Dynamical transitions in a driven diffusive model with interactions
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] J a n epl draft Dynamical transitions in a driven diffusive model with interactions
D. Botto , , A. Pelizzola , and M. Pretti , Dipartimento di Scienza Applicata e Tecnologia, Politecnico di Torino, Corso Duca degli Abruzzi 24, I–10129 Torino,Italy INFN, Sezione di Torino, via Pietro Giuria 1, I-10125 Torino, Italy Consiglio Nazionale delle Ricerche - Istituto dei Sistemi Complessi (CNR-ISC), Via dei Taurini 19, 00185 Roma,Italy
PACS – Markov processes
PACS – Lattice theory and statistics (Ising, Potts, etc.)
PACS – Complex systems
Abstract – We study the dynamics of an asymmetric simple exclusion process with open bound-aries and local interactions using a pair approximation which generalizes the 2–node cluster meanfield theory and the Markov chain approach to kinetics and shares with these approaches theproperty of reproducing exact results for the bulk current–density relation and the steady statephase diagrams. We find that the relaxation rate exhibits a dynamical transition, with no staticcounterpart, analogous to that found without interactions. Remarkably, for some values of themodel’s parameters, we find 2 dynamical transitions in the same low density phase. We study thedynamics of relaxation to the steady state on both sides of these transitions and make an attemptat providing a physical interpretation for this phenomenon. Results from numerical approachesand a modified Domain Wall Theory confirm the picture provided by the pair approximation.
Introduction. –
A fundamental aspect of non–equilibrium statistical physics is the investigation of steadystates (SS) [1], which are not yet as well understood astheir equilibrium counterparts. Driven lattice gases havebeen shown to be excellent model systems for such in-vestigations, and a prominent role in this class of modelsis played by the Asymmetric Simple Exclusion Process(ASEP) and its generalizations, inspired by biological andvehicular traffic phenomena (see [2, 3] for reviews). InASEP, the nodes of a one–dimensional lattice can be oc-cupied by at most one particle, and particles hop to emptynearest–neighbour nodes with asymmetric rates, e.g. hop-ping in the rightward direction is more likely than left-ward hopping. If leftward hopping is forbidden the modelis called Totally Asymmetric Simple Exclusion Process(TASEP). On an open lattice, injection and extractionof particles are allowed at lattice boundaries, and the SSof the models exhibits, as a function of the injection andextraction rates, rich phase diagrams, well described bythe theory of boundary–induced phase transitions [4]. Inthe last decades, many exact results have been obtained (a)
E-mail: [email protected] (b)
E-mail: [email protected] (c)
E-mail: [email protected] [5–8] for these phase diagrams and other properties such asdensity profiles. These models have thus become paradig-matic in non–equilibrium statistical physics, like the Isingmodel in the equilibrium case.In an attempt at moving towards more realistic model-ing of vehicular traffic, Antal and Sch¨utz (AS) considered aTASEP with local interactions [9]. In the AS model, ratesdepend on the occupation of the next–nearest–neighbournode in the direction of motion. The model is an in-stance of a more general one, previously introduced byKatz, Lebowitz and Spohn (KLS) [10]. Both attractiveand repulsive interactions were considered, leading to dif-ferent physical behaviours. Among many results in [9], it isworth mentioning an exact solution for the SS distributionin special cases. In particular, for periodic boundary con-ditions, and also for open boundary conditions with spe-cial (bulk–adapted) values of the boundary rates, the SSdistribution can be written as the equilibrium distributionof a one–dimensional Ising model with nearest–neighbourinteractions, a property shared by several models in theKLS class. Correlations in this SS are therefore richer thanthose exhibited by TASEP, whose SS distribution, underappropriate conditions, factors over nodes in a mean–fieldlike fashion, making certain mean–field results (e.g. thep-1. Botto et al. location of many SS phase transitions) exact. Indeed, ASreported a very poor performance of mean–field for theSS properties of their model. Given that the pair approxi-mation (PA) is exact for the equilibrium one–dimensionalIsing model (see e.g. [11] and refs. therein), various PAshave been recently employed with success [12–19] in thestudy of several models in the KLS class and their gen-eralizations. It is therefore worth investigating how a PAperforms in the case of the AS model, and applying it tothe study of properties of this model for which an exactsolution is not available.In this direction, it is of particular interest to con-sider the possibility of existence of a dynamical transition,which will be the main focus of this work. This transition,found and exactly located by de Gier and Essler [20,21] incertain ASEPs, including TASEP, corresponds to a singu-larity in the relaxation rate which is not associated to anysingularity in the SS. In spite of the fact that the locationof the transition is exactly known, its physical meaning isnot yet well understood [22]. In the case of TASEP, thetransition has been recently found, and located with rea-sonable accuracy, in the framework of different mean–fieldlike approximations of increasing complexity, including aPA [23].The aim of the present paper is therefore to show, us-ing a PA, supported by results from numerical approachesand a modified Domain Wall Theory (mDWT), that thedynamical transition is a robust phenomenon, which is ex-hibited also in the case of the AS model, and to make anattempt at providing a physical interpretation.
Model and pair approximation. –
The AS model[9] is defined on a one–dimensional lattice of N nodes,labeled i = 1 , , . . . , N , with open boundaries. Each nodecan be empty or singly occupied, the occupation numbervariable for node i at time t is n ti = 0 ,
1. In the followingwe will denote by P ti [ n i n i +1 · · · n i + k ] the probability that,at time t , the occupation numbers of nodes from i to i + k take values n i , n i +1 , · · · , n i + k respectively. The average ofan occupation number variable is the local density ρ ti = h n ti i = P ti [1]. In the SS local densities do not depend ontime and are denoted by ρ i , dropping the time index. Ifthe local density SS is also uniform, we denote it simplyby ρ , dropping also the node index. A particle at node i can hop to node i + 1, provided this is empty, with arate which depends on the occupation of node i + 2. Ifnode i + 2 is empty (respectively occupied), the hoppingrate from i to i + 1 is denoted by r (resp. q ). For q < r (respectively q > r ) interactions are said to be repulsive(resp. attractive). The current J ti from node i to node i +1at time t can be written as J ti = h n ti (1 − n ti +1 )[ qn ti +2 + r (1 − n ti +2 )] i = qP ti [101] + rP ti [100] , i = 1 , . . . , N − . (1)It was shown in [9] that a model with the kinetics describedabove and periodic boundary conditions has a SS current– density relation in the thermodynamical limit given by J ( ρ ) = rρ " p − ρ (1 − ρ )(1 − q/r ) − − ρ )(1 − q/r ) . (2)On a lattice with open boundaries, some care is neededin the definition of the boundary rates. In [9] these rateshave been defined in such a way that they would yieldconstant density profiles for semi–infinite systems (theseboundary rates are usually called bulk–adapted [13,14,18],whereas a possible different choice is that of equilibrated–bath [13,14] boundary rates). Consider the left boundary:it is reasonable to assume that the injection rate at node 1depends on the occupation of node 2. This injection rateis denoted by α (respectively α ) if node 2 is occupied(resp. empty). It has been shown in [9] that imposing thecondition that a uniform density ρ L is obtained in the SSof a semi–infinite system ( i = 1 , , . . . , ∞ ) one obtains α = q (cid:20) − J ( ρ L ) rρ L (cid:21) , α = r (cid:20) − J ( ρ L ) rρ L (cid:21) . (3)Consider now the right boundary: here one needs tospecify the hopping rate from node N − N ,which is denoted by β , and the extraction rate from node N , denoted by β . The condition that a uniform den-sity ρ R is obtained in the SS of a semi–infinite system( i = −∞ , . . . , N − , N ) now gives [9] β = J ( ρ R )1 − ρ R (cid:20) − J ( ρ R ) rρ R (cid:21) − , β = J ( ρ R ) ρ R . (4)In order to introduce the PA we will assume, as in pre-vious works based on the Markov chain approach to kinet-ics (MCAK) [12–14], the cluster mean–field (CMF) theory[18, 19] and related ideas [23–26], that k –node marginals( k ≥
3) factor, at any given time t , according to P ti [ n i n i +1 . . . n i + k − ] = Q i + k − l = i P tl [ n l n l +1 ] Q i + k − l = i +1 P tl [ n l ] . (5)The 2–node marginal P ti [ n i n i +1 ] ( i = 1 , . . . , N − ρ ti and ρ ti +1 together with ψ ti = P ti [10]. Asa consequence we have P ti [00] = 1 − ρ ti +1 − ψ ti , P ti [01] = ρ ti +1 − ρ ti + ψ ti and P ti [11] = ρ ti − ψ ti . With the aboveassumptions, we can now write the equation for the timeevolution of ρ ti and ψ ti . For the local densities we obtain˙ ρ ti = J ti − − J ti , i = 1 , . . . , N, (6)where the current in the PA is given by Eq. 1 with Eq. 5and boundary currents are given by J t = α P t [01] + α P t [00] ,J tN − = β P tN − [10] ,J tN = β P tN [1] . (7)p-2ynamical transitions in a driven diffusive model with interactionsFor the 2–node expectations we obtain˙ ψ ti = rP ti − [100] + qP ti [1101] + rP ti [1100] − J ti (8)for i = 2 , . . . , N − ψ t = α P t [00] + qP t [1101] + rP t [1100] − J t , ˙ ψ tN − = rP tN − [100] + β P tN − [110] − J tN − , ˙ ψ tN − = rP tN − [100] + β P tN − [11] − J tN − (9)at the boundaries. Eqs. 8–9 represent an improvementwith respect to the MCAK [12–14], where the dynamicalequations are closed by assuming that the 2–node expec-tations, or correlators, ψ ti depend at any time on the localdensities in the same way as they do in the equilibriumone–dimensional Ising model describing the SS.Notice that Eqs. 6–9 can be viewed, by expressing 3–and 4–node marginals using Eq. 5, as an equation˙ x t = f ( x t ) (10)for the time evolution of the (2 N − x t = ( ρ t , ψ t , . . . , ρ tN − , ψ tN − , ρ tN ) . (11)The SS x = ( ρ , ψ , . . . , ρ N − , ψ N − , ρ N ) will be given bythe condition f ( x ) = 0, and relaxation near the SS will bedescribed by the relaxation matrix M , with elements M ab = − ∂f a ∂x tb (cid:12)(cid:12)(cid:12)(cid:12) x t = x , a, b = 1 , . . . , N − . (12)In particular, its smallest eigenvalue λ is the slowest re-laxation rate, the inverse of the longest relaxation time. Results. –
First of all, we look for bulk solutions inthe SS, where by continuity the current is uniform, J i = J .In more detail, we look for a SS with ρ i = ρ and ψ i = ψ (as a consequence all marginals will be independent ofposition), at least sufficiently far from the boundaries. Inthis case the condition ˙ ψ ti = 0 becomes (dropping indices i and t in the marginals)0 = rP [100] + qP [1101] + rP [1100] − ( qP [101] + rP [100])= rP [1100] − qP [0101]= r ( ρ − ψ ) ψ (1 − ρ − ψ ) ρ (1 − ρ ) − q ψ ρ (1 − ρ ) , (13)which is solved by ψ = 1 − p − ρ (1 − ρ )(1 − q/r )2(1 − q/r ) . (14)The corresponding current is J ( ρ ) = qP [101] + rP [100]= q ψ − ρ + r ψ (1 − ρ − ψ )1 − ρ = rρ (cid:18) − ψ − ρ (cid:19) , (15) which turns out to be exact (see Eq. 2 and [9]). Simplealgebra shows that with the boundary rates defined as inEqs. 3–4 with ρ L = ρ R = ρ , in the bulk SS Eqs. 7 yield J = J N − = J N = J ( ρ ) and the r.h.s. of Eqs. 9 vanish.With this definition of the boundary rates, in the PA wefind a bulk SS with the exact current–density relation atany finite size N . This implies that in the PA the exactlocation of most SS phase transitions is recovered. Theseexact results, and as a consequence the location of mosttransition lines in the phase diagram, can also be obtainedby using the CMF theory in [18,19] or the MCAK [12–14].Let us focus on the SS phase diagram, in the limit oflarge lattice size N , using as parameters, in addition to q and r , the densities ρ L and ρ R . More precisely, in orderto make contact with [9] and the literature on TASEP,our parameters will be ρ L and 1 − ρ R . By studying thelong time behaviour of our time evolution equations wefind the same SS phases as in [9], namely a low–density(LD) phase (with small bulk density ρ L extending to theleft boundary, and a boundary layer, whose characteristiclength remains finite in the large N limit, on the right),a high–density (HD) phase (with large bulk density ρ R extending to the right boundary, and a boundary layeron the left), a maximal current (MC) phase (with bulkdensity ρ ∗ = argmax J ( ρ ) in the central region of the sys-tem and 2 boundary layers) and, for q sufficiently largerthan r (numerically we find q/r &
6) and 1 − ρ R closeto 1, another high–density (labelled HD ′ in the following)phase. Typical SS phase diagrams are reported in Fig. 1for repulsive interactions ( r > q ), in Fig. 2 for weakly at-tractive interactions ( q > r , q/r not too large) and in Fig.3 for strongly attractive interactions ( q > r , q/r large),using solid lines (dashed lines, and the corresponding dis-tinctions between fast and slow phases, will be discussedlater). The (continuous) transition line between the LD(respectively HD) and the MC phase is given by the con-dition ρ L = ρ ∗ (resp. ρ R = ρ ∗ ), while the (discontinuous)transition line between the LD and HD phases is given by J ( ρ L ) = J ( ρ R ). The HD ′ phase appearing in the stronglyattractive case in Fig. 3 has a density profile qualitativelysimilar to the MC phase, with a central bulk region and2 boundary layers, but its bulk density ρ ′ (which dependsonly on ρ R , as in the “ordinary” HD phase) is slightlylarger than ρ ∗ (the largest value found in the case of Fig.3 was 0 . > ρ ∗ ≃ . ′ and the MC phases is given by thecondition ρ ′ = ρ ∗ , while the (discontinuous) transition linebetween the HD ′ and LD phases is given by J ( ρ ′ ) = J ( ρ L ),but since ρ ′ is not equal to any of ρ ∗ , ρ L and ρ R , we cannotexpect its value, and as a consequence the correspondingphase boundaries, to be exact.We now turn our attention to the investigation of dy-namical transitions, which are represented in Figs. 1–3.Dynamical transitions correspond to singularities (in theinfinite size limit) in the relaxation rate λ , without anycorresponding singularity in the SS properties. Consid-ering the HD phase to fix ideas, a dynamical transitionp-3. Botto et al. ρ L - ρ R HD-slow MCLD-fast HD-fastLD-slow
Fig. 1: Typical phase diagram for repulsive interactions, here q = 0 . r = 1. Solid lines denote SS transitions. Thick(respectively thin) dashed lines denoted dynamical transitionsgiven by the PA (resp. mDWT). Phase labels are explained inthe text. ρ L - ρ R MCLD-fastLD-slow HD-fastHD-slow
Fig. 2: Same as Fig. 1 for weakly attractive interactions, here q = 1 and r = 0 . ρ L - ρ R MCHD’-slowLD-fastLD-slow
Fig. 3: Same as Fig. 1 for strongly attractive interactions, here q = 1 and r = 0 .
1. The portion of the phase diagram with theHD ′ phase is shown. ρ L λ - Fig. 4: The bottom part of the spectrum of the relaxationmatrix for N = 100, q = 1, r = 0 . ρ R = 0 . λ − , the line is a guide forthe eye connecting relaxation rates λ . separates a region of the phase diagram (labelled fast forreasons which will become clearer in the following) where λ depends only on ρ R (the parameter fixing the SS bulkdensity) from one or more regions (labelled slow ) where λ depends also on ρ L (the roles of ρ L and ρ R are exchangedin the LD phase). In the case of certain ASEPs, includ-ing TASEP (that is the present model with q = r = 1)the location of the transition is exactly known, as well asthe value of λ on both sides of the transition [20, 21].The physical meaning of the transition is however not yetclear, as remarked in [22] by Proeme, Blythe and Evans.In [23] we have shown numerically that the spectrum of themean–field relaxation matrix at large N has different qual-itative properties in the fast and slow phases of TASEP.In the fast phases, as N → ∞ , the spectrum tends toa continuous band, while in the slow phases an isolatedeigenvalue appears, below the continuous band, which cor-responds to a slowest relaxation mode being much slower than all the other modes. We have recently confirmedanalytically (still at mean–field level) these results [27] inthe case of both simple TASEP and TASEP with Lang-muir kinetics (introduced in [28, 29]) in the so–called bal-anced case. In the present work we observe the same phe-nomenon, illustrated in Fig. 4, where we plot the 9 smallesteigenvalues λ − of the relaxation matrix as a function of ρ L , for N = 100, q = 1, r = 0 . ρ R = 0 .
8, that isin the HD phase in Fig. 2. One can clearly see a regionon the right (the fast phase) where λ takes its maximumvalue, independent of ρ L , and a region on the left (theslow phase) where the relaxation is slower and λ detachesfrom the rest of the spectrum. One might argue that inthe fast phase the left boundary condition is “consistent”with the bulk ( ρ L is sufficiently close to the bulk density ρ R ), so that the relaxation dynamics is dominated by thebulk properties, while in the slow phase ρ L is so differ-ent from the bulk density ρ R that the system exhibits anew, boundary–driven, relaxation mode. All the eigenval-ues in Fig. 4 are real, while going up in the spectrum oneencounters also pairs of complex conjugate eigenvalues.In Fig. 5 the relaxation rate λ is plotted for vari-ous system sizes, for model parameters as in Fig. 4. Itp-4ynamical transitions in a driven diffusive model with interactions ρ L λ Fig. 5: The relaxation rate λ as a function of ρ L for q = 1, r = 0 . ρ R = 0 . N = 100, 200, 400 and 800 from top to bottom. Thinlines: mDWT. Filled circles: extrapolation of exact finite sizeresults. is clear that λ is practically independent of the systemsize in the slow phase, while some weak size dependencecan be observed in the fast phase. This is consistentwith the mean–field results in [27], where for the caseof pure TASEP the mean–field rate was shown to ap-proach its asymptotic value exponentially (respectively as1 /N ) in the slow (resp. fast) phase. In the same figurewe report, for comparison, results from the mDWT by deGier and Essler [21]. These authors compared their ex-act result for the relaxation rate of pure TASEP with theDWT result [30, 31] λ = D R + D L − √ D L D R , where D L,R = J ( ρ L,R ) / ( ρ R − ρ L ). They found that the DWTresult is exact in the slow phase, and the dynamical transi-tion corresponds to a maximum of the DWT rate. In theirmDWT, which is exact by construction for pure TASEP,they take the DWT result in the slow phase and the maxi-mum rate in the fast phase. The mDWT is likely to be nolonger exact for the AS model, but in Fig. 5 we see that itconfirms the occurrence of a dynamical transition, whoselocation is close to the PA one. Notice also (Fig. 1) thatin the repulsive case, as 1 − ρ R →
0, the mDWT predictsthat the HD–phase dynamical transition tends to a value ρ L <
1, at odds with the PA. This behaviour is observedfor sufficiently strong repulsion, namely q/r < .
5. Asa further confirmation, in Fig. 5 we plot results obtainedalong the lines of [31,32] (where very accurate results wereobtained for pure TASEP), that is by extrapolating exactfinite size ( N ≤
24) results with the Bulirsch–Stoer algo-rithm [33, 34]. The parameter ω , characterizing the lead-ing term in the expected size dependence, has been setat 2, based on the exactly known finite size behaviour ofthe relaxation rate for pure TASEP [20,21], after verifyingnumerically that (even for the AS model) this value givesnear–optimal results according to the criterion proposedin [34]. For ρ L ≥ . λ are smaller than2 · − , strongly suggesting that the fast phase is not anartifact of the PA and the mDWT.It is a remarkable novel feature of this model that, forstrongly attractive interactions as in Fig. 3, two dynam-ical transitions are observed in the LD–phase, with theappearance of 2 LD–slow phases, at small (respectively ρ R λ Fig. 6: The relaxation rate λ as a function of 1 − ρ R for q = 1, r = 0 . ρ L = 0 .
6. Thick lines: PA, N = 100, 200, 400 and800 from top to bottom. Thin lines: mDWT. large) values of 1 − ρ R , close to the HD (resp. HD ′ ) phase.In Fig. 3 only a portion of the LD–slow phase close toHD ′ is shown (for the small values of r needed to observethe HD ′ phase, as ρ L gets small, the relaxation matrix be-comes severely ill–conditioned, and the determination ofthe dynamical transition line is affected by progressivelylarger errors). The two dynamical transitions are illus-trated in Fig. 6 by plotting the relaxation rate as a func-tion of 1 − ρ R for various system sizes. The mDWT resultsconfirm the existence of the dynamical transitions also inthis case. Here some care is needed for very small ρ R ,close to the HD ′ phase, because the domain wall modelledby the mDWT is between a low–density region of density ρ L and a high–density, HD ′ –like region whose density isnot given by ρ R , but by a function ρ ′ ( ρ R ). Since in PAthis function is practically linear, we replaced ρ R in themDWT with a linear function fitting the HD ′ density. Forsuch strongly attractive interactions, the estimates of λ obtained by extrapolation of exact finite size results arenot stable, probably much larger sizes would be needed.In order to try to understand the physical meaning ofthe dynamical transition, we have investigated in somedetail the full dynamics of the model in the fast and slowphases. In particular, in the repulsive case (Fig. 1), wehave analyzed a point in the HD-slow phase ( ρ L = 0 . ρ R = 0 .
5) and one in the HD-fast phase ( ρ L = 0 . ρ R = 0 . trajectories, showing that the qualita-tive picture provided by the PA is correct, the main differ-ence being that shocks are too sharp in the PA (a similarbehaviour has been observed in the MCAK results for aslightly more general model [14]).In Fig. 7 the dynamics can be divided into 2 parts.In the first part (analogous to the penetration regime in[14]), until t /N = ρ L /J ( ρ L ) ∼ .
5, particles fill the lat-tice (which is initially almost empty) and form an LD–likeplateau of density ρ L , which occupies the whole latticep-5. Botto et al.
200 400 600 800 1000i00.20.4 ρ it Fig. 7: Density profile as a function of time for N = 1000, q = 0 . r = 1, ρ L = 0 . ρ R = 0 . t/N . t/N = 12is indistinguishable from the SS. Thick smooth lines: PA, thinnoisy lines: KMC simulation (average over 10 trajectories). except for a boundary layer near the right end. The sec-ond part (analogous to the intermediate regime in [14]),from t to the SS, is characterized by the motion of ashock, separating 2 regions of densities ρ L and ρ R , re-spectively. According to the theory of boundary–inducedphase transitions [4], the shock moves leftward with veloc-ity v s = ( J ( ρ R ) − J ( ρ L )) / ( ρ R − ρ L ).In Fig. 8 the dynamics can also be divided into 2 parts.In the first part, however, due to a larger ρ L , the entryrate is so large that the particles do not have time to forma plateau at density ρ L (this would take a time t /N = ρ L /J ( ρ L ) ∼ | v s | was increasing with ρ L . Its speed, which increases withtime, is however smaller than v s (in the limit ρ L → ρ − R ),since the density immediately on the left of the shock issmaller than ρ L . Indeed, a more detailed analysis revealsthat, in the whole parameter region of the HD–fast phase,the shock speed no longer increases with ρ L . Actually, thefull dynamics is practically independent of ρ L , except nearthe left boundary. This is shown in Fig. 9 for the currentprofiles and, more importantly, this is clearly confirmedby KMC simulations. Similar results are obtained if oneconsiders the density, or the 2–node marginals.The dynamical features we have obtained above for theHD phases remain valid for other values of q and r andin the LD phases. The HD ′ phase is characterized by a(very small) relaxation rate which depends on both ρ R and ρ L , as in the HD–slow phase, but no plateau is formed atintermediate times in the dynamics. Discussion. –
We have considered a simple PA, whichextends the 2–node CMF theory and the MCAK by intro-ducing time evolution equations for 2–node expectations,and shares with these techniques the property of repro-ducing certain exact results for the SS of the AS modelwith bulk–adapted boundary rates (in particular the bulkcurrent–density relation and the location of most SS phasetransitions). We have used this approximation to investi-gate the relaxation dynamics of the model, finding dynam-
200 400 600 800 1000i00.20.4 ρ it Fig. 8: Same as Fig. 7 for ρ L = 0 . t/N = 6is indistinguishable from the SS. J it Fig. 9: Current profile as a function of time for N = 100, q = 0 . r = 1, ρ L = 0 . . . ρ R = 0 . t/N . t/N = 6 is indistinguishable from theSS. Thick smooth lines: PA, thin noisy lines: KMC simulation(average over 10 trajectories). ical transitions similar to those found in ASEPs, both inthe LD and HD phases. The existence of these transitionsis confirmed by the mDWT by de Gier and Essler and(at least for not too strongly attractive interactions) byextrapolation of exact finite size results. It is remarkablethat, for sufficiently strong attractive interactions, two dy-namical transitions can be found by PA and mDWT in thesame LD phase.The dynamical transitions separate slow and fastphases. In the slow phases, the relaxation rate dependson both boundary densities ρ L and ρ R , while in the fastphases it depends only on the parameter which determinesthe bulk density, that is ρ L in the LD phase and ρ R in theHD phase. We have shown, confirming results we had al-ready obtained [23, 27] in TASEP with various mean–fieldlike approximations, including the PA, that the spectrumof the relaxation matrix changes qualitatively at a dynam-ical transition. In the fast phase, it tends to a continuousband, while in the slow phase, an isolated eigenvalue, cor-responding to the relaxation rate appears below the con-tinuous band. A natural interpretation is that in the fastphase the boundary condition which does not determinethe bulk density (e.g. ρ L in the HD phase) is “consistent”with the bulk ( ρ L is sufficiently close to the bulk den-sity ρ R ), so that the relaxation dynamics is dominated byp-6ynamical transitions in a driven diffusive model with interactionsthe bulk properties (hence becoming independent of ρ L ),while in the slow phases one boundary density is so differ-ent from the bulk density that a slower, boundary–driven,relaxation mode appears.We have also studied the full relaxation dynamics in theslow and fast phases, looking for qualitative differences.An interesting result is that in the HD–slow (respectivelyLD–slow) phases, with initial conditions corresponding toan almost empty (resp. almost full) lattice, the system de-velops an LD–like (resp. HD–like) plateau before reachingthe SS. This plateau is not a long–lived metastable state,nevertheless, since the slow phases are located near theLD–HD transition lines, on which these phases coexist, itis tempting to view the slow phases as (loose) analoguesof metastability regions in an equilibrium phase diagram.No such plateaus are observed in the fast phases, and an-other remarkable result is that in these phases the full dy-namics, not just the relaxation rate, depends only on theparameter which determines the bulk density, as shownin Fig. 9 considering current profiles. In the same figurewe have also reported Kinetic Monte Carlo simulation re-sults, which confirm that the full dynamics is independentof ρ L . A direct calculation of the relaxation rate withKMC would also be welcome, in order to confirm the re-sults illustrated in Figs. 5 and 6, but unfortunately thisseems not feasible, as discussed in [22], where the authorseventually switched to a density–matrix renormalizationgroup approach. After simulating systems of sizes up to N = 1000 (much larger than in [22]), we similarly observethat it is very difficult to get a clear single–exponentialrelaxation. One can reasonably argue that this is to beascribed to the small separation between the lowest eigen-values of the relaxation matrix. We believe, however, thatthe PA and mDWT results for the relaxation rate, sup-ported by extrapolation of exact finite size results (whichfor pure TASEP is at least as accurate as the density–matrix renormalization group) make a strong case in favorof the existence of dynamical transitions in the AS model.Furthermore, the agreement of the PA with the KMC re-sults, confirming in particular that the full dynamics is in-dependent of ρ L in the HD–fast phase, support the overallreliability of the PA results.Work is in progress to extend these results to other mod-els, and we hope that the present paper will stimulate fur-ther investigations about the possible onset of dynamicaltransitions in non–equilibrium SS, including those exhib-ited by more realistic traffic models like the ones consid-ered in [12–19]. REFERENCES[1] van Kampen N. , Stochastic Processes in Physics andChemistry (Elsevier, Amsterdam) 2007.[2]
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