Pattern formation in flows of asymmetrically interacting particles: Peristaltic pedestrian dynamics as a case study

Figure 1.  Patterns emerging in the pedestrian model. Color indicates pedestrian index. Numerical solution of Eqs. (9) for parameters specified in the text.

Figure 2.  Transverse stationary distance $ b $ between pedestrians in the two-lane zig-zag flow shown in Fig. 1(b). Panel (a): $ b $ vs. density $ \rho $ for fixed $ \nu = 1 $, panel (b): $ b $ vs. interaction strength $ \nu $ for fixed $ \rho = 1 $, $ \rho $ being pedestrian density, $ \nu $ being strength of pedestrian wall interaction. Panel (a): in the region to the left (right) of the curve the flow is single (two-) lane. Panel (b): in the region to the left (right) of the curve the flow is two- (single) lane. Direct numerical simulations (circles) and analytical predictions (curves) are in agreement.

Figure 3.  Panel (a): the growth rate $ \Re(z_2) $ of the linear mode $ \mu = 2 $ vs the wave number $ k $. Panel (b): the growth rate of the first three harmonics of the linear mode $ \mu = 2 $ vs the mean interparticle distance $ a $. In both figures $ \epsilon = 0.5, \nu = 0.05, N = 32 $

Figure 4.  Bifurcation diagrams obtained from the linear stability analysis for $ N = 32 $ and the asymmetry parameter $ \epsilon = 0 $ (panel (a)) and $ \epsilon = 0.5 $ (panel (b)). Insets show details of the diagram in the vicinity of the two-lane regime instability. In the white (orange) area the one- (two-) lane flow is stable. In the blue area we observe the peristaltic regime and the distance between lanes is spatially and time modulated, in the green area the two-lane flow is linearly unstable, and the instability leads to unsorted motion as shown in Fig. 1(d)

Figure 5.  The same as in Fig. 4 for $ N = 128 $

Figure 6.  The order parameter $ R $ of the peristaltic phase vs. mean headway, $ a $, for fixed $ \nu = 0.05 $. Panel (a): $ R $ vs. $ a $, in the case of totally symmetric social interaction $ \epsilon = 0 $: panel (b) $ R $ vs. $ a $, in the case of partially asymmetric social interaction: $ \epsilon = 0.5 $. Panel (a): in the region between arrows a hysteretic behavior takes place: red-dot-curve presents downsweep stable branch, black-dot-curve presents upsweep stable branch. Panel (b): as in panel (a); the inset shows the hysteretic behavior near the right boundary of the peristaltic phase $ a = 3.198 $.

Figure 7.  Staggered transversal coordinates $ (-1)^n y_n $ in the mixed phase state for two different values of the pedestrian headway: $ a = 2.93 $ (panel (a)) and $ a = 3.13 $ (panel (b)). The social interaction is symmetric: $ \epsilon = 0 $. Other parameters are chosen inside the domain of mixed phases state: $ ~\nu = 0.05, ~N = 128 $. The solid lines represent the results obtained in the frame of the analytical approach, the dots represent the results of numerical solutions of Eqs. (9)

Figure 8.  Longitudinal distances between nearest neighbors $ x_{n+1}-x_n $ in the mixed phase state. All parameters are the same as in Fig. 7.

Figure 9.  Energy difference between the mixed state and the spatially homogeneous two-lane state as a function of the mean headway $a$ in the interval $a\in (a_2,a_3)$. The social interaction is symmetric $\epsilon=0$, the pedestrian-wall interaction is fixed: $\nu =0.05$

Figure 10.  The stationary value of the inverse width $\kappa$ vs the mean distance $a$ obtained from Eq. (81). The solid (dashed) curve presents a stable (unstable) solution. The curves are plotted in the mean distance interval $a\in(a_2,a_l)$, where the mixed phase is unstable in the linear anaylsis aproach and it is stable in the frame of the variational approach. The solid line gives the contour, where $\partial^2_\kappa {\mathcal E}_b=0$. The social interaction is symmetric $\epsilon=0$, the pedestrian-wall interaction is fixed: $\nu =0.05$

Figure 11.  Spatio-temporal evolution of the local distance between lanes $ \Delta y_n = |y_{n+1}-y_n| $ (panel (a)) and the excess density $ \Delta \rho_n = \frac{1}{x_{n+1}-x_n}-\frac{1}{a} $ (panel (b)) for totally asymmetric social interaction ($ \epsilon = 1 $). Other parameters are chosen inside the domain of peristaltic motion: $ ~ a = 1.4, ~\nu = 0.65 $. The two profiles are separated by the time difference $ \Delta t = 250 $

Figure 12.  Velocity of the peristaltic pulse as a function of the inverse density. Comparison of the analytical results obtained from Eq. (90) (solid curve) and full scale numerical results (dots). The social interaction is weakly asymmetric $ \epsilon = 0.01 $, the pedestrian-wall interaction is fixed: $ \nu = 0.05 $, the number of pedestrian $ N = 128 $

Figure 13.  Panel (a): The modulus of elliptic function $ m $ vs. mean headway $ a $, dashed line presents an energetically unstable branch. Panel (b): The dimensionless energy difference between the spatially homogeneous two-lane state and the peristaltic state $ \delta E = (E_{per}-E_{two-lane})/|E_{two-lane}| $ vs. mean headway $ a $. The critical headway $ a_r $ gives the right boundary of the peristaltic state stability interval. The two-lane state looses its stability and the peristaltic state is established for $ a<a_r $. The solid and dashed lines correspond to two branches presented in panel (a).

Figure 14.  Two stationary localized solutions $ Y(n) $ of Eq. (93) in the case of symmetric interparticle interaction for the mean headway $ a = a_r-0.0015 $, the pedestrian-wall interaction $ \nu = 0.05 $. The number of pedestrian is $ N = 128 $. The solid line corresponds to the energetically more favorable state.

Figure 15.  Staggered transversal coordinate $ (-1)^n y_n $ profile obtained by numerical simulations (dots) and analytically from Eq. (103). The social interaction is symmetric $ \epsilon = 0 $, the number of particles is $ N = 128 $, the pedestrian-wall interaction is fixed: $ \nu = 0.05 $, the mean headway $ a = a_r-0.001 $ (panel(a)), and $ a = a_r-0.0003 $ (panel(b))

Figure 16.  Pulse velocity in the vicinity of the bifurcation point $ a_r $ obtained from numerical solutions of Eq. (9) (dots) and from analysis (see Eq. (111)) in the case of weakly asymmetric interparticle interaction $ \epsilon = 0.01 $ for the mean headway $ 0<a_r-a\ll 1 $. The pedestrian-wall interaction is $ \nu = 0.05 $. The number of pedestrian is $ N = 128 $. See also Fig. 12