Freezing transition in bi-directional CA model for facing pedestrian traffic
Introduction
Recently, pedestrian and vehicular traffic flows have attracted considerable attention [1], [2], [3], [4], [5]. Many observed dynamical phenomena in pedestrian and traffic flows have been successfully reproduced with physical methods. The pedestrian flow dynamics is closely connected with the self-driven many-particle system [6]. It has also encouraged physicists to study evacuation processes by self-driven many-particle models [7], [8], [9], [10], [11], [12], [13]. The pedestrian and vehicular traffic models have been applied to the traffic flow of such mechanical mobile objects as robots [14], [15].
The typical pedestrian flows have been simulated by the use of a few models in two-dimensional space: the lattice-gas model of biased-random walkers [11], [12], [13], [14], [15], [16], the molecular dynamic model of active walkers [6], [10], [17], and the cellular automaton model [7], [8]. Their models are not deterministic but stochastic. Their models are described in two-dimensional space. The molecular dynamic model of active walkers is described by the behavioral (or generalized) force on two-dimensional off-lattice. The lattice-gas model of biased-random walkers and the CA model are described by stochastic rules on the square lattice. Helbing et al. have found that the “freezing by heating” occurs in the facing pedestrian traffic by the use of the molecular dynamic model of active walkers [17]. By using the lattice gas model of biased-random walkers, Muramatsu et al. have found independently that the freezing transition occurs from the free traffic to the frozen (stopping) state when the pedestrian density is higher than the critical value [16]. The freezing transition in the facing pedestrian traffic has been studied by some researchers [18], [19].
In the jamming transition, pedestrian flow in the crowd changes from the free traffic to the jammed traffic in which pedestrians are distributed heterogeneously and move slowly. In the freezing transition, pedestrian flow change to the frozen state in which all pedestrians cannot move by preventing from going ahead each other. The analytical works are unknown for the facing pedestrian flow. The pedestrian flow has been investigated by the numerical simulation of the stochastic models on two-dimensional space. It is not easy to analyze the two-dimensional stochastic models because the dynamical behavior is complex. However, the one-dimensional deterministic CA models have not been proposed for facing traffic of pedestrians until now.
In this Letter, we present the one-dimensional, deterministic, and bi-directional CA model for the facing pedestrian traffic. We study the dynamical states and dynamical phase transitions in the model of facing pedestrian traffic. We show that there exist four pedestrian states and the jamming and freezing transitions occur when pedestrian density increases. We show that the bi-directional CA model reproduces the Burgers CA model of unidirectional multi-lane traffic in the limit of no facing pedestrians.
Section snippets
Bi-directional CA model
We consider the facing (bi-directional) traffic of pedestrians on a wide passage. There exist two kinds of walkers on the passage: the one is the walkers moving to the east and the other the walkers moving to the west. The walker moving to the east (or west) interacts highly with the other walkers in the front. When the density of walkers ahead is higher, the current decreases more because the movement of walkers will be prevented by other walkers.
We consider the one-dimensional approximation
Simulation and result
We carry out the numerical simulation for bi-directional CA model described by Eqs. (1), (2). The boundaries are periodic. We consider the following initial condition. Walkers to east and to west distribute uniformly on the passage, respectively. The initial condition is described by The densities of walkers to east and to west are defined, respectively, as and .
Summary
We have presented the bi-directional cellular automaton (CA) model for facing pedestrian traffic. The bi-directional CA model is one-dimensional and deterministic. We have studied the dynamical phase transition and traffic states. We have found that two jamming transitions and a freezing transition occur. We have shown that there exist four traffic states: the free traffic, jammed traffic 1 and jammed traffic 2, and frozen state.
The present model is the first one of the deterministic CA models
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