Takashi Nagatani

The physics of traffic jams

Traffic flow is a kind of many-body system of strongly interacting vehicles. Traffic jams are a typical signature of the complex behaviour of vehicular traffic. Various models are presented to understand the rich variety of physical phenomena exhibited by traffic. Analytical and numerical techniques are applied to study these models. Particularly, we present detailed results obtained mainly from the microscopic car-following models. A typical phenomenon is the dynamical jamming transition from the free traffic (FT) at low density to the congested traffic at high density. The jamming transition exhibits the phase diagram similar to a conventional gas-liquid phase transition: the FT and congested traffic correspond to the gas and liquid phases, respectively. The dynamical transition is described by the time-dependent Ginzburg-Landau equation for the phase transition. The jamming transition curve is given by the spinodal line. The metastability exists in the region between the spinodal and phase separation lines. The jams in the congested traffic reveal various density waves. Some of these density waves show typical nonlinear waves such as soliton, triangular shock and kink. The density waves are described by the nonlinear wave equations: the Korteweg-de-Vries (KdV) equation, the Burgers equation and the Modified KdV equation. Subjects like the traffic flow such as bus-route system and pedestrian flow are touched as well. The bus-route system with many buses exhibits the bunching transition where buses bunch together with proceeding ahead. Such dynamic models as the car-following model are proposed to investigate the bunching transition and bus delay. A recurrent bus exhibits the dynamical transition between the delay and schedule-time phases. The delay transition is described in terms of the nonlinear map. The pedestrian flow also reveals the jamming transition from the free flow at low density to the clogging at high density. Some models are presented to study the pedestrian flow. When the clogging occurs, the pedestrian flow shows the scaling behaviour.

Jamming transition in pedestrian counter flow

A lattice gas model with biased random walkers is presented to mimic the pedestrian counter flow in a channel under the open boundary condition of constant density. There are two types of walkers without the back step: the one is the random walker going to the right and the other is the random walker going to the left. It is found that a dynamical jamming transition from the freely moving state at low density to the stopped state at high density occurs at the critical density. The transition point is given by pc=0.45±0.01, not depending on the system size. The transition point depends on the strength of drift and decreases with increasing drift. Also, we present the extended model to take into account the traffic rule in which a pedestrian walks preferably on the right-hand side of the channel.

Modified KdV equation for jamming transition in the continuum models of traffic

Continuum models of traffic are proposed to describe the jamming transition in traffic flow on a highway. They are the simplified versions of the hydrodynamic model of traffic. Two continuum models are presented: one is described by the partial differential equations and the other is the discrete lattice version. The linear stability theory and the nonlinear analysis are applied to the continuum models. The modified Korteweg–de Vries equation (KdV) near the critical point is derived using the reduction perturbation method. It is shown that the jamming transition and the density wave in the congested traffic flow are described by the modified KdV equation. The solutions of the KdV equations obtained from the two models are compared with that of the optimal velocity model (car following model).

Scaling behavior of crowd flow outside a hall

Crowd flow going outside a hall is investigated by using the lattice–gas model of pedestrian flow. Some different dynamical states are distinguished for the crowd flow. It is shown that a dynamical phase transition occurs from the choking flow to the decaying flow at a critical time tc. In the choking-flow region, a scaling behavior is found as follows: the crowd flow rate J scales as J∝W0.88±0.02 and the transition time tc scales as tc∝W−1.16±0.01, where W is the size of door.

Jamming transition in two-dimensional pedestrian traffic

Jamming transitions of pedestrian traffic are investigated under the periodic boundary condition on the square lattice by the use of the lattice gas model of biased random walkers without the back step. The two cases are presented: the one with two types of walkers and the other with four types of walkers. In the two types of walkers, the first is the random walker going to the right and the second is the random walker going up. In the four types of walkers, the first, second, third, and fourth are, respectively, the random walkers going to the right, left, up, and down. It is found that the dynamical jamming transitions occur at the critical densities. The transition points do not depend on the system size for large system but depend strongly on the strength of drift. The jamming transitions are compared with that obtained by the cellular automaton model of car traffic.

TDGL and MKdV equations for jamming transition in the lattice models of traffic

The lattice models of traffic are proposed to describe the jamming transition in traffic flow on a highway in terms of thermodynamic terminology of phase transitions and critical phenomena. They are the lattice versions of the hydrodynamic model of traffic. Two lattice models are presented: one is described by the differential-difference equation where time is a continuous variable and space is a discrete variable, and the other is the difference equation in which both time and space variables are discrete. We apply the linear stability theory and the nonlinear analysis to the lattice models. It is shown that the time-dependent Ginzburg–Landau (TDGL) equation is derived to describe the traffic flow near the critical point. A thermodynamic theory is formulated for describing the phase transitions and critical phenomena. It is also shown that the perturbed modified Korteweg-de Vries (MKdV) equation is derived to describe the traffic jam.

Scaling of pedestrian channel flow with a bottleneck

Pedestrian channel flow at a bottleneck is investigated under the open boundaries by using the lattice-gas model of biased random walkers. It is shown that a dynamical phase transition occurs from the free flow to the choking flow at a critical density pc with increasing density. The flow rate saturates at higher density than the critical density. In the choking-flow region, a scaling behavior is found as follows: the saturated flow rate Js scales as Js∝d0.93±0.02 and the critical density pc scales as pc∝(d/W)1.16±0.02, where d is the width of the bottleneck and W is the width of channel. The plot of the rescaled flow rate against the rescaled density collapses onto a single curve.

Jamming transition of pedestrian traffic at a crossing with open boundaries

Pedestrian traffic at a crossing is investigated under the open boundary condition by the use of the lattice gas model of biased random walkers without the back step. The four types of walkers interact with each other at the crossing where there are random walkers going to the right, left, up, and down. It is found that a dynamical jamming transition from the moving state at low density to the stopped state at high density occurs at the critical density. The transition point depends on the strength of drift and decreases with increasing drift. The transition point does not depend on the length of roads connecting the crossing for the long road. Also, the pedestrian traffic with two types of walkers is studied where there are random walkers going to the right and up. It is compared with the pedestrian traffic with the four types of random walkers.

Clogging transition of pedestrian flow in T-shaped channel

Pedestrian flow is investigated under the open boundaries in a T-shaped channel where the branch flow joins the main flow at the junction. The pedestrian merging flow is simulated by the use of the lattice-gas model of biased random walkers. When the main flow rate increases under the constant value of branch flow rate, the clogging transitions occur at the main flow or branch flow or both flows. It is shown that the dynamical phase transitions depend on both inlet densities. The four distinct phases are found. The phase diagram is presented for the distinct phases. The scaling of saturated flow rate and transition point is shown. The flow rate exhibits the universal scaling form.

Transition and saturation of traffic flow controlled by traffic lights

We study the traffic flow controlled by traffic lights on a single-lane roadway by using the optimal velocity model. The characteristic of traffic flow is clarified for the three different strategies of traffic light control: the simple synchronized, green wave, and random switching strategies. The current–density diagrams are calculated for the three different strategies. It is found that the saturation of current occurs at the critical density. The critical density of the dynamical transition depends on the cycle time of the traffic light and strategy. The value of the saturated current does not depend on the cycle time and the strategies. The density wave propagating backward appears when the current saturates. It is shown that the density wave is consistent with the spontaneous jam. A theoretical analysis is also presented for the dynamical transition.