Exclusion in Junction Geometries
EExclusion in Junction Geometries
Keming Zhang, P. L. Krapivsky, and S. Redner Department of Computer Science, Rice University, Houston, TX, 77005 USA Department of Physics, Boston University, Boston, MA, 02215 USA Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM, 87501 USA
We investigate the dynamics of the asymmetric exclusion process at a junction. When two inputroads are initially fully occupied and a single output road is initially empty, the ensuing rarefactionwave has a rich spatial structure. The density profile also changes dramatically as the initial densitiesare varied. Related phenomenology arises when one road feeds into two. Finally, we determine thephase diagram of the open system, where particles are fed into two roads at rate α for each road,the two roads merge into one, and particles are extracted from the single output road at rate β . I. INTRODUCTION
In exclusion processes, sites can be occupied by atmost one particle and particles hop to empty sites. Thisparadigmatic model sheds much light on non-equilibriumsteady states, large deviations, and other aspects ofstrongly-interacting infinite-particle systems (see, e.g.,[1–7] and references therein). The totally asymmetricsimple exclusion process (TASEP), where particles canhop to neighboring empty sites in one direction, is a min-imalist realization of exclusion processes that is particu-larly tractable and also has a diverse range of applica-tions [8–11].In this work, we investigate the properties of theTASEP at a junction , where a small number of incomingroads, that each carry a TASEP, meet at a single pointand particles leave via an outgoing road (or roads) alsoby the TASEP (Fig. 1). Our initial motivation came fromthe observation of maddening delays that arise when dis-embarking from a passenger plane. Here, the aisle(s) getclogged with passengers who are either slow in retrievingtheir belongings or in walking, leading to a clogging atthe exit door of the plane. Our junction TASEP model isa rough caricature for this disembarkment process. Westudy in detail (Sec. III) the (2 ,
1) junction geometry withtwo roads that start at x = −∞ and merge at x = 0 intoa single outgoing road that extends to x = + ∞ . Wealso analyze the (1 ,
2) junction geometry (Sec. IV) andfinite systems (Sec. V). Our analysis can be generalizedto other junction geometries. (a) (b)
FIG. 1: Illustration of the TASEP at: (a) a (2 ,
1) junction and(b) a (1 ,
2) junction. Shown is the downstep initial conditionin which sites are fully occupied for x <
For the (2 ,
1) junction geometry, one might expect apileup of particles as the junction is approached, remi- niscent of what occurs when highway traffic approaches alane constriction. The role of blockage in the TASEP hasbeen considered previously in a one-dimensional geome-try in which the hopping rate of a single bond is reducedfrom 1 to r < ρ ( x, t ) as thebasic dynamical variable in the long-time limit.For the “downstep” initial condition in the (2 ,
1) junc-tion geometry, in which each site on the two incomingroads are initially occupied while the single outgoing roadis empty, the density profile at long times contains botha constant-density jammed segment upstream from thejunction, as well as a downstream linear rarefaction wave(Fig. 2). As the initial density in the incoming roads isdecreased, the form of the rarefaction wave changes dra-matically and a shock wave can even arise. Similarly richphenomenology arises for the (1 ,
2) junction geometry.Finally, we study the open (2 ,
1) system in which currentis fed in to the system at rate α at each upstream roadfar from the junction and current is extracted at rate β in the single road far downstream from the junction. Wemap out the phase diagram of this system and highlightthe differences with the open TASEP system on the line. a r X i v : . [ c ond - m a t . s t a t - m ec h ] M a y II. SHOCK AND RAREFACTION WAVES
As a preliminary, we recapitulate the well-known (see,e.g., [7]) density profile that arises in the TASEP on theline for the initial density step ρ = (cid:40) ρ L x < ,ρ R x > , (1)where ρ L and ρ R are constant densities to the left andto the right of the step. The hydrodynamic behavior isgoverned by the continuity equation ∂ρ∂t + ∂j∂x = 0 , (2)with the current given by j = ρ (1 − ρ ). The solution toEq. (2) subject to (1) has a remarkably simple scalingform ρ ( x, t ) = f ( z ) , z = x/t . (3)When this scaling form is substituted into (2), two dis-tinct behaviors arise that depend on whether ρ L > ρ R or ρ L < ρ R : • Rarefaction wave ( ρ L > ρ R ). An initial down-step relaxes to the rarefaction wave ρ ( x, t ) = ρ L z < − ρ L , (1 − z ) 1 − ρ L < z < − ρ R ,ρ R z > − ρ R . (4a) • Shock wave ( ρ L < ρ R ). An upstep persists as ashock wave and merely translates: ρ ( x, t ) = (cid:40) ρ L z < c ,ρ R z > c , (4b)with shock speed c = 1 − ρ R − ρ L . III. RAREFACTION AT A (2,1) JUNCTION
We now investigate the evolution of the initial den-sity step (1) at a junction where two roads merge intoone. Here, and in the following section, the systemis unbounded, with the incoming road(s) extending to x = −∞ and the outgoing road(s) extending to x = + ∞ .For simplicity, we treat the special case where the outgo-ing road is empty, ρ R = 0, and where the initial densitiesin the incoming roads are both equal to ρ L . Three dis-tinct behaviors arise for: (a) ρ L > ρ + , (b) ρ + > ρ L > ρ − ,and (c) ρ L < ρ − , where ρ + and ρ − are critical densitieswhose values are given in Eq. (6) below. We discuss thesethree cases in turn. A. High input density: ρ L ≥ ρ + As in the conventional TASEP, a density downstep de-velops into a rarefaction wave in the subrange 0 < x < t .However, for ρ L ≥ ρ + , a density pileup develops just up-stream from the junction, with a sudden density drop at x = 0 (Fig. 2). This same qualitative behavior occurs aslong as the initial input density ρ L is greater than ρ + .Upstream from the pileup, the density profile is againgiven the the classic rarefaction wave.This rich behavior can be readily understood in thehydrodynamic limit. Substituting the scaling form (3)into the continuity equation (2) shows that the scalingfunction satisfies f (cid:48) ( z ) (cid:2) − f ( z ) − z (cid:3) = 0 , (5)where the prime denotes differentiation with respect to z . The only solutions to this equation are either:flat profile f (cid:48) ( z ) = 0 , linear profile f = (1 − z ) . Any solution is a combination of these elementalforms. For the conventional TASEP, the solutions arethe aforementioned rarefaction and shock wave solutions,Eqs. (4a)–(4b). To determine the rarefaction wave at a(2 ,
1) junction, note that the current just to the right ofthe junction cannot exceed the maximum possible valueof j max = . If one starts at z = − x = − t ), and increases z , the density decays from +1 as ρ ( z ) = (1 − z ). Correspondingly, the current increasesas j ( z ) = 2 ρ (1 − ρ ) = 2 × (1 − z ), where the prefac-tor 2 accounts for the two upstream roads. Because thecurrent cannot exceed , j ( z ) must “stick” at this valuewhen first reached, which happens when z = − / √ z in the range − / √ < z <
0, the density alsosticks at a pileup value that corresponds to this maximalcurrent. This maximal current condition is 2 ρ (1 − ρ ) = ,with solutions ρ + = 12 + 1 √ , ρ − = 12 − √ . (6)It is the larger root ρ + ≈ . z = 0 + , there is only a single road and the densitymust suddenly drop to , so that the current at z = 0 + matches the maximal current j max = at z = 0 − . For0 < z <
1, the density decays linearly with z until thedensity reaches 0 at z = 1. Thus in the high-densityregime defined by ρ L ≥ ρ + , we conclude that the scaleddensity profile consists of five distinct segments: f = ρ L z < − ρ L (1 − z ) 1 − ρ L < z < − √ ρ + − √ < z < (1 − z ) 0 < z < z > . (7) j ρ + z f (a)(b) FIG. 2: (a) Schematic, but to scale, scaled density profilefor a rarefaction wave (blue) at a (2 ,
1) junction for an initialdensity downstep with ρ L = 1. The corresponding currentis shown in red. (b) Simulation data for the scaled densityprofile for this same initial condition for t = 125 (red), 250,(green) 500 (brown), and 1000 (blue). Simulation data converge to this five-segment form, withfinite-time corrections that systematically vanish as t in-creases (Fig. 2(b)). For the step initial condition, thesystem length is effectively infinite because the spatialrange over which the density is varying is less than theactual system length. B. Intermediate input density: ρ − ≤ ρ L ≤ ρ + Distinct behavior arises when ρ L lies between the twocritical values ρ + and ρ − . For ρ L in this range, the cur-rent in each incoming road is less than j max , but thesum of the currents in the two roads exceeds j max . Thusthere again must be a pileup of particles upstream fromthe junction point, as the maximum current that canbe transmitted at the junction is j max = . To matchthe outgoing current at the junction, the pileup densitymust equal ρ + . On the other hand, the asymptotic den-sity for z → −∞ is ρ L . As a result, a shock wave mustarise whose speed is given by 1 − ρ + − ρ L . Thus when ρ − < ρ L < ρ + , the asymptotic density profile consists offour segments: f = ρ L z < − ρ + − ρ L ρ + − ρ + − ρ L < z < (1 − z ) 0 < z < z > . (8) Even though this initial density downstep leads to a rar-efaction wave in the classic TASEP, the road constrictionleads to a jam the manifests itself as a left-moving shockwave on the upstream side of the junction. A typical ex-ample profile for the case of ρ L = 2 / +
1− − f z ρ L ρ ρ + L ρ (a)(b) FIG. 3: (a) The scaled density profile in Eq. (8) for the (2 , ρ L = 2 /
3. (b) Simulation data for this sameinitial condition for t = 125 (red dots), 250 (green), 500(brown), and 1000 (blue). The linear rise in the density near z = 1 − ρ L − ρ + gradually steepens for increasing t , showingthat this behavior is a finite-size effect. C. Low input density: ρ L ≤ ρ − Finally, we treat the low input density regime, wherethe incoming density obeys ρ L ≤ ρ − . Now the totalcurrent in the two incoming roads is always less thanor equal to j max = . Consequently, all the incomingcurrent can be accommodated by the single output road.Therefore, there is no pileup at x = 0 and the densityprofile in the incoming roads does not change in time.In the special case of ρ L = ρ − , the total input cur-rent to the junction equals j max = , corresponding tothe maximum current that can be accommodated by theoutput road. Here, the density profile for z > ρ L < ρ − , the inputcurrent to the junction, 2 ρ L (1 − ρ L ), is less than j max . Tohave a consistent scaling solution for z >
0, there mustbe a flat profile immediately to the right of the junction,with density ρ R , that eventually joins to the rarefactionwave ρ ( z ) = (1 − z ). We determine the density in theflat region to the right of the junction by matching theinput and outgoing currents at z = 0. This yields j in = 2 ρ L (1 − ρ L ) = ρ R (1 − ρ R ) (9a)from which ρ R = 1 − (cid:112) − ρ L (1 − ρ L )2 . (9b) f z + ρ (a) ρ f z ρ RL (b) FIG. 4: The scaled density profile for: (a) ρ L = ρ − ≈ . ρ L = . Assembling these results, the scaled density profile con-sists of three segments when ρ L = ρ + (Fig. 4(a)): f = ρ L z < (1 − z ) 0 < z < z > , (10a)and four segments for ρ L < ρ + (Fig. 4(b)): f = ρ L z < ρ R < z < − ρ R (1 − z ) 1 − ρ R < z < z > . (10b) IV. RAREFACTION AT A (1,2) JUNCTION
The same type of arguments as those given above canbe applied to the (1 ,
2) junction (see Fig. 1(b)). It isagain natural to consider the initial condition of ρ = ρ L for z < ρ = 0 for z > ρ L . As in the (2 ,
1) junction, a rich set ofbehaviors arises for varying ρ L (Fig. 5).For the initial state where ρ L = 1, that is, the inputroad is fully occupied and the two downstream roads are (a)(b) FIG. 5: Simulation data for the density profile for the (1 , ρ L = 2 / ρ L = 1 / t = 125 (reddots), 250 (green), 500 (brown), and 1000 (blue). empty, we can exploit particle-hole duality of the TASEPto immediately infer the density profile. In this duality,a particle moving to the right corresponds to a hole (avacancy) moving to the left. The density of holes ρ h isrelated to the particle density ρ by ρ h = 1 − ρ . Thus thedynamics of a right-moving TASEP at a (1 ,
2) junctionwith the downstep initial state of ρ = 1 for z < ρ = 0 for z > ,
1) junction geometrywith ρ h = 1 for z > ρ h = 0 for z <
0. Thelatter is the same as the particle density profile in a right-moving TASEP at a (2 ,
1) junction, after making thereplacements ρ → − ρ and z → − z . Simulations showthat the density profile in this case is the mirror imageof the density profile in Fig. 2(b)An input density ρ L < ,
2) junction geometrycorresponds to a (2 ,
1) junction with density ρ = 1 for z < ρ = 1 − ρ L for z >
0; this correspondenceis obvious after making the replacements ρ → − ρ and z → − z . It is simpler to describe the dynamics in termsof a right-moving TASEP at a (2 ,
1) junction with theinitial condition of ρ = 1 for z < ρ = ρ R > z > t >
0, the density upstream from the junctionagain exhibits a pileup, in which the scaled profile f = 1for z < − f = (1 − z ) for − < z < − / √
2, and finally f = ρ + for − / √ < z <
0. For ρ R < , the incomingcurrent equals its maximum value of . This incomingcurrent can be accommodated by a rarefaction wave for z > ρ R . On theother hand, for ρ R > , the outgoing current is densitylimited and, therefore equal to j R = ρ R (1 − ρ R ). Thismeans that the pileup density ρ L at z = 0 − is determinedby matching the currents at z = 0. This matching gives2 ρ L (1 − ρ L ) = j R , or ρ L = (1+ √ − j R ), in agreementwith the density profile shown in Fig. 5(b). V. OPEN (2,1) JUNCTION GEOMETRY
We now study the (2 ,
1) junction with open boundaryconditions with input rate α and output rate β . Whenthe leftmost sites of the system are empty, particles areinserted with rate α ; one could consider distinct rates α and α for the two roads, but we limit ourselves tothe symmetric case of α = α = α . Similarly, when aparticle reaches the rightmost site, it is extracted withrate β . These rates may take arbitrary positive values,but we limit ourselves to the range 0 < α < <β <
1. This restriction corresponds to the system beingcoupled to reservoirs with particle density α on the leftand density 1 − β on the right.The behavior of this open system can be analyzed us-ing the so-called domain wall theory [30–34]; the basicpredictions of this theory agree with exact analyses (see[5, 34] for reviews). To put our results in context, itis helpful to first summarize the properties of an opensingle-road TASEP. Here, there are three phases (Fig: 6):(i) a low-density (LD) phase, when α < and α < β ;(ii) a high-density (HD) phase, when β > and α > β ;(iii) a maximal-current (MC) phase, when α, β > . Forthe (2 ,
1) junction, the same three phases arise, but thelocations of the phase boundaries are different than in asingle-road system. The new feature for the (2 ,
1) junc-tion geometry, which already arose in the closed system,is that current conservation at the junction leads to dis-tinct densities just to the left and to the right of thejunction.
LD Phase: α < ρ − and β > ρ R ( α ). In the low-densityphase, the exit rate β is relatively fast and the particledensity is limited by the rate at which particles enter thesystem. Thus the density for z < ρ L = α . Thisstatement holds as long as α < ρ − , so that the current isless than . In this case, the right-half of the system cansupport and transmit this incoming current. Using thecurrent conservation statement (9a) across the junction,as well as ρ L = α , we immediately obtain, for the densityin the right half of the system, ρ R ( α ) = (cid:104) − (cid:112) − α (1 − α ) (cid:105) . (11a)The current in this LD phase is j ( α ) = 2 α (1 − α ). HD Phase: α > ρ L ( β ) and β < . In the high-densityphase, the exit rate β is relatively slow and the particledensity is determined by the rate at which particles enterthe system. In this HD phase, the density for z > ρ + ρ −
10 1 β α
LD MCHD
HDLD MC
FIG. 6: Phase diagram for the open (2 ,
1) junction. Insidethe MC phase, α > ρ − and β > , the dashed vertical line at α = ρ + separates a regime where the incoming road densityis low, ρ L = ρ − when α < ρ + , from a high-density regime, ρ L = ρ + when α > ρ + . The inset shows the correspondingphase diagram for the open single-road system. is ρ R = β . We again invoke the current conservationstatement (9a) across the junction and now solve for ρ L to give ρ L ( β ) = (cid:104) − (cid:112) − β (1 − β ) (cid:105) . (11b)The current in this HD phase is j ( β ) = β (1 − β ). MC phase: α > ρ − and β > . The density before thejunction in the MC phase is ρ L = (cid:40) ρ − when ρ − < α < ρ + ρ + when ρ + < α (12)The density in the two input roads can be either ρ − or ρ + to ensure that the total incoming current is the max-imal possible current in a single road. Interestingly, thedensity in the left half of the system changes discontin-uously when the input rate α = ρ + . For both cases thedensity in the right half of the system is ρ R = and thecurrent is j = (see Fig. 7). Coexistence line:
The co-existence line is defined bythe condition ρ R ( α ) = β , with ρ R ( α ) given by Eq. (11),or equivalently ρ L ( β ) = α , with ρ L ( α ) given by Eq. (11b).The additional conditions α < ρ − and β < must alsohold. This line (magenta in Fig. 6) separates the LD andHD phases. In the case of the single-road TASEP, subtlebehaviors occur on the co-existence line 0 < α = β < .It is known [31] that the density profile is a stationaryshock wave with ρ = α near the left end and ρ = 1 − α near the right end. The location of the shock is a uni-formly distributed random variable. By averaging over α ρ L ( α)ρ R ( α) j( α) FIG. 7: The bulk densities ρ L ( α ) , ρ R ( α ), and the current j ( α )when β > . all possible locations of the shock for the open system onthe interval − L < x < L , the density is given by ρ ( x ) = 12 + (cid:18) − α (cid:19) xL . (13)We anticipate a similar behavior on the co-existenceline for the (2 ,
1) junction. The density near the entranceis α and the density near the exit is 1 − β . In the simplestsituation when the shock is located at the junction ρ = (cid:40) α − L < x < − β < x < L . (14)However, the distribution of the position of the shockfront is unknown, and the lengths L and L of the in-coming and outgoing roads may play a significant role. VI. DISCUSSION
We introduced a simple extension of the totally asym-metric simple exclusion process (TASEP), in which mul-tiple roads meet at a junction. We focused on two simplegeometries, namely the (2 ,
1) and the (1 ,
2) junctions, inwhich either two roads merge into one, or one road splitsinto two. We first treated the density downstep initialcondition, which normally leads to a rarefaction wave.For the (2 ,
1) junction geometry, we found much richerbehavior in which there can be a particle pileup upstreamfrom the junction. Additionally a shock wave can arisefor a suitable range of initial densities. These phenom-ena qualitatively resemble what occurs in real traffic thatapproaches a constriction on a highway.We also investigated non-equilibrium steady states inthe (2 ,
1) junction with open boundaries in which par-ticles are continuously fed in at the left end and re-moved from the right end. We analyzed this system usingdomain-wall theory [30–34], which is known to correctlypredict the phase diagram for the TASEP and more com-plicated lattice gases. It would be desirable to study junctions with open boundaries using exact approaches.One possibility is to attempt to extend the matrix prod-uct approach [35] to the junction geometry. If the ma-trix product formulation can be extended to the junctiongeometry, then it should be feasible to provide detailedinsights into the spatial structure of the non-equilibriumsteady states. For example, the matrix product approachshould be able to give the behavior of the density profilein the boundary layers near the left and right ends of thesystem, and in the inner layer near the junction. In thesingle-lane TASEP, these boundary layers exhibit qual-itative changes within the low-density and high-densityphases: the LD phase may be subdivided into LD-I andLD-II, and similarly for the HD phase. A more accuratedescription of the open (2 ,
1) junction phase diagram mayperhaps be richer still due to the potentially different be-haviors in the inner layer on either side of the junction.Apart from generalizations to the ( m, n ) junction ge-ometry and to models with different hopping rates in theincoming and outgoing roads (which could be viewed asdifferent speed limits in the two types of roads), one canprobe the influence of the coupling between the paral-lel lanes in the bulk, in addition to the local interac-tion at the junction site. Multilane models can exhibitrich behaviors even without junctions (see, e.g., [11, 36–38]). Junction-like geometries have been recently inves-tigated in modeling pedestrian traffic [39, 40]; however,the movement rules in these pedestrian movement modelswere significantly different from TASEP dynamics.It would be also interesting to study the TASEP onmore complicated graphs with vertices mimicking junc-tions. One amusing example is the TASEP on a figure-eight geometry, in which a particle can pass through thejunction of the figure eight only when it is clear. Thisgeometry is inspired by the infamous automobile raceson the Islip Figure-Eight Speedway [41] that were heldbetween 1962 and 1984. The course is in the shape of afigure eight, with a collision point where the two loops ofthe figure eight meet. In this figure-eight geometry, weanticipate large collision-induced temporal fluctuationsin the current passing through the junction.Finally we emphasize that our analysis of junction ge-ometries relied on hydrodynamic techniques, which yieldonly average characteristics. Fluctuations in the TASEPhave attracted considerable interest. For example, forthe density downstep initial condition, the total numberof particles N ( t ) that flow to the initially empty half-lineby time t is a random quantity whose fluctuations scaleas t / ; that is, N ( t ) = t + t / ξ . (15)The distribution P ( ξ ) of the random variable ξ was es-tablished by Johansson [42]. Remarkably, the same andrelated Tracy-Widom distributions were derived earlierin the context of random matrices [43], and they arisein a wider range of problems (see [44, 45] and referencestherein). For the (2 ,
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