Figure 1.
Patterns emerging in the pedestrian model. Color indicates pedestrian index. Numerical solution of Eqs. (9) for parameters specified in the text.
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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 |
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.
Keywords:
Nonlinear lattice dynamics,
pedestrian flow,
pattern formation in complex systems,
asymmetry,
general perturbation schemes,
bifurcations and instability,
and nonlinear modes.
Mathematics Subject Classification:
Primary: 37N20, 37K60, 37M05, 70K50; Secondary: 70H09, 70K60, 70K75.
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
![]()
References:
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M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, Dover Publications, Inc., New York, 1966. |
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N. Bellomo and C. Dogbe,
On the modeling of traffic and crowds: A survey of models, speculations, and perspectives, SIAM Review, 53 (2011), 409-463.
doi: 10.1137/090746677. |
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V. J. Blue and J. L. Adler,
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C. Burstedde, K. Klauck, A. Schadschneider and J. Zittartz,
Simulation of pedestrian dynamics using a two-dimensional cellular automaton, Physica A, 295 (2001), 507-525.
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O. Corradi, P. G. Hjorth and J. Starke,
Equation-free detection and continuation of a Hopf bifurcation point in a particle model of pedestrian flow, SIAM Journal on Applied Dynamical Systems, 11 (2012), 1007-1032.
doi: 10.1137/110854072. |
| [6] |
T. Dessup, C. Coste and M. Saint Jean, Subcriticality of the zigzag transition: A nonlinear bifurcation analysis, Physical Review E, 91 (2015), 032917, 1-14.
doi: 10.1103/PhysRevE.91.032917. |
| [7] |
T. Dessup, T. Maimbourg, C. Coste and M. Saint Jean, Linear instability of a zigzag pattern, Physical Review E, 91 (2015), 022908, 1-12.
doi: 10.1103/PhysRevE.91.022908. |
| [8] |
F. Dietrich and G. Köster, Gradient navigation model for pedestrian dynamics, Physical Review E, 89 (2014), 062801, 1-8.
doi: 10.1103/PhysRevE.89.062801. |
| [9] |
J. E. Galván-Moya and F. M. Peeters, Ginzburg-Landau theory of the zigzag transition in quasi-one-dimensional classical Wigner crystals, Physical Review B, 84 (2011), 134106, 1-10. Google Scholar |
| [10] |
D. Helbing,
Traffic and related self-driven many-particle systems, Reviews of Modern Physics, 73 (2001), 1067-1141.
doi: 10.1103/RevModPhys.73.1067. |
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D. Helbing, P. Molnar, I. J. Farkas and K. Bolay,
Self-organizing pedestrian movement, Environment and Planning B-planning and Design, 28 (2001), 361-383.
doi: 10.1068/b2697. |
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D. Helbing and P. Molnar,
Social force model for pedestrian dynamics, Physical Review E, 51 (1995), 4282-4286.
doi: 10.1103/PhysRevE.51.4282. |
| [13] |
D. Helbing, I. Farkas and T. Vicsek,
Simulating dynamical features of escape panic, Nature, 407 (2000), 487-490.
doi: 10.1038/35035023. |
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S. P. Hoogendoorn and W. Daamen,
Pedestrian behavior at bottlenecks, Transportation Science, 39 (2005), 147-288.
doi: 10.1287/trsc.1040.0102. |
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A. Jelić, C. Appert-Rolland, S. Lemercier and J. Pettré, Properties of pedestrians walking in line: Fundamental diagrams, Physical Review E, 85 (2012), 036111, 1-9. Google Scholar |
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doi: 10.1002/9783527610006.ch16. |
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A. Johansson, D. Helbing and P. K. Shukla,
Specification of the social force pedestrian model by evolutionary adjustment to video tracking data, Advances in Complex Systems, 10 (2007), 271-288.
doi: 10.1142/S0219525907001355. |
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B. S. Kerner, The physics of Traffic: Empirical Freeway Pattern Features, Engineering Applications, and Theory, 1st edition, Springer-Verlag, Berlin Heidelberg, 2004.
doi: 10.1007/978-3-540-40986-1. |
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Y. G. Kevrekidis and G. Samaey,
Equation-free multiscale computation: Algorithms and applications, Annual Review of Physical Chemistry, 60 (2009), 321-344.
doi: 10.1146/annurev.physchem.59.032607.093610. |
| [20] |
H.-K. Li, E. Urban, C. Noel, A. Chuang, Y. Xia, A. Ransford, B. Hemmerling, Y. Wang, T. Li, H. Häffner and X. Zhang, Realization of translational symmetry in trapped cold ion rings, Physical Review Letters, 118 (2017), 053001, 1-5.
doi: 10.1103/PhysRevLett.118.053001. |
| [21] |
C. Marschler, J. Starke, M. P. Sørensen, Yu. Gaididei and P. L. Christiansen,
Pattern formation in annular systems of repulsive particles, Physics Letters A, 380 (2016), 166-170.
doi: 10.1016/j.physleta.2015.10.038. |
| [22] |
C. Marschler, J. Starke, P. Liu and Y. G. Kevrekidis, Coarse-grained particle model for pedestrian flow using diffusion maps, Physical Review E, 89 (2014), 013304, 1-11.
doi: 10.1103/PhysRevE.89.013304. |
| [23] |
C. Marschler, J. Sieber, P. G. Hjorth and J. Starke, Equation-free analysis of macroscopic behavior in traffic and pedestrian flow, in: Traffic and Granular Flow '13 (eds. M. Chraibi, M. Boltes, A. Schadschneider and A. X. Armin Seyfried) Springer-Verlag, (2015), 423-439. Google Scholar |
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C. Marschler, J. Sieber, R. Berkemer, A. Kawamoto and J. Starke, Implicit methods for
equation-free analysis: Convergence results and analysis of emergent waves in microscopic
traffic models, SIAM Journal on Applied Dynamical Systems, 13 (2014), 1202-1238, [http://arXiv.org/abs/1301.6044]
doi: 10.1137/130913961. |
| [25] |
J. E. Marsden and M. McCracken, The Hopf Bifurcation and Its Applications, Springer-Verlag, New York, 1976. |
| [26] |
M. Moussaïd, E.G. Guillot, M. Moreau, J. Fehrenbach, O. Chabiron, S. Lemercier, J. Pettré, C. Appert-Rolland, P. Degond and G. Theraulaz, Traffic Instabilities in Self-Organized Pedestrian Crowds, PLoS Computational Biology, 8 (2012), e1002442, 1-10. Google Scholar |
| [27] |
A. Schadschneider and A. Seyfried,
Empirical results for pedestrian dynamics and their implications for modeling, Networks and Heterogeneous media, 6 (2011), 545-560.
doi: 10.3934/nhm.2011.6.545. |
| [28] |
J. P. Schiffer,
Phase transitions in anisotropically confined ionic crystals, Physical Review Letters, 70 (1993), 818-821.
doi: 10.1103/PhysRevLett.70.818. |
| [29] |
W. Tian, W. Song, J. Ma, Z. Fang, A. Seyfried and J. Liddle,
Experimental study of pedestrian behaviors in a corridor based on digital image processing, Fire Safety Journal, 47 (2012), 8-15.
doi: 10.1016/j.firesaf.2011.09.005. |
| [30] |
T. Vicsek and A. Zafeiris,
Collective motion, Physics Reports, 517 (2012), 71-140.
doi: 10.1016/j.physrep.2012.03.004. |
| [31] |
D. E. Wolf, M. Schreckenberg and A. Bachem (Eds.), Traffic and Granular Flow, World Scientific, Singapore, 1996.
doi: 10.1142/9789814531276. |
| [32] |
Z. Xiaoping, Z. Tingkuan and L. Mengting, Modeling crowd evacuation of a building based on seven methodological approaches, Building and Environment, 44 (2009), 437-445. Google Scholar |
show all references
References:
| [1] |
M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, Dover Publications, Inc., New York, 1966. |
| [2] |
N. Bellomo and C. Dogbe,
On the modeling of traffic and crowds: A survey of models, speculations, and perspectives, SIAM Review, 53 (2011), 409-463.
doi: 10.1137/090746677. |
| [3] |
V. J. Blue and J. L. Adler,
Cellular automata microsimulation for modeling bi-directional pedestrian walkways, Transportation Research Part B, 35 (2001), 293-312.
doi: 10.1016/S0191-2615(99)00052-1. |
| [4] |
C. Burstedde, K. Klauck, A. Schadschneider and J. Zittartz,
Simulation of pedestrian dynamics using a two-dimensional cellular automaton, Physica A, 295 (2001), 507-525.
doi: 10.1016/S0378-4371(01)00141-8. |
| [5] |
O. Corradi, P. G. Hjorth and J. Starke,
Equation-free detection and continuation of a Hopf bifurcation point in a particle model of pedestrian flow, SIAM Journal on Applied Dynamical Systems, 11 (2012), 1007-1032.
doi: 10.1137/110854072. |
| [6] |
T. Dessup, C. Coste and M. Saint Jean, Subcriticality of the zigzag transition: A nonlinear bifurcation analysis, Physical Review E, 91 (2015), 032917, 1-14.
doi: 10.1103/PhysRevE.91.032917. |
| [7] |
T. Dessup, T. Maimbourg, C. Coste and M. Saint Jean, Linear instability of a zigzag pattern, Physical Review E, 91 (2015), 022908, 1-12.
doi: 10.1103/PhysRevE.91.022908. |
| [8] |
F. Dietrich and G. Köster, Gradient navigation model for pedestrian dynamics, Physical Review E, 89 (2014), 062801, 1-8.
doi: 10.1103/PhysRevE.89.062801. |
| [9] |
J. E. Galván-Moya and F. M. Peeters, Ginzburg-Landau theory of the zigzag transition in quasi-one-dimensional classical Wigner crystals, Physical Review B, 84 (2011), 134106, 1-10. Google Scholar |
| [10] |
D. Helbing,
Traffic and related self-driven many-particle systems, Reviews of Modern Physics, 73 (2001), 1067-1141.
doi: 10.1103/RevModPhys.73.1067. |
| [11] |
D. Helbing, P. Molnar, I. J. Farkas and K. Bolay,
Self-organizing pedestrian movement, Environment and Planning B-planning and Design, 28 (2001), 361-383.
doi: 10.1068/b2697. |
| [12] |
D. Helbing and P. Molnar,
Social force model for pedestrian dynamics, Physical Review E, 51 (1995), 4282-4286.
doi: 10.1103/PhysRevE.51.4282. |
| [13] |
D. Helbing, I. Farkas and T. Vicsek,
Simulating dynamical features of escape panic, Nature, 407 (2000), 487-490.
doi: 10.1038/35035023. |
| [14] |
S. P. Hoogendoorn and W. Daamen,
Pedestrian behavior at bottlenecks, Transportation Science, 39 (2005), 147-288.
doi: 10.1287/trsc.1040.0102. |
| [15] |
A. Jelić, C. Appert-Rolland, S. Lemercier and J. Pettré, Properties of pedestrians walking in line: Fundamental diagrams, Physical Review E, 85 (2012), 036111, 1-9. Google Scholar |
| [16] |
A. Johansson and D. Helbing, Crowd dynamics, in: Econophysics and Sociophysics. Trends and Perspectives (eds. B.K. Chakrabarti, A. Chakraborti and A. Chatterjee), Wiley-VCH, Weinheim, (2006), 449-472.
doi: 10.1002/9783527610006.ch16. |
| [17] |
A. Johansson, D. Helbing and P. K. Shukla,
Specification of the social force pedestrian model by evolutionary adjustment to video tracking data, Advances in Complex Systems, 10 (2007), 271-288.
doi: 10.1142/S0219525907001355. |
| [18] |
B. S. Kerner, The physics of Traffic: Empirical Freeway Pattern Features, Engineering Applications, and Theory, 1st edition, Springer-Verlag, Berlin Heidelberg, 2004.
doi: 10.1007/978-3-540-40986-1. |
| [19] |
Y. G. Kevrekidis and G. Samaey,
Equation-free multiscale computation: Algorithms and applications, Annual Review of Physical Chemistry, 60 (2009), 321-344.
doi: 10.1146/annurev.physchem.59.032607.093610. |
| [20] |
H.-K. Li, E. Urban, C. Noel, A. Chuang, Y. Xia, A. Ransford, B. Hemmerling, Y. Wang, T. Li, H. Häffner and X. Zhang, Realization of translational symmetry in trapped cold ion rings, Physical Review Letters, 118 (2017), 053001, 1-5.
doi: 10.1103/PhysRevLett.118.053001. |
| [21] |
C. Marschler, J. Starke, M. P. Sørensen, Yu. Gaididei and P. L. Christiansen,
Pattern formation in annular systems of repulsive particles, Physics Letters A, 380 (2016), 166-170.
doi: 10.1016/j.physleta.2015.10.038. |
| [22] |
C. Marschler, J. Starke, P. Liu and Y. G. Kevrekidis, Coarse-grained particle model for pedestrian flow using diffusion maps, Physical Review E, 89 (2014), 013304, 1-11.
doi: 10.1103/PhysRevE.89.013304. |
| [23] |
C. Marschler, J. Sieber, P. G. Hjorth and J. Starke, Equation-free analysis of macroscopic behavior in traffic and pedestrian flow, in: Traffic and Granular Flow '13 (eds. M. Chraibi, M. Boltes, A. Schadschneider and A. X. Armin Seyfried) Springer-Verlag, (2015), 423-439. Google Scholar |
| [24] |
C. Marschler, J. Sieber, R. Berkemer, A. Kawamoto and J. Starke, Implicit methods for
equation-free analysis: Convergence results and analysis of emergent waves in microscopic
traffic models, SIAM Journal on Applied Dynamical Systems, 13 (2014), 1202-1238, [http://arXiv.org/abs/1301.6044]
doi: 10.1137/130913961. |
| [25] |
J. E. Marsden and M. McCracken, The Hopf Bifurcation and Its Applications, Springer-Verlag, New York, 1976. |
| [26] |
M. Moussaïd, E.G. Guillot, M. Moreau, J. Fehrenbach, O. Chabiron, S. Lemercier, J. Pettré, C. Appert-Rolland, P. Degond and G. Theraulaz, Traffic Instabilities in Self-Organized Pedestrian Crowds, PLoS Computational Biology, 8 (2012), e1002442, 1-10. Google Scholar |
| [27] |
A. Schadschneider and A. Seyfried,
Empirical results for pedestrian dynamics and their implications for modeling, Networks and Heterogeneous media, 6 (2011), 545-560.
doi: 10.3934/nhm.2011.6.545. |
| [28] |
J. P. Schiffer,
Phase transitions in anisotropically confined ionic crystals, Physical Review Letters, 70 (1993), 818-821.
doi: 10.1103/PhysRevLett.70.818. |
| [29] |
W. Tian, W. Song, J. Ma, Z. Fang, A. Seyfried and J. Liddle,
Experimental study of pedestrian behaviors in a corridor based on digital image processing, Fire Safety Journal, 47 (2012), 8-15.
doi: 10.1016/j.firesaf.2011.09.005. |
| [30] |
T. Vicsek and A. Zafeiris,
Collective motion, Physics Reports, 517 (2012), 71-140.
doi: 10.1016/j.physrep.2012.03.004. |
| [31] |
D. E. Wolf, M. Schreckenberg and A. Bachem (Eds.), Traffic and Granular Flow, World Scientific, Singapore, 1996.
doi: 10.1142/9789814531276. |
| [32] |
Z. Xiaoping, Z. Tingkuan and L. Mengting, Modeling crowd evacuation of a building based on seven methodological approaches, Building and Environment, 44 (2009), 437-445. Google Scholar |
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 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 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
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