Asymmetric coupling in multi-channel simple exclusion processes

J

ournal of Statistical Mechanics: Theory and Experiment

An IOP and SISSA journal

Asymmetric coupling in multi-channel simple exclusion processes 1

School of Engineering Science, University of Science and Technology of China, Hefei 230026, People’s Republic of China 2 Department of Mathematics and Statistics, Curtin University of Technology, Perth WA6845, Australia E-mail: [email protected], [email protected], [email protected], [email protected], [email protected] and [email protected] Received 12 June 2008 Accepted 3 July 2008 Published 18 July 2008 Online at stacks.iop.org/JSTAT/2008/P07016 doi:10.1088/1742-5468/2008/07/P07016

Abstract. In this paper, we investigate N -channel simple exclusion processes in which particles can move across N parallel lattices and jump fully asymmetrically between the channels. Monte Carlo simulations are carried out to investigate the phase diagrams and density profiles of the system. It is shown that the phase diagram structure changes with increase of N . Moreover, how the phase diagram depends on N is obtained. A vertical cluster mean-field analysis is carried out; it shows good agreement with simulations.

Keywords: phase diagrams (theory), driven diffusive systems (theory), traffic models

c 2008 IOP Publishing Ltd and SISSA

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J. Stat. Mech. (2008) P07016

Zhong-Pan Cai1, Yao-Ming Yuan1, Rui Jiang1, Mao-Bin Hu1,2, Qing-Song Wu1 and Yong-Hong Wu2

Asymmetric coupling in multi-channel simple exclusion processes

Contents 1. Introduction

2

2. Model

3

3. Simulation results

4

4. Theoretical calculations

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Acknowledgments

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References

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1. Introduction Since their first use in 1968 for describing the kinetics of biopolymerization [1], asymmetric simple exclusion processes (ASEPs), which are discrete non-equilibrium models that describe the stochastic dynamics of multi-particle transport across one-dimensional lattices, have attracted great interest from physicists, biologists and chemists [2, 3]. In this model, each lattice site can be occupied by a single particle or it can be empty; the particle can hop to the left or the right with hard-core exclusion. This simple transport model provides the first crucial steps towards the modeling of realistic processes, such as protein synthesis [4]–[6], surface growth [7, 8], the motion of motor proteins along cytoskeleton filaments [9], and vehicular traffic [10]. In the simplest limit of an ASEP, which is called a totally asymmetric simple exclusion process (TASEP), the particles can move only in one direction. Exact solutions for TASEPs exist [11, 12]. With open boundaries, three stationary phases exist, specified by the processes at the entrance, at the exit and in the bulk of the system. We denote the injection and extraction rates at the entrance and exit by α and β respectively. When α < β and α < 0.5, the system has low density (LD). When α > β and β < 0.5, the system has high density (HD). And when α > 0.5 and β > 0.5 the system has maximum current (MC). In order to describe some more complex realistic processes, some two-channel ASEPs have been studied [13]–[24]. Pronina and Kolomeisky have studied a two-channel ASEP, both with symmetric coupling [16] and with asymmetric coupling [17]. Juh´asz studied the weak coupling situation for a two-way two-channel ASEP [18]. Jiang et al investigated both strong and weak coupling in a unidirectional two-channel ASEP [19]. Their model is different from the model of Pronina and Kolomeisky [17], in which the particles change lane first if the corresponding site on the other lane is empty. Jiang et al also studied a two-channel ASEP considering particle attachment and detachment [20]. Popkov and Peschel investigated a two-channel ASEP in which there is no exchange of particles, but the hopping rates in one chain depend on the local configuration in the other [21]. Brankov et al have studied asymmetric exclusion process on chains with a double-chain section in the middle [22]. The two-channel asymmetric exclusion processes with narrow entrances doi:10.1088/1742-5468/2008/07/P07016

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5. Conclusion and discussion

Asymmetric coupling in multi-channel simple exclusion processes

has been investigated in [23, 24]. It is found that two symmetry breaking phases can be observed either under random update or under parallel update. Nevertheless, previous works only focus on the two-channel ASEP and never consider an ASEP with more channels. We believe that a coupling of more than two channels should have a strong effect on the stationary-state phase diagram. As a first step toward understanding the multi-channel ASEP, we extend the study on the asymmetric coupling of two-channel ASEP [17] to the N-channel (N > 2) situation in this paper. Both vertical cluster mean-field analysis and extensive computer Monte Carlo simulations are carried out to investigate the phase diagrams and density profiles of the system; these show that the phase diagram structure does change with increase of N. The paper is organized as follows. In section 2 the model is introduced. The simulation results are presented in section 3 and the analytic results are described in section 4. Finally, the conclusions are given in section 5. 2. Model As illustrated in figure 1, the system consists of N equal-size channels and each channel contains L sites. Random update is adopted and the updating rules are as follows. In an infinitesimal time interval dt, channel i is chosen at random (1 ≤ i ≤ N), then site j is chosen at random on channel i. • If 1 ≤ i < N, then site j will be updated according to the following updating rules. * If j = 1 (entrance of channel i) and this site is empty, a particle is inserted into this site with rate α. If this site is occupied, then the particle jumps to site 1 on channel i + 1 provided that the site is empty. If that site is also occupied, then the particle jumps to site 2 on channel i provided that site 2 is empty. * If j = L (exit of channel i) and this site is occupied, the particle jumps to site L on channel i + 1 provided that the site is empty. If that site is occupied, then the particle is removed with rate β. * If 1 < j < L and this site is occupied, the particle can jump to site j on channel i + 1 provided that the site is empty. If that site is occupied, then the particle can move to site j + 1 on channel i provided that site j + 1 is empty. doi:10.1088/1742-5468/2008/07/P07016

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Figure 1. Schematic picture of the model for N -channel asymmetric simple exclusion processes. Arrows indicate the allowed transitions.

Asymmetric coupling in multi-channel simple exclusion processes

• If i = N, i.e., channel N is chosen, then site j will be updated according to the following updating rules. Note that particles on channel N do not jump to any other channels. * If j = 1 (entrance of channel N) and this site is empty, a particle is inserted into this site with rate α. If this site is occupied, the particle jumps to site 2 on channel N provided that site 2 is empty. * If j = L (exit of channel N) and this site is occupied, the particle is removed with rate β. * If 1 < j < L, and this site is occupied, the particle can move to the site j + 1 provided that the site is empty. To summarize, in the model, a particle first tries to change to the next lane, and if it fails, it moves horizontally. Namely, the model corresponds to a full asymmetry (see [17]). The more general asymmetric coupling will be studied in future work. 3. Simulation results In this section, the simulation results are presented and discussed. In our simulations, we set L = 2000. A transient time of 4 × 108 time steps is discarded. We gather data for 4 × 109 time steps. Figure 2 shows the phase diagram corresponding to N = 3. One can see that 12 stationary states exist. In the phases (HD, 1, 1) and (LD, 1, 1), the densities of channels 2 and 3 equal 1; channel 1 has high density and low density, respectively. The typical density profiles are shown in figures 3(a) and (b). In the phases (0, HD, 1) and (0, LD, 1), the density of channel 1 equals 0 and the density of channel 3 equals 1. Channel 2 has high density and low density, respectively. The typical density profiles are shown in figures 3(c) and (d). doi:10.1088/1742-5468/2008/07/P07016

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Figure 2. Phase diagram when N = 3. The lines are obtained from mean-field analysis. The Monte Carlo simulation results are in good agreement with the analytical results and therefore are not shown here.

Asymmetric coupling in multi-channel simple exclusion processes

J. Stat. Mech. (2008) P07016 Figure 3. Density profiles at (a) (HD, 1, 1)α = 0.10, β = 0.02; (b) (LD, 1, 1) α = 0.10, β = 0.04; (c) (0, HD, 1) α = 0.10, β = 0.08; (d) (0, LD, 1) α = 0.08, β = 0.10; (e) (0, 0, HD) α = 0.04, β = 0.10; (f) (0, 0, LD) α = 0.02, β = 0.10; (g) (10 ) α = 0.60, β = 0.14; (h) (11 ) α = 0.20, β = 0.20; (i) (12 ) α = 0.14, β = 0.60; (j) (20 ) α = 0.60, β = 0.20; (k) (21 ) α = 0.20, β = 0.60; (l) (3) α = 0.60, β = 0.60. The system size is L = 2000. , ×,  represent channels 1, 2, 3, respectively.

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Asymmetric coupling in multi-channel simple exclusion processes

In the phases (0, 0, HD) and (0, 0, LD), the densities of channels 1 and 2 equal 0; channel 3 has high density and low density, respectively. The typical density profiles are shown in figures 3(e) and (f). In the phases 10 , 11 and 12 , maximum current (MC) appears on one of the channels. Phase 10 corresponds to (MC, 1, 1), where channel 1 has MC and channels 2 and 3 are fully occupied (figure 3(g)). Phase 11 corresponds to (0, MC, 1), where channel 1 has zero density, channel 2 has MC and channel 3 is fully occupied (figure 3(h)). Phase 12 corresponds to (0, 0, MC), where channels 1 and 2 have zero density and channel 3 has MC (figure 3(i)). In the phases 20 and 21 , the system is classified into two parts by a domain wall. In phase 20 , the left bulk is in (MC, 1, 1) and the right bulk is in (0, MC, 1) (figure 3(j)). In phase 21 , the left bulk is in (0, MC, 1) and the right bulk is in (0, 0, MC) (figure 3(k)). Here we would like to mention that the domain wall does not necessarily have to be in the middle of the system. It can be anywhere provided that the system is large enough. However, in a small system (e.g., L = 2000), the domain wall is exactly in the middle of the system3 . Finally, in phase 3, the system is classified into three parts by two domain walls. The left bulk is in (MC, 1, 1), the middle bulk is in (0, MC, 1), and the right bulk is in (0, 0, MC) (figure 3(l)). Similarly, the two domain walls do not necessarily have to be exactly at x = 1/3, 2/3 in a large enough system. Figure 4 shows the phase diagram corresponding to N = 4. It can be seen that 18 different regions are classified. The phase diagram structure could be analyzed similarly to figure 2. We would like to mention that when α > 0.5 and β > 0.5, the system is in 3

When N = 2, the domain wall is also not necessarily exactly in the middle of the system provided that the system is large enough, which is not as claimed in [17].

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Figure 4. Phase diagram when N = 4. The lines are obtained from mean-field analysis. The Monte Carlo simulation results are in good agreement with the analytical results and therefore are not shown here.

Asymmetric coupling in multi-channel simple exclusion processes

the phase 4, in which the system is classified into four parts by three domain walls. From left to right, the bulks are in (MC, 1, 1, 1), (0, MC, 1, 1), (0, 0, MC, 1), (0, 0, 0, MC), respectively. The results can be generalized to any value of N and are summarized as follows. (i) There exist 2N + N(N + 1)/2 different stationary states. (ii) When α > 0.5 and β > 0.5, the system is in the phase N in which the system is classified into N parts by N −1 domain walls. (iii) In the phase nm , the system is classified into n parts by n − 1 domain walls. In the next section, an analytical investigation on the phase boundaries is presented.

In this section, we use the vertical cluster mean-field approach to calculate the phase diagram. Firstly, we consider the situation of N = 3. We define Pτ1 τ2 τ3 as a probability for finding a vertical cluster in which the sites belonging to channels 1, 2, 3 are in states τ1 , τ2 , τ3 , respectively. Here τi = 0 or 1 means that the site on channel i is empty or occupied (i = 1, 2, 3). The conservation of probability requires that P111 + P110 + P101 + P100 + P011 + P010 + P001 + P000 = 1.

(1)

In addition, we can use the following equation to express the bulk densities for each channel: ρ1 = P111 + P110 + P101 + P100 ,

(2)

ρ2 = P111 + P110 + P011 + P010 ,

(3)

ρ3 = P111 + P101 + P011 + P001 .

(4)

We can use master equations to describe the evolution of the vertical cluster state, which take into account possible hopping of one particle between two neighboring clusters along the same channel, as well as allowed changes between channels within the same cluster: dP111 = P110 P101 + 2P110 P011 + P110 P001 + P011 P101 dt − 3P111 P000 − 2P111 P100 − 2P111 P010 − 2P111 P001 ,

(5)

dP110 = 2P111 P100 + 2P111 P010 + P111 P000 + P011 P100 − P110 dt − 2P110 P011 − 2P110 P001 − P110 P000 − P110 P101 ,

(6)

dP101 = 2P111 P100 + 2P111 P001 + P111 P000 + P100 P011 + P100 P001 + P110 dt + P110 P001 − P101 − P101 P110 − P101 P010 − P101 P000 − P101 P011 ,

(7)

dP100 = P000 P111 + P000 P110 + P101 P110 + P101 P010 − P100 dt − 2P100 P111 − 2P100 P011 − P100 P001 ,

(8)

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4. Theoretical calculations

Asymmetric coupling in multi-channel simple exclusion processes

dP011 = P010 P111 + P010 P101 + P010 P001 + 2P111 P001 + P111 P010 + P111 P000 dt + P101 − P011 P101 − 2P011 P100 − 2P011 P000 − 2P011 P110 ,

(9)

(10)

dP001 = P000 P111 + P000 P101 + 2P000 P011 + P011 P101 + P011 P100 dt + P010 − 2P001 P110 − P001 P100 − P001 P010 − 2P001 P111 ,

(11)

dP000 = P001 P110 + P001 P100 + P001 P010 − 3P000 P111 dt − P000 P110 − 2P000 P011 − P000 P101 .

(12)

Here terms with ‘+’ denote the generating probabilities of Pτ1 τ2 τ3 , and terms with ‘−’ denote the disappearance probabilities of Pτ1 τ2 τ3 . In the limit of t → ∞ the system reaches a stationary state with dPτ1 τ2 τ3 /dt = 0. Substituting this into equations (5)–(12), together with equation (1), it is straightforward to obtain three solutions: Solution 1: P101 = P100 = P111 = P110 = P010 = P011 = 0,

(13)

Solution 2: P101 = P100 = P111 = P110 = P010 = P000 = 0,

(14)

Solution 3: P101 = P100 = P110 = P000 = P001 = P010 = 0.

(15)

Solution 1 means that channels 1 and 2 are empty, solution 2 means that channel 1 is empty and channel 3 is fully occupied, while solution 3 means that channels 2 and 3 are fully occupied. We can express the stationary currents in the bulk of the system: Jbulk,1 = (P111 + P110 )(1 − P111 − P110 − P101 − P100 ),

(16)

Jbulk,2 = (P111 + P011 )(1 − P111 − P110 − P011 − P010 ),

(17)

Jbulk,3 = (P111 + P101 + P011 + P001 )(1 − P111 − P101 − P011 − P001 ).

(18)

The particle currents at the entrance are given by the following expressions: Jentr,1 = α(1 − P111 − P110 − P101 − P100 ),

(19)

Jentr,2 = α(1 − P111 − P110 − P011 − P010 ),

(20)

Jentr,3 = α(1 − P111 − P101 − P011 − P001 ).

(21)

The particle currents at the exit are given by the following expressions: Jexit,1 = β(P111 + P110 ),

(22)

Jexit,2 = β(P111 + P011 ),

(23)

Jexit,3 = β(P111 + P101 + P011 + P001 ).

(24)

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dP010 = P000 P111 + 2P000 P011 + P100 + P100 P011 + 2P110 P011 + P110 P001 dt + P110 P000 − P010 − 2P010 P111 − P010 P101 − P010 P001 ,

Asymmetric coupling in multi-channel simple exclusion processes

• Firstly, let us consider the situation when the system is described by solution 1 (equation (13)). In this case, the currents can be expressed as follows: Jbulk,1 = 0,

Jbulk,2 = 0,

Jbulk,3 = P001 (1 − P001 ),

(25)

Jentr,1 = α,

Jentr,2 = α,

Jentr,3 = α(1 − P001 ),

(26)

Jexit,1 = 0,

Jexit,2 = 0,

Jexit,3 = βP001 .

(27)

The effective entrance rate αeff can be determined in the following way [22]: Jentr,1 + Jentr,2 + Jentr,3 = Jbulk,1 + Jbulk,2 + Jbulk,3 .

(28)

α + α + α(1 − P001 ) = αeff (1 − P001 ) = P001 (1 − P001 ).

(29)

αeff can be obtained (denoted as αeff,I ):  1 + α − (1 + α)2 − 12α αeff,I = . (30) 2 On the other hand, the effective exit rate βeff can be determined in the following way [22]: Jexit,1 + Jexit,2 + Jexit,3 = Jbulk,1 + Jbulk,2 + Jbulk,3 .

(31)

Substituting equations (25) and (27) into equation (31), we have βP001 = βeff P001 = P001 (1 − P001 ).

(32)

βeff can be obtained (denoted as βeff,I ): βeff,I = β.

(33)

• Now let us consider the situation when the system is described by solution 2 (equation (14)). In this case, the currents can be expressed as follows: Jbulk,1 = 0,

Jbulk,2 = P011 (1 − P011 ),

Jentr,1 = α,

Jentr,2 = α(1 − P011 ),

Jexit,1 = 0,

Jexit,2 = βP011 ,

Jbulk,3 = 0, Jentr,3 = 0,

Jexit,3 = β.

(34) (35) (36)

Substituting equations (34) and (35) into equation (28), we have α + α(1 − P011 ) = αeff (1 − P011 ) = P011 (1 − P011 ).

(37)

αeff can be obtained (denoted as αeff,II ):  1 + α − (1 + α)2 − 8α αeff,II = . (38) 2 On the other hand, substituting equations (34) and (36) into equation (31), we have β + βP011 = βeff P011 = P011 (1 − P011 ). βeff can be obtained (denoted as βeff,II ):  1 + β − (1 + β)2 − 8β βeff,II = . 2 doi:10.1088/1742-5468/2008/07/P07016

(39)

(40) 9

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Substituting equations (25) and (26) into equation (28), we have

Asymmetric coupling in multi-channel simple exclusion processes

• Finally let us consider the situation when the system is described by solution 3 (equation (15)). In this case, the currents can be expressed as follows: Jbulk,1 = P111 (1 − P111 ), Jentr,1 = α(1 − P111 ), Jexit,1 = βP111 ,

Jbulk,2 = 0, Jentr,2 = 0,

Jexit,2 = β,

Jbulk,3 = 0, Jentr,3 = 0,

Jexit,3 = β.

(41) (42) (43)

Substituting equations (41) and (42) into equation (28), we have (44)

αeff can be obtained (denoted as αeff,III ): αeff,III = α.

(45)

On the other hand, substituting equations (41) and (43) into equation (31), we have 2β + βP111 = βeff P111 = P111 (1 − P111 ). βeff can be obtained (denoted as βeff,III ):  1 + β − (1 + β)2 − 12β . βeff,III = 2

(46)

(47)

Having the expressions for αeff and βeff , the boundaries are determined as follows. • The boundary between the (0, 0, LD) phase and the (0, 0, HD) phase is determined by αeff,I = βeff,I and αeff,I < 1/2, i.e.,  1 + α − (1 + α)2 − 12α = β, α < 1/10. 2 • The boundary between the (0, 0, HD) phase and the (0, LD, 1) phase is determined by αeff,II = βeff,I and αeff,II < 1/2, i.e.,  1 + α − (1 + α)2 − 8α = β, α < 1/6. 2 • The boundary between the (0, LD, 1) phase and the (0, HD, 1) phase is determined by αeff,II = βeff,II and αeff,II < 1/2, i.e., α = β,

α < 1/6.

• The boundary between the (0, HD, 1) phase and the (LD, 1, 1) phase is determined by αeff,III = βeff,II and βeff,II < 1/2, i.e.,  1 + β − (1 + β)2 − 8β , β < 1/6. α= 2 • The boundary between the (LD, 1, 1) phase and the (HD, 1, 1) phase is determined by αeff,III = βeff,III and βeff,III < 1/2, i.e.,  1 + β − (1 + β)2 − 12β α= , β < 1/10. 2 doi:10.1088/1742-5468/2008/07/P07016

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α(1 − P111 ) = αeff (1 − P111 ) = P111 (1 − P111 ).

Asymmetric coupling in multi-channel simple exclusion processes

Note that the boundaries are symmetric due to particle–hole symmetry. The analytical results are shown in figures 2 and 4, and they are in good agreement with simulation results. Having obtained these boundaries, other boundaries, which are straight lines, could be determined accordingly. Specifically, the horizontal boundaries are β = 1/2, 1/6, 1/10 and the vertical boundaries are α = 1/2, 1/6, 1/10. The results can be extended to the N-channel situation with N > 3, as listed below.

• The boundary between the (LD, 1, 1, . . . , 1) phase and the (0, HD, 1, . . . , 1) phase is determined by  1 + β − (1 + β)2 − 4(N − 1)β . (49) α= 2 • The boundary between the (0, 0 · · · 0, HD, 1, 1, . . . , 1) phase and the (0, 0 · · · 0, LD, 1, 1, . . . , 1) (the number of zeros is n) phase is determined by   1 + α − (1 + α)2 − 4(n + 1)α 1 + β − (1 + β)2 − 4(N − n)β = . (50) 2 2 • The boundary between the (0, 0 · · · 0, LD, 1, 1, . . . , 1) (the number of zeros is n) phase and the (0, 0, 0 · · · 0, HD, 1, 1, . . . , 1) (the number of zeros is n+1) phase is determined by   1 + β − (1 + β)2 − 4(N − n − 1)β 1 + α − (1 + α)2 − 4(n + 1)α = . (51) 2 2 • The horizontal boundaries are β = 1/2, 1/6, 1/10, . . . , 1/(4N − 2). The vertical boundaries are α = 1/2, 1/6, 1/10, . . . , 1/(4N − 2). 5. Conclusion and discussion In this paper, we have studied N-channel simple exclusion systems in which particles can move across N parallel lattices and jump between the channels fully asymmetrically. We consider the situations N = 3 and 4 in detail. We find that 12 and 18 stationary states exist in the phase diagram, respectively. The density profiles of the phases are presented and the variation rules of the phase diagram are obtained. Then the results are generalized to the N-channel ASEP. The cluster mean-field analysis is also carried out; it shows good agreement with simulation results. In our future work, the studies will be extended to consider more general asymmetric couplings in the multi-channel ASEP. Furthermore, other issues (including weak coupling, particle attachment and detachment, inhomogeneous coupling, etc) will also be investigated. doi:10.1088/1742-5468/2008/07/P07016

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• The boundary between the (HD, 1, 1, . . . , 1) phase and the (LD, 1, 1, . . . , 1) phase is determined by  1 + β − (1 + β)2 − 4Nβ . (48) α= 2

Asymmetric coupling in multi-channel simple exclusion processes

Acknowledgments This work was supported by the National Basic Research Program of China (No. 2006CB705500), the NNSFC under Project Nos 10532060, 70601026, 10672160, the CAS President Foundation, the NCET and the FANEDD. Y-H Wu acknowledges the support of the Australian Research Council through a Discovery Project Grant. References

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