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1 THE DYNAMICAL STRUCTURE FACTOR AND CRITICAL BEHAVIOR OF A TRAFFIC FLOW MODEL 61 L. ROTERS, S. L UBECK, and K. D. USADEL Theoretische Physik, Gerhard-Mercator-Universitat, 4748 Duisburg, Deutschland, flars,sven,usadelg@thp.uni-duisburg.de The Nagel-Schreckenberg trac ow model shows a transition from a free ow regime to a jammed regime for increasing car density. The measurement of the dynamical structure factor oers the chance to observe the evolution of jams without the necessity to dene a car to be jammed or not. Above the jamming transition the dynamical structure factor S(k;!) exhibits, for a given k-value, two maxima corresponding to the separation of the system into the free ow phase and jammed phase. From a nite-size analysis of the smallest jam mode we nd out that correlations of the jams arise, as we approach the transition point. 1 Introduction In 199 Nagel and Schreckenberg 1 introduced a simple cellular automata model, which simulates single-lane one-way trac, in order to study the transition from free ow trac to jammed trac with increasing car density. The behavior of the model is determined by three parameters, the maximal velocity v max, the noise parameter P and the global density of cars = N=L, where N denotes the total number of cars and L the system size, respectively. The variables describing a car n at time t are its position r n f1; ; :::; Lg, its velocity v n f; 1; :::; v max g and the gap g n, which is the number of unoccupied cells in front of the car. For each particle, the following update steps representing the acceleration, the slowing down, the noise, and the motion of the particles are done in parallel: (1) if v n < g n we increase the velocity with respect to the maximal velocity v n! Minfv n + 1; v max g, () to avoid crashes we descrease the velocity v n! g n if v n > g n, (3) for v n > we decrease the velocity v n! v n? 1 with probability P in order to allow uctuations, and (4) the motion of the cars is given by r n! r n + v n. Increasing the global density for xed P and v max jams occur above a certain critical value 1 c. Contrary to the forward movement of all particles the jammed region is characterized by backward moving shock waves, i.e. backward moving density uctuations. This property of jams was found in real trac ow already in the 6's ;3. Investigations of the Nagel-Schreckenberg trac ow model show that crossing the critical point a transition takes place from a homogeneous regime (free ow phase) to an inhomogeneous regime, which is characterized by a coexistence of two phases (free ow and jammed trac). Thereby, the free ow is characterized by a low local density and the jammed phase by a high local density, respectively. Due to the particle conservation of the model, the transition is realized by a separation into low density and high density regions 4. Therefore, the dynamical structure factor is an appropriate tool to investigate the decomposition of the two phases above the transition point. Additionally, the dynamical structure factor has the decisive advantage that both phases can be distinguished in a natural way by the

2 in Fractals and beyond: Complexities in the Sciences, ed. by M. M. Novak (World Scientic, Singapore, 1998). 6 time T l space Figure 1: Space-time plot for v max = 5, P = :5, and > c. Note the separation of the system in high and low density regions. sign of their characteristic velocities. Thus no articial denitions of jams are needed which were used in previous investigations and lead to controversial results (see for instance 4;5;6;7 ). Simulation and Results Recently we presented measurements of the dynamical structure factor of the Nagel- Schreckenberg trac ow model based on the local occupation variable r;t, given by 1 if cell r is occupied at time t r;t = (1) otherwise : The evolution of r;t leads directly to the space-time diagram where the propagation of the particles can be visualized by plotting dots at occupied sites (r,t). Figure 1 shows a space-time diagram of the system above the critical value. Trac jams are characterized by the backward movement of the high density regions. dynamical structure factor S r (k;!) is given by S r (k;!) = * 1 lt X + r;t e i(kr?!t) r;t The ; () where the Fourier transform is taken over a nite rectangle of the space-time diagram of size l T (see Fig. 1). In Fig. we present the dynamical structure factor above the transition. The structure factor exhibits two modes formed by the ridges corresponding to the two coexisting phases. The slopes of the ridges agree with

3 in Fractals and beyond: Complexities in the Sciences, ed. by M. M. Novak (World Scientic, Singapore, 1998). 63 S (k;!) 1 1 1?1 1?3??? k!? Figure : The dynamical structure factor S r(k;!) above the critical value. maximal S r(k;!) indicates the various modes. The ridges with the characteristic velocities (v of the two phases 8. The positive slope corresponds to the forward movement of the particle in the free ow phase and the negative slope corresponds to the backward movement of density uctuations (jams), respectively. A detailed analysis (see 8 ) of the behavior of the two modes shows that the free ow phase is characterized by the velocity v f = v max? P. On the other hand the jam velocity v j is a function of the noise parameter P only. In this work we address the question whether the transition from the free ow regime to the phase coexisting regime (where the system separates into the jammed and free ow phase) can be considered as a phase transition. Approaching the transition point c from lower densities, the jammed particle mode appears and grows with increasing density g. Thus it is tempting to try to relate the order parameter to the number of jammed cars. This requires a denition of a car to be jammed or not. Several attempts where made in previous works. For instance Vilar and Souza used the average number of standing cars (v n;t = ) as an order parameter 9. Without noise (P = ) this quantity vanishes continuously below the transition. But, including noise, the amount of standing cars is nite for any density g >, i.e. the average number of cars does not vanish below the critical density. The crucial point is that below the transition a particle may stop due to the noisy slowing down rule of the dynamics, but this behavior does not coincide with backward moving density uctuations and algebraic correlations which would indicate a critical behavior of the system. At present, a convincing denition of an order parameter which tends to zero for g < c is not known. Several authors even came to the conclusion that the Nagel-Schreckenberg trac ow model exhibits no real critical behavior 6;7. It was argued that for any nite noise parameter P the transition is smeared out 6, i.e. the transition cannot be explained in terms of the

4 in Fractals and beyond: Complexities in the Sciences, ed. by M. M. Novak (World Scientic, Singapore, 1998) v n,t n Figure 3: Snapshot of the velocity v n;t as a function of car number n at a certain time t and P = :5. Jams are characterized by several particles with low velocity. The free ow particles uctuate nearly independently between v n;t = 4 and v n;t = 5. concepts of ordinary phase transitions. On the other hand measurements of the relaxation time, which diverges at the transition point 5;6;7, and measurements of the life time distribution of jams, which are power-law distributed at 1 c, display criticality. In the present paper we rst focus our attention on the density uctuations (jams) measured by the dynamical structure factor. This quantity is an appropriate tool for analyzing a possible phase transition because it naturally distinguishes between the two phases characterized by the positive and negative slopes. We are especially interested in the behavior of S r (k;!) in the limit k! and!! because these values describe the long range and long time correlations of the jams. Unfortunately, the superposition of both modes at this point makes it dicult to consider the pure jammed mode (see Fig. ) and it would be desirable to nd a way which allows to separate both modes. This can be achieved by changing from the occupation function r;t, dened in real space, to the velocity-particle space where the evolution of each particle velocity v n;t is considered. A snapshot of the velocity-particle space is shown in Fig. 3. In the free ow phase, where the cars can be considered as independent particles 4, the velocities uctuate according to the noisy slowing down rule between the two values v n;t = 4 and v n;t = 5, respectively. Extended regions with small or even zero velocities correspond to trac jams. Comparable to the space-time diagram these regions move backward in time. The Fourier transform of the velocity-particle diagram in two dimensions leads to the dynamical structure factor S v (k;!) which is given by S v (k;!) = * 1 NT X n;t v n;t e i(kn?!t) + : (3) Compared to the dynamical structure factor of the ordinary space-time diagram the dynamical structure factor of the velocity-particle space has the advantage that the free ow phase contributes only white noise (S v (k;!)j free ow = const), i.e. the analysis of the occurring trac jams is made easier. Figure 4 shows the structure factor S v (k;!) above the transition. The structure factor displays one ridge with a negative slope, corresponding to the backward moving jams. The notch parallel

5 in Fractals and beyond: Complexities in the Sciences, ed. by M. M. Novak (World Scientic, Singapore, 1998). 65 S v (k;!) ??? k Figure 4: The dynamical structure factor S v(k;!) of the velocity-particle space above the transition for P = :5. Notice that the free ow phase contributes to S v(k;!) only white noise.?! to the k-axis through! = is caused by nite-size eects and disappears with increasing system size. The peak in the center of the diagram [S v (k = ;! = )] describes the uctuations of the average velocity of the whole system. This value yields no useful information about the occurring jams because it is an average over both coexisting phases. In the following we show that above the transition the system exhibits long range correlations of the jammed particles and below the transition the correlations display no algebraic decay. In Fig. 5 we plot the values of the dynamical structure factor S v (k;!) for!=k = v j, i.e. the values along the ridge which correspond to the jam modes. In order to take the nite-size eects into account we use the scaling ansatz S v (k;!)j!=k=vj = N? f(n k): (4) For = :78 and = :34 we obtain a convincing data collapse of all curves. The dynamical structure factor decays algebraically, S v (k;!)j!=k=vj k? ; (5) where the exponent is given by 3:6. This algebraic decay of the dynamical structure factor indicates that the corresponding correlation function is also characterized by an algebraic decay, i.e. the system displays long range correlations above the transitions. In Fig. 6 we plot the values of S v (k;!) for!=k = v j below the transition. The values which correspond to the jams in the free ow phase disappear with increasing N. For small values of the inverse wavelength k the dynamical structure factor exhibits no power-law behavior indicating that the free ow regime displays no long range correlations. This coincides with measurements of the life time distribution

6 in Fractals and beyond: Complexities in the Sciences, ed. by M. M. Novak (World Scientic, Singapore, 1998) S(k) k γ N θ S v (k) γ=3.6±. N=51,14, N ζ k Figure 5: The scaling plot of S v(k;!) along the jam mode for various values of N (with N = T ) and a xed value of the global density with g > c. For = :78 and = :34 we got a good data collapse indicating that the correlation of the jams decay algebraically (dashed line). of jams at the critical point 1 which displays a power-law behavior, i.e. the trac jams occur on all time scales. Below the critical density the life time of the jams is nite and no long range correlations of jams can occur. As mentioned above we are interested in the behavior of S v (k! ;!! ) with!=k = v j, i.e. we consider the smallest positive mode on the ridge of the jammed particles. This value contains the information how the longe range and long time correlations appear when the transition takes place. This value is believed to be closely related to an order parameter (see for instance 11 ). If the transition from the free ow regime to the jammed regime can be described as a critical phenomenon, S v (k! ;!! ) should vanish below the transition ( g < c ). Simulating nite system size this means that the smallest jam mode obeys the nite-size scaling S v (k) 1-6 N k Figure 6: The dynamical structure factor of S v(k;!) along the jam mode for N = 51; 14; 48 and a xed value of the global density with g < c.

7 in Fractals and beyond: Complexities in the Sciences, ed. by M. M. Novak (World Scientic, Singapore, 1998). 67 N y S v (k,ω ) y=1.74 x=.17 ρ c =.88 N N x (ρ-ρ c ) Figure 7: The dynamical structure factor S v(k;!) of the velocity-particle space above the transition for P = :5 and N = T = 18; 56; 51; 14. Notice that the free ow phase contributes to S v(k;!) only white noise. ansatz S v (k! ;!! )j!=k=vj = N?y f[n x ( g? c )]: (6) As one can see from Fig. 7 the nite-size scaling ansatz works quite well for the values of the scaling exponents y 1:74 and x :17, respectively. Note that it is not justied to identify the scaling exponents with the usual critical exponents of second order phase transitions (y = = and x = 1=). First it is not clear if S v (k! ;!! ) equals the order parameter (see 11 and references therein). Second, the usual nite-size scaling ansatz rests on the validity of the hyperscaling relation between the exponents (see for instance 1 ). And the third point is that the system displays an anisotropic scaling behavior, i.e. we expect that the correlations along the space and time axis scale with dierent exponents. It is known that the scaling behavior in strong anisotropic systems is much more complicated than in isotropic systems (see for instance 13 ). Despite of these restrictions, the nite-size scaling analysis shows that the smallest jam mode vanishes below the transition. In the hydrodynamic limit (N! 1, T! 1) no long range correlation in space and time occur for g < c. On the other hand above the critical value the correlation of the jams displays an algebraic decay of the correlation function. Since the correlations of the jams are nite below and innite above c the system displays critical behavior. This is in agreement with measurements of the relaxation time of the system. It was shown 5;6;7 that the relaxation time diverges at the critical point with increasing system size L z with a P dependent exponent z. But one has to mention that above the transition the measurements of the relaxation time yields unphysical result, in the sense that the relaxation time becomes negative 6. We think that the origin of this behavior is caused by the inhomogeneous character of the system above the transition where the system separates into two coexisting phases. The relaxation time measurement does not take this inhomogeneous character into account.

8 in Fractals and beyond: Complexities in the Sciences, ed. by M. M. Novak (World Scientic, Singapore, 1998) Conclusions We studied numerically the Nagel-Schreckenberg trac ow model. The investigation of the dynamical structure factor allowed us to examine the transition of the system from a free ow regime to a jammed regime. Above the transition the dynamical structure factor exhibits two modes corresponding to the coexisting free ow and jammed phase. Due to the sign of their characteristic velocities v f and v j, both phases can be clearly distinguished. The analysis of the dynamical structure factor of the velocity-particle space shows that above the transition the system exhibits long range correlations of the jammed particles and below the transition the correlations display no algebraic decay. Using a nite-size scaling analysis we showed that in the thermodynamic limit the smallest jam mode which corresponds to the long range correlations of jams vanishes below the transition. We think that an extended investigation of this quantity could lead to a convincing denition of an order parameter which describes the transition from the free ow to the jammed regime. References 1. K. Nagel and M. Schreckenberg, J. Phys. I, 1 (199).. M. J. Lighthill and G. B. Whitham, Proc. R. Soc. London, Ser. A 9, 317 (1955). 3. J. Treiterer, Appx IV to the nal Report EES -, Columbus Ohio State University (1965). 4. S. Lubeck, M. Schreckenberg, and K. D. Usadel, Phys. Rev. E 57, 1171 (1998). 5. G. Csanyi and J. Kertesz, J. Phys. A 8, L47 (1995). 6. B. Eisenblatter, L. Santen, A. Schadschneider, and M. Schreckenberg, Phys. Rev. E 57, 139 (1998). 7. M. Sasvari and J. Kertesz, Phys. Rev. E 56, 414 (1997). 8. S. Lubeck, L. Roters and K. D. Usadel, Investigation of the dynamical structure factor of the Nagel-Schreckenberg trac ow model, Proceedings of the conference Trac and Granular Flow 97, edited by M. Schreckenberg and D. E. Wolf, World Scientic, Singapore (1998). 9. L. C. Q. Vilar and A. M.C. Souza, Physica A 11, 84 (1994). 1. K. Nagel and M. Paczuski, Phys. Rev. E 51, 99 (1995). 11. M. Schmittmann and R. K. P. Zia, in Phase Transition and Critical Phenomena, edited by C. Domb and J. L. Lebowitz, (Academic Press, London, 1995), Vol K. Binder and D. W. Heermann, Monte Carlo Simulation in Statistical Physics, Springer Series in Solid-State Sciences 8, Springer Heidelberg (1988). 13. K. Binder and J.-S. Wang, J. Stat. Phys. 55, 87 (1989).