phase diagrams of the katz-lebowitz-spohn process on...
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PHYSICAL REVIEW E 87, 062124 (2013)
Phase diagrams of the Katz-Lebowitz-Spohn process on lattices with a junction
Bo Tian,1 Rui Jiang,1,* Zhong-Jun Ding,2 Mao-Bin Hu,1 and Qing-Song Wu1
1School of Engineering Science, University of Science and Technology of China, Hefei 230026, China2School of Transportation Engineering, Hefei University of Technology, Hefei 230009, China(Received 14 July 2012; revised manuscript received 13 March 2013; published 18 June 2013)
This paper studies the Katz-Lebowitz-Spohn (KLS) process on lattices with a junction, where particles moveon parallel lattice branches that combine into a single lattice at the junction. It is shown that 11 kinds of phasediagrams could be observed, depending on the two parameters ε and δ in the KLS process. We have investigatedthe phase diagrams as well as bulk density analytically based on flow rate conservation and the extremal currentprinciple. Extensive Monte Carlo computer simulations are performed, and it is found that they are in excellentagreement with theoretical prediction.
DOI: 10.1103/PhysRevE.87.062124 PACS number(s): 05.70.Ln, 02.50.Ey, 05.60.Cd
I. INTRODUCTION
The one-dimensional driven diffusive system has been arewarding research topic in recent decades, and has served asa fruitful testing ground for fundamental research in nonequi-librium physics [1–5]. Steady states in driven diffusive systemsexhibit many surprising or counterintuitive features, given ourexperiences with equilibrium systems, e.g., boundary inducedphase transitions, spontaneous symmetry breaking, and phaseseparation [6–10].
The nonequilibrium kinetic Ising model introduced byKatz, Lebowitz, and Spohn (KLS) is one much-studied one-dimensional driven diffusive system [11]. It was introduced in1984 to describe nonequilibrium steady states of a stochasticlattice gas model of fast ionic conductors. This is an exclusionprocess in which each lattice site is either occupied by oneparticle or empty. Particles hop randomly (with some bias)to their nearest-neighbor sites with rates depending on theoccupation of the nearest- and next-nearest-neighbor site. Inthe totally asymmetric case, particles hop only to the right withbulk hopping rates
0100 → 0010 with rate 1 + δ,
1100 → 1010 with rate 1 + ε,
0101 → 0011 with rate 1 − ε,
1101 → 1011 with rate 1 − δ,
where −1 < ε, δ < 1. Here, 1 marks the occupation ofa lattice site by a particle and 0 means that the site isempty. By using standard transfer matrix techniques, thestationary current j (ρ) can be computed exactly fromthe stationary measure of the periodic system. One obtains, inthe thermodynamic limit [11,12],
j (ρ) = λ[1 + δ(1 − 2ρ)] − ε√
4ρ(1 − ρ)
λ3, (1)
with
λ = 1√4ρ(1 − ρ)
+√
1
4ρ(1 − ρ)− 1 + 1 − ε
1 + ε. (2)
As shown in Fig. 1, the two parameters ε and δ determinethe shape of the current-density relation, which could exhibit
either two local maxima (j1,j2) and a local minimum (jmin)or a single maximum. If there are two local maxima, the leftlocal maximum is larger (smaller) than the right one providedδ > 0 (δ < 0). With δ = 0, the system has a symmetricalcurrent-density relation. In particular, in the special caseε = δ = 0, the KLS process reduces to the totally asymmetricsimple exclusion process (TASEP).
The majority of the driven diffusive systems investigatedinvolve particle movement along the one-channel lattices. Amore realistic description of nonequilibrium systems requiresan extension of the driven diffusive system to include thepossibility of transport on lattices with a more complexgeometry. Actually, there have been many works involvingextensions of the one-lane situation to multiple lanes, eitherparallel lanes [13–26] or intersected lanes [27–29]. Junctionis another geometry frequently observed in nonequilibriumsystems such as vehicle traffic and motor traffic, which hasbeen widely investigated in the literature [30–35]. Neverthe-less, these works have been carried out in the framework of theTASEP. This paper extends the investigation of the junction tothe general case of the KLS process. It is shown that 11 kindsof phase diagrams could be observed, depending on the twoparameters ε and δ in the KLS process. We have investigatedthe phase diagram as well as bulk density analytically andnumerically, and it is found that simulation results are inexcellent agreement with theoretical ones.
The paper is organized as follows. The description of themodel is given in Sec. II. In Sec. III, the analytical andnumerical results are presented. Finally, conclusions are givenin Sec. IV.
II. MODEL
Figure 2 shows a sketch of the KLS process on latticeswith a junction. Chains I and II merge at the junction to formchain III. In the bulk, the particles hop according to the KLSprocess rates. The left boundary of chains I and II is coupledto a reservoir with density ρ−; the right boundary of chain IIIis coupled to a reservoir with density ρ+.
We need to pay special attention to particles at site L +1 on chain III, since the particle hopping rate depends onthe occupation of the nearest upstream site. In this paper, wehave studied three different situations: (i) We suppose that theupstream site is empty only when both sites L on chains I and
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TIAN, JIANG, DING, HU, AND WU PHYSICAL REVIEW E 87, 062124 (2013)
FIG. 1. Four different current-density relations of the KLS process in a closed system. (a) ε = 0.995, δ = 0.2; (b) ε = 0.995, δ = −0.2;(c) ε = 0.995, δ = 0; (d) ε = 0.3,δ = 0.1.
II are empty. Otherwise, it is supposed that the upstream site ofsite L + 1 is occupied. (ii) We suppose that the upstream siteis occupied when both sites L on chains I and II are occupied.Otherwise, it is supposed that the upstream site of site L + 1is empty. (iii) We suppose that the hopping rate of a particleon site L + 1 depends on the state of site L on the branch itcomes from. Simulations show that the phase diagram and bulkdensity are the same under the three different situations [38].
The random sequential update rules are adopted as follows.In an infinitesimal time interval dt , one site i is chosen atrandom.
(a) If the site is at the left boundary of chains I and II (i = 1):If the site is empty, then a particle is injected to the site with
rate P or Q, depending on whether the next site 2 is emptyor not,
|00P−→ |10, |01
Q−→ |11,
P = (1 + δ)〈0100〉ρ− + (1 + ε)〈1100〉ρ−
〈00〉ρ−,
Q = (1 − δ)〈1101〉ρ− + (1 − ε)〈0101〉ρ−
〈01〉ρ−.
If the site is occupied and site 2 is empty, then the particlehops with rate P1 or Q1, depending on whether site 3 is emptyor not,
|100P1−→ |010, |101
Q1−→ |011,
P1 = (1 + δ)〈0100〉ρ− + (1 + ε)〈1100〉ρ−
〈100〉ρ−,
Q1 = (1 − δ)〈1101〉ρ− + (1 − ε)〈0101〉ρ−
〈101〉ρ−.
(b) If the site is at the right boundary of chain III (i = 2L)and the site is occupied, then the particle is removed withrate R or S, depending on whether site 2L − 1 is empty
FIG. 2. Schematic of the model. The arrow shows allowedhopping and the cross shows prohibited hopping. The symbols abovethe arrows show the hopping rate. Filled circles indicate that the sitesare occupied by particles.
or not,
01| R−→ 00|, 11| S−→ 10|,
R = (1 + δ)〈0100〉ρ+ + (1 − ε)〈0101〉ρ+
〈01〉ρ+,
S = (1 − δ)〈1101〉ρ+ + (1 + ε)〈1100〉ρ+
〈11〉ρ+.
(c) If i = 2L − 1 and the site is occupied, and the rightboundary site 2L is empty, then the particle hops with rate R1
or S1, depending on whether site 2L − 2 is empty or not,
110| R1−→ 101|, 010| S1−→ 001|,
R1 = (1 − δ)〈1101〉ρ+ + (1 + ε)〈1100〉ρ+
〈110〉ρ+,
S1 = (1 + δ)〈0100〉ρ+ + (1 − ε)〈0101〉ρ+
〈010〉ρ+.
Here, e.g., 〈100〉ρ is a stationary-state probability of aconfiguration 100 in an infinite system with average density ρ.For more details, see appendix B in Ref. [12]. In other cases,particles hop according to the KLS process rates, as mentionedabove.
III. RESULTS
We first recall the phase diagram of the KLS process on asingle open chain in the (ρ−,ρ+) plane. When the KLS processhas only one local maximum, the phase diagram is simple.There exist three phases, i.e., high density (HD), low density(LD), and maximum current (MC).
When the KLS process has two local maxima, the phasediagram becomes complex. In the case that the two localmaxima are not equal, the phase diagram consists of sevenphases, as shown in Figs. 3(a) and 3(b). In the LD phase, thebulk density ρ = ρ− < ρ∗
1 ; in the HD phase, ρ = ρ+ > ρ∗2 ;
in the Y phase, ρmin < ρ = ρ− < ρ∗2 ; and in the X phase,
ρ∗1 < ρ = ρ+ < ρmin. In the min, max1, max2 phases, ρ =
ρmin, ρ∗1 , ρ∗
2 , respectively. Here, ρmin, ρ∗1 , ρ∗
2 correspond tolocal minimum (jmin), the left local maximum (j1), and theright local maximum (j2), respectively (see Fig. 1). We alsonote that when δ > 0 (δ < 0), the left local maximum j1 islarger (smaller) than the right one j2, and the X (Y) phaseseparates phases max1 and max2.
In the case that the two local maxima are equal (δ = 0), onemore phase S emerges in the phase diagram [Fig. 3(c)]. Whenρ− > ρ∗
2 and ρ+ < ρ∗1 , a shock appears in the middle of the
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FIG. 3. Phase diagrams of the KLS model in an open system. (a) ε = 0.995, δ = 0.4; (b) ε = 0.995, δ = −0.2; (c) ε = 0.95, δ = 0.
system. The densities upstream and downstream of the shockare equal to ρ∗
2 and ρ∗1 , respectively.
Now we study the junction. When the KLS process has onlyone local maximum, the phase diagram is similar to that of theTASEP as studied in [30] and is straightforwardly obtained.Details are not shown here. Next we investigate the three casesj1 > j2, j1 < j2, and j1 = j2 in the following sections.
A. j1 > j2
To investigate the problem, following the methodology inRef. [30], we map the system into three coupled subsystems,as shown in Fig. 4. For chains I and II, the effective density ofthe right reservoir of chains I and II is denoted as ρeff
+ , and theeffective density of the left reservoir of chain III is denoted asρeff
− . Let J1, J2, and J3 represent the current in chains I, II, andIII. The flow rate conservation requires
J3 = J1 + J2 = 2J1. (3)
Moreover, the extremal current principle requires [12]
J1 ={
max j (ρ) for ρ− > ρeff+
min j (ρ) for ρ− < ρeff+ ,
(4)
J3 ={
max j (ρ) for ρeff− > ρ+
min j (ρ) for ρeff− < ρ+.
(5)
FIG. 4. The KLS process on the lattice with a junction can bemapped to three homogeneous KLS processes. The effective densityρeff
+ describes the right reservoir of chain I and chain II, and ρeff−
describes the effective density of the left reservoir of chain III.
FIG. 5. Five kinds of phase diagrams in the case j1 > j2.(a) C > D > 2,E < 2 (ε = 0.995, δ = 0.2); (b) C > D > 2,E > 2(ε = 0.995, δ = 0.4); (c) C > 2 > D,E < 2 (ε = 0.97, δ = 0.26);(d) C > 2 > D,E > 2 (ε = 0.995, δ = 0.6); (e) 2 > C > D,E < 2(ε = 0.95, δ = 0.2). Lines are from theoretical calculation andsquares are simulation results of the boundaries, which are in goodagreement.
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FIG. 6. (Color online) Density profiles with (a)–(k) ε = 0.995, δ = 0.2 and (l) ε = 0.995, δ = 0. Red solid lines describe the results of MonteCarlo simulations [with situation (i); see Sec. II], while black dashed lines are theoretical predictions of bulk densities. The insets show detailsof simulation results near the junction. (a) Phase (LD,LD), ρ− = 0.05,ρ+ = 0.3; (b) Phase (LD,LD), ρ− = 0.09,ρ+ = 0.3; (c) Phase (X,max1),ρ− = 0.4,ρ+ = 0.2; (d) Phase (X,X), ρ− = 0.4,ρ+ = 0.35; (e) Phase (min,Y), ρ− = 0.4,ρ+ = 0.6; (f) Phase (LD,Y), ρ− = 0.03,ρ+ = 0.7; (g)Phase (Y,Y), ρ− = 0.52,ρ+ = 0.7; (h) Phase (HD,max1), ρ− = 0.8,ρ+ = 0.2; (i) Phase (HD,X), ρ− = 0.8,ρ+ = 0.35; (j) Phase (HD,max2),ρ− = 0.8,ρ+ = 0.6; (k) Phase (HD,HD), ρ− = 0.8,ρ+ = 0.8; (l) Phase (HD,S), ρ− = 0.8,ρ+ = 0.2.
We would like to mention that as pointed out in Ref. [37], theextremal current principle is valid only for some specified bulk-adapted boundary-reservoir couplings, such as the left andright boundary couplings in the studied model in the presentpaper. For the junction, we use an effective boundary reservoirinstead of specific couplings; therefore, the extremal currentprinciple is valid.
Based on Eqs.(1)–(5), one can derive the phase diagramand the bulk density. Unfortunately, since the current densityrelation j (ρ) is complex, one can only obtain numericalsolutions instead of analytical expressions.
Figure 5 shows that there are five kinds of phase diagramsin the j1 > j2 case. Let us define C = j1/jmin, D = j2/jmin,and E = j1/j2. When C > D > 2 and E < 2, there exist10 stationary phases in the phase diagram, i.e., (LD,LD),(LD,Y), (X,max1), (X,X), (min,Y), (Y,Y), (HD,max1), (HD,X),(HD,max2), and (HD,HD); see Fig. 5(a). Here, (A,B) means
that chains I and II are in phase A, and chain III is in phaseB. Figures 6(a)–6(k) show the density profile in each phase.One can see that the analytical bulk densities are in goodagreement with the Monte Carlo simulation results. In thesimulations, the system size L = 1000. Initially the system isset empty. In the simulations, the first 2.1 × 1010 time steps arediscarded to let the transient time die out. Then the stationarydensity profiles have been obtained over 3 × 106 time averages(with a typical time interval of 3L between each average step).For a definition of the simulation phase transition line, seeAppendix A.
Now we present how to determine the bulk density in thephases. Let us take phase (X,X) for example, and other phasescan be analyzed similarly. In the X phase, the bulk density isdecided by the right boundary. Therefore, the bulk density ρ =ρ+ on chain III, and current J3 = j (ρ+) can be calculated. Thecurrents on chains I and II are thus equal to J3/2. As a result,
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FIG. 7. Three kinds of phase diagrams in the case j1 < j2. (a)2 < C < D (ε = 0.995, δ = −0.2); (b) C < 2 < D (ε = 0.98, δ =−0.3); (c) C < D < 2 (ε = 0.95, δ = −0.1).
ρeff+ can be calculated via Eq. (1), and the bulk density on chains
I and II equals ρeff+ . Here the effective boundary densities ρeff
+and ρeff
− are being fitted from the stationary currents, whichexplains the perfect fit between the Monte Carlo simulationand theoretical results. Since we do not estimate ρeff
+ and ρeff−
in an independent way, only one of them could be obtainedin a specific phase. For example, ρeff
− is not determined in the(X,X) phase. The analysis is valid when the system flow ratecould be determined in advance, which is the case in all phasesin the three versions of our model. Otherwise, one still needsto estimate the effective boundary densities independently toanalyze phases such as (HD, LD), if they exist.
We would like to mention that oscillations occur in thedensity profile near the junction in a wide range of ρ− and ρ+.Specifically, oscillations occur in phases (LD,Y), (X,max1),(X,X), (min,Y), and (Y,Y) [Figs. 6(c)–6(g)]. However, there isno oscillation in the phases where chains I and II are in theHD state [Figs. 6(h)–6(l)]. More interestingly, in the (LD,LD)phase, the oscillations occur near the boundary [Fig. 6(b)],and it gradually disappears when ρ− decreases [Fig. 6(a)]. Theoscillatory behaviors are related to the boundary couplingsnear the junction. As shown in Ref. [37], density profiles couldexhibit oscillations close to the boundaries. Nevertheless, thequantitative relationship between the oscillatory behaviors andthe dynamics near the junction needs to be investigated in afuture work, which might be a challenge.
When E becomes larger than 2, the phase (HD,max1)disappears [Fig. 5(b)] because the line j = j1/2 has nointersection with the high density part of the current-densityrelation j (ρ). Therefore, the flow rate conservation cannot besatisfied in the phase (HD,max1). Similarly, the disappearance
FIG. 8. Two kinds of phase diagrams in the case j1 = j2. (a)C = D > 2 (ε = 0.995, δ = 0); (b) C = D < 2 (ε = 0.95, δ = 0).
of other phases, as shown below, is also due to violation offlow rate conservation.
When C > 2 > D and E < 2, phases (min,Y) and (Y,Y)vanish and eight phases remain [Fig. 5(c)]. When E becomeslarger than 2, the phase (HD,max1) disappears and only sevenphases exist [Fig. 5(d)].
Finally, when 2 > C > D and E < 2, phases (X,max1),(X,X), (min,Y), and (Y,Y) disappear and six phases are left[Fig. 5(e)]. We would like to point out that E cannot be largerthan 2 provided 2 > C > D, which can be easily proved asshown in Appendix B.
We also would like to mention that we compare C,D,E
with 2 because two lattices merge into one. For case m, thelattices merge into n ones (m > n), and it is expected that weneed to compare C,D,E with m/n.
B. j1 < j2
Figure 7 shows that there are three kinds of phase diagramsin the j1 < j2 case. The value of E does not have a qualitativeinfluence on the phase diagram. When 2 < C < D, thephase diagram have eight phases, with phases (HD,X) and(HD,max1) absent [Fig. 7(a)]. When C < 2 < D, the (X,X)and (X,max1) phases vanish and six phases remain [Fig. 7(b)].Finally, when C < D < 2, phases (min,Y) and (Y,Y) disappearand only four phases exist [Fig. 7(c)].
C. j1 = j2
Figure 8 shows that there are two kinds of phase diagramsin the j1 = j2 case. When C = D > 2, there are nine phases[Fig. 8(a)]. While phases (HD,max1) and (HD,X) are absent,a new phase (HD, S) emerges, the density profile of which isshown in Fig. 6(l). When C = D < 2, phases (min,Y), (X,X),(X,max1) and (Y,Y) vanish and five phases are left [Fig. 8(b)].
IV. CONCLUSIONS
This paper has studied the KLS process on lattices with ajunction, where particles move on parallel lattices that combineinto a single lattice at the junction. It is shown that the twoparameters ε and δ in the KLS process have a qualitativeinfluence on the phase diagram. Altogether, 11 kinds of phasediagrams could be observed.
When the KLS process has only one local maximum, thephase diagram consists of three phases, (LD,LD), (HD,HD),
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and (HD,MC) (excluding the phases existing on the bound-aries), which is similar to that of the TASEP as studiedin [30].
When the KLS process has two local maxima and one localminimum, ten kinds of phase diagrams could be observed.When j1 > j2, five kinds of phase diagrams exist dependingon the ratios j1/jmin, j2/jmin, and j1/j2. When j1 < j2, threekinds of phase diagrams exist, which is independent of j1/j2,but depends on j1/jmin and j2/jmin. Finally, when j1 = j2, onlytwo kinds of phase diagrams exist, depending on j1/jmin.
As shown in Fig. 4, the KLS model with a junction can beviewed as three coupled KLS on a single chain. For conditionswhen there are two maxima and one minimum for each KLS,there are seven (or eight) phases. For three independent KLS,one might predict that 73 = 343 (or 83 = 512) phases exist.However, three KLS chains in the system are coupled, and thissignificantly reduces the number of phases. For example, dueto the current conservation in the stationary phase, currents atchains I and II are always smaller than the current at chain III.As a result, maximal current phases are not possible for chainsI and II. The role of the junction is then to serve as a couplingelement that controls possible stationary phases.
We have studied the situation that two lattices merge intoone in detail. The conclusion is valid for the general situationin which m lattices merge into n ones in the case m > n.
We have investigated the phase diagrams as well as bulkdensity analytically based on flow rate conservation and theextremal current principle. Extensive Monte Carlo computersimulations are performed, and it is found that they are inexcellent agreement with theoretical prediction.
ACKNOWLEDGMENTS
We thank Prof. A. B. Kolomeisky for helpful discussion.This work is funded by the National Basic Research Programof China (Grant No. 2012CB725404) and the National NaturalScience Foundation of China (Grants No. 11072239 and No.71171185). R.J. acknowledges the support of FundamentalResearch Funds for the Central Universities (Grant No.WK2320000014).
FIG. 9. (Color online) The plot of the averaged bulk density ineach channel vs ρ−. ρ+ = 0.6 is fixed. The parameters are ε = 0.995,δ = 0.4 [corresponding to Fig. 5(b)].
APPENDIX A
To determine the simulation phase transition line, we fixρ+ (or ρ−) and plot the averaged bulk density in each chainversus ρ− (or ρ+). Here the averaged bulk density is defined asthe average of densities in the 200 sites in the middle in eachchain. For instance, see Fig. 9, in which ρ+ = 0.6 is fixed.One can see that five phases are classified by the four verticaldashed lines. In this way, the phase boundaries are determined.
APPENDIX B
Here we prove that when 2 > C > D, E cannot be largerthan 2. From C = j1
jmin< 2, one has
j1 < 2jmin. (B1)
If we assume E = j1
j2> 2, then one obtains
j1 > 2j2. (B2)
From Eqs. (B1) and (B2), one has 2j2 < j1 < 2jmin, so thatj2 < jmin, which could never be met.
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[38] The three versions are different in that ρeff− and/or ρeff
+ aredifferent; see Sec. III A for the definitions of ρeff
− and ρeff+ .
However, the differences are not large enough to make thephase diagram and bulk density different. This can be understoodvia a comparison with the situation where two TASEP latticesmerge into one. Suppose particles at site L on chains I and IIhop to site L + 1 on chain III with rate q, and at other sites,the hopping rate is 1. It can be easily derived via the meanfield approximation that there exists a critical value qc. Whenq > qc = 1 − √
2/2, the bulk density and the phase diagram areindependent of q. When q < qc, the bulk density and the phasediagram change with q. Similar results are also obtained in thesituation where one TASEP lattice branches into two, as shown inRef. [36].
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