TOTALLY ASYMMETRIC EXCLUSION PROCESSES ON LATTICES...

December 18, 2009 12:3 WSPC/141-IJMPC 01488

International Journal of Modern Physics CVol. 20, No. 12 (2009) 1999–2012c© World Scientific Publishing Company

TOTALLY ASYMMETRIC EXCLUSION PROCESSES ON

LATTICES WITH A BRANCHING POINT

XIONG WANG∗, RUI JIANG∗,†,‡, MAO-BIN HU∗, KATSUHIRO NISHINARI†

and QING-SONG WU∗

∗School of Engineering Science

University of Science and Technology of China

Hefei 230026, China†Department of Aeronautics and Astronautics

School of Engineering, University of Tokyo

Hongo, Bunkyo-ku, Tokyo 113-8656, Japan‡[email protected]

Received 29 May 2009Accepted 6 September 2009

We study totally asymmetric simple exclusion process (TASEP) where particles move ona single-chain lattice which diverges into two parallel lattice branches. At the branchingpoint, the particles move to one of the two branches with equal rate r/2. The phasediagram and density profiles are investigated by using mean-field approximation andMonte Carlo simulations. It is found that the phase diagram can be classified into threeregions at any value of r. However, a threshold rc = 2 −

√2 is identified. In cases

of r > rc and r < rc, the phase diagram exhibits qualitatively different phases. Theanalytical results are in good agreement with the results of Monte Carlo simulations.

Keywords: TASEP; mean-field approximation; phase diagram.

1. Introduction

The analysis of self-driven particle models is a central issue of non-equilibrium sta-

tistical mechanics. These models show a variety of generic non-equilibrium effects,

in particular, in low dimensions. A very prominent example of such a model is the

asymmetric exclusion processes (ASEPs). ASEPs were introduced originally as a

model to explain the processes of transcription (creation of the messenger RNA)

and translation (creation of proteins) in 19681 and now have become an important

tool to investigate various non-equilibrium phenomena in chemistry, physics and

biology. They have been applied successfully to understand polymer dynamics in

dense media,2 diffusion through membrane chains,3 gel electrophoresis,4 dynam-

ics of motor proteins moving along rigid filaments,5 the kinetics of synthesis of

proteins,6,7 and traffic flow analysis.8,9

ASEPs are one-dimensional models where particles hop along discrete lattice.

The only interaction between particles is the hard-core exclusion that prevents any

1999


2000 X. Wang et al.

site from being occupied by more than one particle, i.e. each lattice site can be

either empty or occupied by a single particle. It is called totally asymmetric simple

exclusion process (TASEP) when particles can only move in one direction. There

are several exact solutions for the stationary state properties of ASEP for periodic10

and open boundary conditions11 and different update rules.12–14

Most previous work of ASEP analyze the multi-particle dynamics along a single-

chain lattice. Although the single-chain exclusion processes can describe many non-

equilibrium processes, the necessity to analyze more realistic complex phenomena,

such as car traffic and the biological transport of motor proteins,5,15 stimulates the

investigation of ASEP on lattices with a more complex geometry. Recently, multi-

chain ASEP have been studied. Brankov et al.16 investigate TASEP on chains

with a double-chain section in the middle. Several stationary phases are found

unexpectedly. Pronina and Kolomeisky17 and Wang et al.18 studied the system of

TASEP on lattices with a junction. The two-chain ASEP, where particles can jump

between the lattice chains, has also been investigated.19 Mitsudo and Hayakawa

extend the two-lane model to a case incorporating the asymmetric lane-change

rule and study the synchronization of kinks.20 Jiang et al. investigate the coupling

of two-lane TASEP with the creation and annihilation of particles.21 Popkov et

al. study the two-lane ASEP in which the hopping rates of particles on one lane

depend on the configuration on the other lane.22 The two-way traffic on two-lane

TASEP23 as well as two-lane TASEP with particles of two different charges24 has

also been investigated. Recently, Cai et al. studied N -lane ASEP (N ≥ 3) and the

general phase diagram structure is derived.25

In this paper we investigate an exclusion process with a particular geometry: a

single-chain lattice which diverges into two equivalent branches. At the branching

point, the particles move to one of the two branches with equal rate r/2. Here r < 1

is a kind of local inhomogeneity,7,26 which might be relevant to (i) the delay of vehi-

cle drivers at the divergence because they need to slow down to consider which route

to choose; (ii) the slow down of leading molecular motors due to additional frictions

exerted on them.27 Note that in the special case r = 1, the branching dynamics

become the same as that in Ref. 16. For simplicity, we choose the simplest model in

terms of TASEP with random sequential update rules. The effective injection and

removal rates for each chain are introduced and the model can be mapped into three

coupled exactly solved single-chain TASEP. We study the possible phase structures

of the system in terms of these rates by using mean-field approximation and Monte

Carlo simulation. It is shown that the phase diagram can be classified into three

regions at any value of r. However, there exists a critical value rc = 2 −√

2. In

cases of r > rc and r < rc, the phases corresponding to these regions and the phase

boundaries between these regions are quite different.

The paper is organized as follows. In Sec. 2, we outline our model. Section 3

presents the theoretical calculation results and Monte Carlo simulation results. Sec-

tion 4 studies the density profiles on phase boundaries from domain wall approach.

We summarize our paper in Sec. 5.


Totally Asymmetric Exclusion Processes on Lattices 2001

(a)

(b)

Fig. 1. (a) Schematic picture of the model for a TASEP on a lattice with a branching point.Arrows indicate the allowed transitions, while crossed arrows correspond to the prohibited move-ments. (b) Schematic picture of the model for a TASEP on a lattice with a branching point afterintroducing in effective injection and removal rates.

2. Model

In this section, the rules of the model are introduced. The schematic picture is shown

in Fig. 1(a), particles move along the lattices with a branching point positioned in

the middle of the system. The system consists of chains I, II and III which start

from sites 1, L+1, 2L+1 and end in sites L, 2L, 3L, respectively. Random update

rules are adopted. In an infinitesimal time interval dt, a site i (1 ≤ i ≤ 2L) is chosen

randomly.

(i) If i = 1 (entrance of chain I), a particle is inserted with rate α provided the

site is empty. If site 1 is occupied, the particle hops into site 2 with rate 1 provided

site 2 is empty.

(ii) If 1 < i < L and the site is occupied, the particle hops into site i + 1 with

rate 1 provided the target site is empty.

(iii) If i = L (the branching point) and the site is occupied, then chains II or III

are chosen with equal probability 1/2. Suppose chain II (III) is chosen, the particle

moves to site L + 1 (2L + 1) with rate r provided site L + 1 (2L + 1) is empty.

(iv) If L + 1 ≤ i ≤ 2L, i.e. chain II is chosen, we choose another site 2L + 1 ≤j ≤ 3L on chain III at random. Then, sites i and j are updated (see also Ref. 16),

in terms of the following updating rules.

If i = 2L (exit of chain II) or j = 3L (exit of chain III) and the site is occupied,

the particle is removed with rate β. If L + 1 ≤ i < 2L (2L + 1 ≤ j < 3L) and the


2002 X. Wang et al.

site is occupied, the particle will move into site i + 1 (j + 1) with rate 1 provided

the target site is empty.

3. Theoretical Calculation and Monte Carlo Simulation

3.1. Mean-field analysis

In this subsection, we present the approximate stationary solutions of this model by

using the mean-field approximation. First let us briefly recall the results of TASEP

on a single-chain with open boundaries. In this case there are three stationary

phases, specified by the processes at the entrance, at the exit and in the bulk of the

system.

At small values of injection rates α < 1/2 and β, the entrance of the particles

is the limiting process, the system is in a low-density (LD) phase with

J = α(1 − α) , ρbulk = α , ρ1 = α , ρN =α(1 − α)

β(1)

where ρ1, ρN and ρbulk are the densities at the entrance, exit and the bulk of the

lattice far away from the boundaries, respectively. J denotes the flux.

At small values of removal rates β < 1/2 and α, the overall dynamics is governed

by the particle exit, the system is in a high-density (HD) phase with

J = β(1 − β) , ρbulk = 1 − β , ρ1 = 1 − β(1 − β)

α, ρN = 1 − β . (2)

Moreover, at large values of the injection rate α ≥ 1/2 and removal rate β ≥ 1/2,

the dynamics of the system is determined by processes in the bulk, the system is

in a maximal-current (MC) phase with

J =1

4, ρbulk =

1

2, ρ1 = 1 − 1

4α, ρN =

1

4β. (3)

There is a first-order phase transition along the phase boundary α = β < 1/2 and

particle density changes linearly from α to 1 − α.

As shown in Fig. 1(b), our system can be viewed as three coupled single-channel

TASEP. Since each chain will be in one of the three stationary phases, there are

33 = 27 possible phases in the system. However, chains II and III must be in the

same phase because of the symmetry. Due to the conservation of the current of the

system

Joverall = JI = JII + JIII = 2JII (4)

the maximal-current phase cannot exist in chains II and III. Consequently, there

are six possible stationary phases: (LD, LD), (HD, LD), (MC, LD), (LD, HD), (HD,

HD), (MC, HD), where (X, Y) denotes that chains I, II and III are in the X phase,

Y phase and Y phase, respectively.

In the mean-field approximation, the correlations around the branching point

are neglected. We introduce effective injection and removal rates for each chain



segment and study the possible phase structures of the system in terms of these

rates.16,17,26 As shown in Fig. 1(b), chain I has the injection rate α and the effective

removal rate βeff . Similarly, chain II (or III) has the effective injection rate αeff and

the removal rate β. Thus, the overall current of the system could be calculated by

Joverall =

{

JI = βeffρL

2JII = 2αeff(1 − ρL+1). (5)

On the other hand, we introduce τi as occupation variable of site i where τi = 1

(or τi = 0) indicates that the state of the ith site is occupied (or vacant) and the

mean-field expression for the current can be written as

Joverall 'r

2〈τL(1 − τL+1)〉 +

r

2〈τL(1 − τ2L+1)〉

' r

2〈τL〉(1 − 〈τL+1〉) +

r

2〈τL〉(1 − 〈τ2L+1〉)

' r〈τL〉(1 − 〈τL+1〉)

= rρL(1 − ρL+1) (6)

where 〈· · ·〉 denotes a statistical average. Here 〈τL+1〉 = 〈τ2L+1〉 = ρL+1 stands for

the stationary particle density at site L + 1 on chain II or at site 2L + 1 on chain

III and 〈τL〉 = ρL is the stationary particle density at site L.

Thus, from Eqs. (5) and (6), one can obtain

αeff =rρL

2, βeff = r(1 − ρL+1) . (7)

Now we can investigate the existence of different stationary phases. First, we

consider the (LD, LD) phase. It is fulfilled when

α <1

2, α < βeff ; αeff <

1

2, αeff < β . (8)

From Eq. (4), we obtain

αeff =1 −

√

1 − 2α(1 − α)

2<

1

2. (9)

In LD phase, the density on the entrance L + 1 equals the bulk density, so that

ρL+1 = ρII,bulk = αeff . (10)

Substituting Eqs (9) and (10) into Eq. (7), we have βeff = (r(1 +√

1 − 2α(1 − α)))/2. Thus, from α < βeff we obtain

r >1 −

√

(1 − 2α(1 − α))

1 − α. (11)

Therefore, for the (LD, LD) phase to exist, one has

α <1

2, β >

1 −√

1 − 2α(1 − α)

2, r >

1 −√

(1 − 2α(1 − α))

1 − α. (12)


2004 X. Wang et al.

Now we discuss the (LD, HD) phase, which exists if

α <1

2, α < βeff ; β < αeff , β <

1

2. (13)

From Eqs. (4), (5), (7) and (13), we can obtain

β =1 −

√

1 − 2α(1 − α)

2, α < βeff = r(1 − ρL+1) , β < αeff =

β(1 − β)

1 − ρL+1

.

Then, we have

α

r< 1 − ρL+1 < 1 − β , i.e. r >

α

1 − β=

1 −√

(1 − 2α(1 − α))

1 − α.

Thus, the possible conditions for the existence of the (LD, HD) phase are

β =1 −

√

1 − 2α(1 − α)

2, α <

1

2, r >

1 −√

(1 − 2α(1 − α))

1 − α(14)

which means it corresponds to phase boundary.

The (HD, LD) phase is specified by

βeff <1

2, βeff < α ; αeff <

1

2, αeff < β . (15)

The density on the exit site equals the bulk density in HD phase while the density

on the entrance site equals the bulk density in LD phase, so that

ρL = ρI,bulk = 1 − βeff ; ρL+1 = ρII,bulk = αeff . (16)

Substituting Eq. (16) into Eq. (7), we have

αeff =r(1 − βeff)

2, βeff = r(1 − αeff) .

Therefore,

αeff =r − r2

2 − r2, βeff =

2r − r2

2 − r2. (17)

From Eqs. (15), (16) and (17), we find the system is in the (HD, LD) when

α >2r − r2

2 − r2, β >

r − r2

2 − r2, r < 2 −

√2 . (18)

The (HD, HD) phase is defined by the following expression

βeff <1

2, βeff < α ; β < αeff , β <

1

2. (19)

Taking into account Eqs. (4) and (7), we obtain

βeff =1 −

√

1 − 8β(1 − β)

2

αeff =rρL

2=

r(1 − βeff)

2=

r(1 +√

1 − 8β(1 − β))

4.

(20)



The requirement that the expression in the square root should be positive leads to

a conclusion that β < (1/2)− (√

2/4). From Eqs. (19) and (20), we find the system

is in the (HD, HD) phase when

β <1

2−

√2

4, α >

1 −√

1 − 8β(1 − β)

2, r >

1 −√

1 − 8β(1 − β)

2(1 − β). (21)

The (MC, LD) phase is determined by

α >1

2, βeff >

1

2; αeff <

1

2, αeff < β . (22)

From Eqs. (4) and (7), we could obtain

αeff =1

2−

√2

4<

1

2(23)

βeff = r(1 − ρL+1) = r(1 − αeff) = r

(

1

2+

√2

4

)

, ρL =JI

βeff

. (24)

Combining Eqs. (22) and (24), the parameters for existence of (MC, LD) phase can

be written as

β >1

2−

√2

4, α >

1

2, r > 2 −

√2 . (25)

Finally, the (MC, HD) phase exists if

α >1

2, βeff >

1

2; β < αeff , β <

1

2. (26)

From Eqs. (4), (5), (7) and (26), we find

β =1

2−

√2

4<

1

2, βeff =

1

4ρL

>1

2, αeff =

rρL

2> β .

Then, we have

2 −√

2

2r< ρL <

1

2, i.e. r > 2 −

√2 .

Hence, the possible conditions for the existence of the (MC, HD) phase are

β =1

2−

√2

4, α >

1

2, r > 2 −

√2 (27)

which means it also corresponds to phase boundary.

Since (MC, LD) and (HD, LD) are excluded to each other, the structure of the

phases could be expressed much clearer.

• For r > rc, three phases are

— (LD, LD): α < 1/2, α(1 − α) < 2β(1 − β).

— (HD, HD): β < rc/4, α(1 − α) > 2β(1 − β).

— (MC, LD) α > 1/2, β > rc/4.

• For r < rc, three phases are


2006 X. Wang et al.

— (LD, LD): α < (2r − r2)/(2 − r2), α(1 − α) < 2β(1 − β). Note that α < 1/2

is automatically satisfied.

— (HD, HD): β < (r − r2)/(2 − r2), α(1 − α) > 2β(1 − β). Note that β < rc/4

is automatically satisfied.

— (HD, LD): α > (2r − r2)/(2 − r2), β > (r − r2)/(2 − r2).

As r decreases, (MC, LD) will transit into (HD, LD). Such a transition can

be taken as a bottleneck effect at the branching point to trigger the congestion in

chain I.

3.2. Simulation results

In this section, the Monte Carlo simulation results are presented and compared

with the analytical results. In the simulation, the system size is set to L = 1000

(a)

(b)

Fig. 2. Phase diagram of the system. (a) r > rc; (b) r = 0.3, 0.4, 0.5 < rc. Symbols are fromMonte Carlo simulations while lines are from mean-field calculations.



unless otherwise mentioned. A phase diagram related to α and β is shown in Fig. 2,

which can be classified into three regions.

Region I corresponds to the (LD, LD) phase, all three chains are in the low-

density phase. The typical density profiles with α = 0.2, β = 0.8, r = 0.7 are

illustrated in Fig. 3(a). As expected from Eqs. (9), (10) and (12), the analytical

results are ρanI,bulk = α = 0.2, ρan

II,bulk = αeff = 0.0877. The simulation results are

ρsimI,bulk ' 0.2005, ρsim

II,bulk ' 0.0870. The comparison of the two results shows that

they agree very well. The bulk densities of chains I, II and III are independent of r

while ρL decreases with the increase of r.

Region II corresponds to the (HD, HD) phase of the system where chains I, II

and III are in the high-density phase. The density profiles with parameters α = 0.8,

β = 0.07, r = 0.7 are shown in Fig. 3(b). By comparing the analytical results

ρanI,bulk = 1 − βeff = 0.8461, ρan

II,bulk = 1 − β = 0.93 with the simulation results

ρsimI,bulk ' 0.8450, ρsim

II,bulk ' 0.9297, it clearly implies that the two results agree very

(a) (b)

(c) (d)

Fig. 3. Density profiles in (a) the (LD, LD) phase; (b) the (HD, HD) phase; (c) the (MC, LD)phase; (d) the (HD, LD) phase. Symbols are obtained from the Monte Carlo simulations whilelines are our theoretical predictions.


2008 X. Wang et al.

well. The bulk densities of chains I, II and III are also independent of r while ρL+1

increases with the increase of r.

The phase in region III depends on r. The threshold rc = 2−√

2 can be obtained

from Eqs. (9), (10) and (12). When r > rc, region III corresponds to the (MC, LD)

phase. The density profiles corresponding to α = 0.8, β = 0.3, r = 0.7 are shown

in Fig. 3(c). The simulation results agree with the analytic ones. When r < rc,

region III corresponds to the (HD, LD) phase. The density profiles corresponding

to α = 0.8, β = 0.3, r = 0.3 are shown in Fig. 3(d). From Eqs. (16) and (17), we

can obtain the analytic results of density profiles ρanI,bulk = 0.7329, ρan

II,bulk = 0.1099,

which agree well with the simulation results: ρsimI,bulk ' 0.7192, ρsim

II,bulk ' 0.1122.

ρI,bulk increases with the decrease of r. In contrast, ρII,bulk decreases with the

decrease of r. Moreover, region III expands and regions I and II shrink with the

decrease of r provided r < rc.

The (HD, LD) phase is also found in Refs. 7 and 26 (note that it is called

maximum-current phase in Ref. 26), which appears when the hopping rate is smaller

than 1. Nevertheless, in our model, the critical value rc at which the (HD, LD) phase

appears changes to rc = 2 −√

2 due to the branching.

4. Density Profiles on Phase Boundaries

Now we focus on the phase boundaries. Firstly, phase transition from (LD, LD) to

(MC, LD) is of second-order and thus does not need to be considered. On the other

hand, boundary 1 between (MC, LD) and (HD, HD), boundary 2 between (HD,

LD) and (HD, HD), boundary 3 between (LD, LD) and (HD, LD) are of first-order

due to density jump on chain I (boundary 3) or on chains II and III (boundaries

1 and 2). Consequently, we can expect a linear density profile on chain I (from α

to 1− α) or on chains II and III (from β to 1 − β). As shown in Fig. 4, the Monte

Carlo simulation results are in good agreement with the theoretical predictions.

The boundary between (LD, LD) and (HD, HD) is also of first-order. However,

different from boundaries 1–3, density jump happens on both chain I and chains

II and III. This introduces non-negligible correlations near the branching point. As

a result, the linear density profile of mean-field expectation strongly deviates from

simulations, which can be seen from Fig. 5. In this case, the domain wall approach

presented by Pronina and Kolomeisky16 takes correlations into account and could

correctly predict the density profiles as shown below.

Let us denote the moving rates of the domain wall in chains I and II (or III) as

uI and uII, which can be determined by utilizing the expression

uk =Jk

ρk+ − ρk

−

for k = I, II (28)

where J is bulk stationary state value of the current and ρ+,− denote density in

the right (+) and left (−) of the domain wall. It is clear



(a) (b)

(c)

Fig. 4. Density profiles on phase boundaries. (a) Boundary between (MC, LD) and (HD, HD);(b) boundary between (HD, LD) and (HD, HD); (c) boundary between (LD, LD) and (HD, LD).Symbols are obtained from the Monte Carlo simulations while lines are our theoretical predictions.

Fig. 5. Density profiles for the boundary between (LD, LD) and (HD, HD) with parametersα = 0.3, β = 0.119. Symbols are obtained from the Monte Carlo computer simulations. Dashedlines are from the mean-field predictions. Solid lines are theoretical predictions from Eqs. (36) and(37).


2010 X. Wang et al.

JI = α(1 − α) , JII =JI

2ρI−

= α , ρI+ = 1 − α , ρII

−

= β , ρII+ = 1 − β .

(29)

Thus,

uI =α(1 − α)

1 − 2α, uII =

α(1 − α)

2(1− 2β)=

α(1 − α)

2√

1 − 2α(1 − α). (30)

Since the domain wall moves between different chains with equal probabilities, we

have

uIPI

L=

uIIPII

L, PI + PII = 1 (31)

here PI and PII are the probabilities of finding the domain wall at any position in

chainss I and II (or III) respectively.

Solving Eq. (31), we have

PI =uII

uI + uII

, PII =uI

uI + uII

(32)

which implies that the domain wall has different probabilities of being found in

chain I and in chain II (or III).

We introduce relative coordinate x = i/L, where i is the site index. Thus, the

case of 0 < x ≤ 1 and 1 < x ≤ 2 describe chains I and II (or III), respectively. xDW

denotes position of the domain wall. Then we can obtain the probability distribution

function of the domain wall

Prob(xDW < x) = PIx , 0 < x ≤ 1 (33)

Prob(xDW < x) = PI + PII(x − 1) , 1 < x ≤ 2 . (34)

Therefore, the density at any position is given by

ρ(x) = ρk+Prob(xDW < x) + ρk

−

Prob(xDW > x) , k = I, II . (35)

Substituting Eqs. (30), (32), (33) and (34) into Eq. (35), we have

ρ(x)I = α +(1 − 2α)2

1 − 2α + 2(1 − 2β)x (36)

ρ(x)II = β +(1 − 2α)(1 − 2β)

1 − 2α + 2(1 − 2β)+

2(1− 2β)2

1 − 2α + 2(1 − 2β)(x − 1) . (37)

The linear density profiles are independent of r. The result from the domain wall

approach is shown in Fig. 5, and it agrees well with simulations.



5. Conclusion

In this paper, we investigate TASEP on lattices with a branching point using a

simple mean-field approximation theory and Monte Carlo simulations. According

to this theory, we neglect the correlations around the branching point and the

model can be mapped into three coupled exactly solved single-chain TASEP. It is

shown that the phase diagram can be classified into three regions at any value of

r. However, a threshold rc = 2 −√

2 is identified. When r > rc, the three regions

correspond to the (LD, LD), (HD, HD) and (MC, LD) phases respectively and

the phase boundaries are independent of r. In the case of r < rc, the (MC, LD)

phase turns into the (HD, LD) phase. Moreover, the phase boundaries become

dependent of r. The analytical results of phase structure and density profiles are

in good agreement with simulations except on the phase coexistence line between

the (LD, LD) phase and the (HD, HD) phase. This reveals that the correlations

is non-negligible in this case. The domain wall theory is adopted to calculate the

density profile in this case and the results agree well with Monte Carlo simulations.

The mean-field approximation theory and the domain wall theory thus provide a

reasonable way of analyzing TASEP with inhomogeneous geographic structure.

Our studies suggest similarities with defect systems, see e.g. Refs. 26 and 28.

Actually, our model can be effectively mapped into the one with a single defect by

choosing the defect strength appropriately, which should increase with the decrease

of r. Note that r > rc corresponds to no defect.

Finally we would like to mention the particle-hole symmetry. Since the transport

of particles from the left to the right is identical to the motion of holes in the

opposite direction, our model can be mapped into a system where two branches

merge into one and particles at the junction hop with rate r/2, which is different

from the model studied in Ref. 17 in the special case r = 1 by a coefficient 1/2.

By mapping the model studied in Ref. 17 (and assuming particles at the junction

hop with rate r) into a system where one branch diverges into two, we can obtain

a different model from the one studied in this paper at the site L in the following

case: If the first site on one chain is occupied and on the other chain is empty,

then the particle at site L hops to the empty site with rate r (instead of r/2 in our

model).

Acknowledgment

We acknowledge the support of National Basic Research Program of China

(No. 2006CB705500), the NSFC (Nos. 10532060, 70601026, 10672160, 10872194),

the NCET and the FANEDD. R. J. acknowledges the support of JSPS.

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