Instability of pedestrian flow and phase structure in a two-dimensional optimal velocity model

A two-dimensional optimal velocity model was proposed for the study of pedestrian and granular flow. We investigate the stability of homogeneous flow in the linear approximation and show the phase diagram of the model. We also investigate the property of the model by numerical simulation in the cases of unidirectional and counter flow. From these results, we present a unified understanding of the behavior of pedestrians and other related systems.

Figure 1

A snapshot of the homogeneous flow. A triangular structure is observed in a numerical simulation with periodic boundary condition. All particles represented by black disks are moving rightward.

Figure 2

A wave propagating in the $\phi (=\pi ∕3)$ direction.

Figure 3

Indices of six particles are shown. The positions of the particle 1, 2, 3, 4, 5, 6 are $(s,u),(s,-u),(0,2u),(0,-2u),(-s,u),(-s,-u)$, respectively.

Figure 4

The phase diagram obtained by the linear analysis. Solid and dashed curves represent the critical curves given by Eqs. (22, 23), respectively. Dotted lines represent the critical lines $r=1.05,r=0.94$, and $r=0.59$. There are four phases that can be characterized by the type of unstable modes: longitudinal mode, transverse mode, and their mixture.

Figure 5

Black disks $P_1\text{–}{P}_4$ and black squares $P_5\text{–}{P}_8$ represent parameters used in simulations for unidirectional flow and for counter flow, respectively.

Figure 6

A typical pattern of the flow at the point $P_1$ in the phase $B$. Sensitivity $a$ is set to 3.0, and the distance among particles $r$ is 1.06. The initial state is the homogeneous flow (a) and the transverse wave is observed after relaxation (b).

Figure 7

A typical pattern of the flow at the early stage (a) and the flow after relaxation (b) at the point $P_2(a=0.5,r=1.3)$ in the phase $C$.

Figure 8

A snapshot of the flow after relaxation. Parameters are $a=3.0$ and $r=1.0(P_3$ in Fig. 5). Particles move randomly but a triangular structure emerges partly.

Figure 9

A snapshot of the flow after relaxation. Parameters are $a=1.0$ and $r=0.5(P_4$ in Fig. 5). Particles move randomly and no structure is observed.

Figure 10

Typical stationary patterns of counter flow at the points (a) $P_5(a=1.0,r=2.0)$ and (b) $P_6(a=2.0,r=1.2)$ in the region $A$. White or black disks represent particles moving leftward and rightward, respectively. Boundary conditions are periodic in both $x$ and $y$ directions.

Figure 11

Typical blocking states of counter flow at the points (a) $P_7(a=1.0,r=1.24)$ and (b) $P_8(a=3.0,r=1.04)$. White or black disks are particles moving leftward and rightward, respectively. Boundary conditions are periodic in both $x$ and $y$ directions.

Figure 12

Parameters $l_1,l_2$, and $l_3$ are graphically shown.