Strong asymmetric coupling of multilane PASEPs

Physics Letters A 376 (2012) 2640–2644

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Physics Letters A www.elsevier.com/locate/pla

Strong asymmetric coupling of multilane PASEPs Qi-Hong Shi, Rui Jiang ∗ , Mao-Bin Hu, Qing-Song Wu State Key Laboratory of Fire Science and School of Engineering Science, University of Science and Technology of China, Hefei 230026, People’s Republic of China

a r t i c l e

i n f o

Article history: Received 25 February 2012 Received in revised form 17 May 2012 Accepted 6 June 2012 Available online 27 July 2012 Communicated by A.R. Bishop Keywords: ASEP Phase diagram Vertical cluster mean field theory

a b s t r a c t This Letter has firstly investigated three-lane PASEPs in which particles on lane i could move forward with rate p i and backward with rate qi . Particles could also jump fully asymmetrically from lane i to lane i + 1. It is found that in the case that p i > qi and p i − qi are not equal to each other, the phase diagram in three-lane system has three different geometric structures. The principles of phase diagram structure have been generalized to multilane PASEPs. The case that p i − qi could be equal has also been studied. Unfortunately, the phase diagram structure is complex and generalization to multilane PASEPs is difficult. A vertical cluster mean-field analysis has been carried out, which shows good agreement with simulations. © 2012 Elsevier B.V. All rights reserved.

In the past years, asymmetric simple exclusion process (ASEP) has attracted interests of many researchers because of its critical role for understanding various non-equilibrium phenomena in chemistry, physics and biology [1–4]. The ASEP is defined on a one-dimensional lattice of size L, on which each site can be either empty or occupied by one particle. In the bulk, if the target site is empty, particles can hop from site i with rate p to site i + 1 or with rate q to site i − 1. The ASEP is called totally ASEP (TASEP) if p = 0 or q = 0, otherwise it is named partially ASEP (PASEP). Mechanisms of many non-equilibrium processes, such as biological transport, kinetics of protein synthesis and biopolymerization, car traffic, and hopping of quantum dots, have been investigated due to the successful description via ASEP [5–13]. Recently many works have been carried out to extend single lane ASEP to two-lane and multilane ASEPs to describe, e.g., the motion of molecular motors along protofilament, the extraction of membrane tubes by molecular motors, car traffic on multilane road, pedestrian traffic in channel, macroscopic clustering phenomena, spin transport and various systems of oppositely moving particles [14–40]. Among them, Cai et al. have studied asymmetric coupling in multiple TASEPs [23], and have obtained how the phase diagram structure depends on the lane number. In this Letter we extend the work of Ref. [23] to study asymmetric coupling in multiple PASEPs. We consider a system of N parallel one-dimensional lattices of length L (Fig. 1). At every time step a site is randomly chosen. The particle dynamics in the bulk of the system are as follows. Particles on lane i (i < N) will jump

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Corresponding author. Tel.: +86 551 3600127; fax: +86 551 3606459. E-mail address: [email protected] (R. Jiang).

0375-9601/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physleta.2012.06.040

Fig. 1. Schematic picture of the N-lane PASEPs. The arrows show allowed hopping.

to lane i + 1 with rate 1 if the corresponding site on lane i + 1 is empty, otherwise they will move on lane i with rate p i toward right and rate q i toward left. Particles on lane N move with rate p N toward right and rate q N toward left without lane changing. At the entrances, particles are injected with rate α . At the exits, particles are removed with rate β . Note that in the special case N = 2, the model reduces to two-lane model, which has been studied in Ref. [40]. Firstly we consider a three-lane PASEP. We carry out a vertical cluster mean field (MF) analysis of the system, which was firstly proposed by Kolomeisky group [24–27] and explicitly takes into account the correlations inside the vertical cluster of lattice site and is proved to be an appropriate and convenient theoretical tool for analyzing the multilane ASEPs. Let us define P τ1 τ2 τ3 as a probability to find a vertical cluster in which the sites belonging to lanes 1, 2, 3 are in states τ1 , τ2 , τ3 , respectively. Here

Q.-H. Shi et al. / Physics Letters A 376 (2012) 2640–2644

τi = 0 or 1 means that the site on lane i is empty or occupied. We have

P 111 + P 110 + P 101 + P 100 + P 011 + P 010

+ P 001 + P 000 = 1

(1)

due to the conservation of probability. Assuming that far away from the boundaries, the system is uniform along the lattices, the dynamics of the bulk vertical clusters are thus governed by master equations. We give one example of master equations as following and other equations could be obtained similarly

d P 111 dt

2641

Firstly, we consider the situation when the system is described by solution 1. The expressions for the particle currents are significantly simplified,

J bulk,1 = 0,

J bulk,2 = 0,

J bulk,3 = P 001 (1 − P 001 )( p 3 − q3 ), J entr,1 = α ,

J entr,2 = α ,

J exit,1 = 0,

J exit,2 = 0,

J entr,3 = α (1 − P 001 ), J exit,3 = β P 001 .

(9)

− ( p 1 + q1 + p 2 + q2 + p 3 + q3 ) P 111 P 000

If we view the vertical clusters (001) as “particles” and vertical clusters (000) as “holes”, the three-lane system can be simplified into one-lane PASEP with forward rate p 3 and backward rate q3 . Naturally there will be three phases LD3 , HD3 and MC3 . Here X i means that lane i is in phase X , lane j with j < i is empty, and lane j with j > i is fully occupied. In the low-density LD3 phase, the total current in the system is determined by the entrance,

− ( p 2 + q2 + p 3 + q3 ) P 111 P 100

J entr,1 + J entr,2 + J entr,3 = J bulk,1 + J bulk,2 + J bulk,3 .

= ( p 3 + q3 ) P 110 P 101 + ( p 1 + q1 + p 3 + q3 ) P 110 P 011 + ( p 3 + q3 ) P 110 P 001 + ( p 1 + q1 ) P 011 P 101

− ( p 1 + q1 + p 3 + q3 ) P 111 P 010

Substituting Eq. (9) into Eq. (10) we have

− ( p 1 + q1 + p 2 + q2 ) P 111 P 001 .

(2) dPτ τ τ

1 2 3 When the system reaches stationary state, we have = 0. dt Solving the master equations and considering particle conservation, it is straightforward to obtain three solutions:

Solution 1:

P 101 = P 100 = P 111 = P 110

= P 010 = P 011 = 0, Solution 2:

(3)

(4)

P 101 = P 100 = P 110 = P 000

= P 001 = P 010 = 0.

αeff ,3 (1 − P 001 ) = 2α + α (1 − P 001 ) = P 001 (1 − P 001 )( p 3 − q3 ).

(5)

Solution 1 means that lane 1 and lane 2 are empty, solution 2 means that lane 1 is empty and lane 3 is fully occupied, while solution 3 means that lane 2 and lane 3 are fully occupied. The stationary currents in the bulk of the system are expressed as:

J bulk,1 = ( P 111 + P 110 )(1 − P 111 − P 110 − P 101 − P 100 )

=

p 3 − q3 + α 2



−

( p 3 − q3 + α )2 − 12α ( p 3 − q3 ) 2

J bulk,2 = ( P 111 + P 011 )(1 − P 111 − P 110 − P 011 − P 010 )

For the high density HD3 phase the exit process determines the particle dynamics. Thus

βeff ,3 = β.

(13)

The LD3 phase is restricted by

( p 3 − q3 )/2, which leads to p 3 − q3 + α −

αeff ,3 < βeff ,3 and αeff ,3 <

 ( p 3 − q3 + α )2 − 12α ( p 3 − q3 ) 2

  < min β, ( p 3 − q3 )/2

ρ3 = P 001 = 

× ( p 2 − q2 ), −

J bulk,3 = ( P 111 + P 101 + P 011 + P 001 )( p 3 − q3 )

(14)

(6)

p 3 − q3 + α 2( p 3 − q 3 )

( p 3 − q3 + α )2 − 12α ( p 3 − q3 ) . 2( p 3 − q 3 )

  β < min αeff ,3 , ( p 3 − q3 )/2 .

J entr,1 = α (1 − P 111 − P 110 − P 101 − P 100 ),

The density on lane 3 of this phase are given by

J entr,2 = α (1 − P 111 − P 110 − P 011 − P 010 ),

ρ3 = P 001 = 1 − β/( p 3 − q3 ). (7)

(15)

This phase (HD3 ) exists under the condition

The particle currents at the entrance of the two lanes are given by the following expressions:

J entr,3 = α (1 − P 111 − P 101 − P 011 − P 001 ).

. (12)

and the density on lane 3

× ( p 1 − q1 ),

× (1 − P 111 − P 101 − P 011 − P 001 ).

(11)

Thus we can obtain

αeff ,3 = P 001 ( p 3 − q3 )

P 101 = P 100 = P 111 = P 110

= P 010 = P 000 = 0, Solution 3:

(10)

(16)

(17)

The MC3 phase is specified by conditions

At the exit, the currents are

αeff ,3 > ( p 3 − q3 )/2,

J exit,1 = β( P 111 + P 110 ),

with ρ3 = 1/2. For solutions 3 and 2, we can assume clusters (111) (or (011)) as “particles” and clusters (011) (or (001)) as “holes”, respectively. Similarly we could have

J exit,2 = β( P 111 + P 011 ), J exit,3 = β( P 111 + P 101 + P 011 + P 001 ).

(8)

β > ( p 3 − q3 )/2

(18)

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Q.-H. Shi et al. / Physics Letters A 376 (2012) 2640–2644

Fig. 2. (Color online.) Density profiles of typical phases under values of p 1 = 0.8, p 2 = 0.6, p 3 = 1, q1 = q2 = q3 = 0. (a) Phase LD1 , α = 0.35, β = 0.1; (b) phase HD1 , α = 0.5, β = 0.05; (c) phase MC1 , α = 0.6, β = 0.15; (d) phase LD2 , α = 0.06, β = 0.1; (e) phase HD2 , α = 0.1, β = 0.07; (f) phase MC2 , α = 0.2, β = 0.15; (g) phase LD3 , α = 0.05, β = 0.8; (h) phase HD3 , α = 0.3, β = 0.4; (i) phase MC3 , α = 0.6, β = 0.7. Symbols correspond to Monte Carlo computer simulations, lines describe the theoretical bulk density profiles in each lane.

αeff ,1 = α , βeff ,1 =

αeff ,2 = βeff ,2 =

p 1 − q1 + β 2 p 2 − q2 + α 2 p 2 − q2 + β 2

 − −

(19)

( p 1 − q1 + β)2 − 12β( p 1 − q1 ) 2

 ( p 2 − q2 + α )2 − 8α ( p 2 − q2 ) 

−

2

( p 2 − q2 + β)2 − 8β( p 2 − q2 ) 2

,

(20)

,

(21)

.

(22)

Then regions of phases LD1 , HD1 , MC1 , LD2 , HD2 , and MC2 can be determined based on αeff ,1 , αeff ,2 , βeff ,1 , βeff ,2 . To summarize, the vertical cluster mean-field analysis indicates that there are nine stationary phases when p 1 − q1 = p 2 − q2 = p 3 − q3 , namely LD1 , HD1 , MC1 , LD2 , HD2 , MC2 and LD3 , HD3 , MC3 . The boundary between phase LDi and HDi (i = 1, 2, 3) is determined by

αeff ,i = βeff ,i and αeff ,i , βeff ,i < 1/2.

(23)

The boundary between phase HDi +1 and LDi (i = 1, 2) is determined by

J HDi+1 = J LDi

(24)

and J i is calculated by

J i = ( p i − qi )ρi (1 − ρi ).

(25)

We carry out extensive Monte Carlo computer simulations to test our theoretical results. In the simulation, we perform 1 × 1011

Monte Carlo time steps and the previous 2.5 × 1010 Monte Carlo time steps are abandoned in order to obtain stationary states. We simulated system of size L = 1000 for each lane, and in several cases we have also checked our simulations for lattices with L = 10 000. The phase diagram structure of system is determined by values p 1 − q1 , p 2 − q2 , p 3 − q3 based on our analysis. With “particle–hole” symmetry, we only need to consider three cases including (p 1 − q1 < p 2 − q2 < p 3 − q3 ), (p 3 − q3 < p 1 − q1 < p 2 − q2 ), and (p 2 − q2 < p 1 − q1 < p 3 − q3 ). Density profiles and phase diagrams are shown in Figs. 2 and 3. One can see that simulation results are in good agreement with theoretical predictions. We note that in Fig. 3(a), phases LD2 , HD2 , MC2 are embedded in region HD3 , and phases LD3 , HD3 , MC3 are embedded in region HD2 . In Fig. 3(b), phases LD1 , HD1 , MC1 (LD3 , HD3 , MC3 ) are embedded in HD2 (LD2 ) respectively. The geometric structure of diagram Fig. 3(c) is more complicated. Firstly phases LD2 , HD2 , MC2 are embedded in region LD1 , then phases LD1 , HD1 , MC1 are embedded in region HD3 . We generalize the principles of phase diagram structure to multilane PASEPs. For a randomly chosen lane i, search within (i + 1 → N ) to find the nearest lane j 1 with p j 1 − q j 1 > p i − qi and search within (1 → i − 1) to find the nearest lane j 2 with p j 2 − q j 2 > p i − qi . If p j 1 − q j 1 < p j 2 − q j 2 , phases LDi , HDi , MCi are embedded in region HD j 1 , otherwise if p j 1 − q j 1 > p j 2 − q j 2 , phases LDi , HDi , MCi are embedded in region LD j 2 . If j 2 ( j 1 ) does not exist, phases LDi , HDi , MCi are embedded in region HD j 1 (LD j 2 ). If p i − qi is larger than p j − q j of any other lane j, phases LDi , HDi , MCi are not embedded in other phase.

Q.-H. Shi et al. / Physics Letters A 376 (2012) 2640–2644

2643

Fig. 4. Phase diagram of a six-lane system obtained by MF analysis. Monte Carlo simulation shows good agreement with our analysis. (a) p 1 = 0.9, p 2 = 0.5, p 3 = 0.8, p 4 = 1, p 5 = 0.7, p 6 = 0.6, q1 = q2 = q3 = q4 = q5 = q6 = 0; (b) is the amplified graph of area marked by dashed line in (a).

Fig. 3. Phase diagrams obtained by MF analysis. Monte Carlo simulation shows good agreement with our analysis. (a) p 1 = 0.4, p 2 = 0.7, p 3 = 1, q1 = q2 = q3 = 0; (b) p 1 = 0.8, p 2 = 1, p 3 = 0.6, q1 = q2 = q3 = 0; (c) p 1 = 0.8, p 2 = 0.6, p 3 = 1, q1 = q2 = q3 = 0.

Furthermore, similar as in three-lane system, multilane PASEPs can be calculated by

αeff ,i = βeff ,i =

p i − qi + α 2 p i − qi + β

 −

 −

αeff ,i , βeff ,i in

( p i − qi + α )2 − 4i α ( p i − qi ) 2

,

(26)

2

( p i − qi + β)2 − 4( N − i + 1)β( p i − qi ) 2

(27)

and boundaries could be obtained as prescribed in Eqs. (23) and (24). We have checked our analysis via simulations, which is found to be correct. As an example, Fig. 4 shows the phase diagram of a six-lane PASEP.

Next we consider the case that p i > qi but p i − qi could be equal. In the subcase that p 1 − q1 = p 2 − q2 = p 3 − q3 , the situation is the same as that discussed in Ref. [23], via a rescaling of time. Now we study the subcase that two of p i − qi are equal. Due to particle–hole symmetry, we need to consider four situations, i.e., (i) p 1 − q1 > p 2 − q2 = p 3 − q3 ; (ii) p 1 − q1 < p 2 − q2 = p 3 − q3 ; (iii) p 2 − q2 < p 1 − q1 = p 3 − q3 ; (iv) p 2 − q2 > p 1 − q1 = p 3 − q3 . Fig. 5 shows four different phase diagram structures corresponding to the four situations. As expected, when p i − qi = p j − q j , coexistence phase MCi , j might be observed, which means that the left bulk is in MCi and the right bulk is in MC j , see Fig. 6. However, we would like to mention that there is no coexistence phenomenon in situation (iv). Due to the complexity of the phase diagram structure, generalization to multilane PASEPs is difficult. In conclusion, this Letter has studied three-lane PASEPs with asymmetric strong coupling. A vertical cluster mean-field analysis has been carried out, which shows good agreement with simulation results. In the case that p i > qi and p i − qi are not equal to each other, three phase diagram structures have been observed. We have generalized our discussion to N-lane system, which is found to be correct via verification of simulation. When p i − qi could be equal, the situation is different. The case that all p i − qi are equal to each other is the same as discussed in Ref. [23] via rescaling of time. When two of them are equal, four phase diagram structures are identified, in which phase coexistence phenomenon has been observed in three of them. Due to complexity of phase diagram structures, generalization to multilane PASEPs is difficult and needs to be investigated carefully in future work.

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Q.-H. Shi et al. / Physics Letters A 376 (2012) 2640–2644

Fig. 5. Phase diagram of a three-lane system obtained by MF analysis. Monte Carlo simulation shows good agreement with our analysis. (a) p 1 = 0.8, p 2 = p 3 = 0.6, q1 = q2 = q3 = 0; (b) p 1 = 0.8, p 2 = p 3 = 1, q1 = q2 = q3 = 0; (c) p 1 = 0.9, p 1 = p 3 = 1, q1 = q2 = q3 = 0; (d) p 2 = 1, p 1 = p 3 = 0.8, q1 = q2 = q3 = 0.

Fig. 6. (Color online.) Density profiles of typical coexistence phases. (a) Phase MC1,2 , α = 0.6, β = 0.2, p 3 = 0.8, p 1 = p 2 = 0.6, q1 = q2 = q3 = 0; (b) phase MC2,3 , 0.2, β = 0.4, p 1 = 0.8, p 2 = p 3 = 0.6, q1 = q2 = q3 = 0; (c) phase MC1,3 , α = 0.7, p 1 = 0.9, p 1 = p 3 = 1, q1 = q2 = q3 = 0.

Acknowledgements This work is supported by the National Basic Research Program of China (No. 2012CB725404), the NNSFC (Nos. 11072239 and 71171185), and the Fundamental Research Funds for the Central Universities (No. WK2320000014). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]

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