Analytical investigation of the faster-is-slower effect with a simplified phenomenological model

We investigate the mechanism of the phenomenon called the “faster-is-slower”effect in pedestrian flow studies analytically with a simplified phenomenological model. It is well known that the flow rate is maximized at a certain strength of the driving force in simulations using the social force model when we consider the discharge of self-driven particles through a bottleneck. In this study, we propose a phenomenological and analytical model based on a mechanics-based modeling to reveal the mechanism of the phenomenon. We show that our reduced system, with only a few degrees of freedom, still has similar properties to the original many-particle system and that the effect comes from the competition between the driving force and the nonlinear friction from the model. Moreover, we predict the parameter dependences on the effect from our model qualitatively, and they are confirmed numerically by using the social force model.

Figure 1

Numerical results for the stationary flow rate obtained from the SF model simulation. A higher flow rate corresponds to a faster evacuation time. (a) $v_0$ and $N$ dependence of the flow rate. The flow rate was not calculated for small values of $v_0$ and $N$. (b) The flow rate with a fixed $N$ of 200.

Figure 2

Schematic phase diagram. The area below the rough additional line is the “free state,”where the flow rate increases as the driving force increases. Each point ($\times$) shows the maximum point for each $N$. The area above the line is the “jamming state,”where the “faster-is-slower”effect occurs. The critical $v_0$ value depends on $N$, and this suggests that the effect begins to slow down evacuation when $N$ is relatively large.

Figure 3

Conceptual images of the model. (Left) The typical formation of the arch near the exit. (Right) The forces acting on the virtual particles.

Figure 4

Force applied to the particle at the center of the semicircle for each $N$. The results are obtained from the SF simulation with zero-width exit. Dashed lines represent Eq. (2) with the dimensionless parameter $b=0.625$.

Figure 5

Stationary flow velocity obtained from the model. (a) The $v_0$ and $N$ dependence of the flow velocity. (b) The flow velocity with a fixed $N$ of 200.

Figure 6

Explicit representation of the function $g\left[l(v_0,N=300)\right]$. (Solid) The analytical expression based on Eq. (6), (dotted) the numerical result calculated directly from Eq. (4), and (chain) the approximation curve $g\left[l(v_0,N)\right]\sim v_0^n$.

Figure 7

Numerical results of the SF model simulation corresponding to each conjecture. Only (iv) shows a maximum.