Keep-Left Behavior Induced by Asymmetrically Profiled Walls

We show, computationally and analytically, that asymmetrically shaped walls can organize the flow of pedestrians driven in opposite directions through a corridor. Precisely, a two-lane ordered state emerges in which people always walk on the left-hand side (or right-hand side), controlled by the system’s parameters. This effect depends on features of the channel geometry, such as the asymmetry of the profile and the channel width, as well as on the density and the drift velocity of pedestrians, and the intensity of noise. We investigate in detail the influence of these parameters on the flow and discover a crossover between ordered and disordered states. Our results show that an ordered state only appears within a limited range of drift velocities. Moreover, increasing noise may suppress such flow organization, but the flow is always sustained. This is in contrast with the “freezing by heating” phenomenon according to which pedestrians tend to clog in smooth channels for strong noise [Phys. Rev. Lett. 84, 1240 (2000)]. Therefore, the ratchetlike effect proposed here acts on the system not only to induce a “keep-left” behavior but also to prevent the freezing by heating clogging phenomenon. Besides pedestrian flow, this new phenomenon has other potential applications in microfluidics systems.

Popular Summary

The flow of pedestrians through corridors such as those found in London’s Tube system during rush hour is typically controlled by signs asking everyone to “keep left.” However, this flow is often disordered and subject to clogging. Could there be a way to ensure that everyone keeps left automatically? Indeed, certain shapes of the corridors’ walls might help. Using numerical simulations and analytical calculations of the motion of self-propelling particles through channels with differently shaped walls, we precisely determine the conditions in which self-organized lanes break the left-right symmetry and attach to the left wall because of a ratchetlike effect.

We model a system of self-driven particles traversing a channel, and we impose a zigzag shape to the channel walls. The zigzag walls are characterized by an asymmetry parameter; this parameter modulates the deflection of particles by the walls and results in the particle flows exhibiting preferential sides. We employ molecular dynamics, and we break up our time interval into 1000 steps. Our work furthermore takes into account “stubborn” particles that insist on walking on the right-hand side. We find that the particles are able to spontaneously “keep left” thanks to the zigzag shape of the walls. This transition occurs as a function of density, disorder, and channel width and exhibits unexpected reentrant and inversion effects.

Our findings could inform the design of both microfluidics channels and human-scale transit stations.

Figure 1

Channel geometry and final configurations of pedestrians for different asymmetry parameters. (a) Channel, of depth $H$ and width $W$, through which a preferred flow orientation of particles is induced. The meaning of geometric parameters is indicated. (b–d) Typical configurations in the steady state, where the particles are keeping left in asymmetric wall channels but are mixed in symmetric ones. Upward movers (downward movers) are represented by full (hollow) particles. The velocity of each particle is indicated by arrows. The asymmetry parameter $b=L/2-L_1$ takes the values $b=8$ in (b), $b=-8$ in (c), and $b=0$ in (d), for $H=2$, $v_d=6$, and $\rho =0.6$. The length unit is written in terms of particle diameter. Noise was not included in these simulations.

Figure 2

Average order parameter $⟨\mathrm{\Phi }⟩$ as a function of the asymmetry parameter $b$, for three different values of the drift velocity. We see that order is only induced for $|b|>0$. The slower $v_d$, the higher $|b|$ should be in order to achieve a two-lane pattern. The remaining parameters, including the lack of noise, are the same as in Fig. 1.

Figure 3

Dependence of density on drift velocity for ordered states. In panel (a), the average order parameter $⟨\mathrm{\Phi }⟩$ is shown as a function of the density for $\theta =0$, $b=8$, $H=2$, and several values of the drift velocity $v_d$. In panel (b), we show the maximum (red squares) and minimum (black circles) densities for which $⟨\mathrm{\Phi }⟩=1$, as functions of the drift velocity. A power-law fit for the minimum density yields ${\rho }_{\min }=17.6v_d^{-2.03}$, whereas a linear fit for the maximum density gives ${\rho }_{\max }=0.919-0.032v_d$. The equations extracted from the fits meet at two points, namely, $v_d\approx 4.67$ and $v_d\approx 28.09$.

Figure 4

Average order parameter as a function of the particle density, for $\theta =0$, $b=8$, $v_d=6$, and various values of the depth $H$ of the serrated region.

Figure 5

Average order parameter as a function of $\theta$, for $\rho =0.6$, $b=8$, $v_d=6$, and $H=2$. The inset shows the crossover from $⟨\mathrm{\Phi }⟩=-1$ to $⟨\mathrm{\Phi }⟩=1$ as $b$ changes from $-8$ to 8, for different values of $\theta$.