Power laws and phase transitions in heterogenous car following with reaction times
PPower laws and phase transitions in heterogenous car following with reaction times
A. Sai Venkata Ramana ∗ New York University Abu Dhabi, Saadiyat Island,P.O. Box 129188, Abu Dhabi, United Arab Emirates
Saif Eddin Jabari † New York University Abu Dhabi, Saadiyat Island,P.O. Box 129188, Abu Dhabi, United Arab Emirates andNew York University Tandon School of Engineering,Brooklyn New York 11201, United States of America
We study the effect of reaction times on the kinetics of relaxation to stationary states and oncongestion transitions in heterogeneous traffic using simulations of Newell’s model on a ring. Het-erogeneity is modeled as quenched disorders in the parameters of Newell’s model and in the reactiontime of the drivers.We observed that at low densities, the relaxation to stationary state from a homo-geneous initial state is governed by the same power laws as derived by E. Ben-Naim et al., Kineticsof clustering in traffic flow, Phys. Rev. E 50, 822 (1994). The stationary state, at low densities,is a single giant platoon of vehicles with the slowest vehicle being the leader of the platoon. Weobserved formation of spontaneous jams inside the giant platoon which move upstream as stop-gowaves and dissipate at its tail. The transition happens when the head of the giant platoon startsinteracting with its tail, stable stop-go waves form, which circulate in the ring without dissipat-ing. We observed that the system behaves differently when the transition point is approached fromabove than it does when approached from below. When the transition density is approached frombelow, the gap distribution behind the leader has a double peak and is fat-tailed but has a boundedsupport and thus the maximum gap in the system and the variance of the gap distribution tend tosize-independent values. When the transition density is approached from above, the gap distributionbecomes a power law and, consequently, the maximum gap in the system and the variance in thegaps diverge as a power law, thereby creating a discontinuity at the transition. Thus, we observea phase transition of unusual kind in which both a discontinuity and a power law are observed atthe transition density. These unusual features vanish in the absence of reaction time, i.e., whenthe vehicles react instantaneously to a perturbation ahead (e.g., automated driving). Overall, weconclude that the nonzero reaction times of drivers in heterogeneous traffic significantly change thebehavior of the free flow to congestion transition while it doesn’t alter the kinetics of relaxation tostationary state.
I. INTRODUCTION
Traffic systems are nonequilibrium driven systems thatexhibit rich collective phenomena in the kinetics of re-laxation to a nonequilibrium stationary state and in thephase transition to the congestion regime. The problembecomes even more complex when a quenched disorderis introduced into the system. A quenched disorder, ina physical sense, implies heterogeneous traffic in whicheach driver-vehicle unit has a different set of parameters.At a fundamental level, two basic kinds of approaches areused to model traffic: car-following models [1] and cellu-lar automata [2]. Physicists have studied the effects ofquenched disorder on the collective phenomena using thetotally asymmetric simple exclusion process (TASEP)and the Nagel-Schkrekenberg (NS) models [3–5], whichare cellular automata, while such studies are rare in thecar-following literature. At the outset it might appearthat the conclusions of the car-following model would ∗ Email:[email protected] † Corresponding author
Email:[email protected] be the same as those in the cellular automata. How-ever, it is important to understand if the subtle differ-ences in basic assumptions between the methods mightlead to differences in the emergent phenomena. It is alsoimportant to characterize these phenomena in the jar-gon of car-following methods, which are widely used intransportation engineering applications [6–8]. In previ-ous work, we studied heterogeneous traffic by introducingquenched disorders into the parameters of Newell’s model[9]. In this work, we study the further effect of the re-action time of drivers by means of numerical simulationsof Newells’ model. We observe that the presence of reac-tion times and quenched disorders in them significantlychanges the picture from one with no reaction times andreveals interesting phenomena that are not observed inthe cellular automata. We first give a brief review of re-lated works that use cellular automata, specifically theNS model in Sec. II and highlight points that promptedthe present study. In Sec. III, we present our car follow-ing model and explain briefly the role played by reactiontimes in the formation of spontaneous stop-go waves. InSec. IV, we discuss the approach to stationary state. InSec. V, the giant platoon that forms in the stationarystate is characterized. In Sec. VI, the phase transition a r X i v : . [ phy s i c s . s o c - ph ] F e b from the platoon forming phase to the congestion phaseis analyzed. The results are summarized in Sec. VII. II. COLLECTIVE PHENOMENA IN NS MODEL
The NS model for a one dimensional lattice of L siteson a ring is as follows: The speed of each vehicle is as-sumed to be discrete with allowed integer values between0 and v max . The time step is taken to be unity and dimen-sionless. Thus, the space gap ( d ) and speed ( v ) have thesame units. Starting from a given initial configuration,the positions and speeds of all the vehicles are updated ateach time step according to the following rules [2]: (i) Thespeed v i of the i th vehicle is updated to min { v max , v i +1 } if v i < d i , where d i is the gap ahead of the i th vehicle.(ii) If v i ≥ d i , v i is updated to d i −
1. (iii) The speed of avehicle is reduced by unity ( v i (cid:55)→ v i −
1) with a probabil-ity p to account for randomness in hopping, also calledas random deceleration. (iv) Each vehicle advances v i sites. The NS model is very similar to TASEP exceptthat in TASEP v max = 1 and the positions of particlesare updated in a random sequential manner.The traffic system as described by the TASEP or theNS model is intrinsically a driven non-equilibrium sys-tem. The evolution of such system towards its station-ary state reveals its dynamical universality class whichmay be distinguished based on the dynamical exponent z related to the emerging length scale ξ ( t ) in the systemas ξ ∼ t /z . In the case of traffic, ξ ( t ) is the length ofthe platoon of vehicles moving as a cluster. While therehave been early numerical studies in the NS literature re-garding the dynamical universality class of the NS model[10, 11], it has been only recently proven by Gier et al. [12]using non-linear fluctuating hydrodynamics that the NSmodel belongs to the super-diffusive Kardar-Parisi-Zhanguniversality class with dynamical exponent z = 3 / p .The presence of a quenched disorder in the systemmakes it even more complex. Krug and Ferrari [22, 23]studied a version of TASEP with quenched disorder in p with probability distribution f ( p ) ∼ ( p − p min ) n when p → p min and conjectured that the dynamical exponent z depends on the exponent of the quenched disorder as z = ( n +2) / ( n +1). Krug and Ferrari also argued that the phase transition from platoon forming phase to a lami-nar phase without platoon formation would be of secondorder if n ≤ n >
1. Evans [24] in-dependently solved for the steady state of the TASEPwith quenched disorder in jump rates and showed thatthe phenomenon of bunching of vehicles behind the slow-est vehicle is analogous to the Bose-Einstein condensa-tion. Ktitarev et al. [25] did simulations of the NS modelwith quenched disorder ( f ( p )) in p and concluded thatthe dynamical exponent z and the exponent for the gapdistribution near the critical point are same as those con-jectured by Krug and Ferrari. Bengrine et al. [26] sim-ulated NS model for open boundary conditions. Theirconclusions also corroborated Krug’s conjectures regard-ing the order of the transition and the exponent z .It can be seen from the above discussion that thestochasticity induced by the random deceleration (whichoccurs with probability p as discussed above) plays a cen-tral role in the collective phenomena exhibited by the NSmodel. The random hopping probability p has been in-troduced to account for spontaneous traffic jams (alsocalled stop-go waves) and various other aspects like non-deterministic acceleration by drivers etc. However, itis not directly related to any physically observable phe-nomenon in traffic flow. There is no analogous parameterin car-following models also as can be seen in Newell’smodel explained in the next section. The main effect ofthe random deceleration in the NS model (i.e., the for-mation of stop-go waves) is captured by driver reactiontimes in car-following models, and this basically modelsthe delay in response of a driver-vehicle unit to a per-turbation ahead of it. The process by which the stop-gowaves form due to reaction times has its origin in theflow instability, which is a deterministic process and thuscould be quite different from the way stop-go waves formin the NS model, which is based on stochasticity inducedby p . This raises the question of whether the collectivephenomena exhibited by car-following models would bethe same as those observed in the NS model. Our presentstudy aims to address this point. III. CAR-FOLLOWING MODEL FORHETEROGENEOUS TRAFFIC
Newell’s model [27] is a simple physical car-followingmodel that is known to reasonably capture the dynamicsof car-following. It has been empirically validated in anumber of studies [28–31]. The equation of motion foran i th vehicle in the model isd x i ( t )d t = V ( s i ( t − τ i )) , (1)where x i ( t ) is the position of vehicle i at time t and V ( · )is a speed relation that takes the spacing between vehicle i and their leader ( s i ≡ x i − − x i ) as input. The as-sumptions of Newell’s model are embedded in V , whichbasically couples the dynamics of the i th vehicle with
The delayed reaction by drivers as modeled by the cou-pled DDEs in Eq. (1) introduces oscillations in the gapsbetween vehicles and in their speeds. This is consideredas a form of instability in traffic flow, similar to insta-bilities in fluid dynamics. Instabilities in traffic flow arebroadly classified as local and string instabilities. A sys-tem of vehicles is locally unstable if the gap and speedfluctuations of each vehicle do not decay with time. Astring instability, as the name suggests, is that in which aperturbation in the gap (and the speed) travels upstreamin a manner similar to a traveling wave in a string. If theconditions in the system are such that the amplitude ofthe perturbation increases as it travels upstream, the per-turbation eventually transforms into a jam, within whichthe vehicle(s) come so close to each other that they ei-ther move very slowly or halt momentarily. The jam frontthus formed continues to move upstream forming what iscalled a stop-go wave . In the appendix, we illustrate therole played by the reaction time in inducing oscillationsin the gap (and hence in the speed) by deriving an ap-proximate analytical expression for the gap and speed forthe case of a follower equilibrating their speed to that ofa slow moving leader. Here, we explain in simple terms,the way in which a perturbation in the speed of one ve-hicle gets amplified into a stop-go wave as it spreads tothe vehicles upstream. We refer the reader to Chapter15 in Treiber and Kesting’s book [35] for a more detaileddiscussion on instabilities in traffic flow.The speed versus time plot for a platoon of seven carswith the i th car following car i − B. Relation to three-phase theory
Before we conclude our brief introduction of car-following theory, we briefly present three-phase theory and its relation to the present context. Newell’s model, aspresented above, is considered to be a two-phase model;the two phases being the free-flow phase ( F ) and the con-gested phase, which we will refer to as the jam phase ( J )to be consistent with the nomenclature used in three-phase theory. The two phases can be seen in the twoparts of Newell’s hypothecized speed relation in Eq. (2).Three-phase theory was developed by Kerner [36, 37]who analyzed data from German autobahns and observeda third phase sandwiched in between F and J , whichhe dubbed synchronized traffic ( S ). The S -phase is acongested phase with no “wide-moving jams” (stop-gowaves). The kinetics of the transition are similar to anucleation process; the S -phase forms (or nucleates) neara traffic bottleneck (e.g., off-ramps and on-ramps on ahighway) and if the conditions are favorable, the size ofthe ‘nucleus’ keeps growing in the upstream traffic direc-tion (from the bottleneck). However, as the size of theregion of the S -phase increases, the flow becomes unstableand stop-go waves emerge. The formation of the stop-gowaves is called an S → J transition. While the abovephenomenon occurs during a transient state, the systemmay reach a non-equilibrium stationary state with thesame flow pattern. Thus, one finds a F -phase ahead ofthe bottleneck and as one goes upstream, first an F → S transition is observed near the bottleneck and then an S → J transition is observed at a point further upstream.Proponents of the three-phase theory have sharply crit-icized two-phase theories on the grounds that they all failto capture the nucleation process described above (themost recent criticism appeared in Appendix A in [38]).This criticism has been extensively debated in the traf-fic flow literature. A number of works showed that theempirical observations mentioned above can be simulatedusing standard car-following models with a proper choiceof parameters. See, for example [39–41]. For example,in the simulations performed in this study depicted inFig. 3, the dark patterns in the plot that form somewherein the middle of the platoon and move upstream are thestop-go waves (phase J ). The leader of the platoon is theslowest vehicle in the system. It experiences free-flowbut being the slowest vehicle in the system it plays therole of a moving bottleneck for the faster vehicles behindit. It can be seen from the figure that the immediatefollowers of the slowest vehicle experience synchronizedflow i.e., the S -phase. We can also see from the figurethat the S -phase doesn’t spread indefinitely in space; itbecomes unstable as it moves upstream where stop-gowaves appear in the system. Therefore, starting fromthe leader of the platoon and moving upstream, one seesa F → S transition near the leader (the bottleneck in ourcase) and a S → J transition at a point further upstream.Fig. 3 may be compared to Figure 1.3 in Ref. [37]. Wenote that the presence of quenched disorders in the pa-rameters of our two-phase model is a unique feature inour model. Further, it was argued by the defenders ofthe two-phase theories that the classification of the S -phase and the J -phase separately and the introduction of F → S and S → J transitions was just qualitative with adifferent interpretation being possible. It was also arguedthat the observed pattern of transitions, F → S followedby S → J , does not always occur in real traffic.The purpose of the present work is to investigate theimpact of a quenched disorder in the reaction times.We make no claims of addressing the ongoing two-phaseand/or three-phase debate. While we elected to usea two-phase model for simplicity of exposition, we be-lieve that quenched disorders, particularly in the reac-tion times, can have profound impacts on emergent phe-nomena in traffic independent of whether a two-phase orthree-phase theory is used. IV. APPROACH TO STATIONARY STATE
We observed that the flow instability induced by thechosen values of τ doesn’t hinder the formation of a singleplatoon at low densities; see Fig. 3. As explained above,some small perturbations in the gaps of the vehicles inplatoon get amplified as they go upstream of the platoonand form stop-go waves. However, the strength of the in-stability doesn’t grow indefinitely. We observed that thestop-go waves may get totally dissolved or the number ofvehicles participating in the stop-go waves keep fluctuat-ing. This may happen because of various factors, e.g., alarge gap between the leader and the follower or an agilefollower with small reaction time and small critical gap.Thus the phase-ordering due to the quenched disorder inspeed wins over the instability due to the reaction timewhen the single platoon forms.As explained in the introduction, the NS model witha quenched disorder in hopping rates is understood tobe belonging to a general dynamical universality classwith z = ( n + 2) / ( n + 1) where n is the exponent ofthe distribution of the quenched disorder. Ben Naim etal. [42] analytically derived the same z value for the caseof one dimensional ballistic aggregation which may berelated to a one dimensional car following model witha quenched disorder in free-flow speed. In this case, n was the exponent of the distribution for v f close to v minf .Thus, for the beta distribution for v f used here, n = a v f −
1. In our previous work, we simulated Newell’smodel with quenched disorders in the v f , S j and w b withzero reaction time and obtained the z numerically andusing finite size scaling, which matched with that of BenNaim et al. and of the NS model. In the present case,where we include a reaction time for each driver, it is not
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FIG. 2. (Top) Position ( x ) versus time ( t ) plot illustratingthe formation of a stop-go wave. The vehicles move from leftto right as depicted in the cartoon. The curves correspondsto the cars with matching colors. (Bottom) Speed ( v ) versustime ( t ) plot for the vehicles in the top figure. Dotted arrowsare guide to the eye. A small trough in the speed of vehicle 1induces a larger trough in vehicle 2 and so on. As this reachesvehicle 7, the perturbation develops into a stop-go wave. Thedamped oscillations occur as the vehicle equalizes its speedwith that of its leader. clear whether the system belongs to the same universalityclass as the collective effects due to the string instabilityinduced by the reaction time oppose the formation of aplatoon and thus we determine it below.The fluctuations in the gaps between the vehicles madeit extremely difficult to identify the size (or length) of theplatoon after the string instability grows strong. Thus,we determined the size of the largest gap ( L g ) in thesystem as a function of time instead of the average pla-toon size. As the growth of L g implies the growth of theplatoon, both should follow the same power-law. Theaverages have been calculated typically over few tens ofindependent simulations each done with a different ran-dom seed for the quenched disorders. From Fig. 4, it canbe seen that a typical gap grows as a power-law. Clearly, FIG. 3. A plot of typical positions of the N vehicles versustime after the stationary state is reached when reaction timeis included. Each line corresponds to a vehicle. The vehiclesat the front of the platoon move smoothly without any stop-go waves. Some small oscillations generated in the vehicles atthe front develop into stop-go waves as they move upstream.The stop-go waves (dark patterns moving upstream in thefigure) get dissipated at the end of the platoon. -3 -2 -1 -4 -3 -2 -1 FIG. 4. Gap size as a function of time. Scaled gap lengthversus scaled time is shown in the inset. Collapse of curvesupon the scaling implies L g ∼ t / . the finite size effects are quite dominant. A finite sizescaling form is also depicted in the inset of Fig. 4 andconfirms the power-law exponent to be z = 3 / (cid:104) v − v minf (cid:105) turns out to be the same as theprevious case i.e., α s = − / (cid:104) v − v minf (cid:105) and its finite size scaling form are shownin Fig. 5. Thus we see that while the string instabilitycomplicates the platoon dynamics, it doesn’t alter thedynamical exponent for platoon formation which may bea confirmation of the kinematic wave criterion as arguedby Tripathi and Barma [43, 44]. -3 -2 -1 -4 -3 -2 FIG. 5. Average speed as a function of time for various tracklengths. Scaled average relative speed is shown in the inset.Collapse of curves with scaling implies that (cid:104) ¯ v (cid:105) ∼ t − / . V. CHARACTERIZATION OF THE GIANTPLATOON
As seen in Fig. 3, at low densities, the system whichwas initially spatially homogeneous drifts into a station-ary state where all the vehicles segregate into a singleplatoon with the slowest vehicle leading it and a largesystem size dependent gap ahead of the slowest vehicle.The difference between the present case with reactiontime and the case with no reaction time is the presenceof stop-go waves. For a finite system, the stop-go wavesare not observed at very low densities. However, whenthe system size is increased keeping the density constant,the stop-go waves emerge. Therefore we note that thestop-go waves exist at all non-zero densities in the largesystem limit (or in the thermodynamic limit).The stationary gap distribution p ( s ) helps in charac-terizing the state of the system. We calculated p ( s ) usingthe binning method. To coarse-grain the fluctuations atsmall time scales, we took averages over sufficiently longtime which is typically few tens of thousands of stepsafter the stationary state is reached. Although the sys-tem is expected to be ergodic (having a unique stationarystate) and self-averaging in the thermodynamic limit, toavoid any initial state dependence because of finite sys-tem size and to smooth the fluctuations further, we doan ensemble average. To perform calculations, we set abin size of ∆ s and compute the probability density as p ( s ) = 1 E T E (cid:88) e =1 T (cid:88) t =1 N et ( s ) N ∆ s (6)where E is the number of ensemble copies, T is the num-ber of time steps over which averaging is done and N et ( s )is the number of vehicles in ensemble e during time step t that have a gap between s and s + ∆ s . We determined -2 -1 -4 -2 FIG. 6. Stationary state probability of gap p ( s ); one part ofit is the gap distribution behind the leader of the platoon p p (the left curve) and other part is the gap distribution aheadof the leader p g (the right curve). the distributions for track lengths L = 5 , , ,
50 and100 kms. For L = 5 km, the ensemble averaging isdone over 200 copies while for L = 100 km, averaging isdone over 24 copies. The number of copies for remaininglengths are between 200 and 24. The copies are chosenas a compromise between the smoothness of the obtainedcurves and the computational time.A typical p ( s ) at a low density where the platoon for-mation happens is depicted in Fig. 6 for various tracklengths. p ( s ) has two distinct components: the probabil-ity of gap behind the slowest vehicle ( p p ( s )) and p g ( s ),which is the probability of gap ahead of the slowest vehi-cle. When there is no reaction time [9], we showed that p p ( s ), in the thermodynamic limit, is identical to p c ,l ( s )which is the critical gap distribution of the leader. In thepresent case, p p ( s ) significantly differs from p c ,l ( s ). How-ever, a peak in the distribution still appears at the gapwhere the p c ,l ( s ) has a peak. In addition, another peakcan be seen close to the jam-gap S j . This peak appearsas a result of stop-go waves. The broadening of the p p ( s )on the right side is because of the gaps of various sizesthat get created due to the stop-go waves. It can be seenfrom the Fig. 6 that the upper bound of the distributionincreased with the increase in track length but tends to-wards a converged value. Thus it becomes a fat-taileddistribution in the thermodynamic limit. The p p part isfound to be independent of density until the phase tran-sition point is reached. The p g ( s ) can also be seen in thefigure. The distribution is much broader than the casewith no reaction time. Thus the p ( s ) is dominated bythe flow instability and the stop-go waves. As density isincreased, p g ( s ) shifts closer to p p ( s ) and merges with iton approaching the phase transition point. VI. THE DYNAMICAL PHASE-TRANSITION
As one goes from low density to high density, a den-sity point is reached after which the giant platoon doesn’tform in the stationary state. In simple terms, one mayanticipate the transition to happen when the head of theplatoon starts interacting with the tail of the platoon.For the case with no reaction time, we showed that thetransition is always of first order following the conjec-ture by Krug et al. [22] that there is no divergence in thevariance of the stationary gap distribution at the tran-sition point for any choice of parameters describing thequenched disorders. We also showed that, in the ther-modynamic limit, the density ( ρ c ) at which the transi-tion happens is actually the reciprocal of the expectationvalue of the gap distribution p p ( s ) behind the slowestvehicle. In the present case, when a reaction time foreach driver is considered, we observed that this is stillapproximately valid i.e.,1 ρ c ≈ (cid:104) s (cid:105) p ≈ (cid:90) d sp p ( s ) s, (7)which becomes exact in the thermodynamic limit. Wenoted from numerical calculations that the ρ c in thepresent case is less than that with zero reaction time.Because of the finite size effects in p p , the calculated the ρ c also has a length dependence. However, we foundthat the tail of the distribution has much lesser weightand therefore the expectation values calculated for tracklengths of 50 km and 100 km were pretty close. To deter-mine the transition density in the thermodynamic limit( ρ c ∞ ), we fitted the the numerically observed ρ c for var-ious lengths to the below form: ρ c ( L ) = ρ c ∞ + BL ν (8)and obtained ρ c ∞ ≈ . B ≈ .
22 and ν ≈ .
03 forthe present case.To illustrate the phenomenon, we plot in Fig. 7 the po-sitions of the N vehicles versus time just below and abovethe predicted ρ c . For ease of visualization, the L = 10 kmcase is depicted in the plot. The following points may benoted by observing the plots. Below ρ c , the collectiveeffect of platoon formation is dominant and a single gi-ant platoon is formed. Some stop-go waves generated inthe middle of the platoon are stable and travel upstreamto the end of it where they get dissipated. Above ρ c ,the collective effect due to flow instability becomes dom-inant and the formation of the giant platoon is hinderedby strong stop-go waves, which move uninterrupted up-stream all around the ring. As a result, there is a con-tinuous process of formation and destruction of platoons.Therefore, as the density is increased from a low value to ρ c , the transition occurs when the stop-go waves start todominate over the platoon formation due to the quencheddisorder in v f . Thus, the phenomenon happening at thephase transition is more complicated than mere interac-tion of the head of the platoon with its tail. FIG. 7. Positions of the N vehicles versus time on a 10 kmtrack. For this case, ρ c lies between 49 and 50 veh/km. Topplot is for ρ = 49 veh/km, which is below ρ c . Bottom plot isfor ρ = 50 veh/km, which is above ρ c . To characterize the transition, we determined somephysical quantities in the stationary state over a rangeof densities above and below ρ c for various track lengths,which we analyze below. The flow-density diagram isplotted for various track lengths in Fig. 8. The finite sizeeffects appearing in the diagram (see inset) are similar tothose observed by Balouchi and Browne [45] and becomenegligible for track length of 100 km and the diagramconverges to a triangular shape. It can be observed thatthe free-flow to congestion transition also happens at thedensity predicted by Eq. (7).The average maximum gap (cid:104) s max (cid:105) observed in the sys-tem after the stationary state is reached is plotted againstaverage density ¯ ρ = N L − in Fig. 9 (scaled by ρ c ∞ ). Forsmall values of ¯ ρρ − ∞ , (cid:104) s max (cid:105) is proportional to L . As ¯ ρ approaches ρ c from below, (cid:104) s max (cid:105) decreases and reachesa minimum value at ρ c . As ¯ ρ approaches ρ c from above, (cid:104) s max (cid:105) increases and resembles a power-law. A numericalfit of (cid:104) s max (cid:105) to (¯ ρρ − − − γ using data for 100 km roadgave γ ≈ .
8. Assuming the below finite-size scaling form (cid:104) s max (cid:105) (cid:39) L − γν f ( XL − ν ) (9)where X = (¯ ρρ − −
1) and f ( X ) ∼ X − γ , the curvesfor various lengths collapsed when ν = 1 and γ = 3 / ν agrees with that determined from Eq. (8)and the value of γ agrees well with the above numerically FIG. 8. Flow versus average density when reaction time isincluded -4 -3.5 -3 -2.5 -200.20.40.60.811.21.4 FIG. 9. Maximum gap versus density when the reaction timeis included. Inset: Plot depicting the scaling (cid:104) s max (cid:105) ∼ (cid:0) ¯ ρρ − − (cid:1) − γ with γ = 3 / determined value. The gap variance∆ = (cid:10) ( s − (cid:104) s (cid:105) ) (cid:11) = (cid:90) d sp ( s ) s − (cid:16) (cid:90) d sp ( s ) s (cid:17) (10)is plotted as a function of ¯ ρ in Fig. 10 for various L .Similar to (cid:104) s max (cid:105) , ∆ is proportional to L for a given ¯ ρ and tends to a minimum size-independent value as ρ c isapproached from below. As ρ c is approached from above,∆ increases as shown in Fig. 10. A numerical fit of ∆ to (¯ ρρ − − − η using data for 100 km road gave η ≈ . (cid:39) L − ην f ( XL − ν ) (11)where f ( X ) ∼ X − η and found that the curves for differ-ent lengths collapse in the power-law regime as shown in -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08-3.5-3-2.5-2-1.5 -5 -4 -3 -2-2.5-2-1.5-1 FIG. 10. Variance of the gap distribution as function of den-sity. Inset: Plot depicting the scaling ∆ ∼ (¯ ρρ − − − η with η = 1 the inset of Fig. 10 when ν = 1 and η = 1 .Thus, in both (cid:104) s max (cid:105) and ∆ , we notice power-law be-havior as the ρ c is approached from above. There is lackof data for much larger track lengths to clearly visualizethe power-law over an extended domain in the log-plotsas simulations become highly computationally costly be-cause of the power-law relaxation time required to reachthe stationary state. However, we believe that the finite-size scaling clearly revealed the exponents γ and η . As aconsequence of the different behaviors of (cid:104) s max (cid:105) and ∆ as ρ c is approached from above and below, a kink canbe observed in these quantities in the neighborhood of ρ c whose sharpness increases with an increase in tracklength L . We attribute the kinks observed in Figs. 9and 10, when L gets large, to the emergence of vehicleclusters in between the stop-go waves . When ρ is onlyslightly larger than ρ c , the stop-go waves start to be-come pronounced but occur infrequently. For large L ,the distances separating the stop-go waves also get large,allowing for large (cid:104) s max (cid:105) to emerge in between the stop-go waves. In this regime ( ρ → ρ +c ) large L also allows forhigher variability in the cluster sizes to emerge near ρ c ,hence the increase in ∆ observed above. These kinks be-come less pronounced (and start to vanish) when L getssmall. In the thermodynamic limit ( L → ∞ ) we expectthe kinks to become infinite discontinuities.In addition, we observed that the gap distribution p ( s )decays as a power-law just above ρ c . A finite size scalingplot of p ( s ) for various L s is shown in Fig. 11, from whichwe deduced that p ( s ) ∼ s − α with α = 3 asymptoticallybefore the finite size effects takeover. The power-law dis-tribution of gaps just above ρ c indicates presence of mul-tiple length-scales in the system, which correspond to thegaps of various sizes that form between the platoons ofvarious sizes with the largest gap being proportional tothe size of the system. This is related to the power-law -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1024681012 FIG. 11. Scaled gap distributions near ρ c for various { L, N } pairs. divergence seen in ∆ and (cid:104) s max (cid:105) .The existence of both the power-law divergence and thediscontinuity in (cid:104) s max (cid:105) and ∆ in the neighborhood of thephase-transition is not reported in the traffic flow theoryliterature to our knowledge. The power-law divergencein ∆ and the power-law tail of p ( s ) have been observedin TASEP and NS models. Krug and Ferrari [22] found,using heuristic arguments, that η = (1 − n ) /n for n ∈ (0 ,
1) and it logarithmically diverges when n = 1 where n = a v f −
1. They also found that α = n +2. These resultshave also been observed in simulations of the NS model.For the present study, a v f = 2 and therefore, accordingto Krug et al., ∆ should have diverged logarithmicallybut we have shown above that η = 1 which is a differentvalue. However, the value of the exponent α we got agreeswith those predicted for TASEP and NS models. Theabove results and analysis show the complex nature of thetransition. The power-law divergence of ∆ as ρ → ρ +c resembles a second order transition. However, the infinitediscontinuity in the ∆ is unexpected of a second ordertransition. On the contrary, the transition may not becalled first order owing to the power-law divergence of∆ . Thus we see the transition to be of unusual sort andits properties do not match with that observed in theNS model except for the power-law gap distribution justabove ρ c . VII. SUMMARY AND DISCUSSION
In this work we have studied the effect of reaction timeon the emergent phenomena in a heterogeneous traffic us-ing numerical simulations of a version of Newell’s model.The heterogeneity is incorporated using quenched disor-ders in each of the parameters of the model and in thereaction time. The dynamical exponents describing theplatoon forming phenomenon at low density are noted0to be the same as those derived by Ben Naim et al. forballistic aggregation and those observed in the Nagel-Schkrekenberg (NS) model.In the single giant platoon that forms in the stationarystate at low densities, we observed that spontaneous stop-go waves form somewhere inside the platoon and moveupstream until the tail of the jam is reached where theydissipate. The phase transition happens when the headof the giant platoon interacts with its tail and the stop-gowaves circulate continuously all through the ring withoutdissipation. The transition density closely matches withthe reciprocal of the expectation value of the gap distri-bution in the giant platoon in the thermodynamic limitand it is numerically observed to be lower than the tran-sition density when there is no reaction time. To under-stand and characterize the transition, we determined thegap distribution ( p ( s )), the maximum gap in the system( (cid:104) s max (cid:105) ) and the variance (∆ ) of the gap distributionfor various densities across the transition density ρ c .The following picture emerges from the observationswe made: First, it may be noted that the phase-transitionin the present system happens because of a competitionbetween the phase-ordering due to the quenched disorderin the free-flow speed and the formation of stable stop-gowaves which destroy the phase-ordering. Below ρ c , thephase ordering effect wins and the system segregates intoa single giant platoon and a large gap ahead of it, (cid:104) s max (cid:105) simply represents the gap ahead of the platoon leader,which diverges in the thermodynamic limit. As densityis increased, keeping the system size constant, the gapahead of the leader reduces and in the limit ρ → ρ − c the gap ahead of the leader becomes less than their criti-cal gap, thus (cid:104) s max (cid:105) becomes finite and size-independent.This may also be seen from the fact that p g ( s ) mergeswith p p ( s ) as ρ → ρ − c , thus making p ( s ) normalizablewith a finite variance ∆ . On the other hand, as thecritical density is crossed ( ρ → ρ +c ), the stop-go wavesbecome dominant and obstruct the formation of a giantplatoon. However, between two stop-go waves, the ve-hicles form clusters of various possible sizes because ofthe phase-ordering effect. As the system size becomesvery large, it is possible to have a large enough distancebetween two stop-go waves that allows the formation oflarge platoons and, thus, gaps proportional to the size ofthe system. Evidence of this happening in the system isthe power-law distribution of the gaps p ( s ) ∼ s − whichrenders the variance ∆ and (cid:104) s max (cid:105) to diverge. Thus, wehave a complex situation in which the quantities ∆ and (cid:104) s max (cid:105) become finite as ρ → ρ − c and diverge as ρ → ρ +c thereby creating a discontinuity at ρ c . Thus the phasetransition observed here is of unusual sort with propertiesof both first and second order transitions. We note thatsuch transitions with properties of both first and secondorder transition are observed in other systems like gran-ular media and various other systems like foams and col-loids. However, a unified picture of all these transitionsis still an open question and with our present study, per- haps heterogeneous traffic joins this class of systems. Overall, we find that the present work reveals somenovel aspects of phase transitions in heterogeneous traf-fic flow in the context of car-following models. Insightsfrom our study may be useful in developing continuumtheories for heterogeneous traffic flow, which have appli-cations in transportation engineering and traffic manage-ment. Further, the unusual nature of the phase transitionmay have implications on fuel economy and pollution asthere would be frequent breaking and acceleration ma-neuvers. Modeling of these aspects and applications thataim to avoid stop-go maneuvers are gaining traction inthe engineering literature; see, e.g. [46] and referencestherein. The power-laws concerning kinetics would givean idea about timescale of build-up of traffic on a high-way. This is of particular importance in the context ofnetwork control techniques that aim to “stabilize” traf-fic networks, which are gaining a lot of popularity in theengineering literature (see, e.g., the work of the secondauthor on the subject [47, 48]), and are even being testedin the real-world for feasibility (see, e.g., [49]).
ACKNOWLEDGMENTS
This work was supported by the NYUAD Center for In-teracting Urban Networks (CITIES), funded by Tamkeenunder the NYUAD Research Institute Award CG001 andby the Swiss Re Institute under the Quantum Cities TM initiative. The views expressed in this paper are those ofthe authors and do not reflect the opinions of CITIES orthe funding agencies. Appendix A: Equilibration of a follower’s speed tothat of a slower leader
Below we discuss, in mathematical terms, the effect ofthe delay induced by the reaction time ( τ ) in the Newell’scar-following model when a follower vehicle moving at ahigh speed equilibrates its speed with that of a slow mov-ing leader. Denote v f ,φ , S j ,φ and w b ,φ and v f ,λ , S j ,λ and w b ,λ as the free-flow speed, jam gap and the backwardwave speed of the follower ( φ ) and the leader ( λ ), respec-tively. Suppose v f ,φ > v f ,λ and assume that both vehicleswere far apart and moving at their respective free flow(or desired) speeds, and that the follower reaches theircritical gap S c ,φ at time t . The follower then begins toadapt to the speed of the leader. For simplicity, let’s as-sume that the follower remains in the congestion regimei.e., at a gap S j ,φ < s < S c ,φ for all t > t . The leadercontinues to coast at their initial speed even after t asthere is no vehicle ahead of it. Thus, x λ ( t ) = x λ ( t ) + ( t − t ) v f ,λ . (A1) We thank an anonymous reviewer for bringing this to our atten-tion. x φ ( t ) = x φ ( t ) + (cid:90) tt d t (cid:48) V (cid:0) s φ ( t (cid:48) − τ ) (cid:1) . (A2)Equation (A2) can be integrated analytically in a piece-wise manner in intervals of reaction time. In the interval[ t , t + τ ), since s φ ( t − τ ) > S c ,φ , the follower doesn’tchange their speed because of the delay due to reactiontime. Therefore, x φ ( t ) = x φ ( t ) + ( t − t ) v f ,φ (A3)for t ∈ [ t , t + τ ). Thus, s φ ( t ) = x λ ( t ) − x φ ( t ) = S c ,φ − ( t − t ) δv f (A4)for t ∈ [ t , t + τ ), where δv f = v f ,φ − v f ,λ . In the next timeinterval [ t + τ, t + 2 τ ), the follower starts responding tothe reduction in gap in the previous time interval andreduces their speed in accord with Eq. (2). CombiningEq. (A3) and Eq. (A4) with Eq. (A2) and integrating, weget for t ∈ [ t + τ, t + 2 τ ) s φ ( t ) = S c ,φ − ( t − t ) δv f + δv f w b ,φ S j ,φ (cid:0) t − ( t + τ ) (cid:1) . (A5)Similarly, for t ∈ [ t + 2 τ, t + 3 τ ): s φ ( t ) = S c ,φ − ( t − t ) δv f + δv f w b ,φ S j ,φ (cid:0) t − ( t o + τ ) (cid:1) − δv f (cid:16) w b ,φ S j ,φ (cid:17) (cid:0) t − ( t + 2 τ ) (cid:1) . (A6)In general, we obtain s φ ( t ) = S c ,φ + ∞ (cid:88) n =0 (cid:18) ( − n +1 δv f ( n + 1)! (cid:16) w b ,φ S j ,φ (cid:17) n × (cid:0) t − ( t + nτ ) (cid:1) n +1 I [ t + nτ, ∞ ) ( t ) (cid:19) . (A7)Eq. (A7) is essentially a polynomial with new termsadded as time evolves. It can be easily checked usingthe ratio test that the series converges. Below we ana-lyze the equation in the limit of small τ and obtain someinsights regarding the effect of reaction time.First, we investigate the limiting (in time) behavior of s φ when τ = 0. When τ = 0, we have that s φ ( t ) = S c ,φ + ( t − t ) v f ,λ − (cid:90) tt d t (cid:48) V (cid:0) s φ ( t (cid:48) ) (cid:1) . (A8)Since s φ ( t ) < S c ,φ for t > t , it can be shown thatEq. (A8) has the following solution: s φ ( t ) = S c ,φ e − w b ,φS j ,φ ( t − t ) + S j ,φ w b ,φ ( v f ,λ + w b ,φ ) (cid:16) − e − w b ,φS j ,φ ( t − t ) (cid:17) , (A9) which tends to the equilibrium gap exactly as dictatedby Eq. (2) in the long time limit. This is also the casein Eq. (A7), when τ →
0. To demonstrate this, we firstassume without loss of generality that t = 0 and writeEq. (A7) as s φ ( t ) = S c ,φ + B ∞ (cid:88) n =1 ( − n n ! A n (cid:0) t − ( n − τ (cid:1) n , (A10)where A ≡ w b ,φ S − ,φ and B ≡ δv f A − . As τ → s φ ( t ) → S c ,φ + (cid:0) e − At − (cid:1) B, (A11)which converges to the equilibrium gap given by Eq. (2)in the long time limit. Eq. (A9) indicates the presence ofa characteristic time scale S j ,φ w − ,φ for relaxation of thespeed of the follower to that of the leader. This shouldimply that if τ (cid:28) S j ,φ w − ,φ , the reaction time will havelittle to no effect on the speed equilibration process. Tosee this, we approximate the series Eq. (A10) as follows s φ ( t ) = S c ,φ + B ∞ (cid:88) n =1 ( − A ) n n ! n (cid:88) r =0 (cid:18) nr (cid:19) t n − r ( − ( n − τ ) r ≈ S c ,φ + B ∞ (cid:88) n =1 ( − A ) n n ! (cid:16) t n − n ( n − t n − τ + n ( n − t n − ( n − τ (cid:17) , (A12)where we have truncated the inner binomial expansionafter the second term. Next, using the series expansionof exponential and after some algebra, we get s φ ( t ) ≈ S c ,φ − B + B (cid:16) − tA τ + 12 A τ (cid:0) A t − tA (cid:1)(cid:17) e − At (A13)The above expression for s φ clearly illustrates the effect -3 FIG. 12. Gap ahead of the follower ( φ ) as a function of timefor a typical case calculated using Eq. (A13). For small τ , therelaxation to stationary gap is monotonic. As τ approaches S j ,φ w − ,φ , s φ ( t ) becomes non-monotone. τ (cid:28) S j w − ,φ , the quadratic polyno-mial multiplying the exponential term in Eq. (A13) hasno real roots and converges monotonically to the station-ary value S c ,φ − B . In fact Eq. (A13) tends to Eq. (A11) inthe limit τ →
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