Two-species TASEP model: from a simple description to intermittency and travelling traffic jams
Pierre Bonnin, Ian Stansfield, M. Carmen Romano, Norbert Kern
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] F e b Two-species TASEP model: from a simple description to intermittency and travellingtraffic jams
Pierre Bonnin,
1, 2
Ian Stansfield, M. Carmen Romano,
1, 2 and Norbert Kern Institute for Complex Systems and Mathematical Biology,Department of Physics, Aberdeen AB24 3UE, UK Institute of Medical Sciences, School of Medicine, Medical Sciences and Nutrition,University of Aberdeen, Aberdeen, AB25 2ZD, UK Laboratoire Charles Coulomb (L2C), University of Montpellier, CNRS, Montpellier, France (Dated: February 5, 2021)We extend the paradigmatic and versatile TASEP (Totally Asymmetric Simple Exclusion Process)for stochastic 1d transport to allow for two different particle species, each having specific entry andexit rates. We offer a complete mean-field analysis, including a phase diagram, by mapping thismodel onto an effective one-species TASEP. Stochastic simulations confirm the results, but indicatedeviations when the particle species have very different exit rates. We illustrate that this is due toa phenomenon of intermittency, and formulate a refined ’intermittent’ mean-field (iMF) theory forthis regime. We discuss how non-stationary effects may further enrich the phenomenology.
I. INTRODUCTION
The Totally Asymmetric Simple Exclusion Process(TASEP) is a paradigmatic process for studying directedstochastic transport in constrained, quasi-1d geometriessubject to excluded volume interactions. One may arguethat the force of the model lies in its simplicity, whichallows it to shed light generically, with implications formany different transport processes. At the same time,it has been successfully adapted to account for the com-plexity of many specific transport situations. In this pa-per we study an extension of the TASEP, exploring theadditional features which arise when different kinds oftransported particles are discriminated by the rates withwhich they enter, and then ultimately leave the system.The TASEP model has been studied extensively in theliterature. It has become a key model of non-equilibriumstatistical physics, exhibiting a rich phenomenology, suchas boundary induced phase transitions [1] and shockwaves [2], to cite but two examples. The TASEP wasoriginally introduced to describe the process of proteinsynthesis [3], and it is still the basis of a large number ofmodels of translation of mRNA into proteins [4–7]. Atthe same time it has found applications in many otherfields, such as transcription [8, 9], intracellular trans-port of molecular motors [10], molecular transport acrossmembrane channels [11, 12], fungal growth [13] and evenvehicular and pedestrian traffic [14, 15].In the majority of considered models all particles be-have identically, i.e. they all share the same microscopicrates at which the enter the system, hop from one site tothe next, and finally exit the system. In some of thesesetups, however, it is more realistic to distinguish differ-ent types of particles travelling through the lattice. Onestraightforward example is vehicular transport, whereclearly motorbikes, cars and lorries would be expectedto enter a main road with very different dynamics, wouldtravel at different speeds, and might also differ in theprocess by which they exit onto a side road. In pedes- trian traffic, different age groups may be described. Sim-ilar considerations are expected to hold, on a microscopicscale, for molecular motors: for example, the rates gov-erning their dynamics may vary between different typesof motors advancing along microtubules [16]. On a yetsmaller scale, in mRNA translation, it is known that twotypes of ribosomes can be distinguished, the dynamicsof which differ according to whether they have boundcertain protein complexes (RAC/NAC complexes) thatassist in the process of polypeptide folding [17].Multi-species TASEP models have been consideredpreviously. Most such models have been constructed with’classes’ of particles in mind, which do not simply differin their dynamics but also weaken the excluded volumeinteractions. In these models, particles pertaining to aclass of a higher ’rank’ in the class hierarchy are allowedto ’overtake’ those of a lower rank, and it is this distinc-tion which leads to different particle dynamics [18–20].These systems have great fundamental interest, as theylead to a rich stochastic process; tracing a small num-ber of such particles of a different class has furthermoreproven a useful approach for dynamically locating theedges of high and low density zones in a TASEP trans-port process [21].A more direct distinction in terms of microscopic dy-namics has been studied in [22], where two different typesof particles share a one-dimensional lattice on which theyadvance, while overtaking is not permitted. This modelhas been introduced to describe the traffic of differenttypes of molecular motors along microtubules. In thiswork, the two types of particles are considered to differin their entry and bulk hopping rates, which complexifiesthe transport dynamics with respect to a single-speciesTASEP model. Their exit rates, however, were assumedto be identical.In this paper we focus on the opposite, complementaryscenario. We take the bulk hopping rates of all parti-cles to be the same, but distinguish two particle speciesthrough particle-specific entry and exit rates. Althoughthis does not complexify the bulk dynamics, we showthat in particular the specificity in exit rates is a funda-mentally new ingredient, which leads to rich behaviour.We first elaborate a full mean-field description for thissystem, by mapping it onto an effective simple-speciesmodel. We construct a comprehensive phase diagram ac-counting for various scenarios, according to the values ofthe four input and exit rates. We show, based on stochas-tic simulations, that this description correctly reproducessimulation results as long as entry and exit rates are ofthe same order of magnitude.In the second part of the paper, we analyse the limitingcase where the exit rates differ greatly between particlespecies. We show that this regime can lead to intermit-tent dynamics, for which the mean-field approach fails topredict both the particle current and the profile of parti-cle density along the lattice. We then introduce a modi-fied mean-field approach for this intermittent dynamics,and show that it yields a good match to numerical simu-lations when intermittency is present. We end the paperby discussing the results and the scope of the proposedapproach, pointing out further interesting features whichthe model exhibits in the intermittent regime, as perspec-tives for further studies.
II. MODEL AND APPROACH
To represent transport of two different types of objectsalong a one-dimensional track, with two sub-populationsof particles, we build on the Totally Asymmetric SimpleExclusion Process model. The standard, single-speciesTASEP consists of a one-dimensional lattice of L sites,along which particles of a single type are transported[23]. Particles attempt to enter the lattice at site i = 1with rate α and they leave the lattice at site i = L withrate β . At the bulk sites ( i = 2 , . . . , L −
1) particles hopstochastically from site i to site i +1 with rate γ , providedthat site i + 1 is not occupied.One way to summarise the key features of the TASEPin a condensed way is by thinking in terms of which pro-cess limits the flow. The hopping process in the bulksets an upper limit to the current. Indeed, a simplemean-field argument suggests a bulk current of γ ρ (1 − ρ ),where ρ is the bulk density. The maximum current (MC)phase therefore corresponds to a current of J MC = γ/ ρ MC = 1 /
2, whenever thelimiting rate is the bulk hopping rate. In contrast, whenparticles enter at a small rate, then this process limitsthe current. In this case, a low density (LD) phase ariseswith a density set as ρ LD = α/γ , with a correspond-ing current of J LD = α (1 − α/γ ). The opposite casearises when the exit rate limits the transport. In thatcase, we are dealing with a high density (HD) phase, forwhich a bulk density of ρ HD = 1 − β/γ leads to a cur-rent of J HD = β (1 − β/γ ). The beauty of the TASEPmodel is underpinned by two observations. First, this β A α A α B β B γ γ AB B B A A A A
FIG. 1: Sketch illustrating the model. Particles areinjected onto the first site with an entry rate dependingon the species ( α A and α B , respectively). Particlesstochastically advance one site at a time, with rate γ ,subject only to the next site being free. The ’bulk’hopping rate γ is the same for both types of particles.The exit rates ( β A and β B ) on the last site are againspecific to the species.straightforward analysis is indeed key to understandingthe transport features, or at least so once a ’phase dia-gram’ is established, which we return to in the following.Second, somewhat surprisingly, the simplified mean-fieldarguments sketched above turn out to reproduce the ex-act results in the limit of an infinite lattice ( L → ∞ ), ashas been show by a variety of arguments [24–26].Here we study an extension of the TASEP model.We consider two categories of particles, which we label A and B , to which a given particle belongs for itsentire journey along the segment. Both types of particlestep along the lattice stochastically, at the same rate γ , according to the exclusion process. Thus particleshave excluded volume interactions, implying they canneither occupy the same site nor overtake one another.However, particle species are distinguished by their entryrates ( α A and α B ) as well as their exit rates ( β A and β B ). This two-population TASEP model is illustratedschematically in Fig.1. III. TWO-SPECIES TASEP: MEAN-FIELDAPPROACH
Essentially, we are dealing with a TASEP in which twodifferent species compete on a given segment. As bothhopping rates are identical, we can thus think of an ef-fective single-species TASEP, for which the mean-fieldapproach makes it possible to establish the correspond-ing effective entry/exit rates. We follow up this simpleapproach, showing that it captures the process in manycases, before addressing its failure when in/out-rates dif-fer greatly between species.
A. Effective entry and exit rates
We take the bulk hopping rate, which is the same toboth species, to be equal to one ( γ = 1). Both speciescontribute indifferently to the bulk dynamics. The totaldensity of particles ρ is therefore ρ = ρ A + ρ B , (1)which is a sum of the partial densities of each species, i.e. ρ A = N A /L for N A particles of type A on a latticeof length L , and similar for particles of type B. The totalcurrent can be written as J = J A + J B (2)in a similar fashion. It is important to note that, sinceparticles do not drop off the lattice, currents for bothpopulations are preserved along the segment, from thefirst to the last site: J (0) A = J ( L ) A = J A , and J (0) B = J ( L ) B = J B . (3)Here J A,B denote the bulk currents (crossing any siteon the lattice), which are thus also equal to theircorresponding currents entering the segment J (0) A,B andto their corresponding currents leaving the segment J ( L ) A,B .At any given site, the mean-field expression for eitherof the partial currents is the product of two probabili-ties: the probability of having a particle of the consid-ered species and the probability of having an empty siteahead: J A,B = ρ A,B (1 − ρ ) , (4)where we have taken the density profile ρ i to be flat, as isknown to be justified in the bulk region [1]. The expres-sion (4) for the partial currents directly implies that, irre-spective of what happens at the boundaries, their ratio issimply identical to that of the partial densities: J A J B = ρ A ρ B .Summing the partial currents furthermore shows that thetotal current obeys the classical expression for TASEP: J = J A + J B = ρ (1 − ρ ) . (5)Now, equating the expressions for the partial in-currents and out-currents, J (0) A,B = α A,B (1 − ρ (1) ) and J ( L ) A,B = β A,B ρ ( L ) A,B (6)leads to the observation that the ratio of partial densi-ties ρ ( L ) A,B at the exit site is directly set by the partialentry/exit rates α A and α B : α A α B = β A ρ ( L ) A β B ρ ( L ) B . (7)This relation will prove central in the following.We now turn to establishing the effective entry/exitrates. Writing the total current at the first site as J (0) = J (0) A + J (0) B = ( α A + α B )(1 − ρ (1) ) (8) directly arises in the form of an inflowing current, J (0) = α eff (1 − ρ (1) ), where the effective entry rate α eff is thusgiven as α eff = α A + α B . (9)To derive an expression for the effective exit rate β eff we start from Eq. 7 and obtain: ρ ( L ) A ρ ( L ) B = β B α A β A α B . (10)Eliminating ρ ( L ) A via Eq. (1) yields an en explicit ex-pression for the density of B particles at the exit site ρ ( L ) B : ρ ( L ) B = β A α B β B α A + β A α B ρ ( L ) . (11)Analogously, we have ρ ( L ) A = β B α A β B α A + β A α B ρ ( L ) , (12)as is seen either from applying Eq. (1) to the last latticesite, or simply from symmetry permuting indices A andB.The exit current can now be written, from Eq. (5)evaluated at the last site, J ( L ) = β A ρ ( L ) A + β B ρ ( L ) B = ( α A + α B ) β A β B β B α A + β A α B ρ ( L ) (13)and, again by analogy to the TASEP mean-field expres-sion, the effective exit rate thus is β eff = ( α A + α B ) β A β B β B α A + β A α B . (14)Note that Eqs. (9) and (14) imply a simple relationfor the ratio between the effective rates: α eff β eff = α A β A + α B β B . (15)It will also be useful to underline that the relations set-ting the effective rates, Eqs. 9 and 14, have direct phys-ical significance. Indeed, the abundance of each type ofparticles can be characterised by the partial densities χ A and χ B = 1 − χ A , defined as χ A = ρ A ρ A + ρ B and χ B = ρ B ρ A + ρ B . (16)Using successively Eqs. (4) and (3), as well as currentconservation, these can be expressed as χ A = J A J A + J B = J (0) A J (0) A + J (0) B , (17)and similar for χ B . From Eq. (6) we then have χ A = α A α A + α B ∈ [0 ,
1] and χ B = α B α A + α B ∈ [0 , , (18)and therefore the partial density of each species is setdirectly by the percentage with which it contributes tothe total input rate, as it is of course expected.Regarding the effective exit rate, we can now re-writeEq. (15) by dividing out α eff = α A + α B to obtain1 β eff = χ A β A + χ B β B . (19)Since an inverse exit rate corresponds to the average timerequired for a particle of a given type to exit from thelast site, the effective exit rate thus corresponds to thepopulation-weighted average of these exit rates.Note that all results stated so far are valid for anychoice of parameters, to the extent that the mean-fieldapproach holds. We will first explore this mean-field be-haviour, and then show how it breaks down under specificconditions. B. Mean-field phase diagram
Based on the expressions for the effective rates in themean-field approximation we can now establish the phasediagram for the model with two types of particles. Welabel the phases, just as in the standard, single-speciesTASEP, as LD, HD or MD, according to whether their(total) density in the bulk is inferior, superior or equalto 1/2. The conditions fixing the well-established single-species phase diagram, summarised in Fig. 2a, are:(LD) (i) α < β and (ii) α < / β < α and (ii) β < / α > / β > / α A = α A β A and ˜ α B = α B β B . (21)Phases can now be delimited by the transition linesof the phase diagram, as each condition in Eq. (20) ex-cludes a certain phase for a particular zone. For thepurpose of illustration, consider the standard TASEP, asrepresented in Fig. 2a, and focus on identifying the HDphase. We proceed in two steps. First, according tocondition (20-LD-i) the LD phase cannot be present if α > β , and therefore the region below the line α = β canonly pertain to an HD or an MC phase. Second, fromcondition (20-HD-ii) the HD phase cannot occur when β > /
2, and therefore below the line β = 1 / LD or a M C phase. Combin-ing these conditions thus identifies the zone in the ( α, β )plane which corresponds to the HD phase. This way ofconstructing the phase diagram is graphically representedin Fig. 2(b).For the full model we can proceed similarly in the( α eff , β eff ) plane, as we know that the transitions be-tween phases fall onto (part of) the following relations(colours refer to Fig. 3):(i) LD-HD: α eff = β eff (blue line), which in terms ofthe rescaled rates is given by˜ α B = 1 − ˜ α A . (22)(ii) LD-MC: α eff = (green line) or, equivalently,˜ α B = 12 β B − β A β B ˜ α A . (23)(iii) HD-MC: β eff = (red line), or equivalently˜ α B = − − β A − β B ˜ α A . (24)As a direct conclusion, all phase boundaries are straightlines in the (˜ α A , ˜ α B ) plane. Fig. 3 shows how by combin-ing these conditions we can assign a zone to each phasein the (˜ α A , ˜ α B ) plane. First of all, the LD-HD line isfixed in this representation, and goes through (0 ,
1) and(1 , / (2 β A ) ,
0) and (0 , / (2 β B )). Finally, the point whereall these lines cross is located at(˜ α ∗ A , ˜ α ∗ B ) = (cid:16) / − β B β A − β B , β A − / β A − β B (cid:17) . (25)In the following we assume the B particles to be theones with a slow exit rate, i.e. we take β B < β A (with-out restricting generality, as the opposite case would becovered by exchanging particle species). Three differentscenarios can now easily be identified, according to howthe exit rates compare to the threshold of 1 /
2. This yieldsthree cases: • (a) β B < β A < / • (b) β B < / < β A , and • (c) 1 / < β B < β A .These cases differ in the relative positions at which theLD-HD and LD-MC separation lines intersect the ˜ α B axis, while the LD-HD line remains fixed. The construc-tion of the phase diagram can be visualised most clearlyif one first admits negative values for the rates ˜ α A and ˜ α A before restricting our interpretation to the physically rel-evant area. With this in mind, the triple point ( α ∗ A , α ∗ B )may be localised in various quandrants of the plane, andit is this which distinguishes the three scenarios. Thethree cases are illustrated in which correspond to subfig-ures 3 (a-c) .From this construction it follows that the LD phase isassigned to the area which is both below the blue andthe green lines, and this can be achieved in all three sce-narios. HD is delimited by the blue and the red lines.In scenario (a) and (b) it corresponds to the zone aboveboth the blue and the red line. In scenario (c) howeverit corresponds to the one above the blue and below thered line: these conditions cannot be met in the physicaldomain (positive rates), such that there is no HD phasewhenever 1 / < β B < β A . Finally, the area correspond-ing to MC is delimited by the green and the red lines. Inscenarios (a) and (b) this is the zone above the green lineand below the red line. In case (a), however, this zoneis not part of the physical region, and so there is no MCphase whenever β B < β A < /
2. In scenario (c), the areacorresponding to MC is above the green and the red lines.These results are compared to numerical simulations inFig. 4, as discussed in the next subsection.
C. Interpretation
Figs 3a, 3b and 3c illustrate the different scenariosof the mean-field diagram representing two-populationTASEP. The arguments that follow are generic, no cor-respondence with data is sought at this stage. The sce-narios can be distinguished based on the location of thecritical point (Eq. (25)) in the plane of rescaled in-rates(˜ α A , ˜ β B ): this essentially fixes all the phase boundariesmeeting here, all of which are straight lines (see Eqs. (22,23 and 24). The LD phase is present in any scenario, asexpected, since this is the ’default’ phase which can al-ways be reached by sufficiently lowering all input rates.All three phases are observable if the critical point fallsinto the physically accessible parameter domain (i.e. intothe first quadrant, panel (b)). If it falls into one of theadjacent quadrants, however, only two phases are observ-able (LD and HD in panel (a), or LD and (MD) in panel(c)). The colour code for the phase boundaries is thatof Fig. 2b; the solid part of the lines indicate the actualphase boundaries. Focusing on Fig. 3a first shows thatan HD phase is possible, and it is in fact the only otherphase in the physical region (positive rates), if the fastexit rate is sufficiently small ( β A < / β B > / β B < /
2) while the fast exitrate is sufficiently large (1 / < β A ), both an HD and anMC phase are possible, see Fig. 3b. Indeed, increasingthe proportion of fast exiting particles (i.e. increasing α A ) will push the system into an MC phase, whereashaving more slow particles (i.e. increasing α B ) will favouran HD phase.A slightly contrasting statement is to be made in termsof which of the parameters are decisive. Indeed, withineach scenario (a, b or c), the phase can be identifiedbased solely on the reduced entry rates ˜ α A and ˜ α B , asillustrated in the phase diagrams. However, it is worthnoting that all rates ( α A and α A as well as β A and β B )are required explicitly in order to determine which regimethe system finds itself in. D. Numerical validation
Stochastic numerical simulations were performed withthe Gillespie algorithm [27]. A lattice of length L = 500sites was used and, unless stated otherwise, measure-ments were cumulated over 10 Gillespie iterations, afterhaving discarded a transient of 4 × iterations.In order to confront the analytical mean-field charac-terisation of phases to data from simulations we plot inFig. 4 the numerically obtained density (averaged bothover time and the segment), from which the phases canbe deduced ( ρ = 1 / β α HDLD MC (a) β α HDLD MC (b) 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β α HD or MC ? 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HD or LD ? (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) β α HD (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (c) FIG. 2: Single-species TASEP. The phase diagram (panel (a)) is constructed based on three conditions (panel (b)): α = β (blue) at the boundary between LD and HD, α = 1 / β = 1 / α B ∼ α A ∼ B α B ∼ * 1/ A α A ∼ *11 LDHD (a) β B < β A < / α B ∼ α A ∼ α B ∼ * α A ∼ *1/ B A HDLD MC (b) β B < / < β A α A ∼ A α B ∼ α B ∼ * α A ∼ * 1/ B MCLD (c) 1 / < β B < β A FIG. 3: Mean-field diagram representing two-population TASEP in 3 different scenarios depending the rescaledparameters ˜ α A and ˜ α B when β B < β A . Scenarios for the phase boundaries are determined by first positioning thecritical point (seen in Eq. ( 25)). We therefore have three cases, which lead to the following phases: (a) LD and HDarise where β B < β A < /
2, (b) LD, MC and HD arise where β B < / < β A and finally (c) LD and MC arise where1 / < β B < β A . (a) β A = 0 . β B = 0 . (b) β A = 0 .
625 and β B = 0 . (c) β A = 0 .
666 and β B = 0 . FIG. 4: Scenarios for the phase diagram from numerical data, plotted in the same rescaled parameter plane(˜ α A , ˜ α B ) as used in the previous figure. The colour gradient plot represents the average density throughout thelattice, calculated on a grid (˜ α A , ˜ α B ) ∈ [0 . , . × [0 . , .
5] with a step of 0 .
01. The black lines in panels (a), (b)and (c) correspond to the analytical expressions for the phase boundaries, as given by Eqs. (22), (23) and (24).FIG. 5: Numerically acquired density profiles (colouredlines), contrasted to the mean-field predictions (blackdashed line). The following phases are shown: LD( α A = 0 . β A = 0 . α B = 0 . β B = 0 .
2) (blue line);MC ( α A = 1, β A = 2 ./ α = 0 . β B = 0 . α A = 0 . β A = 0 . α B = 0 . β B = 1 /
3) (green line). We obtain good agreement forthese choices of parameters, except for the boundarylayers which are expected from the regularsingle-species TASEP. For other parameter choicesdeviations arise, which are discussed below andanalysed in the following section.
E. Mean-field discrepancies
However, as illustrated in Fig. 6, this agreement isnot fully general: density profiles can differ significantlyfrom the prediction of the effective single-species TASEPmodel. The 2-species model can therefore display newfeatures, which are not captured by the mean-field de-scription in terms of an effective single-species TASEPwhich we have elaborated so far, at least for certainchoices of parameters. In order to qualify these differ-ences we focus again on density profiles, rather than onthe entire phase diagram, which will also help to establisha strategy for improving the theoretical approach.In principle there are 4 independent parameters to themodel (two rates for each species), and therefore, theparameter space to be explored is vast. Varying any ofthese rates may affect current and density, and may alsopush the system across a phase boundary. However, theeffective one-species model suggests that it is exclusivelythe two effective rates α eff and β eff which determine thebehaviour, or at least so long as the mean-field analysisremains valid. We therefore choose, in a first instance,to vary all independent rates, α A and β A as well as α B and β B , jointly , such as to preserve the total effectiverates. In this way we can compare results from numericalsimulations to analytical mean-field predictions without modifying the position in the mean-field phase diagram,i.e. no mean-field phase transition can be triggered bysuch a change.We thus need to pick two additional parameters, inadition to α eff and β eff , to define our system. Beforechoosing how to do this, consider two limiting cases of oursystem, one where α B = 0 (i.e. the case where we recoverthe single-species model, since B-particles are absent),and another one where β B → β A (an equivalent scenario,since both particle species behave identically). In bothlimits the single-species TASEP model must hold. Thissuggests choosing the two remaining parameters in sucha way that they characterise (i) the fraction of (slow)B particles in the system and (ii) the ’slowness’ of Bparticles as compared to A particles.A natural choice for the first parameter thus is the frac-tion of B particles, χ B ∈ [0 , χ B di-rectly implies the individual input rates as (see Eq. (16))as α A = (1 − χ B ) α eff and α B = χ B α eff . (26)Varying χ B ∈ [0 ,
1] maps out this degree of freedom at afixed total effective rate α eff , as desired.We now pick a second parameter, say s ∈ [0 , s to make B particles slower, by setting β B = (1 − s ) β A , (27)while requiring that the effective exit rate β eff remainunaffected. According to Eq. (19) this implies1 β eff = χ A β A + χ B (1 − s ) β A which can be solved to yield β A = β eff (cid:20) − χ B s − s (cid:21) (28) β B = β eff [1 − (1 − χ B ) s ] . (29)In essence, we can thus use the parameters χ B and s to vary the abundance of B particles and their ’slowness’independently, while leaving the effective entry/exit ratesunchanged.Figure 6 shows density profiles for two examples forwhich the mean-field prediction is an HD phase. For allgraphs in each panel, the effective in/out rates as wellas the particle distribution have been kept constant: theonly parameter which is varied is s , which regulates theslowness of B particles. Deviations from the mean-fieldprediction (black dashed line) become increasingly signif-icant as the slowness s increases: since all other parame-ters have been maintained constant, we can conclude thatthe mean-field theory fails as B particles become too slowto exit.Deviations concern not only the average density value,but also the shape of the density profile can deviate sig-nificantly from what is expected from an effective single-species description. In Fig. 6 (a) the shape of the densityprofile for the two largest values of s resembles a densityprofile in the maximal current (MC) phase, despite thelattice being in HD phase (average density on the lat-tice is above 0.5). In panel (b) the systematic positiveslope in the profile makes it qualitatively different froma single-species profile.In essence, these examples show that the effectivesingle-species model is no longer appropriate as one ofthe particle species becomes significantly slower to leavethan its counterpart. The intuition at this point is thatthose particles provoke temporary blockages, leading to’intermittent’ flow with entirely new characteristics. Wepursue this thought further in the following section, andshow how intermittency may be used to construct an im-provement to the mean-field predictions. IV. INTERMITTENCY
In the previous section numerical evidence has exposedthe fact that an effective single-species model no longerdoes justice to the traffic in our two-species model whenone particle species becomes very slow to leave. Thissuggests a considerable alteration of the traffic, the na-ture of which becomes clear by considering the limitingcase where B particles become extremely slow to leave(slowness s ≃ intermittency inthe following, and show how mean-field arguments canbe amended to account for this phenomenon.In order to better appreciate the phenomenon we showa series of snapshots of density profiles in Fig. 7. Eachof these is a quasi-instantaneous density profile, obtainedby averaging the occupancy of each lattice site over 2,600Gillespie iterations. The 12 graphs are presented inchronological order (from left to right, top to bottom),thus illustrating the time evolution of the density profile.Panel (a) shows a blockage at the exit, with a ’jammed’region (highlighted in green). As time progresses, this’jammed’ region ’travels upstream’, i.e., to the left, asshown in panel (b). More precisely, it ’grows’ to the left,as particles within the jammed region are of course es-sentially stuck and therefore static, but further particlesjoin the jam from the left. In panel (c) the blocking par-ticle has finally exited the lattice; therefore the particlesfrom the right boundary of the jammed region can startmoving ahead, and eventually leave the lattice. Thus thejammed region decreases in size from its right boundary.The net effect is that the jammed region appears to travelupstream, as particles join at its left boundary and oth-ers leave the jammed region at its right boundary. Somemore complicated effects can occur, as shown in panel (a) (cid:0) ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✂✄☎✆✝ ✞✟✠✡☛✡✡☞✌✍✎✏✑✒✄☎✓✔ ✕✟✠✡☛✡✡✡✖✗✎✏✑✒✘✟✠✡☛✡✡✡☞✌✎✏✑✒✙✟✠✡☛✡✡✡✗✍✎✏✑✒✚ ✛ ✜✢✄☎✓ ✣✤✤✥✤✤✤✦✤✤✧✤✤✤★✤✤✩✪ (cid:0)✪✪ ✫✄✄ ✬✪✪ ✭✄✄ ✮✄✄ (b) FIG. 6: HD density profiles from numerical simulation,illustrating the breakdown of mean-field predictions inthe case where the exit rates β A and β B differ greatly.For all simulations parameters were chosen as tomaintain identical effective rates, just as the proportionof B particles ( χ B ). The plots represent several choicesfor the slowness parameter, s = 1 − β B /β A . Thesingle-species prediction is thus seen to break down asthe species B becomes increasingly slow to leave. Panel(a): α eff = 1, β eff = 9 . × − , χ B = 1 × − .Panel (b): α eff = 1 . × − , β eff = 9 . × − , χ B = 1 × − .(c), where the jammed region breaks into two parts as it’travels’ upstream. In panels (d)-(f) the jammed regionis dissolved as it reaches the left boundary of the lattice,and the bulk density relaxes to the one determined bythe faster, non-blocking A particles. Ultimately, we cansee another jam forming in panel (h), caused by the ar-0rival of a B particle to the last site of the lattice (jammedregion highlighted again in green).These figures illustrate that the presence of intermit-tency in the two-population model is the root causeof discrepancy with respect to the effective single-population model, as we will show now.The simplest case to think about when developing theargument is when the B particles are both very slow toleave ( s ≃
1) and sparse ( χ B ≪ A particles almost always,except when one of the rare B particles reaches the exitsite. Then a jam is created at the exit, corresponding to astretch of density 1 in front of the exit. However, as soonas the blockage is resolved, the jam evacuates and thesystem returns to its original phase. As B particles arevery sparse, this is essentially a process involving A parti-cles only, and thus the flow phase is in fact characterisedby the underlying ’pure’ single-species phase (obtainedasymptotically as χ B → M C → HD ∗ phase. What we mean by this no-tation is that the underlying ’pure’ system of A particlesswould be found in an MC phase, as is indeed seen dur-ing the periods without blockage. However, due to thepresence of slow-to-leave B particles, the resulting phaseis more apparent of an HD phase. The star thus labelsthose phases which already are the result of intermittentbehaviour.With this picture in mind we now focus on the effectof intermittency in the density profile for 3 different sce-narios, corresponding to 3 different choices for the under-lying ’pure’ single-species phase. These are: LD → HD ∗ (Fig. 8a), MC → HD ∗ (Fig. 8c) and HD → HD ∗ (Fig. 8e).For each of these scenarios we fix the entry and exit ratesof the A particles, as well as the exit rate of the B parti-cles, but we vary the proportion χ B of particles of typeB by changing α B [31].Remarkably, in both the LD → HD ∗ and MC → HD ∗ cases, the density profiles are qualitatively different fromthe ones of an effective single species TASEP in an HDphase. In the LD → HD ∗ scenario, the density profilesexhibit a positive slope from the left to the right bound-ary of the lattice. In the MC → HD ∗ scenario the shapeof the density profiles resembles the ones of an MC sin-gle species TASEP, but with an average density higherthan 0.5. The only density profiles that remain qualita-tively the same are the ones in the HD → HD ∗ scenario,although they are quantitatively different from the mean-field predictions. In the next subsection we introduce anextended mean-field approach that accounts some extentfor the results obtained in this intermittent regime. A. Intermittent Mean Field (iMF) Approximation
We propose to pursue the picture established above topropose an ’intermittent mean-field theory’ (iMF) as anextension to the mean-field arguments presented above,designed to account for intermittency in the case whereB particles are very slow ( β B ≪ β A ) and very sparse( χ B ≪ χ A ). We are thus dealing with entire stretchesof A particles, say n A of them on average, separated byisolated B particles. The average number n A of A par-ticles in each stretch can be then estimated as α A /α B .To a first approximation we can therefore think of thecurrent as being the one corresponding to the underly-ing ’pure’ phase (i.e. the phase which corresponds tovanishing χ B ). It is interrupted every so often, for thetime it takes for the occasional slow B particle to free theexit site. During this period the exit current is zero. Aslong as these intermittent jams persist, and as an HD-likestretch builds up close to the exit site, we can estimatethis average ’blockage’ time interval as τ HD = 1 β B . (30)From this, a prediction for the total current followsfor each of the three different scenarios introduced above: • LD → HD ∗ : in this scenario, the single-phase cur-rent is J LD = α A (1 − α A ). Using Eq. 30 we there-fore have J LD → HD ∗ = J LD τ LD τ LD + τ HD , which is es-sentially a weighted average of the current, since weexpect to have current J LD during the time inter-val τ LD and zero current the rest of the time, with τ LD = n A J LD . Thus we have J LD → HD ∗ = α A β Bα A β B α A (1 − α A ) + α B . (31) • MC → HD ∗ : following the same approach as above,we obtain J MC → HD ∗ = 14 α A α B α A α B + β B . (32) • HD → HD ∗ : in this case, we obtain: J HD → HD ∗ = α A β Bα A β B β A (1 − β A ) + α B . (33)Notice that in Eq. (33), as both exit rates β A and β B vanish, J HD − HD ≈ β eff , in agreement with the conven-tional TASEP current in HD: J HD = β (1 − β ) ≈ β . (34)Predictions from this ’intermittent’ mean-field (iMF)theory for the current are shown in Figs. 8b, 8d and 8f1
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FIG. 7: Time evolution of the density profiles for the
M C → HD ∗ scenario, showing a temporary blockage of theexit site and its evolution for a lattice of N = 1000 sites. Panels (a) to (h) represent the succession ofinstantaneous-like density profiles along the same simulation. Parameters are α A = 1; α B = 0 . β A = 1; β B = 0 . → HD ∗ andMC → HD ∗ scenarios the iMF approach performs betterthan MF in the limit of very small values of χ B . Forintermediate χ B values, both approaches are compara-2 numMFiMF (a) LD → HD ∗ -4 -3 -2 -1 J numMFiMF -4 -3 -2 -1 -1-0.500.51 ( J ana l y t - J nu m ) / J nu m MFiMF (b) LD → HD ∗ numMFiMF (c) MC → HD ∗ -5 -4 -3 -2 -1 J numMFiMF -5 -0.6-0.4-0.200.20.4 ( J ana l y t - J nu m ) / J nu m MFiMF (d) MC → HD ∗ numMFiMF (e) HD → HD ∗ -5 -4 -3 -2 -1 J numMFiMF -5 -0.5-0.4-0.3-0.2-0.100.1 ( J ana l y t - J nu m ) / J nu m MFiMF (f) HD → HD ∗ FIG. 8: Effect of intermittency on the density profile ρ ( i ) (panels a,c,e) and on the current (panels b,d,f), as theproportion χ B of slow particles is progressively increased. The parameters used in the sub-figures are as follows:(a,b) LD → HD ∗ phase, with α A = 0 . β A = 1, β B = 1 × − ; (c,d) MC → HD ∗ phase, with α A = 1; , β A = 1, β B = 1 × − ; (e,f) HD → HD ∗ phase, with α A = 1, β A = 0 . β B = 1 × − . The insets show the results from thenumerical simulations (solid line), MF approach (dashed line) and iMF approach (dotted line) for χ B = 2 × − (panel a); χ B = 1 × − (panel c); χ B = 2 × − (panel e). The plots of current J (panels b,d,f) show numericalresults (red), MF prediction (blue) and an iMF prediction (black) as a function of χ B . The insets show the relativeerror of the current for both the MF and iMF predictions.3ble. For larger values of χ B , the MF performs betterthan the iMF: this is as expected, since for larger valuesof χ B , where the proportions of A and B particles arecomparable, there should be no intermittent behaviour.A special case is the HD → HD ∗ scenario, for which iMFand MF perform indistinguishably well for both smalland intermediate values of χ B . This is because in theHD → HD ∗ scenario both β A and β B are very similar, andtherefore the difference in the MF and iMF expressionsis very small, as pointed out in Eq. 34.In Figs. 8a, 8c and 8e the insets show the comparisonof the average density predicted by the MF (dashed line),iMF (dotted line) and the numerically obtained densityprofile (solid line) for a fixed value of χ B . Parametersare chosen to be well within the intermittent regime,and the average density in the iMF has been attributedby equating the current to the mean-field expression ρ (1 − ρ ) and solving for ρ . Comparing the differentapproximations for the density and the correspondingnumerical simulations thus mirrors what is observed forthe currents.The iMF approach therefore successfully takes overfrom the standard MF description when B particles arevery slow to leave, as far as they are remain sparse com-pared to A particles. The limitations of the approach liein assuming that the current of A particles is stationary.In reality, we know that whenever a slow B particle freesthe exit, a highly non-stationary process will ensue, dur-ing which the density profile relax from a totally jammedstate at the exit to the stationary density profile corre-sponding to a ’pure’ flow of A particles in the appropriatephase (LD, MC or HD). The iMF approach as outlinedhere therefore assumes that these non-stationary phasesremain sufficiently short so that they do not affect thetime-averaged density. This assumption must fail whenblockages become too frequent for this to be true, or eventoo frequent for the density to return to its stationarydensity before the next blockage: clearly, this is the rea-son why iMF predictions perform poorly as the fractionof B particles becomes significant. V. DISCUSSION
In this paper we have analysed an extension of theTASEP model where we consider two different types ofparticles. No overtaking is allowed, in contrast to mostmulti-species TASEP previously introduced in the liter-ature [18, 19, 21]. Hence, our model is similar to theone introduced in [22], which describes different typesof molecular motors moving along microtubules. At thesame time our model is complementary, as we focus onan entirely different scenario, considering that particlespecies have the same bulk hopping rate but differ bothin their entry and exit rates with which enter and leavethe lattice, respectively.We have shown that a standard mean-field theory can be formulated by mapping the two-species TASEP modelonto an effective single-species TASEP, via appropriatelydefined effective entry/exit rates. A comprehensive phasediagram can be established based on the ensemble of allentry/exit rates for both particle species, according towhich different scenarios arise. In these several, but notnecessarily all, possible TASEP phases are present. Forexample, when the exit rates of both particles are inferiorto half of the bulk hopping rate, then there cannot be amaximum current (MC) phase. Comparison to stochasticsimulations has shown that this approach yields excellentresults for the current and for the density profile along thelattice as long as the entry and exit rates of the differenttypes of particles are of the same order of magnitude.The mapping onto an effective single-species modefails, however, when one type of particles is much rarerand has a much slower exit rate than the other one. Wehave shown that the origin of this discrepancy lies in theemergence of intermittent dynamics caused by temporaryblockages of the exit by the slow-to-leave particles. Thekey to the intermittent regime is to analyse the trans-port process in terms of traffic jams which form close tothe exit, whenever a slow-to-leave particle arrives at thelast lattice site. This picture constitutes a valid repre-sentation, based on which we have introduced a modi-fied mean-field approach. This ’intermittent mean-field’(iMF) description takes into account the intermittent be-haviour, and we have shown that it provides good pre-dictions when compared to simulations.Further dynamic features of traffic jams reveal addi-tional questions in their own right. For example, as ablocking particle ultimately leaves the lattice, the trafficjam it has created then ’moves’ upstream (Fig. 9). This isin fact the apparent result of an ongoing evolution of theblocked region: this region extends to the upstream side(as new particles join the traffic jam), while simultane-ously shrinking at the downstream side (where particlesare shed from the jam as they become free to move aheadand ultimately leave the lattice). Indeed, we may viewsuch a ’jammed’ region as being delimited by two abruptchanges in the density, also known as ’shocks’ or ’domainwalls’, and it would be interesting to specifically analysethe process in terms of the diffusive dynamics of thesediscontinuities [29, 30].A important question to be explored further concernsthe time scales at which shocks form, evolve and dis-appear, and at which the system relaxes to its steady-state transport regime. Indeed, consider the case wherea (rare) slow-to-leave B particle ultimately leaves thesystem, allowing the intermittent traffic jam to dissolve.Then the bulk density in the lattice is expected to relaxto that corresponding to that of a ’pure’ phase of non-blocking particles. Our iMF analysis may thus be ex-pected to be valid if this can typically be achieved beforethe next blocking particle initiates another traffic jam.The opposite case, though, is intrinsically non-stationary,and much harder to analyse. One may speculate whethersuch a description would also provide a full rationalisa-4
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FIG. 9: Time evolution of the density profiles for the
M C → HD ∗ scenario, showing the evolution of 3 temporaryblockages of the exit site and their evolution. Panels (a) to (h) represent the succession of instantaneous-like densityprofiles along the same simulation. Parameters are α A = 1; α B = 0 . β A = 1; β B = 0 . Acknowledgments