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Phase transitions in two-dimensional traffic flow models

arXiv:hep-lat/9305015v1 20 May 1993MA/UC3M/4/93Phase transitions in two-dimensional traffic flow modelsJos´e A. Cuesta, Froil´an C. Mart´ınez, Juan M. Molera, and Angel S´anchezEscuela Polit´ecnica Superior, Universidad Carlos III de Madrid,Avda. Mediterr´aneo 20, E-28913 Legan´es, Madrid, Spain(May 6, 2019)AbstractWe introduce two simple two-dimensional lattice models to study traffic flowin cities.

We have found that a few basic elements give rise to the characteristicphase diagram of a first-order phase transition from a freely moving phaseto a jammed state, with a critical point. The jammed phase presents newtransitions corresponding to structural transformations of the jam.

We discusstheir relevance in the infinite size limit.Ms. numberPACS numbers: 05.40.+j, 05.70.Fh, 64.60.Cn, 89.40.+kTypeset using REVTEX1

Car displacements through large cities, data exchange among processors in a massivelyparallel computer, or communications in computer networks, are examples of situationswhere avoiding undesirable traffic jams is extremely important [1]. It is therefore necessaryto achieve a comprehensive understanding of the mechanisms leading to the nucleation,growth, and evolution of these traffic jams, as they can be responsible for decrease or evenfull suppression of flow between different parts of the system.

To this end, models basedon cellular automata (CA) seem to be a suitable tool to approach the problem becauseof both their nature and their computational efficiency. Some one-dimensional [1–4] andtwo-dimensional [5] models have been proposed in the past to study different traffic flowproblems.

In the one-dimensional case, there is good agreement between CA [4] and fluid-dynamical [6] results, as well as to data obtained from actual traffic in highways [1,4]. Thus,the main aspects of car flow in freeways or individual streets seem to be at least qualitativelyunderstood.

As regards 2D CA models, the only ones we are aware of are based upon thefundamental assumption that cars never turn [5]. The traffic-jam transition reported in[5] can be a direct consequence of this unrealistic hypothesis.

In this Letter we show thatallowing cars to turn (as in actual situations) brings along the appearance of a complexphase diagram.Our model consists of the following basic ingredients. We have cars moving inside atown.

The town is made of one-way perpendicular (L horizontal and L vertical) streetsarranged in a square lattice with periodic boundary conditions. The way of every streetis fixed independently according to a certain rule that depends on the particular choice ofthe model (see below).

Cars sit at the crossings, and they can move to one of its nearestneighbors (allowed by the direction of the streets) every time step. Two cars cannot be atthe same crossing simultaneously.

Each car is assigned a trend or preferred direction, ruledby a variable wi(r), which we define as the probability that car i, located at node r, jumpsto the allowed neighbor in the horizontal street (accordingly, 1 −wi(r) is the probability tomove vertically). Finally, there are traffic lights, that permit horizontal motion at even timesteps and vertical motion at odd time steps.We now define the dynamics of the model.

Every time step wi(r) is evaluated and thedirection where to move next is chosen accordingly. Then, it is checked that the chosen siteis empty and that the motion is allowed by the light; otherwise the car will not be moved.Finally, all cars that can be moved are placed at their destination site and the next timestep starts.

We want to stress that the whole process is carried out simultaneously for allcars. The fact that traffic lights allow motion alternatively in vertical and horizontal streetsprevents two cars from colliding at any crossing.In this Letter, we concern ourselves with two of the simplest versions of the model, thatwe name model A and model B.

Both models are characterized by a unique parameter γwhich we call randomness. This parameter allows us to control the trend of the motion ofevery car by defining wi(r) through it.

Model A has streets pointing only up and left. Halfof the cars are given a trend wi(r) = γ, and the other half are set to wi(r) = 1 −γ.

Thisamounts to having half of the cars moving preferently upwards and the other half leftwards.For simmetry reasons, it is enough to study the range 0 ≤γ ≤1/2. It has to be noticedthat, in case we fix γ = 0, the cars are deterministic and always move along their preferreddirection, and Model A becomes Model I of Ref.

[5]. Model B is defined in the following way.It has streets that point alternatively up and down, and right and left.

We work with four2

equal-number groups of cars: Each of them is assigned one of the four possible directionsas its trend. This is accomplished by the following definition: Upward or downward boundcars have wi(r) = γ if a street with the same direction as the trend of the i-th car passesthrough site r, and wi(r) = 1 −γ otherwise; left or right cars behave the other way around.On the above described models, we have carried out an extensive simulation program,simulating the corresponding CA’s on towns of 32×32, 64×64, and 128×128 sites.

A typicalrun consists of the evolution along 106 time steps of a randomly chosen initial condition fora given density (number of cars/number of lattice sites). For every time step we monitor themean velocity, defined as the number of moved cars divided by the total number of cars.

Bymeans of this magnitude, we distinguish when the system reaches a steady state. Once in thisstate, we perform time averages on magnitudes of interest until the end of the simulation.

Wehave also studied the outcome of different randomly chosen initial configurations. Althoughin general, these outcomes are similar, a very particular dependence is found in some partsof the phase diagram (see below).

We have also checked different random number generators[7] and all of them lead to the same results. We have studied these models for a numberof car densities ranging from n = 0 to n = 1, and for randomness values on the range0 ≤γ ≤1/2.

In addition, we have recorded any possible structure of the traffic jam in thesteady state by measuring the average occupation time per site, defined as the number oftime steps during which a site is occupied by a stopped car divided by the total averagingtime. All simulations were performed in workstations HP 720, and DEC 3100 and 5100; atypical simulation for a given car density on a 64×64 town takes about 2 hours of CPU time,and 12 hours for a 128×128 town (this is for model A, for model B times are approximatelya 25% higher).Results for model A are summarized in Fig.

1. Such a figure can be understood as thephase diagram of a first order phase transition [9] from a freely moving to a jammed phase.The curves v(n) undergo a discontinuous transition of magnitude ∆v(γ) at density nt(γ).As γ increases, nt(γ) shifts to higher densities and ∆v(γ) decreases, eventually vanishingfor some randomness γc.

The point of density nc = nt(γc) and average velocity vc = v(nc),belonging to the curve for γc, will correspond to a critical point. This conclusion is furthersupported by the large increase of the fluctuations of v observed in the vicinity of that point.As can be inferred from Fig.

1, the location of the critical point lies somewhere in the range0.45 ≤γc ≤0.5, but it cannot be more accurately determined from our simulations first,because γ is an input parameter, and second, due to the strong size-dependence of γc.The part of the curves v(n) corresponding to the free phase (which for γ = 1/2 means thewhole curve) fits rather well the linear law v(n) = (1 −n)/2. It can be proven analyticallyfor the infinite system [8] that this is precisely the asymptotic behavior of v(n) when n →0,for any value of γ.

It is remarkable that the agreement with the simulations is rather goodeven far from this limit.The jammed phase, and in fact the nature of the transition, can be better understoodby analyzing the average distribution of cars on the lattice. Fig.

1 shows in this phase, forthe lowest values of γ, a few small jumps in which the value of v increases. The explanationof these jumps is the following: Before the jamming transition occurs, cars are distributedhomogeneously, whereas after the transition cars always order along broad diagonal stripsextending throughout the whole system (see Fig.

2), with the two types of cars roughlyseparated in two halves.These strips do not trap empty sites (holes) inside; thus, the3

observed remnant average velocity is due only to the movement of the cars on the borders ofthe strips. Different number of strips characterize different ordered phases.

A given initialconfiguration goes to one of this phases with a certain probability. For a given density, wecompute this probability by taking a large number of initial configurations and countinghow many of them go to each phase.

The stable phase will be that of maximum probability,the rest of them being metastable. Accordingly, in Fig.

1 we plot the velocity of the stablephase. The small jumps correspond to a exchange of stability between two phases.

In thesejumps v increases since every new strip provides two more borderlines along which cars canmove. As this remnant movement is just a “surface” effect, it should vanish when L →∞;however, at the same time the number of strips increases, hence supplying extra movingcars.

The resulting value in the infinite system will depend on this competition of effects;we will return to this point later on.For γ = 0 the results are similar to those reported in [5] (the only difference being thatour velocities are, by definition, half theirs). Since cars do not change direction, any initialconfiguration ends up either in a periodic or in a stuck (v = 0) state.

The predominanceof each kind of state decides in which phase is the system. According to the authors of [5],their results do not allow them to exclude the possibility that nt(0) →0 as L →∞.

Incontrast, though in our simulations the values of nt(γ) also decrease as L increases, we canclearly see that for the lowest values of γ (say, γ = 0.1, 0.2 and 0.3) it already converges toa nonzero value, even for the relatively small sizes we are dealing with. Besides, as we havecommented on above, we have proven [8] that the slope of v(n), for infinite L, is exactly−1/2, independently of γ, in the limit n →0.

This result also holds for γ = 0; however,the simulations of [5] indicate that the value of the slope in the free phase is 0 for L up to512. This fact supports the idea that when L →∞, nt(0) →0, though the possibility of achange of slope from 0 to −1/2 at much larger system sizes is not excluded, but seems veryunlikely.From the size-dependence of the parameters of our simulation we can draw an image ofwhat happens in model A when L →∞.

On the one hand, as we have already pointed out,nt(γ) converges to a nonzero value in the infinite system. On the other hand, the γ = 0.5curve does not change with L, and the critical point moves towards this curve as L increases(it cannot be inferred from our results whether γc finally reaches the value 1/2).

Accordingly,even though the transition densities, nt(γ), for the rest of the values of γ still decrease, theyshould reach a nonzero value when L →∞.This part of the phase diagram will thusnot qualitatively change at infinite size. Regarding the structure of the jammed phase, asstrips never trap holes, the only possible way that such structures survive in an infinite citywith n ̸= 0 and n ̸= 1 is that an infinite number of strips appear.

Consequently, those“transitions” between jammed phases with different number of strips are a finite systemsize effect; a result that is further supported by the fact that such transitions move quicklytowards nt(γ) as L increases.The infinite system will then be formed by infinite stripswith a typical size and a typical separation (which will in general depend on n), and thevalue of v will simply be the ratio of the average number of moving cars per strip to theaverage number of cars per strip. Nevertheless, as the average separation between stripsseems comparable to the sizes used in our simulations, the values of v obtained are stillaffected by strong finite-size effects.The phase diagram of model B is similar to that of model A.

There is also a phase4

transition from the “free” to the jammed regime, with diagonal strip structure right afterthe transition. This is indicated as before, by a sharp decrease of the velocity for a certainvalue of the density nt(γ) (smaller than in model A).

The dependence of the parameterscharacterizing the transition, nt(γ) and ∆v(γ), on γ and the size of the city L is qualitativelythe same as in model A. The main differences of this model are in the structure of thejammed states appearing after the transition.

Having this model four different types of carsand streets, it has a symmetry (absent in model A) under 90◦rotations, and the jammedstrip can appear with equal probability along each diagonal direction. Besides that, thestrip has inside some holes that allow the diffusion through the jam of the different cartypes.

They also contribute to the remnant velocity in the jam state and could have somesignificant effect in the infinite size case. The other important departure from the behaviorof model A is the type of stable phases present in the jammed region.

First of all, whilein model A the jammed phase can show multiple strips, in model B we only see one strip.This strip is composed of two longitudinal halves, each containing a mixture of two types ofcars. For example, if the strip runs from the lower-left to the upper-right, the upper half ofit is mainly composed of cars of the types trying to go right and down, and the bottom halfby the ones trying to go left and up.

Secondly, as density is increased, a point is reachedwhere the form of the stable jammed phase suddenly changes. The majority of the holes,that before this point were forming a paralell strip to the cars, now arrange themselves in aclosed square-like regions (see Fig.

3). Meanwhile, the cars in the jam have separated in fourregions according to their type.

The cars trying to go up are above each empty region, tothe left the ones trying to go left, and so on. This change of structure produces a noticeable,though small, change in the slope of the velocity curves v(n) in the phase diagram.

Oursimulations do not allow us to conclude whether this transition is continuous or weakly firstorder. More work on this point is in progress [8].From our data we can only conjecture what the structure of the jammed phase in modelB will be, as the size of system goes to infinity.

We have already seen, in some preliminaryruns, that the number of those empty regions increases, as we simulate, at constant density,in larger cities. Keeping the parallelism with model A, we can think that in the limit ofinfinity size an infinite number of strips will appear (though the existence of holes insidethem weakens this conclusion).

However, due to the 90◦-rotation symmetry present in modelB, the strips may appear in both diagonal directions simultaneously. If this happened thestable state would be one in which there would be an infinite number of square-like emptyregions, as described above, arranged in a kind of lattice structure.

We hope that we willbe able to settle this question in the future.In summary, we have studied models incorporating what we think are the essentialingredients (excluded volume and turn capability) of urban car movement in cities withrealistic structures (defined by the arrangement of the streets and the organization of trafficlights) and we have found a first order phase transition from a freely moving regime to ajammed state. It is important to check whether this striking feature still stands when moreelements are added (say, disorder via forbidden streets or non-synchronized traffic-lights,preferred streets, rush hours, etc).

If this happens, it will be possible to conclude that, as aconsequence of car interaction, any city will always have a saturation density of cars afterwhich the average velocity falls sharply. This could be an important issue to consider inthe design of city and traffic policies.

If a city, with a given density of cars, were saturated,5

local improvements would have little effect on the average velocity, and only global changesin the city capacity or the number of cars will be able to solve the problem. More analyticaland numerical work is currently being developed [8] along this line.Two of us (J.A.C.

and A.S.) acknowledge financial support of two projects (PB91-0378 and MAT90-0544, respectively) of the Direcci´on General de Investigaci´on Cient´ıficay T´ecnica (Spain).6

REFERENCES[1] D. L. Gerlough and M. J. Huber, Traffic Flow Theory (NRC, Washington D. C., 1975);Y. Sheffi, Urban Transportation Networks: Equilibrium Analysis with Mathematical Pro-gramming Methods (Prentice-Hall, New Jersey, 1985); N. H. Gartner and N. H. M.Wilson, editors, Transportation and Traffic Theory (Elsevier, New York, 1987); W.Leutzbach, Introduction to the Theory of Traffic Flow (Springer, Berlin, 1988). [2] M. J. Lighthill and G. B. Whitham, Proc.

Roy. Soc.

Lond. A229, 317 (1955).

[3] R. K¨uhne, in Highway Capacity and Level of Service, U. Brannolte, editor (Balkema,Rotterdam, 1991). [4] K. Nagel and M. Schreckenberg, J. Physique I (Paris) 2, 2221 (1992).

[5] O. Biham, A. A. Middleton, and D. Levine, Phys.

Rev. A 46, R6124 (1992).

[6] B. S. Kerner and P. Konhauser, in Europhysics Conference Abstracts 17A (13th GeneralConference of the Condensed Matter Division of the European Physical Society), W.Heinicke, editor (EPS, Geneve, 1993). [7] W. H. Press, S. A. Teukolsky, B. P. Flannery, and W. T. Vetterling, Numerical Recipesin C, 2nd Edition (Cambridge University, New York, 1992), chapter 3.

[8] J. A. Cuesta, F. Mart´ınez, J. M. Molera, and A. S´anchez, unpublished.

[9] H. E. Stanley, Introduction to Phase Transitions and Critical Phenomena (Oxford, 1971).7

FIGURESFIG. 1.

Average velocity, v, for cars in model A as a function of car density, n, for differentrandomness values, γ, in a city of 64× 64 streets. The full lines are a guide to the eye.

The dashedline is an approximate fit to the points where the transition occurs.FIG. 2.

Average site density of stopped cars in model A (128 × 128 streets) in the jammedphase, for a car density n = 0.7 and a randomness γ = 0.1. Different values are represented bydifferent grey levels ranging from black (site always empty) to white (site always occupied).

Thispicture illustrate the multistrip structure of the jammed phase in model A.FIG. 3.

Same as Fig. 2, for model B and car density n = 0.9 and randomness γ = 0.2.

Thisparameters correspond to the second ordered phase (see text).The existence of closed emptyregions surrounded by cars is illustrated.8

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