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On a C-integrable equation for second sound propagation in heated dielectrics
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.
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. |
[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. |
[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. |
[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. |
[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. |
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. |
[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. |
[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. |
[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. |
[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. |














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