Two-lane totally asymmetric simple exclusion process with extended Langmuir kinetics
Hiroki Yamamoto, Shingo Ichiki, Daichi Yanagisawa, Katsuhiro Nishinari
aa r X i v : . [ n li n . C G ] J a n Two-lane totally asymmetric simple exclusion process with extended Langmuirkinetics
Hiroki Yamamoto , ∗ Shingo Ichiki , Daichi Yanagisawa , , and Katsuhiro Nishinari , School of Medicine, Hirosaki University, 5 Zaifu-cho Hirosaki city, Aomori, 036-8562, Japan Research Center for Advanced Science and Technology, The University of Tokyo,4-6-1 Komaba, Meguro-ku, Tokyo 153-8904, Japan Department of Aeronautics and Astronautics, School of Engineering, The University of Tokyo,7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan (Dated: January 27, 2021)Multi-lane totally asymmetric simple exclusion processes with interactions between the lanes haverecently been investigated actively. This paper proposes a two-lane model with extended Langmuirkinetics on a periodic lattice. Both bidirectional and unidirectional flows are investigated. Inour model, the hopping, attachment, and detachment rates vary depending on the state of thecorresponding site in the other lane. We obtain a theoretical expression for the global density of thesystem in the steady state from three kinds of mean-field analyses (1-, 2-, and 4-cluster cases). Weverify that the 4-cluster mean-field analysis approximates well the results of computer simulationsfor the two directional flows and reproduces the differences between them. We expect these findingsto contribute to a deeper understanding of the dynamic features of actual traffic systems.
I. INTRODUCTION
The asymmetric simple exclusion process (ASEP),which is a stochastic process involving particles on a lat-tice, has been applied in many fields [1] since it wasfirst proposed by MacDonald and Gibbs [2, 3]. A spe-cial version of ASEP, in which particles on a lattice canhop unidirectionally, is referred to as a totally asymmet-ric simple exclusion process (TASEP). Researchers haveapplied TASEPs to traffic flows of self-driven particles,as in biological transport [4–7], vehicular traffic [8, 9],and pedestrian flow [10–12]. Recently, one-lane TASEPswith varying hopping probabilities [13, 14] and multi-laneTASEPs with interactions between lanes [15–21] have be-gun to be investigated. For example, in Refs. [13, 14], thehopping probability of a particle varies depending on thestates of the sites surrounding it, whereas in Refs. [16–18], the hopping rate of a particle varies depending onthe state of the other lane.One of the extensions of TASEP, a TASEP with Lang-muir kinetics, which we refer to as LK-TASEP in thepresent paper, has begun to be actively investigated [22–36]. In the original LK-TASEP, a particle attaches (de-taches) at a certain rate ω A ( ω D ) when the targetedsite on the lattice is empty (occupied). The LK-TASEPwas first proposed in Ref. [22] and was investigated us-ing mean-field theory in Refs. [23, 24]. Recently, multi-lane LK-TASEPs have been studied [25–28]. In addition,Refs. [29–33] changed the attachment and detachmentrates depending on the occupancy of the adjacent sites.We note that the models of Refs. [30–32] are more gen-eralized versions of those discussed in Ref. [23].In the present paper, we consider a two-lane extendedLK-TASEP on a periodic lattice. The proposed model ∗ [email protected]; considers the interaction between particles in each lanewithout lane changing. Specifically, the hopping rate p and the attachment (detachment) rate ω A ( ω D ) varydepending on the state of the corresponding site in theother lane. We stress that the hopping rule has alreadybeen employed in Refs. [16–18]; however, the attachmentand detachment rule has not been considered in previousstudies. In the presnt paper, we investigate both unidi-rectional and bidirectional flows. We conduct computersimulations and perform three kinds of mean-field anal-yses (1-, 2-, and 4-cluster cases) to investigate the globaldensity of the system in the steady state. We find thatthe 4-cluster mean-field analysis reproduces the resultsof the simulations well.From a practical point of view, LK-TASEP has beenused extensively for analyzing the motions of motor pro-teins in Refs. [34–36], in which the investigators deter-mined the parameters from experimental data. For ap-plications to traffic flow, a TASEP model with an absorb-ing lane, which can be classified into the same categoryas LK-TASEP, has been investigated in Refs. [19] (park-ing lots) and [20, 21] (airport transportation systems).Our proposed model can also be applied to traffic flow;e.g., to crowd dynamics in the situations where multiplelanes are formed in a narrow passage, such as in trains,airplanes, and concert halls. In those situations, we ob-serve two important phenomena. First, we often observethat people decrease their walking speed to avoid colli-sions while walking side-by-side in unidirectional flows(passing each other in bidirectional flows). Second, deci-sion making during inflow to (outflow from) a passage isinfluenced by the local state. For example, people tendto hesitate to enter a passage when it is congested locallyat the inflow point. The former phenomenon correspondsto a change of the hopping rate, whereas the latter onecorresponds to changes of attachment and detachmentrates.This paper is organized as follows. Section II describesthe details of our proposed model. In Sec. III, the nu-merical results from mean-field analyses are presented.Section IV compares the results from mean-field analy-ses and simulation results. Finally, the paper concludesin Sec. V. II. MODEL
The model consists of two L -site lanes, labeled i =1 , , ..., L , as shown in Fig. 1. Each site can either beempty or be occupied by one particle. The state of a siteis represented by 1 if a particle occupies that site; other-wise, its state is represented by 0. We employ periodicboundary conditions, i.e., site L and site 1 are connected,and we use random updating. Changing lanes is prohib-ited in this model. In the present paper, we consider twocases: unidirectional flows (particles in both lanes all hopin the same direction) and bidirectional flows (the parti-cles in the two lanes hop in opposite directions).Next, we describe the update scheme for the case of aunidirectional flow. In the proposed model, the hoppingrate and attachment (detachment) rates depend on theoccupancy of the corresponding site in the other lane.For the hopping rate, a particle at site i in one lanehops to site ( i + 1) with rate 1 if site i in the other laneis vacant; otherwise, it hops with rate p (0 ≤ p ≤ i in one lane is empty and site i in the other laneis empty (occupied), the particle attaches with rate ω A1 ( ω A2 ), whereas (ii) if site i in one lane is occupied andsite i in the other lane is empty (occupied), the particledetaches with rate ω D1 ( ω D2 ). For bidirectional flows,only the hopping rule for lane 2 differs from the case ofunidirectional flows. In this case, a particle at site i inlane 2 hops to site ( i −
1) with rate 1 if the correspondingsite in lane1 is vacant; otherwise, it hops with rate p (0 ≤ p ≤ p = 1, ω A1 = ω A2 , and ω D1 = ω D2 . III. MEAN-FIELD ANALYSES
In this section, we investigate the density profile inthe steady state using three kinds of mean-field analyses.We hereafter write the probability finding configuration τ as P ( τ ). The configuration τ , which is one elementof the set S which consists of all possible configurations,contains (2 × L ) figures. The top (bottom) row representsthe state of the sites in lane 1 (2). For example, when L = 5 and the occupied site numbers in lane 1 are 1, 2,and 3, and those in lane 2 are 4 and 5, τ can be writtenas τ = . (1) p (cid:1) (cid:1)(cid:1) (cid:1) lane 2lane 1 (cid:2)
11 1 ω (cid:1)(cid:2) ω (cid:1)(cid:2) ω (cid:3)(cid:2) ω (cid:1)(cid:4) ω (cid:1)(cid:4) ω (cid:3)(cid:2) ω (cid:3)(cid:4) ω (cid:3)(cid:4) pp (cid:1) (cid:1)(cid:1) (cid:1) lane 2lane 1 (cid:2) ω (cid:1)(cid:2) ω (cid:1)(cid:2) ω (cid:3)(cid:2) ω (cid:1)(cid:4) ω (cid:1)(cid:4) ω (cid:3)(cid:2) ω (cid:3)(cid:4) ω (cid:3)(cid:4) pL sites i = 1 i = 2 i = L UB Langmuir kinetics
FIG. 1. (Color Online) Schematic illustration of the presentmodel. The upper and lower panels represent unidirectional(U) and bidirectional (B) flows, respectively. Red (green)circles represent particles in lane 1 (2). The left and rightboundaries are connected (periodic boundary conditions).We note that this figure shows the case L = 10. The master equation for this system can be written as dP ( τ ) dt = X τ ′ ∈ S W ( τ ′ → τ ) P ( τ ′ ) − X τ ∈ S W ( τ → τ ′ ) P ( τ ) , (2)where W ( τ ′ → τ ) is the transition weight to go fromstate τ ′ to state τ . However, it is very difficult to analyze(2 × L )-site configurations. We therefore consider clusterapproximations; specifically, 1-, 2-, and 4-cluster approx-imations. In the following subsections, we consider bothunidirectional and bidirectional flows. A. One-cluster mean-field analysis
In this subsection, we consider the 1-cluster mean-fieldanalysis, which is identical to the normal mean-field anal-ysis used in ASEP investigations. Translational invari-ance leads to spatial homogeneity, i.e., the probability isindependent of the site number; therefore, we can abbre-viate the site number in the following discussions (simi-larly in Subsec. III B and III C).For the probability P ( ∗ ), where ∗ represents either 0or 1, the master equation for unidirectional flows can bewritten as dP ( ∗ ) dt = (cid:2) P ( ∗ ) + pP ( ∗ ) + ω A1 P ( ) + ω A2 P ( ) (cid:3) − (cid:2) P ( ∗ ) + pP ( ∗ ) + ω D1 P ( ) + ω D2 P ( ) (cid:3) , (3)where the underlined sites in the right hand correspondsto the sites in the left hand (and similarly hereafter). Wenote that this equation does not change for bidirectionalflows, although that of P ( ∗ ) changes.Performing the mean-field analysis, i.e., ignoringhigher correlations in Eq. (3), we have dP ( ∗ ) dt = (cid:2) P ( ∗ ) P ( ∗ ) P ( ∗ ) + pP ( ∗ ) P ( ∗ ) P ( ∗ )+ ω A1 P ( ∗ ) P ( ∗ ) + ω A2 P ( ∗ ) P ( ∗ ) (cid:3) − (cid:2) P ( ∗ ) P ( ∗ ) P ( ∗ ) + pP ( ∗ ) P ( ∗ ) P ( ∗ )+ ω D1 P ( ∗ ) P ( ∗ ) + ω D2 P ( ∗ ) P ( ∗ ) (cid:3) , (4)where P ( ∗ ) + P ( ∗ ) = 1 and P ( ∗ ) + P ( ∗ ) = 1.Given the obvious symmetry between lanes 1 and 2,after a long enough time we obtain ρ = P ( ∗ ) = P ( ∗ );therefore, Eq. (4) reduces to dρdt =( ω A1 + ω D1 − ω A2 − ω D2 ) ρ + ( − ω A1 + ω A2 − ω D1 ) ρ + ω A1 , (5)where all the terms including p disappear. We note thatEq. (5) does not change for bidirectional flows.Because dρdt = 0 in the steady state, we obtain aρ + bρ + c = 0 , (6)where a = ω A1 − ω A2 + ω D1 − ω D2 , (7) b = − ω A1 + ω A2 − ω D1 , (8)and c = ω A1 . (9)For a = 0, the solution of Eq. (6) can be written as ρ = − b − √ b − ac a , (10)where b − ac = ( ω A2 − ω D1 ) + 4 ω A2 ω D2 > . (11)We discuss the exclusion of the other solution of Eq. (6),i.e., ρ = − b + √ b − ac a , in Appendix A.However, when a = 0, we have ρ = ω A1 ω A1 + ω D1 . (12) B. Two-cluster mean-field analysis
In this subsection, we consider the 2-cluster mean-fieldanalysis, where 2(= 2 ×
1) sites are regarded as one clus-ter. We note that this analysis is called “simple mean-field method” in Refs. [16–18]; however, we do not use this terminology in order to clarify the difference fromthe analysis in the previous subsection and to avoid mis-understanding.For the 2-cluster probability, the following two masterequations can be derived for the case of unidirectionalflows: dP ( ) dt = (cid:2) P ( ) + P ( ) + P ( ) + P ( )+ ω D1 P ( ) + ω D1 P ( ) (cid:3) − (cid:2) P ( ) + P ( ) + 2 pP ( )+ 2 ω A1 P ( ) (cid:3) (13)and dP ( ) dt = (cid:2) P ( ) + pP ( ) + pP ( ) + pP ( )+ ω A1 P ( ) + ω D2 P ( ) (cid:3) − (cid:2) P ( ) + pP ( ) + P ( ) + P ( )+ ω A2 P ( ) + ω D1 P ( ) (cid:3) . (14)Again performing the mean-field analysis, i.e., ignoringthe higher correlations in Eq. (13)–(14), we have dP ( ) dt = (cid:2) P ( ) P ( ) + P ( ) P ( ) + P ( ) P ( )+ P ( ) P ( ) + ω D1 P ( ) + ω D1 P ( ) (cid:3) − (cid:2) P ( ) P ( ) + P ( ) P ( )+ 2 pP ( ) P ( ) + 2 ω A1 P ( ) (cid:3) =2 P ( ) P ( ) − pP ( ) P ( )+ ω D1 P ( ) + ω D1 P ( ) − ω A1 P ( ) (15)and dP ( ) dt = (cid:2) P ( ) P ( ) + pP ( ) P ( ) + pP ( ) P ( )+ pP ( ) P ( ) + ω A1 P ( ) + ω D2 P ( ) (cid:3) − (cid:2) P ( ) P ( ) + pP ( ) P ( ) + P ( ) P ( )+ P ( ) P ( ) + ω A2 P ( ) + ω D1 P ( ) (cid:3) =2 pP ( ) P ( ) − P ( ) P ( ) + ω A1 P ( )+ ω D2 P ( ) − ω A2 P ( ) − ω D1 P ( ) . (16)We stress here that Eqs. (13) and (14) change for thecase of bidirectional flows; however, Eqs. (15) and (16)do not change and thus yields the same results for themean-field analysis.Again given the obvious symmetry between lanes 1 and2 after a long enough time, we obtain P ( ) = P ( ) . (17)Moreover, P ( ij ) ( i, j ∈ { , } ) must satisfy the normal-ization condition: P ( ) + P ( ) + P ( ) + P ( ) = 1 . (18)Because ddt P ( ij ) = 0 in the steady state, we obtain thefollowing expression from Eqs. (15)–(18): A { P ( ) } + BP ( ) + C = 0 , (19)where A = 2 + 4 pα + 2 pα , (20) B = 4 pαβ +4 pβ + ω A2 + ω D1 +2 ω D2 + ω D2 α − pα − ω A1 α, (21) C = 2 pβ + ω D2 β − pβ − ω A1 β − ω D2 , (22) α = ω D1 − ω A2 − ω D2 ω A1 + ω D2 , (23)and β = ω D2 ω A1 + ω D2 . (24)Solving Eq. (19) yields P ( ) in the form P ( ) = − B + √ B − AC A . (25)We note that because A = 2 + 4 pα + 2 pα = 2(1 − p ) + 2 p ( α + 1) > C = 2 pβ + ω D2 β − pβ − ω A1 β − ω D2 (27)= − pω A1 − ω D2 ω A1 − ω A1 ω D2 ω A1 + ω D2 < , (28)we have B − AC > . (29)We discuss the exclusion of the other solution of Eq. (19),i.e., P ( ) = − B −√ B − AC A in Appendix B.Because the density is defined as ρ = P ( ) + P ( ) , (30)we finally have ρ = 1 − β − (1 + α )( − B + √ B − AC )2 A . (31)
C. Four-cluster mean-field analysis
This subsection presents the 4-cluster mean-field anal-ysis, where 4(= 2 ×
2) sites are regarded as one cluster.We note that this analysis is called the “2-cluster mean-field method” in Refs. [16–18]; however, as with Subsec.III B, we do not use this terminology in order to clarifythe difference from the analyses in the last two subsec-tions and to avoid misunderstanding.Unlike the two previous mean-field analyses, in thiscase the final results are different for the two directionalflows. Therefore, in this subsection we consider the twoflows separately.
1. Unidirectional flows
The master equation for P ( ) can be expressed as dP ( ) dt = (cid:2) P ( ) + P ( ) + P ( )+ P ( ) + ω D1 P ( ) + ω D1 P ( )+ ω D1 P ( ) + ω D1 P ( ) (cid:3) − (cid:2) P ( ) + P ( ) + 2 pP ( )+ 4 ω A1 P ( ) (cid:3) . (32)Utilizing the concept of conditional probability, we canexpress P ( i k mj l n ) in this mean-field analysis in the form P ( i k mj l n ) = P ( i kj l ) P ( k ml n ) P m ∈{ , } P n ∈{ , } P ( k ml n ) , (33)where i, j, k, l, m, n ∈ { , } .Inserting Eq. (33) into Eq. (32) and noting that in thesteady state ddt P ( i kj l ) = 0, we obtain P ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) + P ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j )+ P ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) + P ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j )+ ω D1 P ( ) + ω D1 P ( ) + ω D1 P ( ) + ω D1 P ( ) − P ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) − P ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) − pP ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) − ω A1 P ( ) = 0 . (34)We can obtain the eight other master equations forthe 4-cluster probabilities in the steady state similarly,as shown in Appendix C.In addition, given the obvious symmetry between lanes1 and 2 after a long enough time, we obtain P ( ) = P ( ) , (35) P ( ) = P ( ) , (36) P ( ) = P ( ) , (37) P ( ) = P ( ) , (38) P ( ) = P ( ) , (39)and P ( ) = P ( ) . (40)Moreover, P ( i kj l ) must satisfy the normalization con-dition X i ∈{ , } X j ∈{ , } X k ∈{ , } X l ∈{ , } P ( i kj l ) = 1 . (41)From 16 independent Eqs. (34)–(41) and (C9)–(C16),we obtain all the probabilities P ( i kj l ).Finally, the definition of density gives ρ = X i ∈{ , } X j ∈{ , } X k ∈{ , } P ( ji k ) . (42)
2. Bidirectional flows
The master equation for P ( ) can be written in theform dP ( ) dt = (cid:2) P ( ) + P ( ) + P ( )+ P ( ) + ω D1 P ( ) + ω D1 P ( )+ ω D1 P ( ) + ω D1 P ( ) (cid:3) − (cid:2) P ( ) + pP ( ) + P ( )+ pP ( ) + 4 ω A1 P ( ) (cid:3) . (43)Inserting Eq. (33) into Eq. (43) and noting that in thesteady state ddt P ( i kj l ) = 0, we obtain P ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) + P ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j )+ P ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) + P ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j )+ ω D1 P ( ) + ω D1 P ( ) + ω D1 P ( ) + ω D1 P ( ) − P ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) − pP ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) − P ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) − pP ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) − ω A1 P ( ) = 0 . (44)We can again obtain the eight other master equationsfor the 4-cluster probabilities in the steady state similarly,as shown in Appendix D.Given the obvious symmetry between lanes 1 and 2after a long enough time, we obtain P ( ) = P ( ) , (45) P ( ) = P ( ) , (46) P ( ) = P ( ) , (47) P ( ) = P ( ) , (48) P ( ) = P ( ) , (49)and P ( ) = P ( ) . (50)From 16 independent Eqs. (41), (44)–(50), and (D9)–(D16), we obtain all the probabilities P ( i kj l ).Finally, the definition of density gives Eq. (42). IV. COMPARISON OF NUMERICAL RESULTSWITH MEAN-FIELD ANALYSES ANDSIMULATION RESULTS
In this section, we compare numerical results from thethree (1-, 2-, and 4-cluster) mean-field analyses with sim-ulation results. Although analytical solutions can bederived for 1- and 2-cluster mean-field analyses, similarsolutions cannot be obtained explicitly for the 4-clustermean-field analysis (see the last section). We thereforeobtain numerical solutions for this case using Newton’siteration method. In this study, we use the function Find-Root, which is based on the Newton’s method, in the soft-ware package Mathematica 12.0. In all the simulationsbelow, we set L = 100 and calculate the steady-statevalue of ρ for 10 time steps after evolving the system for10 time steps, unless otherwise specified. A. Special case: ω A1 = ω A2 and ω D1 = ω D2 In this special case, the results of all three mean-fieldanalyses for the two directional flows give us the followingequation for the steady-state value of ρ : ρ = ω A ω A + ω D . (51)Eq. (51) shows that all the mean-field analyses give aresult independent of p for this special cases.Figure 2 compares the simulation and mean-field val-ues of ρ as functions of p for various ( ω A , ω D ) ∈{ (0 . , . , (0 . , . , (0 . , . } for (a) uni-directional and (b) bidirectional flows. In both fig-ures, the simulations show very good agreement with ourmean-field analyses. B. General case
In this subsection, we consider the more general casewith ω A1 = ω A2 or ω D1 = ω D2 .Table I summarizes the three kinds of mean-field anal-yses (1-, 2-, and 4-cluster cases), which were discussed indetail in Sec. III. In the 1-cluster mean-field analysis, ρ (a) (b) p p (cid:1) ( ω A , ω D )=(0.004,0.008)( ω A , ω D )=(0.005,0.005)( (cid:1) A , (cid:1) D )=(0.008,0.004) (cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:2)(cid:8)(cid:9)(cid:10)(cid:11)(cid:12)(cid:13)(cid:4)(cid:5)(cid:7)(cid:13) (cid:14)(cid:12)(cid:6)(cid:9)(cid:15)(cid:16)(cid:2)(cid:12)(cid:5)(cid:17)(cid:10)(cid:18)(cid:4)(cid:11)(cid:19)(cid:12)(cid:13) FIG. 2. (Color Online) Simulation (circles) and mean-field values (curves) of ρ for (a) unidirectional and (b) bidirectionalflows as functions of p with ( ω A , ω D ) ∈ { (0 . , . , (0 . , . , (0 . , . } . is a function of ω independent of p ; therefore, the influ-ence of p cannot be captured. In contrast, the 2-clustermean-field analysis gives ρ as a function of both ω and p ; however, the function is the same for unidirectionaland bidirectional flows. Finally, in the 4-cluster mean-field analysis, ρ is a function of both ω and p , but thefunction is different for unidirectional and bidirectionalflows. From this discussion, we expect that the 1-clustermean-field analysis to approximate the simulation resultswell in cases where p ∼
1, whereas the 2-cluster mean-field analysis roughly captures the influence of p , and the4-cluster mean-field analysis reproduces the difference be-tween two directional flows. TABLE I. Comparison of the three kinds of mean-fieldanalyses.Mean-field analysis Direction p We next consider eight fundamental cases; specifically, (a) ( ω A1 , ω A2 , ω D1 , ω D2 ) = (0 . , . , . , . . (b) ( ω A1 , ω A2 , ω D1 , ω D2 ) = (0 . , . , . , . . (c) ( ω A1 , ω A2 , ω D1 , ω D2 ) = (0 . , . , . , . . (d) ( ω A1 , ω A2 , ω D1 , ω D2 ) = (0 . , . , . , . . (e) ( ω A1 , ω A2 , ω D1 , ω D2 ) = (0 . , . , . , . . (f) ( ω A1 , ω A2 , ω D1 , ω D2 ) = (0 . , . , . , . . (g) ( ω A1 , ω A2 , ω D1 , ω D2 ) = (0 . , . , . , . . (h) ( ω A1 , ω A2 , ω D1 , ω D2 ) = (0 . , . , . , . . (52)These cases enable us to investigate the influence ofchanges in ω A or ω D depending on the state of the corre-sponding site in the other lane. Specifically, Cases (a)–(d) exhibit the influence of ω A , whereas Cases (e)–(h)show that of ω D . Figure 3 plots the simulation and mean-field curves of ρ for the parameter sets of Cases (a)–(h).As expected, Fig. 3(a)–(h) shows that (i) the 1-cluster mean-field analysis agrees very well with the simulationresults only near p = 1, (ii) the 2-cluster mean-field re-sults capture well the qualitative change of ρ with ( p, ω )for both directions, and (iii) the 4-cluster case not onlyimproves the accuracy of the approximations but alsosucceeds in reproducing the difference between the twodirectional flows. We note that the deviation of the nu-merical results from simulations is the smallest for 1-cluster mean-field analysis, depending on ω , e.g., Cases(a) and (f), although the 1-cluster analysis does not cap-ture the p -dependence.In addition, we observe an interesting phenomenon inFig. 3(a)–(h). Specifically, there are smaller discrep-ancies between the simulations and numerical results forthe 2- and 4-cluster mean-field analyses for unidirectionalflows than for bidirectional flows. In other words, the nu-merical results from the 2-cluster mean-field analysis arealready good approximations for unidirectional flows.In the following subsections, we discuss in detail thediscrepancies between the numerical results for the 2- and4-cluster mean-field analyses and for ( p, ω )-dependence of ρ for the two directional flows.
1. Discrepancies between the numerical results from the 2-and 4-cluster mean-field analyses
Figure 3 shows that the discrepancies are smaller forunidirectional flows than for bidirectional flows.To investigate this phenomenon, we first define the fol-lowing correlation between two adjacent clusters, each ofwhich consists of 2(= 2 ×
1) sites: µ ( i kj l ) = P (cid:0) i kj l (cid:1) − P (cid:0) ij (cid:1) P (cid:0) kl (cid:1) , (53)where i, j, k, l ∈ { , } . From the definition, | µ | = 0 indi-cates that there is no correlation between the two adja-cent clusters, whereas µ > µ <
0) indicates that thepossibility of the spontaneous appearance of two adjacentclusters is larger (smaller) than the possibility under the p pp pp p (cid:1) (cid:1)(cid:1) (cid:1)(cid:1) (cid:1) simulation for Bsimulation for U1-cluster2-cluster4-cluster for B4-cluster for U (cid:1) (c) (d)(e) (f)(g) (h) (a) (b) p p (cid:1)
FIG. 3. (Color Online) Values of ρ from simulations (symbols) and mean-field calculations (blue solid/green dashed/reddash-dotted/orange dotted curves) as functions of p for (a) ( ω A1 , ω A2 , ω D1 , ω D2 ) = (0 . , . , . , . ω A1 , ω A2 , ω D1 , ω D2 ) = (0 . , . , . , . ω A1 , ω A2 , ω D1 , ω D2 ) = (0 . , . , . , . ω A1 , ω A2 , ω D1 , ω D2 ) = (0 . , . , . , . ω A1 , ω A2 , ω D1 , ω D2 ) = (0 . , . , . , . ω A1 , ω A2 , ω D1 , ω D2 ) = (0 . , . , . , . ω A1 , ω A2 , ω D1 , ω D2 ) = (0 . , . , . , . ω A1 , ω A2 , ω D1 , ω D2 ) = (0 . , . , . , . -0.08-0.06-0.04-0.0200.020.040.060.080.10.12-0.08-0.06-0.04-0.0200.020.040.060.080.10.12 µ (cid:1)(cid:1)(cid:1)(cid:1) µ (cid:2)(cid:2)(cid:2)(cid:2) µ (cid:1)(cid:2)(cid:1)(cid:1) µ (cid:1)(cid:1)(cid:1)(cid:2) µ (cid:1)(cid:2)(cid:1)(cid:2) µ (cid:2)(cid:1)(cid:1)(cid:1) µ (cid:2)(cid:2)(cid:1)(cid:1) µ (cid:2)(cid:1)(cid:1)(cid:2) µ (cid:2)(cid:2)(cid:1)(cid:2) µ (cid:1)(cid:1)(cid:2)(cid:1) µ (cid:1)(cid:2)(cid:2)(cid:1) µ (cid:1)(cid:1)(cid:2)(cid:2) µ (cid:1)(cid:2)(cid:2)(cid:2) µ (cid:2)(cid:1)(cid:2)(cid:1) µ (cid:2)(cid:2)(cid:2)(cid:1) µ (cid:2)(cid:1)(cid:2)(cid:2) -0.08-0.06-0.04-0.0200.020.040.060.080.10.12 µ (cid:1)(cid:1)(cid:1)(cid:1) µ (cid:2)(cid:2)(cid:2)(cid:2) µ (cid:1)(cid:2)(cid:1)(cid:1) µ (cid:1)(cid:1)(cid:1)(cid:2) µ (cid:1)(cid:2)(cid:1)(cid:2) µ (cid:2)(cid:1)(cid:1)(cid:1) µ (cid:2)(cid:2)(cid:1)(cid:1) µ (cid:2)(cid:1)(cid:1)(cid:2) µ (cid:2)(cid:2)(cid:1)(cid:2) µ (cid:1)(cid:1)(cid:2)(cid:1) µ (cid:1)(cid:2)(cid:2)(cid:1) µ (cid:1)(cid:1)(cid:2)(cid:2) µ (cid:1)(cid:2)(cid:2)(cid:2) µ (cid:2)(cid:1)(cid:2)(cid:1) µ (cid:2)(cid:2)(cid:2)(cid:1) µ (cid:2)(cid:1)(cid:2)(cid:2) µ (cid:1)(cid:1)(cid:1)(cid:1) µ (cid:2)(cid:2)(cid:2)(cid:2) µ (cid:1)(cid:2)(cid:1)(cid:1) µ (cid:1)(cid:1)(cid:1)(cid:2) µ (cid:1)(cid:2)(cid:1)(cid:2) µ (cid:2)(cid:1)(cid:1)(cid:1) µ (cid:2)(cid:2)(cid:1)(cid:1) µ (cid:2)(cid:1)(cid:1)(cid:2) µ (cid:2)(cid:2)(cid:1)(cid:2) µ (cid:1)(cid:1)(cid:2)(cid:1) µ (cid:1)(cid:2)(cid:2)(cid:1) µ (cid:1)(cid:1)(cid:2)(cid:2) µ (cid:1)(cid:2)(cid:2)(cid:2) µ (cid:2)(cid:1)(cid:2)(cid:1) µ (cid:2)(cid:2)(cid:2)(cid:1) µ (cid:2)(cid:1)(cid:2)(cid:2) -0.08-0.06-0.04-0.0200.020.040.060.080.10.12-0.08-0.06-0.04-0.0200.020.040.060.080.10.12 -0.08-0.06-0.04-0.0200.020.040.060.080.10.12 µ (cid:1)(cid:1)(cid:1)(cid:1) µ (cid:2)(cid:2)(cid:2)(cid:2) µ (cid:1)(cid:2)(cid:1)(cid:1) µ (cid:1)(cid:1)(cid:1)(cid:2) µ (cid:1)(cid:2)(cid:1)(cid:2) µ (cid:2)(cid:1)(cid:1)(cid:1) µ (cid:2)(cid:2)(cid:1)(cid:1) µ (cid:2)(cid:1)(cid:1)(cid:2) µ (cid:2)(cid:2)(cid:1)(cid:2) µ (cid:1)(cid:1)(cid:2)(cid:1) µ (cid:1)(cid:2)(cid:2)(cid:1) µ (cid:1)(cid:1)(cid:2)(cid:2) µ (cid:1)(cid:2)(cid:2)(cid:2) µ (cid:2)(cid:1)(cid:2)(cid:1) µ (cid:2)(cid:2)(cid:2)(cid:1) µ (cid:2)(cid:1)(cid:2)(cid:2) µ (cid:1)(cid:1)(cid:1)(cid:1) µ (cid:2)(cid:2)(cid:2)(cid:2) µ (cid:1)(cid:2)(cid:1)(cid:1) µ (cid:1)(cid:1)(cid:1)(cid:2) µ (cid:1)(cid:2)(cid:1)(cid:2) µ (cid:2)(cid:1)(cid:1)(cid:1) µ (cid:2)(cid:2)(cid:1)(cid:1) µ (cid:2)(cid:1)(cid:1)(cid:2) µ (cid:2)(cid:2)(cid:1)(cid:2) µ (cid:1)(cid:1)(cid:2)(cid:1) µ (cid:1)(cid:2)(cid:2)(cid:1) µ (cid:1)(cid:1)(cid:2)(cid:2) µ (cid:1)(cid:2)(cid:2)(cid:2) µ (cid:2)(cid:1)(cid:2)(cid:1) µ (cid:2)(cid:2)(cid:2)(cid:1) µ (cid:2)(cid:1)(cid:2)(cid:2) µ (cid:1)(cid:1)(cid:1)(cid:1) µ (cid:2)(cid:2)(cid:2)(cid:2) µ (cid:1)(cid:2)(cid:1)(cid:1) µ (cid:1)(cid:1)(cid:1)(cid:2) µ (cid:1)(cid:2)(cid:1)(cid:2) µ (cid:2)(cid:1)(cid:1)(cid:1) µ (cid:2)(cid:2)(cid:1)(cid:1) µ (cid:2)(cid:1)(cid:1)(cid:2) µ (cid:2)(cid:2)(cid:1)(cid:2) µ (cid:1)(cid:1)(cid:2)(cid:1) µ (cid:1)(cid:2)(cid:2)(cid:1) µ (cid:1)(cid:1)(cid:2)(cid:2) µ (cid:1)(cid:2)(cid:2)(cid:2) µ (cid:2)(cid:1)(cid:2)(cid:1) µ (cid:2)(cid:2)(cid:2)(cid:1) µ (cid:2)(cid:1)(cid:2)(cid:2) µµµ µµµ (a) (b)(c) (d)(e) (f)UnidirectionBidirection p= p= p= p= p= p= FIG. 4. (Color Online) Calculated values of 16 kinds of µ of unidirectional (blue) and bidirectional (red) flows for various p ∈ { , . , . , . , . , } , fixing ( ω A1 , ω A2 , ω D1 , ω D2 ) = (0 , , ,
0) and ρ = 0 .
5. We note (i) that those values are calculatedby averaging 100 different initial configurations for p = 0 and (ii) that they are calculated for 10 time steps after evolving thesystem with p = 0 .
01 for 10 time steps because it takes more time for the system to evolve into the steady state. assumption that clusters appear randomly in the system.This explains why the 4-cluster mean-field analysis im-proves the approximate accuracy more than the 2-clusteranalysis for relatively large | µ | . In contrast, the 2-clusteranalysis is already a good approximation for relativelysmall | µ | .Figure 4 compares 16 kinds of µ for unidi-rectional and bidirectional flows for various p ∈{ , . , . , . , . , } , fixing ( ω A1 , ω A2 , ω D1 , ω D2 ) = (0 , , ,
0) and ρ = 0 .
5. We set ( ω A1 , ω A2 , ω D1 , ω D2 ) =(0 , , ,
0) to observe the pure correlations caused by p because Langmuir kinetics reduce the correlations. Wenote that—strictly speaking—we cannot discuss the case p = 0 in the same way as for the cases with p > p = 0 inFig. 5).We observe two important phenomena in Fig. 4. First,for the two directional flows | µ | approaches to 0 as p in-creases; i.e., the correlation becomes smaller with larger p . This indicates that for larger p , the discrepancies be-tween the numerical results from the 2- and 4-clustermean-field analyses become smaller, which explains theoverlap of the 2- and 4-cluster theoretical lines in Fig. 3.Second, we confirm that in most cases many values of | µ | for unidirectional flows are smaller than those for bidi-rectional flows with the same values of p [Fig. 4(b)–(e)];i.e., the correlations for unidirectional flows are smallerthan those for bidirectional flows. This indicates thatthere are smaller discrepancies between the numericalresults from the 2- and 4-cluster mean-field analyses forunidirectional flows compared with those for bidirectionalflows. In contrast, for p = 0 [Fig. 4(a)], some values of | µ | become large even for unidirectional flows, and therefore,the discrepancies also become large (see Fig. 3). ( p, ω ) -dependence of ρ This subsection discusses the ( p, ω )-dependence of ρ ,observed from Fig. 3, for the numerical results from the2- and 4-cluster mean-field analyses and for the simula-tion results.First, to investigate the influence of p on ρ , weinvestigate P ( ), P ( ) + P ( ), and P ( ), fixing( ω A1 , ω A2 , ω D1 , ω D2 ) = (0 , , ,
0) and ρ = 0 .
5, as in Fig.5. p for U P (cid:1)(cid:1) for B P (cid:1)(cid:1) for B P (cid:1)(cid:2) +P (cid:2)(cid:1) P (cid:1) (cid:2) + P (cid:2) (cid:1) P (cid:1) (cid:1) (cid:1) for U P (cid:1)(cid:2) +P (cid:2)(cid:1) FIG. 5. (Color Online) Simulation results for steady-statevalues of P ( ) + P ( ), and P ( ) as functions of p , fixing( ω A1 , ω A2 , ω D1 , ω D2 ) = (0 , , ,
0) and ρ = 0 . P ( ) = P ( )when ρ = 0 .
5. The points between p = 0 and p = 0 .
05 are for p = 0 .
01. We note (i) that the values for p = 0 are calculatedby averaging 100 different initial configurations and (ii) thatthey are calculated steps for 10 time steps after evolvingthe system with p = 0 .
01 for 10 time steps because it takesmore time for the system to evolve into the steady state. Figure 5 shows (i) that P ( ) and P ( ) increase and P ( ) + P ( ) decreases with smaller p for the two di- rectional flows, (ii) that the degrees of those changesare greater for unidirectional flows than for bidirec-tional flows, excluding the case with p = 0 for unidi-rectional flows, and (iii) that P ( ) and P ( ) decreaseand P ( ) + P ( ) increases abruptly from p = 0 .
01 to p = 0 for unidirectional flows.Three effects can qualitatively explain these phenom-ena: (i) the trapping, (ii) jamming, and (iii) blockingeffects. First, in the trapping effect, for small p , particlescan become trapped in the state ( ), which leads to an in-crease in P ( ) and P ( ), and a decrease in P ( )+ P ( ),as shown in Fig. 6. We stress that the trapping effectworks in common between the two directional flows. UB (cid:1) (cid:1)(cid:1) (cid:1)(cid:2) small p small p small p small p small p small p (cid:1) (cid:1)(cid:1) (cid:1)(cid:2) small p small p small p small p small p small p trapped particles FIG. 6. (Color Online) Schematic illustration of thetrapping effect. Langmuir kinetics are not considered in thisfigure, which shows the case L = 10. Contrarily, in the jamming effect, for small p , the par-ticles following a trapped particle can become involvedin a jam, as shown in Fig. 7. For unidirectional flows,this effect works the same as the trapping effect; specifi-cally, P ( ) and P ( ) increase and P ( )+ P ( ) decreaseswith smaller p . In contrast, for bidirectional flows, thiseffect tends to counter the trapping effect; specifically, P ( ) and P ( ) decrease, and P ( )+ P ( ) increases withsmaller p . We note that the jamming effect is smallerthan the trapping effect because jams occur only aftermany trapped particles appear. Because the jamming ef-fect influences each P oppositely in the two directionalflows, the degrees of those changes are greater for unidi-rectional flows than for bidirectional flows, excluding thecase with p = 0.Finally, the blocking effect appears only for p = 0,as shown in Fig. 8. Once the particles are trapped inthe state ( ), this state never changes for p = 0 and ω D2 = 0; therefore, the isolated particles between thetwo clusters of ( ) must finally make the state ( ) or( ). Because this effect counters the trapping and jam-ming effects, P ( ) and P ( ) decrease and P ( ) + P ( )0 UB (cid:1) (cid:1)(cid:1) (cid:1)(cid:2) (cid:1) (cid:1)(cid:1) (cid:1)(cid:2) small p small p small p small p jam FIG. 7. (Color Online) Schematic illustration of thejamming effect. Langmuir kinetics are not considered in thisfigure, which shows the case L = 10. increases from p = 0 .
01 to p = 0 for unidirectional flows.We note (i) that this effect is virtually the same as thejamming effect for bidirectional flows and (ii) that theblockages can be dismantled if ω D2 > (cid:1) (cid:1)(cid:1) (cid:1)(cid:2) blockage FIG. 8. (Color Online) Schematic illustration of the blockingeffect for unidirectional flows. Langmuir kinetics are notconsidered in this figure, which shows the case L = 10. We can also confirm the existence of those three effectsin Fig. 4. Table II summarizes the kinds of µ , whichbecome larger with smaller p . We note that Table IIexcludes the case p = 0 for unidirectional flows becauseof its singularity.For unidirectional flows, the trapping and jamming ef-fects explain why all the listed µ values become largerwith smaller p . Contrarily, for bidirectional flows, thefact that µ ( ), µ ( ), and µ ( ) become larger withsmaller p can be explained by the trapping effect, whereasthe fact that µ ( ) and µ ( ) become larger withsmaller p can be explained by the jamming effect. Con-versely, for p = 0, µ ( ), µ ( ), µ ( ), and µ ( )also soar for unidirectional flows, indicating the blockingeffect.On the basis aforementioned discussions, we finally TABLE II. Kinds of µ , which become larger with smaller p ,for each direction (see Fig. 4).Direction Corresponding µ Unidirection µ ( ), µ ( ), µ ( ), µ ( )Bidirection µ ( ), µ ( ), µ ( ), µ ( ), µ ( ) summarize the influences of the three effects on P ( ), P ( ) + P ( ), and P ( ) in Tab. III. We again stressthat the results shown in Fig. 5 can be explained fromTab. III, noting that the trapping effect is stronger thanthe jamming effect and that the blocking effect only ap-pears for p = 0. TABLE III. Influence of the three effects on P ( ), P ( ) + P ( ), and P ( ). The sign “+” (“ − ”) indicates thatthe corresponding effect causes increases (decreases) in each P . Langmuir kinetics are not considered in this table. Wenote that “++” (“ −− ”), used for unidirectional flows,represents the fact that the effect is larger than that forbidirectional flows.Direction Effect P ( ) P ( ) + P ( ) P ( )Unidirection Trapping effect + − +Jamming effect + − +Blocking effect − + − Trapping + Jamming ++ −− ++Bidirection Trapping effect + − +Jamming effect − + − Blocking effect − + − Trapping+Jamming + − + Next, we consider the influences of ω on ρ , P ( ), P ( ) + P ( ), and P ( ). P (cid:1)(cid:1) P (cid:2)(cid:2) P (cid:2)(cid:1) P (cid:1)(cid:2) + (cid:1) A1 (cid:1) A2 (cid:1) D1 (cid:1) D2 FIG. 9. State-transition diagram for P ( ), P ( ) + P ( ),and P ( ). From the definition of ω and the state-transition of dia-gram, as shown in Fig. 9, we can summarize the increase(+) and decrease ( − ) of each value by the change of ω in Tab. IV. Blank cells indicate that the effect is indi-rect and the sign is not apparent; however, those indirecteffects are small enough to be ignored in the followingdiscussion.On the basis of the aforementioned discussions of( p, ω )-dependence of ρ , we consider the following fourphenomena in Fig. 3. We stress that those phenomenacan be confirmed not only for the simulation results butalso for the numerical results with 4-cluster mean-fieldanalysis.1 TABLE IV. Increase (+) and decrease ( − ) of each valuecaused by the change in ω (∆ ω > ρ indicates that ρ increases withlarger ω A1 .Change of ω ρ P ( ) P ( ) + P ( ) P ( ) ω A1 → ω A1 + ∆ ω + − + ω A2 → ω A2 + ∆ ω + − + ω D1 → ω D1 + ∆ ω − + − ω D2 → ω D2 + ∆ ω − + − a. Increase/decrease of ρ , depending on ω , in the re-gion where p ' . P ( ) and P ( ) increaseand P ( ) + P ( ) decreases for smaller p , the effects of ω A1 and ω D2 ( ω A2 and ω D1 ) become smaller. Therefore,considering Fig. 9 and Tab. IV, ρ increases (decreases)for smaller p for Cases (a), (d), (e), and (h) [(b), (c),(f), and (g)] in the region where p ' .
1. For exam-ple, for Case (a), P ( ) increases, and the effect of ω A1 isenhanced for smaller p , resulting in an increase in ρ forsmaller p . b. Differences in the degree of change between the twodirectional flows — Because the effect of p on each P islarger for unidirectional flows than for bidirectional ones(see Fig. 5 and Tab. III), the degree of the change in ρ with p becomes greater for unidirectional flows than forbidirectional ones. c. Change of the trend in unidirectional flows for p = 0 — For unidirectional flows, due to the block-ing effect, the configuration changes drastically from verysmall p ( = 0) to p = 0, although Langmuir kinetics reducethat effect. This results in a trend change when p = 0,the extent of which depends on ω . We note that for uni-directional flows the influence of p on P is large enoughso that it is not necessary to consider the influence of ω . d. Change of the trend in bidirectional flows when p becomes smaller — Unlike unidirectional flows, we can-not ignore the effect of ω on P for bidirectional flows.Therefore, for Cases (a), (d), (f), and (g), where the in-fluence of ω on P counters that of smaller p , the trendchanges in the change-easing direction with smaller p . Incontrast, for Cases (b), (c), (e), and (h), where the in-fluence of ω on P reinforces that of smaller p , the trendchanges in the change-accelerating direction with smaller p . V. CONCLUSION
In the present paper, we have investigated a two-laneextended LK-TASEP on a periodic lattice, where thehopping rate p and the attachment (detachment) rate ω A ( ω D ) vary depending on the state of the correspond-ing site in the other lane. The proposed model is new inthat it introduces a varying rule for the attachment anddetachment rate. We have investigated the steady-stateglobal density for unidirectional and bidirectional flows using both computer simulations and mean-field analy-ses.We have conducted three kinds of mean-field analyses(1-, 2-, and 4-cluster cases) and have compared them withsimulation results. In the 1-cluster mean-field analysis,the calculated value of ρ is in good agreement with thesimulation results only in the region where p is near 1. Incontrast, the 2-cluster analysis can reproduce the roughtrend of the simulation results for ρ as functions of p ,even though it cannot distinguish between unidirectionaland bidirectional flows. Finally, the 4-cluster analysiscan not only approximates better the simulation resultsfor ρ as functions of p but also reproduces the differencebetween the two directional flows.We have therefore considered further the discrepan-cies between the numerical results from 2- and 4-clustermean-field analyses for unidirectional flows—which aresmaller than those for bidirectional flows—by calculatingthe correlations between two adjacent (2 ×
1) clusters. Wehave also discussed the ( p, ω )-dependence of ρ in termsof three effects (the trapping, jamming, and blocking ef-fect). Those three effects by p and ω determine the trendof ρ .We again emphasize that, despite its simplicity, theproposed model has a potential for applications to real-world phenomena. For example, for crowd dynamics(traffic flow) in a narrow passage (road), the proposedmodel can consider the velocity and inflow/outflow ofpedestrians (vehicles). In particular, unlike previousmodels, our model makes it possible to consider of thedependence of the changes of the inflow/outflow on thelane state. ACKNOWLEDGMENTS
This work was partially supported by JST-Mirai Pro-gram Grant Number JPMJMI17D4, Japan, JSPS KAK-ENHI Grant Number JP15K17583.
Appendix A: Exclusion of the other solution of Eq.(6)
In this Appendix, we discuss the exclusion of the othersolution of Eq. (6); specifically, ρ = − b + √ b − ac a . (A1)First, we consider the case where a >
0; specifically, ω A1 + ω D1 > ω A2 + ω D2 . (A2)In this case, we have b = − ω A1 + ω A2 − ω D1 (A3)= − ( ω A1 + ω D1 ) − ω A1 + ω A2 (A4) < − ( ω A2 + ω D2 ) − ω A1 + ω A2 (A5)= − ω A1 − ω D2 , (A6)2resulting in b <
0. Noting that a > b < c > − b + √ b − ac a (A7) > − b − b a = − ba (A8)= 2 ω A1 − ω A2 + ω D1 ω A1 − ω A2 + ω D1 − ω D2 (A9)= ω A1 − ω A2 + ω D1 + ω A1 ω A1 − ω A2 + ω D1 − ω D2 > . (A10)Second, we consider the case where a <
0. In this case,because − b + p b − ac > − b + | b | ≥ , (A11)we have − b + √ b − ac a < . (A12)Because ρ lies in the range 0 ≤ ρ ≤
1, this solution isinappropriate, and because the system must evolve intoonly one steady state, we obtain Eq. (10).
Appendix B: Exclusion of the other solution of Eq.(19)
In this Appendix, we discuss the exclusion of the othersolution of Eq. (19); specifically, P ( ) = − B − √ B − AC A . (B1)Noting Eq. (26) and (28), we have − B − p B − AC < − B − | B | ≤
0; (B2)therefore, − B − √ B − AC A < . (B3)Because P ( ) lies in the range 0 ≤ P ( ) ≤
1, thissolution is inappropriate, and because the system mustevolve into only one steady state, we obtain Eq. (25).
Appendix C: Other independent master equations for the 4-cluster mean-field analysis for unidirectional flows
In this Appendix, we summarize the other independent master equations for 4-cluster probabilities for unidirectionalflows.For P ( ), P ( ), P ( ), P ( ), P ( ), P ( ), P ( ), and P ( ), the master equations can be expressed as dP ( ) dt = (cid:2) P ( ) + pP ( ) + P ( ) + pP ( ) + P ( ) + P ( ) + P ( ) + P ( )+ ω A2 P ( ) + ω A2 P ( ) + ω D1 P ( ) + ω D1 P ( ) (cid:3) − (cid:2) pP ( ) + 2 ω D2 P ( ) + 2 ω A1 P ( ) (cid:3) , (C1) dP ( ) dt = (cid:2) P ( ) + pP ( ) + P ( ) + pP ( ) + ω A2 P ( ) + ω A2 P ( ) + ω A2 P ( ) + ω A2 P ( ) (cid:3) − (cid:2) pP ( ) + pP ( ) + pP ( ) + 4 ω D2 P ( ) (cid:3) , (C2) dP ( ) dt = (cid:2) P ( ) + pP ( ) + P ( ) + P ( ) + P ( ) + P ( ) + ω A1 P ( ) + ω D1 P ( )+ ω D1 P ( ) + ω D2 P ( ) (cid:3) − (cid:2) P ( ) + P ( ) + pP ( ) + 2 ω A1 P ( ) + ω A2 P ( ) + ω D1 P ( ) (cid:3) , (C3) dP ( ) dt = (cid:2) P ( ) + pP ( ) + pP ( ) + ω D1 P ( ) + ω A1 P ( ) + ω D1 P ( ) + ω D2 P ( ) (cid:3) − (cid:2) P ( ) + P ( ) + 2 pP ( ) + P ( ) + P ( ) + 2 ω A1 P ( ) + ω D1 P ( ) + ω A2 P ( ) (cid:3) , (C4) dP ( ) dt = (cid:2) P ( ) + pP ( ) + pP ( ) + pP ( ) + ω A1 P ( ) + ω D2 P ( ) + ω A1 P ( ) + ω D2 P ( ) (cid:3) − (cid:2) P ( ) + P ( ) + P ( ) + pP ( ) + 2 ω D1 P ( ) + 2 ω A2 P ( ) (cid:3) , (C5)3 dP ( ) dt = (cid:2) P ( ) + pP ( ) + pP ( ) + pP ( ) + pP ( ) + ω A1 P ( ) + ω D2 P ( ) + ω D2 P ( )+ ω A1 P ( ) (cid:3) − (cid:2) P ( ) + P ( ) + pP ( ) + P ( ) + P ( ) + 2 ω D1 P ( ) + 2 ω A2 P ( ) (cid:3) , (C6) dP ( ) dt = (cid:2) P ( ) + pP ( ) + P ( ) + pP ( ) + pP ( ) + pP ( ) + ω A2 P ( ) + ω A2 P ( )+ ω A1 P ( ) + ω D2 P ( ) (cid:3) − (cid:2) P ( ) + P ( ) + pP ( ) + 2 ω D2 P ( ) + ω D1 P ( ) + ω A2 P ( ) (cid:3) , (C7)and dP ( ) dt = (cid:2) P ( ) + pP ( ) + pP ( ) + ω A1 P ( ) + ω D2 P ( ) + ω A2 P ( ) + ω A2 P ( ) (cid:3) − (cid:2) P ( )+ pP ( ) + 2 pP ( ) + pP ( ) + pP ( ) + ω D1 P ( ) + ω A2 P ( ) + 2 ω D2 P ( ) (cid:3) . (C8)Inserting Eq. (33) into Eqs. (C1)–(C8) and noting that in the steady state, ddt P ( i kj l ) = 0 ( i, j, k, l ∈ { , } ), we get P ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) + pP ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) + P ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) + pP ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) P ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) + P ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) + P ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) + P ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j )+ ω A2 P ( ) + ω A2 P ( ) + ω D1 P ( ) + ω D1 P ( ) − pP ( ) − ω A1 P ( ) − ω D2 P ( ) = 0 , (C9) P ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) + pP ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) + P ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) + pP ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j )+ ω A2 P ( ) + ω A2 P ( ) + ω A2 P ( ) + ω A2 P ( ) − pP ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) − pP ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) − pP ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) − ω D2 P ( ) = 0 , (C10) P ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) + pP ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) + P ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) + P ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j )+ P ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) + P ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) + ω A1 P ( ) + ω D1 P ( ) + ω D1 P ( ) + ω D2 P ( ) − P ( ) − P ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) − pP ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) − ω A1 P ( ) − ω A2 P ( ) − ω D1 P ( ) = 0 , (C11) P ( ) + pP ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) + pP ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) + ω D1 P ( ) + ω A1 P ( ) + ω D1 P ( )+ ω D2 P ( ) − P ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) − P ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) − pP ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) − P ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) − P ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) − ω A1 P ( ) − ω D1 P ( ) − ω A2 P ( ) = 0 , (C12)4 P ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) + pP ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) + pP ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) + pP ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j )+ ω A1 P ( ) + ω D2 P ( ) + ω A1 P ( ) + ω D2 P ( ) − P ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) − P ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) − P ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) − pP ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) − ω D1 P ( ) − ω A2 P ( ) = 0 , (C13) P ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) + pP ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) + pP ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) + pP ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j )+ pP ( ) + ω A1 P ( ) + ω D2 P ( ) + ω D2 P ( ) + ω A1 P ( ) − P ( ) − P ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) − pP ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) − P ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) − P ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) − ω D1 P ( ) − ω A2 P ( ) = 0 , (C14) P ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) + pP ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) + P ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) + pP ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j )+ pP ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) + pP ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) + ω A2 P ( ) + ω A2 P ( ) + ω A1 P ( ) + ω D2 P ( ) − pP ( ) − P ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) − P ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) − ω D2 P ( ) − ω D1 P ( ) − ω A2 P ( ) = 0 , (C15)and pP ( ) + P ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) + pP ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) + ω A1 P ( ) + ω D2 P ( ) + ω A2 P ( )+ ω A2 P ( ) − P ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) − pP ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) − pP ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) − pP ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) − pP ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) − ω D1 P ( ) − ω A2 P ( ) − ω D2 P ( ) = 0 , (C16)respectively. Appendix D: Other independent master equations for the 4-cluster mean-field analysis for bidirectional flows
In this Appendix, we summarize the other independent master equations for 4-cluster probabilities for bidirectionalflows.For P ( ), P ( ), P ( ), P ( ), P ( ), P ( ), P ( ), and P ( ), the master equations can be expressed as dP ( ) dt = (cid:2) P ( ) + pP ( ) + pP ( ) + pP ( ) + pP ( ) + pP ( ) + P ( ) + pP ( )+ ω A1 P ( ) + ω A1 P ( ) + ω D2 P ( ) + ω D2 P ( ) (cid:3) − (cid:2) P ( ) + 2 ω A2 P ( ) + 2 ω D1 P ( ) (cid:3) , (D1) dP ( ) dt = (cid:2) P ( ) + pP ( ) + P ( ) + pP ( ) + ω A2 P ( ) + ω A2 P ( ) + ω A2 P ( ) + ω A2 P ( ) (cid:3) − (cid:2) pP ( ) + pP ( ) + pP ( ) + pP ( ) + 4 ω D2 P ( ) (cid:3) , (D2)5 dP ( ) dt = (cid:2) P ( ) + pP ( ) + pP ( ) + pP ( ) + P ( ) + P ( ) + ω A1 P ( ) + ω D1 P ( )+ ω D1 P ( ) + ω D2 P ( ) (cid:3) − (cid:2) P ( ) + P ( ) + pP ( ) + ω D1 P ( ) + 2 ω A1 P ( ) + ω A2 P ( ) (cid:3) , (D3) dP ( ) dt = (cid:2) P ( ) + P ( ) + P ( ) + ω A1 P ( ) + ω D2 P ( ) + ω D1 P ( ) + ω D1 P ( ) (cid:3) − (cid:2) P ( )+ P ( ) + P ( ) + pP ( ) + P ( ) + pP ( ) + ω D1 P ( ) + ω A2 P ( ) + 2 ω A1 P ( ) (cid:3) , (D4) dP ( ) dt = (cid:2) P ( ) + pP ( ) + pP ( ) + pP ( ) + ω A1 P ( ) + ω A1 P ( ) + ω D2 P ( ) + ω D2 P ( ) (cid:3) − (cid:2) P ( ) + 2 P ( ) + pP ( ) + 2 ω D1 P ( ) + 2 ω A2 P ( ) (cid:3) , (D5) dP ( ) dt = (cid:2) P ( ) + P ( ) + pP ( ) + P ( ) + P ( ) + ω A2 P ( ) + ω A2 P ( ) + ω D1 P ( ) + ω D1 P ( ) (cid:3) − (cid:2) pP ( ) + pP ( ) + pP ( ) + P ( ) + pP ( ) + 2 ω D2 P ( ) + 2 ω A1 P ( ) (cid:3) , (D6) dP ( ) dt = (cid:2) pP ( ) + P ( ) + pP ( ) + ω A2 P ( ) + ω A2 P ( ) + ω A1 P ( ) + ω D2 P ( ) (cid:3) − (cid:2) pP ( ) + pP ( ) + P ( ) + 2 P ( ) + pP ( ) + ω D1 P ( ) + 2 ω D2 P ( ) + ω A2 P ( ) (cid:3) , (D7)and dP ( ) dt = (cid:2) P ( ) + pP ( ) + P ( ) + pP ( ) + pP ( ) + pP ( ) + ω A2 P ( ) + ω A2 P ( )+ ω A1 P ( ) + ω D2 P ( ) (cid:3) − (cid:2) pP ( ) + pP ( ) + pP ( ) + 2 ω D2 P ( ) + ω D1 P ( ) + ω A2 P ( ) (cid:3) . (D8)Inserting Eq. (33) into Eqs. (D1)–(D8) and noting that in the steady state ddt P ( i kj l ) = 0 ( i, j, k, l ∈ { , } ), we get P ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) + pP ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) + pP ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) + pP ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j )+ pP ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) + pP ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) + P ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) + pP ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j )+ ω A1 P ( ) + ω D2 P ( ) + ω D2 P ( ) + ω A1 P ( ) − P ( ) − ω D1 P ( ) − ω A2 P ( ) = 0 , (D9) P ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) + pP ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) + P ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) + pP ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j )+ ω A2 P ( ) + ω A2 P ( ) + ω A2 P ( ) + ω A2 P ( ) − pP ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) − pP ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) − pP ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) − pP ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) − ω D2 P ( ) = 0 , (D10)6 P ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) + pP ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) + pP ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) + pP ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j )+ P ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) + P ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) + ω A1 P ( ) + ω D1 P ( ) + ω D1 P ( ) + ω D2 P ( ) − P ( ) − P ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) − pP ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) − ω A1 P ( ) − ω A2 P ( ) − ω D1 P ( ) = 0 , (D11) P ( ) + P ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) + P ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) + ω A1 P ( ) + ω D2 P ( ) + ω D1 P ( )+ ω D1 P ( ) − P ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) − P ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) − pP ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) − P ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) − pP ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) − ω D1 P ( ) − ω A2 P ( ) − ω A1 P ( ) = 0 , (D12) P ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) + pP ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) + pP ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) + pP ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j )+ ω A1 P ( ) + ω D2 P ( ) + ω A1 P ( ) + ω D2 P ( ) − P ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) − P ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) − pP ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) − ω D1 P ( ) − ω A2 P ( ) = 0 , (D13) P ( ) + P ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) + pP ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) + P ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j )+ P ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) + ω A2 P ( ) + ω A2 P ( ) + ω D1 P ( ) + ω D1 P ( ) − pP ( ) − pP ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) − pP ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) − P ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) − pP ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) − ω A1 P ( ) − ω D2 P ( ) = 0 , (D14) pP ( ) + P ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) + pP ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) + ω A2 P ( ) + ω A2 P ( ) + ω A1 P ( )+ ω D2 P ( ) − P ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) − pP ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) − pP ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) − pP ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) − P ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) − ω D2 P ( ) − ω D1 P ( ) − ω A2 P ( ) = 0 , (D15)and P ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) + pP ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) + P ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) + pP ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j )+ pP ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) + pP ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) + ω A2 P ( ) + ω A2 P ( ) + ω A1 P ( ) + ω D2 P ( ) − pP ( ) − pP ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) − pP ( ) P ( ) P i ∈{ , } P j ∈{ , } P ( i j ) − ω D2 P ( ) − ω D1 P ( ) − ω A2 P ( ) = 0 , (D16)7respectively. 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