i.e., 0=1/2, the total number ofparticles is no longer conserved. Nevertheless, we expect thatthe stationary density profiles averaged over configurationswith a fixed global density , x, and x, can still beconstructed from the solutions 8 and 9 of the hydrody-namic equations. These profiles are similar to those of theclosed system and the only difference is that thecomplementary-density segments fit to the altered boundaryconditions 0=1= and 1=0=1−. This isindeed the case in the limit →0. Here, the injections andremovals of particles at the boundary sites, which modify theglobal density, are infinitely rare, such that the system hasalways enough time to relax, i.e., to adjust the density pro-files to the slightly altered global density. As long as theshocks are not in the vicinity of the boundaries, the densitiesat the boundary sites are independent from the global density,which influences only the position of the equal density seg-ment. Therefore, the stochastic variation of the global densityt is described by a homogeneous, symmetric randomwalk in the interval 0, 1 with reflective boundary condi-tions. For finite , we can give only a heuristic argumentwhy we expect that the fluctuations of the global density arequasistationary in the above sense. In the stationary state, thecenter of the mass of a small instantaneous local perturbationpropagates with a velocity vx=1−2x 29,43, which

0

1

ρ(x)

,π(x

)

a)

0

1

ρ(x)

,π(x

)

a)

0

1

ρ(x)

,π(x

)

a)

0

1

ρ(x)

,π(x

)

a)

0

1

ρ(x)

,π(x

)

a)

0

1

ρ(x)

,π(x

)

a)

0

1

ρ(x)

,π(x

)

a)

0

1

ρ(x)

,π(x

)

a)

0

1

ρ(x)

,π(x

)

a)

0

1

ρ(x)

,π(x

)

a) b)b)b)b)b)b)b)b)b)b)

0

1

0 1

ρ(x)

,π(x

)

x

c)

0

1

0 1

ρ(x)

,π(x

)

x

c)

0

1

0 1

ρ(x)

,π(x

)

x

c)

0

1

0 1

ρ(x)

,π(x

)

x

c)

0

1

0 1

ρ(x)

,π(x

)

x

c)

0

1

0 1

ρ(x)

,π(x

)

x

c)

0

1

0 1

ρ(x)

,π(x

)

x

c)

0

1

0 1

ρ(x)

,π(x

)

x

c)

0 1x

d)

0 1x

d)

0 1x

d)

0 1x

d)

0 1x

d)

0 1x

d)

0 1x

d)

0 1x

d)

0 1x

d)

0 1x

d)

FIG. 2. Density profiles of the closed system ==0 with=0.5 for different global densities: a =0.26, b =0.44, c=0.5, and d =0.74. The thin solid and dashed lines representthe flow field of the differential equation 5 corresponding to thecomplementary-density and equal-density solutions, respectively.The thick solid and dashed lines are the density profiles x andx, respectively.

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changes sign at =1/2. In the complementary-density seg-ments, the perturbations in the density, which come from thefluctuations of the boundary reservoirs, are thus driven to-ward the equal-density segment with a finite velocity. Thecharacteristic traveling time of the perturbation, as well asthe time scale related to the lane change processes in a finitesystem of size L is OL. The relaxation time of the pertur-bation is thus expected to be OL. However, the randomwalk dynamics of the global density implies that the timescale of a finite change in the global density is OL2, whichis large compared to the relaxation time, thus the densityprofile has enough time to follow the instantaneous globaldensity. The fluctuating global density t is thus expectedto be a symmetric random walk with reflective boundaries at and 1−. In the stationary state, the global density istherefore homogeneously distributed in the interval ,1−.

On the other hand, one can easily calculate that if theposition of the shock in lane A is xs, the global density ofparticles in the system is xs= 1−xsxs++ for xs

1/2, where we have introduced the position dependentheight of the shock: xs=2xs−1+1−2. Note that, asopposed to the single lane TASEP, this relation is no longerlinear, therefore the probability distribution of the position ofthe shock is not uniform in the steady state.

With these prerequisites, the steady-state density profile inlane A can be easily calculated by averaging x over thesteady-state distribution of the global density: x= 1

1−21−xd. Skipping the details of the straightfor-

ward calculations, we shall give the profile x in the inter-val 1

2 x1, whereas for 0x12 , it is obtained by the help

of the relation x=1−1−x, that follows from Eqs. 6and 7. The density profile in lane B can be calculated bymaking use of Eq. 7, which implies x=1−x. Thecases corresponding to the different signs of 1

2 must be

treated separately. For 12

0, we obtain

x =1

2+

x − 122x

1 − 2, 1

2 0, 10

which is a third-degree polynomial of x. If 12

0, the sec-ond derivative of x is discontinuous at x= 1

2 . In the limit→0, the linear profile of the TASEP at the coexistence lineis recovered. If 1

2=0, Eq. 10 simplifies to

x = 12 + 4x − 1

23, 12 = 0, 11

which is everywhere analytic. For 12

0, the profile isconstant in the interval 1

2 x 12

/ 2+ 12 , where x

=1/2, while it is given by Eq. 10 in the interval 1

2 / 2+ 1

2 x1. These curves, as well as results ofMonte Carlo simulations for finite systems of size L=64,128, and 256 are shown in Fig. 3. In the numerical simula-tions, after waiting a period of 106 Monte Carlo steps inorder to reach the steady state, we have measured the localoccupancies every 10 Monte Carlo steps during a period of5109 steps. For increasing L, the properly scaled profilesseem to tend to the analytical curves expected to be valid inthe continuum limit.

IV. ASYMMETRIC LANE CHANGE

In this section, the stationary properties of the model areinvestigated in the case K1. Due to the symmetries of thesystem, we may restrict ourselves to the investigation of thepart K1, of the parameter space, which is related tothe remaining part through Eqs. 6 and 7. Analyzing thesolutions to the hydrodynamic equation 5, one can con-struct the phase diagram in the four-dimensional parameterspace spanned by , , A, and B. Two representative two-dimensional cross sections of the parameter space at fixedlane change rates are shown in Figs. 4 and 5. As can be seen,the phase boundaries are symmetric to the diagonal, which isa consequence of Eq. 7. On the other hand, the phase dia-grams are richer compared to that of the symmetric model:Besides the phases where the profiles are continuous in theinterior of the system, the asymmetry in the lane change

0

1

⟨niA

⟩

a)

Ω=0.4

L=64L=128L=256

0

1

⟨niA

⟩

a)

Ω=0.4

continuum

0

1

⟨niA

⟩

a)

Ω=0.4

0

1

⟨niA

⟩

a)

Ω=0.4

0

1

⟨niA

⟩

a)

Ω=0.4

0

1

⟨niA

⟩

b)

Ω=0.8

L=64L=128L=256

0

1

⟨niA

⟩

b)

Ω=0.8

continuum

0

1

⟨niA

⟩

b)

Ω=0.8

0

1

0 1

⟨niA

⟩

(i-0.5)/L

c)

Ω=1.2

L=64L=128L=256

0

1

0 1

⟨niA

⟩

(i-0.5)/L

c)

Ω=1.2

continuum

0

1

0 1

⟨niA

⟩

(i-0.5)/L

c)

Ω=1.2

0

1

0 1

⟨niA

⟩

(i-0.5)/L

c)

Ω=1.2

0

1

0 1

⟨niA

⟩

(i-0.5)/L

c)

Ω=1.2

0

1

0 1

⟨niA

⟩

(i-0.5)/L

c)

Ω=1.2

FIG. 3. Density profiles in lane A at the coexistence line at ==0.1, obtained by numerical simulation for different systemsizes and for different values of : a =0.4, b =0.8, and c=1.2. In the case =0.8, the height of the shock at xs=1/2 iszero. The solid curves are the analytical predictions in the hydro-dynamic limit. The thick solid lines represent the profile x atglobal density =1/2.

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kinetics leads to the appearance of phases where one of thelanes contains a localized shock in the bulk. This is reminis-cent of the shock phase of the single lane TASEP with Lang-muir kinetics. As a new feature, the position of the shockmay vary discontinuously with the boundary rates here,when the so-called discontinuity line is crossed. The coexist-ence line, where coherently moving delocalized shocksemerge in both lanes, is still present, however, it is shorterthan in the symmetric case and the shocks walk only ashrunken domain. The subsequent part of the section is de-voted to the detailed analysis of these findings.

A. Density profiles

We start the presentation of the results with the descrip-tion of the density profiles in the phases below the diagonal= of the two-dimensional phase diagrams.

If the exit rate is small enough, the densities in the bulkexceed the value 1/2 in both lanes Fig. 6a. Both profilesare continuous in the bulk and at the exits, i.e., limx→1 x=limx→0 x=1−b, but they are discontinuous at the en-trances, i.e., limx→0 xa and limx→1 xa, which sig-

nals the appearance of boundary layers on the microscopicscale. The profiles x and x, as well as the current,depend exclusively on , while influences only the bound-ary layers at the entrances. This situation is observed also inthe high-density phase of the TASEP with Langmuir kinetics16,18; therefore we call this phase H-H phase, referring tothe high density in both lanes.

In the phase denoted by S-H in the phase diagram, xand x are continuous at the exits, similarly to the H-Hphase, however, the discontinuity in x is no longer at x=0 but it is shifted to the interior of the system Fig. 6b.Thus limx→0 x=a holds and a shock is located in lane A atsome xs 0xs1. Therefore this phase is termed S-Hphase, where the letter S refers to the shock in lane A andletter H refers to the high density x1/2 in lane B. Thefunction x is discontinuous at x=1, i.e., limx→1 xaand it is not differentiable although continuous at xs. Sinceboth x and x are continuous at x=0, the total current isgiven by

J = a1 − a − b1 − b 12

in this phase. The profile x at fixed and can be com-puted by substituting the current calculated from Eq. 12into the differential equation 5. Then, the solutions propa-gating from the left-hand and the right-hand boundary, i.e.,the solutions lx and rx fulfilling the boundary condi-tions l0=a and r1=1−b, respectively, are calculatednumerically. Finally, the position of the shock xs is obtainedfrom the relation lxs=1−rxs, which is implied by thecontinuity of the current in lane A. Once x is at our dis-posal, x can be calculated from Eq. 4.

Apart from the discontinuity line to be discussed in thenext section, the position of the shock xs varies continuously

0

ρ1

0.5

1

0 ρ1 0.5 1

α

β

L-H

H-H

S-H

L-S

L-L

0

ρ1

0.5

1

0 ρ1 0.5 1

α

β

L-H

H-H

S-H

L-S

L-L

0

ρ1

0.5

1

0 ρ1 0.5 1

α

β

L-H

H-H

S-H

L-S

L-L

0

ρ1

0.5

1

0 ρ1 0.5 1

α

β

L-H

H-H

S-H

L-S

L-L

0

ρ1

0.5

1

0 ρ1 0.5 1

α

β

L-H

H-H

S-H

L-S

L-L

0

ρ1

0.5

1

0 ρ1 0.5 1

α

β

L-H

H-H

S-H

L-S

L-L

FIG. 4. Phase diagram at A=2, B=0.4. Phase boundaries areindicated by solid lines. Letters L, H, and S refer to low-density,high-density, and localized shock phase, respectively; the firstsec-ond letter refers to lane AB. The thick solid line indicates thecoexistence line. At the dashed lines, the function J , isnonanalytic.

0

ρ1

0.5

1

0 ρ1 0.5 1

α

β

L-H

H-H

S-H

L-S

L-L

0

ρ1

0.5

1

0 ρ1 0.5 1

α

β

L-H

H-H

S-H

L-S

L-L

0

ρ1

0.5

1

0 ρ1 0.5 1

α

β

L-H

H-H

S-H

L-S

L-L

0

ρ1

0.5

1

0 ρ1 0.5 1

α

β

L-H

H-H

S-H

L-S

L-L

0

ρ1

0.5

1

0 ρ1 0.5 1

α

β

L-H

H-H

S-H

L-S

L-L

0

ρ1

0.5

1

0 ρ1 0.5 1

α

β

L-H

H-H

S-H

L-S

L-L

0

ρ1

0.5

1

0 ρ1 0.5 1

α

β

L-H

H-H

S-H

L-S

L-L

0

ρ1

0.5

1

0 ρ1 0.5 1

α

β

L-H

H-H

S-H

L-S

L-L

0

ρ1

0.5

1

0 ρ1 0.5 1

α

β

L-H

H-H

S-H

L-S

L-L

FIG. 5. Phase diagram at A=2, B=1. The dotted curves arethe discontinuity lines, which terminate at the points indicated bythe full circles.

0

1

0 1

b)

1-β1-β

α

0

1

0 1

b)

1-β1-β

α

0

1

0 1

b)

1-β1-β

α

0

1

0 1

b)

1-β1-β

α

0

1

0 1

b)

1-β1-β

α

0

1

0 1

b)

1-β1-β

α

0

1

0 1

ρ(x)

,π(x

)

x

c)

1-β

α

0

1

0 1

ρ(x)

,π(x

)

x

c)

1-β

α

0

1

0 1

ρ(x)

,π(x

)

x

c)

1-β

α

0

1

0 1

ρ(x)

,π(x

)

a)

1-β 1-β

0

1

0 1

ρ(x)

,π(x

)

a)

1-β 1-β

0

1

0 1

ρ(x)

,π(x

)

a)

1-β 1-β

0

1

0 1x

d)

1-αα

0

1

0 1x

d)

1-αα

FIG. 6. Density profiles in lane A lower line and lane B upperline for the parameters A=2, B=0.4, =0.45 and for variousexit rates: a =0.15, b =0.25, c =0.35, and d =0.45.The profiles in panels a–c were obtained by numerically solvingEq. 5, whereas those in panel d are the analytical curves in Eq.14. Results of numerical simulations dotted lines obtained forsystem size L=10 000 by averaging the occupancies in a period of107 Monte Carlo steps in the steady state are hardly distinguishablefrom these curves.

RÓBERT JUHÁSZ PHYSICAL REVIEW E 76, 021117 2007

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with the boundary rates in the S-H phase. Fixing and re-ducing , xs is decreasing and at a certain value of , =H, the shock reaches the left-hand boundary at x=0. Atthis point the right-hand solution rx extends entirely to theleft-hand boundary and a further increase in drives thesystem to the H-H phase. The phase boundary H be-tween the S-H and the H-H phase is thus determined fromthe condition xs=0 or, equivalently, r1=1−a. When isincreased along a vertical path in the phase diagram at afixed , xs increases and for 1, where the constant 1will be determined later, the path hits the coexistence linebefore the shock would reach the right-hand boundary seeSec. 3D. Increasing along a path at some 1, theshock reaches the right-hand boundary at x=1 for a certainvalue of , =L and the path leaves the S-H phase. Atthe phase boundary, the left-hand solution lx extends tothe whole system and l1=b must hold.

Crossing the phase boundary L, the L-H phase is en-tered, where letter L refers to the low density in lane A sincehere, x1/2 and x1/2 hold in the bulk Fig. 6c.In this phase, x and x are discontinuous at x=1,whereas they are continuous at x=0 hence the current isgiven by Eq. 12. In the part of the L-H phase where thecurrent is zero, i.e., if = or , 1/2, the profiles can becalculated analytically. The equal-density solutions of Eq. 5are

lnex1 − ex = AK − 1x + const,

ex = ex , 13

whereas the complementary-density solutions read as

1

2lncx − 1 −

2

1lncx − 2 = 2A

Kx + const,

cx = 1 − cx , 14

where the constants 11/ 1+K−1/2 and 21/ 1−K−1/2are the roots of the equation SA ,1−=0. There is, further-more, a special complementary-density solution with con-stant densities,

cx = 1,

cx = 1 − 1. 15

In the part of the L-H phase where J=0, the profiles aregiven by the complementary-density solution which fulfillsthe boundary conditions c0=a and c0=1−a Fig.6d.

B. Phase boundaries and the discontinuity line

In the S-H phase and the L-H phase, the profiles xand x, as well as the current are independent of if 1/2. Here, limx→0 x=1/2 and influences only the mi-croscopic boundary layer, as we argued in Sec. II. As a con-sequence, the phase boundaries H and L are hori-zontal lines in the domain 1/2 see Figs. 4 and 5 and we

may restrict the investigation of the phase boundaries to thedomain 1/2.

Although we cannot give an analytical expression for thedensity profiles in general, some information can be gainedon the phase boundaries of the S-H phase by investigatingthe constant solutions of the hydrodynamic equations. A con-stant solution x=r, x= p must obey SAr , p=0, other-wise the spatial derivatives xx and xx would not van-ish in Eq. 2. On the other hand, the constants satisfy theequation J=r1−r− p1− p, where J is determined by theboundary rates via Eq. 12. Eliminating p yields that r isgiven by the implicit equation

rr − 11 −K

K + 1 − Kr2 = J, . 16

In the S-H phase, this equation has two roots r1(J , ,K)and r2(J , ,K) shortly r1 and r2 in the interval 0, 1.One can check that for the larger root r2, the relation 1/21−r2 holds, whereas the smaller one, r1, may be largeror smaller than 1/2.

First, we examine the phase boundary separating the L-Hphase and the S-H phase. One can check that at this bound-ary line, r11/2 holds. Moreover, it follows from Eq. 2that

dx

dx 0 if 0xr1, anddx

dx 0 if r1x1/2.Thus, the line x=r1 behaves as an attractor for the solu-tions lx propagating from the left-hand boundary x=0 ifl0=1/2, meaning that lx approaches monotonouslyto r1 as x increases and limx→ lx=r1. Since lx is mo-notonous and l0=, as well as l1= hold at the phaseboundary, at the common point of the boundary line and thediagonal =, lx must be a constant function lx=.This, however, implies that must coincide with r1. Theendpoint of the boundary line L is therefore at ==r1J=0,K=1/ 1+K−1/2, which depends only on K.

As opposed to this point, the whole function L de-pends both on A and B. Nevertheless, we can find ananalytical expression for L in the limit K=const, A

→. We can see from Eq. 5 that the derivativedx

dx isproportional to A for a fixed K. As a consequence, thelarger A is the more rapidly lx tends to r1. Thus, in thelimit specified above, limA→l1−r1=0, which leads to=r1. Substituting this into Eq. 16, we obtain for the in-verse of the boundary curve L in the limit K=const,A→,

L−1

=1

21 −1 − 41 − 2 −

K

K + 1 − K2 .

17

This curve is plotted in Fig. 7. The phase boundaries ob-tained by integrating Eq. 5 numerically for finite lanechange rates tend rapidly to this limiting curve.

Next, we turn to examine the boundary curve between theS-H and the H-H phase, H. Along this line, r0=1− and r1=1− hold, and the roots of Eq. 16 are ar-

WEAKLY COUPLED, ANTIPARALLEL, TOTALLY… PHYSICAL REVIEW E 76, 021117 2007

021117-7

ranged as r11−1−r2. One can show that the linex=r2 is an attractor for the solutions lx which start atx=0 if l0max1/2 ,r1. Moreover, if r11/2, the linex=r1 repels the solutions lx starting from x=0 if r1

l0r2, or, in other words, the solutions rx propagat-ing from the right-hand boundary, for which r1r1r2,are attracted by the line x=r1. When the diagonal is ap-proached along the phase boundary L, the profile rx,being monotonous, must tend to the constant function rx=1−, as well as x since the current is zero at the diag-onal. However, equal densities in the two lanes are possiblefor K1 only if the density is one or zero, therefore theboundary line must approach the diagonal at =0. This is inaccordance with the fact that r2=1 if J=0. Thus, we obtainlim→0 L=0, independently of the lane change rates.

Similarly to L, the location of the whole boundaryline H depends on both lane change rates see Figs. 7and 8 and we can give an analytical expression for Hagain only in the limit K=const, A→. As aforesaid, theprofile given by rx lies between two attractors, x=r1

and x=r2, to which the solutions tend in the limit x→− and x→, respectively. When A is increased such that

K is fixed, then L at a fixed decreases. Thus, thecurrent is increasing and the two roots of Eq. 16, r1 and r2are coming closer. Keeping in mind that r11−1−r2, there are now two possible cases. Depending on thevalue of , either the gap between 1− and r2 or the gapbetween r1 and 1− vanishes first. In other words, in theformer case, the profile is attracted to the line x=r2 at x=1 in the limit A→, i.e., limA→r1−r2=0, whereasin the latter case it is attracted to the line x=r1 at x=0provided that r11/2, i.e., limA→r0−r1=0. In thefirst case, substituting r=1− into Eq. 16, we obtain forthe inverse of the limiting curve H

b in terms of L,

Hb −1 = L−11 − , 18

whereas in the second case, r=1− yields

Ha =

1

21 −1 −

41 − KK + 1 − K1 − 2 . 19

These curves are plotted in Fig. 8. The phase boundary in thelimit K=const, A→ is given by

H = maxHa ,H

b . 20

The value * at which the functions Ha and H

b inter-sect varies with K. If K→1, * tends to

3−123

=0.211 32. . .,while *=1/2 if K is equal to

K* = 1 + 2 − 21 + 2 = 0.216 84 ¯ . 21

Thus, for KK*, the limiting curve of H is given by Eq.18 alone, otherwise it is composed of Eq. 18 and Eq. 19as given by Eq. 20.

Although the curve H gives the phase boundary lineonly in the limit K=const, A→, we show that for KK* and for large enough A, A A

*K, the phase tran-sition point at =1/2 is given exactly by Eq. 19, i.e.,H1/2=H

a 1/2= K1+K .

In order to see this, we discuss first the possible appear-ance of a discontinuity line in the S-H phase, at which theposition of the shock xs changes discontinuously. If r1=1/2,one can see from Eq. 2 that for the spatial derivative of the

profile, lim→1/2

dx

dx 0 holds and the left-hand and right-hand solutions lx and rx may propagate as far as theline x=1/2. If lx1=1/2 and rx2=1/2 hold for somex1 and x2, such that 0x1x21, then the left-hand andright-hand solutions are connected by a constant segmentx=1/2 in the interval x1 ,x2 and the profile is continuousFig. 9c. Substituting r=1/2 into Eq. 16 we obtain theequation of the discontinuity line, 1−−1−= 1−K

21+K 2, along which the current is constant. Solving this

equation for , we obtain

d =1

21 −1 − 41 − + 4 1 − K

21 + K 2 .

22

This curve is shown in Fig. 5. When at an arbitrary pointof this line, is infinitesimally decreased, then r1 exceeds

0

ρ1

0.5

0 ρ1 0.5

α

β

0

ρ1

0.5

0 ρ1 0.5

α

β

0

ρ1

0.5

0 ρ1 0.5

α

β

0

ρ1

0.5

0 ρ1 0.5

α

β

ΩA=0.5ΩA=1ΩA=2ΩA=4ΩA=8

0

ρ1

0.5

0 ρ1 0.5

α

β

FIG. 7. Phase boundaries of the S-H phase obtained by integrat-ing Eq. 5 numerically for K=1/5 and for various A. The solidlines are the limiting curves: L, given by Eq. 17 and H,given solely by Eq. 18 since KK*.

0

ρ1

0.5

0 ρ1 0.5

α

β

↑β

H∞(α)a

↑β

H∞(α)b

0

ρ1

0.5

0 ρ1 0.5

α

β

↑β

H∞(α)a

↑β

H∞(α)b

0

ρ1

0.5

0 ρ1 0.5

α

β

↑β

H∞(α)a

↑β

H∞(α)b

ΩA=0.2ΩA=0.5

ΩA=0.719ΩA=1ΩA=2ΩA=4

FIG. 8. Phase boundaries between the S-H phase and the H-Hphase obtained by numerical integration of Eq. 5 for K=1/2K* and for several values of A. The solid lines are the limitingcurves given in Eq. 18 and Eq. 19.

RÓBERT JUHÁSZ PHYSICAL REVIEW E 76, 021117 2007

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the value 1/2 and a shock appears at x1 with an infinitesimalheight Fig. 9a. Conversely, an infinitesimal increase in decreases r1 below 1/2 and an infinitesimal shock appears atx2 Fig. 9b. Thus, when this line is crossed, the position ofthe shock jumps from x1 to x2. At the point of the disconti-nuity line at =1/2, x1=0 holds and an infinitesimal in-crease decrease in drives the system to the H-H S-Hphase. Therefore the point of this curve at =1/2 coincideswith the phase boundary, i.e., d1/2=H1/2. On theother hand, comparing Eq. 19 and Eq. 22, we see thatd1/2=H

a 1/2= K1+K . The function H

a thus givesthe phase transition point at =1/2 exactly, provided thecurve d is located in the S-H phase. That means if KK* and A A

*K, where A*K is the value of A at

which H1/2 first reaches the value K1+K , when A is in-

creased from zero. For example, for K=1/2, we have foundA

*1/20.719 Fig. 8. We emphasize, however, that thediscontinuity line is lacking if KK* or KK* but AA

*K.If KK* and AA

*K, then, at the transition point at=1/2, we have x1=0; A influences only x2 and x2→0 ifA→A

*K. Moving away from the point at =1/2 alongthe discontinuity line, the length of the constant segment x2−x1 is decreasing and vanishes at a certain point. Here, thedensity profile becomes analytical and the discontinuity lineterminates see Fig. 9d and Fig. 5. The position of thisendpoint depends on A and it is moving toward smallervalues of for increasing A; in the large A limit, it tendsto the point of intersection of d and H

b , whereas in thelimit A→A

*K, the abscissa of the endpoint tends to 1/2.Finally, we mention that an other special curve in the S-H

phase is defined by the equation SA ,1−=0. At thiscurve the left-hand solution lx is constant, therefore bothx and x are constant in the interval 0,xs.

C. Current

We have seen that, apart from the H-H and the L-Lphase, the current is given by Eq. 12. It is zero at the line

= and in the domain , 1/2 and it is independent of in the H-H L-L phase. It is a continuous function of theboundary rates everywhere, although nonanalytic at theboundaries of the H-H phase and the L-L phase, as well as atthe lines =1/2 and =1/2 outside the H-H phase and theL-L phase, where the effective boundary rates a and b satu-rate at 1 /2.

In the following, we concentrate on the current in the H-Hphase. Since it is continuous at the boundary line H, itcan be expressed in the H-H phase in terms of H asJ=H

−11−H−1−1−. According to numerical

results see Fig. 10, the current J has a maximum in theH-H phase, and for large A, the location of the maximumtends to =H1/2 for KK*, while for KK*, it tends to* where the curves H

a and Hb intersect. Contrary

to this, the current is a monotonously decreasing function of in the S-H phase. We now turn to the question, at whichparameter combination the total current is maximal in thesteady state. For fixed K, the maximal current is realized inthe limit A→ at H1/2 for KK* and at * for KK*. Since the current is growing faster in the S-H phase fordecreasing than in the H-H phase above * if KK*, weconclude that the parameter combination that maximizes thecurrent is found in the domain KK*. For KK*, the maxi-mal current is thus JmaxK=1/4−H1/21−H1/2.Since H1/2 decreases monotonously with decreasing K,the current is maximal at K=0, where H1/2= 2−2

4 andJmax0=1/8. The value 1/8 is thus an upper bound for thetotal current and J→1/8 in the limit A→ if 1/2,

= 2−24 and B=0.We close this section with the discussion of the current in

the H-H phase when the lane change rates are small, i.e., A,B1. If the vertical hopping processes are switched off,the current is zero, hence we expect that for small lanechange rates the current is small, as well. Assuming that J 1

2 −2 and expanding the right-hand side of Eq. 5 in aTaylor series up to first order in J, then solving the resultingdifferential equation yields finally

J =1 − K

1 + K1 − 1 − 2

1 + 21 + KA

1 − 2− 1 + OA

3 . 23

The details of the calculation are presented in Appendix A.

0

1

0 xs 1

b)

0

1

0 xs 1

b)

0

1

0 xs 1

b)

0

1

0 x1 x2 1

ρ(x)

x

c)

0

1

0 x1 x2 1

ρ(x)

x

c)

0

1

0 x1 x2 1

ρ(x)

x

c)

0

1

0xs 1

ρ(x)

a)

0

1

0xs 1

ρ(x)

a)

0

1

0xs 1

ρ(x)

a)

0

1

0 1x

d)

FIG. 9. Density profiles for parameters A=2, B=1 and a=0.4, =0.29, where r1 is slightly above 1/2; b =0.4, =0.32, where r1 is slightly below 1/2; c =0.4, 0.305, wherer1=1/2. d The endpoint of the discontinuity line at 0.297, 0.237.

0

0.1

0 βH∞(1/2) 0.5

J(β)

β

a)

0

0.1

0 βH∞(1/2) 0.5

J(β)

β

a)

0

0.1

0 βH∞(1/2) 0.5

J(β)

β

a)

0 βH∞(1/2)β 0.5β

b)

*0 βH∞(1/2)β 0.5β

b)

*0 βH∞(1/2)β 0.5β

b)

*0 βH∞(1/2)β 0.5β

b)

*

FIG. 10. The current as a function of along the line =1/2 forK=1/5 a and K=1/2 b. The solid line is the current in the limitA→, the other curves from bottom to top correspond to A

=0.25,0.5,1 ,2 ,4, respectively.

WEAKLY COUPLED, ANTIPARALLEL, TOTALLY… PHYSICAL REVIEW E 76, 021117 2007

021117-9

This expression is compared to the current calculated by in-tegrating Eq. 5 numerically in Fig. 11. Expanding this ex-pression for small A, we obtain

J = 1 − 1 − KA1 −1 + K

2

A

1 − 2 + OA

3 .

24

The current is thus in leading order proportional to A. Ex-amining the higher-order terms in the series expansion of theright-hand side of Eq. 5, one can show that for arbitraryA, the current vanishes as J1−K when K→1.

D. Coexistence line

Let us assume that a given point of the section =1 is approached along a path in the S-H phase. In thiscase, the position of the shock in lane A, xs, tends to somexmin=xmin ,A ,B, which is somewhere in the bulk, i.e.,0xmin1. When the same point is approached from theL-S phase, the position of the shock in lane B tends to thesame xmin according to Eq. 7. Thus, when the boundary line=1 is passed from the S-H phase, the shock in lane Ajumps from xmin to the right-hand boundary at x=1, where adiscontinuity appears, while the discontinuity at x=1 in laneB, which can be regarded as a shock localized there, jumps toxmin. So, the density profile changes discontinuously. Strictlyat =, the shocks in both lanes are delocalized and performa stochastic motion in the domain xmin,1, similarly to thesymmetric model with K=1 at the line =1/2.

Now, this phenomenon is investigated in detail in the caseof asymmetric lane change. Since the current is zero if =, the solutions of the hydrodynamic equations are thosegiven in Eq. 13 and Eq. 14 see Fig. 12. The argumen-tation about the quasistationarity of the fluctuations of theglobal density presented in the case K=1 apply to the caseK1, as well. Thus, in the open system, the density profilesaveraged over configurations with a fixed global density,x and x, can be constructed from the solutions 13and 14.

The structure of these profiles is identical to those ob-tained in the case K=1 see Fig. 13. Generally, they consist

of three segments see Fig. 13b: An equal-density segmentis located in the middle part of the system in the domainx0 , x0. This region is connected with the left-hand boundaryby a complementary-density segment, which is continuous atx=0, i.e., c,l0=, c,l0=1−, and with the right-handboundary by another complementary-density segment, whichis continuous at x=1, i.e., c,r1=1−, c,r1=. Eachlane contains a shock, which are at the opposite ends of theequal-density segment one at x0, the other one at x0. Thelocation of the equal-density segment, as well as x0 and x0are determined by the actual global density. If x0= x0 x, theequal-density segment is lacking and c,lx and c,rx aredirectly connected by a shock at x Fig. 13c. Since x0 andx0 are not independent, the shocks move in a synchronizedway, and their motion is confined to the range xmin,1. Ifone of the shocks is at x=1, the other one is at xmin, thus, thelower bound xmin is determined by the equation c,lxmin=1−exmin, where ex is the equal-density solution whichfulfills e1=1− see Fig. 13a. The lower bound xmin is

0

0.01

0.02

0 0.4

J(β)

β

0

0.01

0.02

0 0.4

J(β)

β

0

0.01

0.02

0 0.4

J(β)

β

0

0.01

0.02

0 0.4

J(β)

β

ΩA=0.2ΩA=0.1

ΩA=0.05

FIG. 11. The current as a function of in the H-H phase ob-tained by numerical integration for K=1/2 and for different valuesof A. The analytical approximation given in Eq. 23 is indicatedby solid lines.

0

ρ1

0.5

1

0 0.5 1

ρ(x)

x

0

ρ1

0.5

1

0 0.5 1

ρ(x)

x

0

ρ1

0.5

1

0 0.5 1

ρ(x)

x

0

ρ1

0.5

1

0 0.5 1

ρ(x)

x

0

ρ1

0.5

1

0 0.5 1

ρ(x)

x

0

ρ1

0.5

1

0 0.5 1

ρ(x)

x

0

ρ1

0.5

1

0 0.5 1

ρ(x)

x

FIG. 12. The flow field of the differential equation 5 for A

=2, B=0.5, and J=0. The solid curves represent thecomplementary-density solutions given in Eq. 14, whereas dashedcurves represent the equal-density solutions given in Eq. 13.

0

1

xmin

ρ ρ(x)

,πρ(

x)

a)

0

1

xmin

ρ ρ(x)

,πρ(

x)

a)

0

1

xmin

ρ ρ(x)

,πρ(

x)

a)

0

1

xmin

ρ ρ(x)

,πρ(

x)

a)

0

1

xmin

ρ ρ(x)

,πρ(

x)

a)

0

1

xmin

ρ ρ(x)

,πρ(

x)

a)

0

1

xmin

ρ ρ(x)

,πρ(

x)

a)

0

1

xmin

ρ ρ(x)

,πρ(

x)

a)

0

1

xmin

ρ ρ(x)

,πρ(

x)

a)

x0

x0

b)

∼x0

x0

b)

∼x0

x0

b)

∼x0

x0

b)

∼x0

x0

b)

∼x0

x0

b)

∼x0

x0

b)

∼x0

x0

b)

∼x0

x0

b)

∼x0

x0

b)

∼x0

x0

b)

∼

0

1

0 x 1

ρ ρ(x)

,πρ(

x)

x

c)

−0

1

0 x 1

ρ ρ(x)

,πρ(

x)

x

c)

−0

1

0 x 1

ρ ρ(x)

,πρ(

x)

x

c)

−0

1

0 x 1

ρ ρ(x)

,πρ(

x)

x

c)

−0

1

0 x 1

ρ ρ(x)

,πρ(

x)

x

c)

−0

1

0 x 1

ρ ρ(x)

,πρ(

x)

x

c)

−0

1

0 x 1

ρ ρ(x)

,πρ(

x)

x

c)

−0

1

0 x 1

ρ ρ(x)

,πρ(

x)

x

c)

−0

1

0 x 1

ρ ρ(x)

,πρ(

x)

x

c)

− 0 x0

x0 1

x

d)

∼0 x0

x0 1

x

d)

∼0 x0

x0 1

x

d)

∼0 x0

x0 1

x

d)

∼0 x0

x0 1

x

d)

∼0 x0

x0 1

x

d)

∼0 x0

x0 1

x

d)

∼0 x0

x0 1

x

d)

∼0 x0

x0 1

x

d)

∼0 x0

x0 1

x

d)

∼0 x0

x0 1

x

d)

∼

FIG. 13. Density profiles in lane A thick solid lines and in laneB thick dashed lines which belong to a fixed global particle den-sity , at four different values of . The flow field of Eq. 5 isindicated by thin lines. The rates are A=0.5, B=0.25, and ==0.1.

RÓBERT JUHÁSZ PHYSICAL REVIEW E 76, 021117 2007

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an increasing function of see Fig. 14. In the limit →0,xmin tends to zero and if →1, xmin tends to 1, thus, at =1, the shock becomes localized at x=1 and the systementers the L-H phase. At this point, the density profile isgiven by the special complementary-density solution, cx=, cx=1−.

Similarly to the case K=1, the stochastic variation of theglobal density t is described by a bounded symmetricrandom walk. The stationary density profile x can be ob-tained from the profile x at a fixed global density byaveraging it over . For K1, we could not carry out theaveraging analytically, nevertheless, we can gain some infor-mation on the stationary density profiles at x without theknowledge of the entire profile. At the point x, the relationx=x holds for all . This, together with the relationx=1−x following from Eq. 7 implies that x=x=1/2. As it is shown in Appendix B, the ratio of thefirst derivatives of the stationary density profile in lane A onthe two sides of the point x is

dx − dx dx +

dx=

K + 1 − K0

1 − 1 − K0, 25

where 0c,x. In the case K1, 011/2 alwaysholds, hence this ratio is smaller than 1 and the first deriva-tive of the density profile is discontinuous at x. Furthermore,it is clear that the stationary density profile is identical to thecomplementary-density solution in the interval 0,xmin,since this domain is forbidden for the shocks. We have per-formed numerical simulations for finite systems of size L=64, 128, and 256 and measured the density profiles in thesame way as in the symmetric case at the coexistence line.Results are shown in Fig. 15.

V. DISCUSSION

In this work, a two-lane exclusion process was studied,where the particles are conserved in the bulk and each lanecan be thought of as a totally asymmetric simple exclusionprocess with nonconserving kinetics in the bulk. As a conse-quence, the model unifies the features of the particle conserv-ing and bulk nonconserving exclusion processes, as far as thedynamics of the shock is concerned. Namely, it exhibits

phases with a localized shock in one lane, while the otherone acts as a nonhomogeneous bulk reservoir, the position-dependent density of which is determined by the dynamicsitself in a self-organized manner. On the other hand, themodel undergoes a discontinuous phase transition at the co-existence line, where delocalized shocks form in both lanesand move in a synchronized way. Here, the global density ofparticles behaves as an unbiased random walk, similarly tothe TASEP at the coexistence line, however, the density pro-files in the coexisting phases are not constant here.

Although we considered throughout this work exchangerates proportional to 1/L, one may imagine other types ofscaling. For the TASEP with Langmuir kinetics, shock local-ization is observed at the coexistence line when the creationand annihilation rates vanish proportionally to 1/La with 1a2 20. It might be worth examining whether the syn-chronization of shocks in the present model persists for lanechange rates vanishing faster than 1/L.

In the limit of large lane change rates, K=const, A→,the density profiles and the boundaries of the phases exhib-iting a localized shock are related to the zeros of the sourceterm in the hydrodynamic equation, which is generallyvalid for systems with weak bulk nonconserving kinetics16,25,42. However, different behavior is observed in gen-eral when the lane change rates A and B are finite in thelimit L→. In this case, the particle current from one lane tothe other may be finite at the boundary layers. These currentsadd to the incoming and/or outgoing currents at the bound-aries, which may lead to nontrivial bulk densities even in thecase K=1, where the profiles are constant.

Possible extensions of the present model are obtainedwhen different exit and entrance rates or different hop ratesin the two lanes are taken into account. Nevertheless, thesegeneralized versions are more difficult to treat because of thereduced symmetry compared to that of the present model.

ACKNOWLEDGMENTS

The author thanks L. Santen and F. Iglói for useful dis-cussions. This work has been supported by the National Of-fice of Research and Technology under Grant No. ASEP1111

0

ρ1

0.5

1

0 0.5 1

ρ ρ(x)

x

0

ρ1

0.5

1

0 0.5 1

ρ ρ(x)

x

0

ρ1

0.5

1

0 0.5 1

ρ ρ(x)

x

0

ρ1

0.5

1

0 0.5 1

ρ ρ(x)

x

0

ρ1

0.5

1

0 0.5 1

ρ ρ(x)

x

0

ρ1

0.5

1

0 0.5 1

ρ ρ(x)

x

0

ρ1

0.5

1

0 0.5 1

ρ ρ(x)

x

0

ρ1

0.5

1

0 0.5 1

ρ ρ(x)

x

0

ρ1

0.5

1

0 0.5 1

ρ ρ(x)

x

0

ρ1

0.5

1

0 0.5 1

ρ ρ(x)

x

0

ρ1

0.5

1

0 0.5 1

ρ ρ(x)

x

0

ρ1

0.5

1

0 0.5 1

ρ ρ(x)

x

0

ρ1

0.5

1

0 0.5 1

ρ ρ(x)

x

0

ρ1

0.5

1

0 0.5 1

ρ ρ(x)

x

0

ρ1

0.5

1

0 0.5 1

ρ ρ(x)

x

0

ρ1

0.5

1

0 0.5 1

ρ ρ(x)

x

FIG. 14. The shock in lane A at the leftmost possible positionfor A=0.5, B=0.25, and for different rates =.

0

1

0 1

⟨niA

⟩

(i-0.5)/L0

1

0 1

⟨niA

⟩

(i-0.5)/L0

1

0 1

⟨niA

⟩

(i-0.5)/L0

1

0 1

⟨niA

⟩

(i-0.5)/L

L=64L=128L=256

FIG. 15. Density profiles in lane A obtained by numerical simu-lations with rates ==0.1, A=0.5, B=0.25, and for differentsystem sizes thin lines. The thick line represents the density pro-file x at global density =1/2 in the continuum limit.

WEAKLY COUPLED, ANTIPARALLEL, TOTALLY… PHYSICAL REVIEW E 76, 021117 2007

021117-11

and by the Deutsche Forschungsgemeinschaft under GrantNo. SA864/2-2.

APPENDIX A

We derive here an approximative expression for the cur-rent in the H-H phase in the limit of small lane change rates.Assuming that J 1

2 −2, we may expand the right-handside of the differential equation 5 in a Taylor series up tofirst order in J. Integrating the differential equation obtainedin this way yields

F 1

K − 1ln − 2 + J K

1 − K2ln

1 − −

1

+1

1 − K2ln

1 − +

1

1 − = Ax + const,

A1

where the first term on the left-hand side is just the equal-density solution for J=0. The solution which obeys theboundary condition 1=1− is F−F1−=Ax−1.From this equation, we get for the density at the left-handboundary, 0 limx→0 x, the implicit equation,

F0 − F1 − = − A. A2

On the other hand, we have another relation between J and0,

J = 01 − 0 − 1 − , A3

thus, we have closed equations for current. Note that theleading order term on the left-hand side of Eq. A2, whichcomes from the difference of the leading terms of Fevaluated at 0 and 1−, is OJ, while the next-to-leadingcontribution in Eq. A2 is OJ2. Expressing 0 from Eq.A3 and expanding it for small J, we get 0=1−− J

1−2

− J2

1−23 +OJ3. Substituting this expression into Eq. A2gives an implicit equation for J. Assuming that J andexpanding the terms containing J in this equation in Taylorseries, we obtain finally

+1 + K

22 + O3 =

A

1 − 2, A4

where J1−21−1−K . For small , which amounts to

A1−2, we get a good approximation for the current bysolving this quadratic equation and arrive at Eq. 23.

APPENDIX B

As we have seen, the positions of the shocks in lane A andlane B are not independent, thus, we may define thereby afunction x0x0, which is given implicitly by the equationex0=1−c,rx0, where ex is the equal-density solutionwhich satisfies the condition c,lx0=ex0. In the follow-ing, we shall denote the density in lane A at x0, if the shockis located at xs, by xs

x0. First, we notice that at the refer-ence point x0 x0 x, xs

x0=c,lx0 holds whenever theshock in lane A resides between x0 and x0x0, i.e., x0xs

x0x0. Similarly, xs(x0x0)=c,rx0 if x0xs x0x0.

When the shock is outside this interval xs x0 , x0, thenxs

x0=ex0, where ex is some equal-density solutiondetermined by the global density and xs

x01/2 orxs

x01/2 if xsx0 or xs x0x0, respectively. As a con-sequence of the particle-hole symmetry, the relation xs

x0=1−xsxs

x0 holds if xs x0 , x0. Furthermore, in the sta-tionary state, the probability that xsx0 is equal to the prob-ability that xs x0x0 for any x0 x. Therefore the contribu-tion to the average profile is 1 /2 when xs x0 , x0. We canthus write for the average densities at x0 x and x0x0 x,

x0 = px0c,lx0 + 1 − px0 12 ,

x0 = p„x0x0…c,rx0 + 1 − p„x0x0… 12 , B1

respectively, where px0 is the probability that the shock inlane A resides in the interval x0 , x0x0, and x0x0 is theinverse function of x0x0. For the spatial derivatives of thedensities at x0 and x0x0, we get

x0 = px0c,lx0 − 12 + px0c,l x0 ,

x0 = px0x0x0c,lx0 − 12 + p„x0x0…c,r x0 ,

B2

where the prime denotes derivation. Using that px=0 andc,lx=1−c,rx, we obtain for the ratio of the left- andright-hand side derivatives at x,

x − /x + = x0x . B3

Expanding the functions c,lx, c,rx, and ex in Taylorseries up to first order in x around x= x, we obtain

x0x =c,l x − exex − c,r x

. B4

Using Eq. 2, these derivatives can be given in terms of0c,x and we arrive at Eq. 25.

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