# American Institute of Mathematical Sciences

March  2019, 8(1): 73-100. doi: 10.3934/eect.2019005

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

 1 Bogolyubov Institute for Theoretical Physics, Metrologichna str. 14B, 01413 Kiev, Ukraine 2 Continental AG, Vahrenwalder Strasse 9, D-30165 Hanover, Germany 3 Department of Applied Mathematics and Computer Science, Technical University of Denmark, DK-2800 Kongens Lyngby, Denmark 4 Department of Physics and Department of Applied Mathematics and Computer Science, Technical University of Denmark, DK-2800 Kongens Lyngby, Denmark 5 Department of Physics, Technical University of Denmark, DK-2800 Kongens Lyngby, Denmark

* Corresponding author: Mads Peter Sørensen

Received  July 2017 Revised  November 2017 Published  March 2019 Early access  January 2019

Fund Project: This work is supported by Civilingeniør Frederik Leth Christiansens Almennyttige Fond, the Otto Mønsteds Fond and a special program of the National Academy of Sciences of Ukraine.

The influence of asymmetry in the coupling between repulsive particles is studied. A prominent example is the social force model for pedestrian dynamics in a long corridor where the asymmetry leads to anisotropy in the repulsion such that pedestrians in front, i.e., in walking direction, have a bigger influence on the pedestrian behavior than those behind. In addition to one- and two-lane free flow situations, a new traveling regime is found that is reminiscent of peristaltic motion. We study the regimes and their respective stabilityboth analytically and numerically. First, we introduce a modified social forcemodel and compute the boundaries between different regimes analytically bya perturbation analysis of the one-lane and two-lane flow. Afterwards, theresults are verified by direct numerical simulations in the parameter plane ofpedestrian density and repulsion strength from the walls.

Citation: Yuri B. Gaididei, Christian Marschler, Mads Peter Sørensen, Peter L. Christiansen, Jens Juul Rasmussen. Pattern formation in flows of asymmetrically interacting particles: Peristaltic pedestrian dynamics as a case study. Evolution Equations & Control Theory, 2019, 8 (1) : 73-100. doi: 10.3934/eect.2019005
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Patterns emerging in the pedestrian model. Color indicates pedestrian index. Numerical solution of Eqs. (9) for parameters specified in the text.
. 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 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.
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)
for $N = 128$">Figure 5.  The same as in Fig. 4 for $N = 128$
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 $a = 3.198$.
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.
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: $\nu =0.05$
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: $\nu =0.05$
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 $\Delta t = 250$
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 $N = 128$
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).
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.
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))
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
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