Phase transitions in two-dimensional traffic-flow models

Empirical evidence for a jamming transition in urban traffic

Understanding the mechanisms leading to the formation and the propagation of traffic jams in large cities is of crucial importance for urban planning and traffic management. Many studies have already considered the emergence of traffic jams from the point of view of phase transitions, but mostly in simple geometries such as highways for example or in the framework of percolation where an external parameter is driving the transition. More generally, empirical evidence and characterization for a congestion transition in complex road networks are scarce, and here, we use traffic measures for Paris (France) during the period 2014-2018 for testing the existence of a jamming transition at the urban level. In particular, we show that the correlation function of delays due to congestion is a power law (with exponent η ≈ 0.4) combined with an exponential cut-off ξ. This correlation length is shown to diverge during rush hours, pointing to a jamming transition in urban traffic. We also discuss the spatial structure of congestion and identify a core of congested links that participate in most traffic jams and whose structure is specific during rush hours. Finally, we show that the spatial structure of congestion is consistent with a reaction-diffusion picture proposed previously.

Maximizing the Strength of Fiber Bundles under Uniform Loading

The collective strength of a system of fibers, each having a failure threshold drawn randomly from a distribution, indicates the maximum load carrying capacity of different disordered systems ranging from disordered solids, power-grid networks, to traffic in a parallel system of roads. In many of the cases where the redistribution of load following a local failure can be controlled, it is a natural requirement to find the most efficient redistribution scheme, i.e., following which system can carry the maximum load. We address the question here and find that the answer depends on the mode of loading. We analytically find the maximum strength and corresponding redistribution schemes for sudden and quasistatic loading. The associated phase transition from partial to total failure by increasing the load has been studied. The universality class is found to be dependent on the redistribution mechanism.

A new cellular automaton model for urban two-way road networks

A new cellular automaton (CA) model is proposed to simulate traffic dynamics in urban two-way road network systems. The NaSch rule is adopted to represent vehicle movements on road sections. Two novel rules are proposed to move the vehicles in intersection areas, and an additional rule is developed to avoid the “gridlock” phenomenon. Simulation results show that the network fundamental diagram is very similar to that of road traffic flow. We found that the randomization probability and the maximum vehicle speed have significant impact on network traffic mobility for free-flow state. Their effect may be weak when the network is congested.

Kinetic Monte Carlo simulations of one-dimensional and two-dimensional traffic flows: comparison of two look-ahead rules

We employ an efficient list-based kinetic Monte Carlo (KMC) method to study traffic flow models on one-dimensional (1D) and two-dimensional (2D) lattices based on the exclusion principle and Arrhenius microscopic dynamics. This model implements stochastic rules for cars' movements based on the configuration of the traffic ahead of each car. In particular, we compare two different look-ahead rules: one is based on the distance from the car under consideration to the car in front of it, and the other one is based on the density of cars ahead. The 1D numerical results of these two rules suggest different coarse-grained macroscopic limits in the form of integro-differential Burgers equations. The 2D results of both rules exhibit a sharp phase transition from freely flowing to fully jammed, as a function of the initial density of cars. However, the look-ahead rule based on the density of the traffic produces more realistic results. The KMC simulations reported in this paper are compared with those from other well-known traffic flow models and the corresponding empirical results from real traffic.

Effect of overpasses in the Biham-Middleton-Levine traffic flow model with random and parallel update rule

The effect of overpasses in the Biham-Middleton-Levine traffic flow model with random and parallel update rules has been studied. An overpass is a site that can be occupied simultaneously by an eastbound car and a northbound one. Under periodic boundary conditions, both self-organized and random patterns are observed in the free-flowing phase of the parallel update model, while only the random pattern is observed in the random update model. We have developed mean-field analysis for the moving phase of the random update model, which agrees with the simulation results well. An intermediate phase is observed in which some cars could pass through the jamming cluster due to the existence of free paths in the random update model. Two intermediate states are observed in the parallel update model, which have been ignored in previous studies. The intermediate phases in which the jamming skeleton is only oriented along the diagonal line in both models have been analyzed, with the analyses agreeing well with the simulation results. With the increase of overpass ratio, the jamming phase and the intermediate phases disappear in succession for both models. Under open boundary conditions, the system exhibits only two phases when the ratio of overpasses is below a threshold in the random update model. When the ratio of the overpass is close to 1, three phases could be observed, similar to the totally asymmetric simple exclusion process model. The dependence of the average velocity, the density, and the flow rate on the injection probability in the moving phase has also been obtained through mean-field analysis. The results of the parallel model under open boundary conditions are similar to that of the random update model.

Dynamical traffic light strategy in the Biham-Middleton-Levine model

In this paper, we study dynamical traffic light strategies in the Biham-Middleton-Levine traffic flow model. The strategies use local vehicular information to control urban traffic, which take into account the interaction of vehicles traveling in different directions via considering their dynamical spatial configuration. Simulations find out two strategies, in which local information at nearby sites is used. The two strategies perform much better than the alternating strategy. Under these two strategies, vehicles can self-organize into a new intermediate state with band structure. The analytical solutions of velocity of this state have been presented, which are in good agreement with simulations.

Formation of density waves in traffic flow through intersecting roads

The formation of density waves in two intersecting roads, with a traffic circle at the intersection, is studied. It is found that, depending on the traffic densities in the two roads, density waves can form in the traffic circle and in one or both of the roads. Depending on the expression chosen for the optimal velocity, either the congestion moves entirely to the traffic circle or the congestion becomes confined to the traffic circle and a part of the road approaching the traffic circle.

Characteristics of vehicular traffic flow at a roundabout

We construct a stochastic cellular automata model for the description of vehicular traffic at a roundabout designed at the intersection of two perpendicular streets. The vehicular traffic is controlled by a self-organized scheme in which traffic lights are absent. This controlling method incorporates a yield-at-entry strategy for the approaching vehicles to the circulating traffic flow in the roundabout. Vehicular dynamics is simulated and the delay experienced by the traffic at each individual street is evaluated. We discuss the impact of the geometrical properties of the roundabout on the total delay. We compare our results with traffic-light signalization schemes, and obtain the critical traffic volume over which the intersection is optimally controlled through traffic-light signalization schemes.

Anisotropy effect on two-dimensional cellular-automaton traffic flow with periodic and open boundaries

Using computer simulations we investigate, in a version of the Biham-Middleton-Levine model with random sequential update on a square lattice, the anisotropy effect of the probabilities of the change of the motion directions of cars, from up to right (p(ur)) and from right to up (p(ru)), on the dynamical jamming transition and velocities under periodic boundaries on one hand and the phase diagram under open boundaries on the other hand. However, in the former case, the sharp jamming transition appears only for p(ur)=0=p(ru)=0 (i.e., when the cars alter their motion directions). In the open boundary conditions, it is found that the first-order line transition between jamming and moving phases is curved. Hence, by increasing the anisotropy, the moving phase region expands as well as the contraction of the jamming and maximal current phases takes place. Moreover, in the anisotropic case, the transition between the jamming phase (or moving phase) and the maximal current phase is of second order while in the isotropic case, and when each car changes its direction of motion at every time step (p(ru)=p(ur)=1), the transition is of first order. Furthermore, in the maximal current phase, the density profile decays with an exponent gamma approximately 1/4.

Enhancement and stabilization of traffic flow by moving in groups

We study the traffic behavior of vehicles moving in groups analytically and numerically. A car-following model of traffic is extended to take into account a binary mixture of vehicles. It is shown that the movement in groups stabilizes the traffic flow. The jamming transition among the free traffic, the inhomogeneous traffic, and the homogeneous congested traffic occurs at a higher density than the threshold of the original model. The traffic current is highly enhanced at a high-density region by keeping a short headway without jam. The jamming transition is analyzed by using the linear stability method. It is found that the theoretical neutral stability curve agrees with the transition line obtained by the simulation.

Bunching transition in a time-headway model of a bus route

A time-headway model is presented to mimic bus behavior on the bus route. The motion of a bus is described in terms of the time headway between its bus and the bus in front. We study the bunching behavior of buses induced by interacting with other buses and passengers. It is shown that the dynamical phase transitions among the inhomogeneous bunching phase, the homogeneous free phase, the coexisting phase, and the homogeneous congested phase occur with varying the initial time headway. We study the effect of not stopping at bus stops on the time-headway profile. It is found that the bunching transition lines are consistent with the neutral stability curves obtained by the linear stability analysis.

Density waves in traffic flow

Density waves are investigated in the car-following model analytically and numerically. This work is a continuation of our previous investigation of traffic flow in the metastable and unstable regions [Phys. Rev. E 58, 4271 (1998); 60, 180 (1999)]. The Burgers equation is derived for the density wave in the stable region of traffic flow by the use of nonlinear analysis. It is shown, numerically, that the triangular shock wave appears as the density wave at the late stage in the stable region. The decay rate of the shock wave is calculated and compared with the analytical result. It is shown that the density waves out of the coexisting curve, near the spinodal line, and within the spinodal line appear, respectively, as the triangular shock wave, the soliton, and the kink-antikink wave. The density waves are described, respectively, by the Burgers, Korteweg-de Vries, and modified Korteweg-de Vries equations.

Traffic jams induced by fluctuation of a leading car

We present a phase diagram of the different kinds of congested traffic triggered by fluctuation of a leading car in an open system without sources and sinks. Traffic states and density waves are investigated numerically by varying the amplitude of fluctuation using a car following model. The phase transitions among the free traffic, oscillatory congested traffic, and homogeneous congested traffic occur by fluctuation of a leading car. With increasing the amplitude of fluctuation, the transition between the free traffic and oscillatory traffic occurs at lower density and the transition between the homogeneous congested traffic and the oscillatory traffic occurs at higher density. The oscillatory congested traffic corresponds to the coexisting phase. Also, the moving localized clusters appear just above the transition lines.

Stabilization and enhancement of traffic flow by the next-nearest-neighbor interaction

The car-following model of traffic is extended to take into account the car interaction before the next car ahead (the next-nearest-neighbor interaction). The traffic behavior of the extended car-following model is investigated numerically and analytically. It is shown that the next-nearest-neighbor interaction stabilizes the traffic flow. The jamming transition between the freely moving and jammed phases occurs at a higher density than the threshold of the original car-following model. By increasing the maximal velocity, the traffic current is enhanced without jam by the stabilization effect. The jamming transition is analyzed with the use of the linear stability and nonlinear perturbation methods. The traffic jam is described by the kink solution of the modified Korteweg-de Vries equation. The theoretical coexisting curve is in good agreement with the simulation result.

Chaotic jam and phase transition in traffic flow with passing

The lattice hydrodynamic model is presented to take into account the passing effect in one-dimensional traffic flow. When the passing constant gamma is small, the conventional jamming transition occurs between the uniform traffic and kink density wave flows. When passing constant gamma is larger than the critical value, the jamming transitions occur from the uniform traffic flow, through the chaotic density wave flow, to the kink density wave flow, with an increasing delay time. The chaotic region increases with passing constant gamma. The neutral stability line is derived from the linear stability analysis. The neutral stability line coincides with the transition line between the uniform traffic and density wave flows. The modified Korteweg-de Vries equation describing the kink jam is derived for small values of gamma by use of a nonlinear analysis.

Soliton and kink jams in traffic flow with open boundaries

Soliton density wave is investigated numerically and analytically in the optimal velocity model (a car-following model) of a one-dimensional traffic flow with open boundaries. Soliton density wave is distinguished from the kink density wave. It is shown that the soliton density wave appears only at the threshold of occurrence of traffic jams. The Korteweg-de Vries (KdV) equation is derived from the optimal velocity model by the use of the nonlinear analysis. It is found that the traffic soliton appears only near the neutral stability line. The soliton solution is analytically obtained from the perturbed KdV equation. It is shown that the soliton solution obtained from the nonlinear analysis is consistent with that of the numerical simulation.

Jamming transition in a two-dimensional traffic flow model

Phase transition and critical phenomenon are investigated in the two-dimensional traffic flow numerically and analytically. The one-dimensional lattice hydrodynamic model of traffic is extended to the two-dimensional traffic flow in which there are two types of cars (northbound and eastbound cars). It is shown that the phase transition among the freely moving phase, the coexisting phase, and the uniformly congested phase occurs below the critical point. Above the critical point, no phase transition occurs. The value a(c) of the critical point decreases as increasing fraction c of the eastbound cars for c<or=0.5. The linear stability theory is applied. The neutral stability lines are found. The time-dependent Ginzburg-Landau (TDGL) equation is derived by the use of nonlinear analysis. The phase separation lines, the spinodal lines, and the critical point are calculated from the TDGL equation.