Abstract
Diagonal stripe formation is a well-known phenomenon in the pedestrian traffic community. Here we define a minimal model of intersecting traffic flows. It consists in an M × M space-discretized intersection on which two types of particles propagate towards east (\(\mathcal{E}\)) and north (\(\mathcal{N}\)), studied in the low density regime. It will also be shown that the behaviour of this model can be reproduced by a system of mean field equations. Using periodic boundary conditions the diagonal striped pattern is explained by an instability of the mean-field equations, supporting both the correspondence between equations and particle model and the generality of this pattern formation. With open boundary conditions, translational symmetry is broken. One then observes an asymmetry between the organization of the two types of particles, leading to tilted diagonals whose angle of inclination slightly differs from 45∘ both for the particle system and the equations. Even though the chevron effect does not appear in the linear stability analysis of the mean-field equations it can be understood in terms of effective interactions between particles, which enable us to isolate a macroscopic nonlinear propagation mode which accounts for it. The possibility to observe this last chevron effect on real pedestrians is then quickly discussed.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
S. Hoogendoorn, P.H. Bovy, Simulation of pedestrian flows by optimal control and differential games. Optim. Control Appl. Methods 24, 153–172 (2003)
K. Yamamoto, M. Okada, Continuum model of crossing pedestrian flows and swarm control based on temporal/spatial frequency, in 2011 IEEE International Conference on Robotics and Automation, Shanghai, 2011, pp. 3352–3357
S.P. Hoogendoorn, W. Daamen, Self-organization in walker experiments, in Traffic and Granular Flow ’03, ed. by S. Hoogendoorn, S. Luding, P. Bovy et al. (Springer, Berlin/New York, 2005), pp. 121–132
S.-I. Tadaki, Two-dimensional cellular automaton model of traffic flow with open boundaries. Phys. Rev. E 54, 2409–2413 (1996)
H. Hilhorst, C. Appert-Rolland, A multi-lane TASEP model for crossing pedestrian traffic flows. J. Stat. Mech. 2012, P06009 (2012)
O. Biham, A. Middleton, D. Levine, Self-organization and a dynamic transition in traffic-flow models. Phys. Rev. A 46, R6124–R6127 (1992)
Z.-J. Ding, R. Jiang, B.-H. Wang, Traffic flow in the Biham-Middleton-Levine model with random update rule. Phys. Rev. E 83, 047101 (2011)
J. Cividini, C. Appert-Rolland, H.J. Hilhorst, Diagonal patterns and chevron effect in intersecting traffic flows. Europhys. Lett. 102, 20002 (2013)
J. Cividini, C. Appert-Rolland, Wake-mediated interaction between driven particles crossing a perpendicular flow. J. Stat. Mech. 2013, P07015 (2013)
Acknowledgements
I thank C. Appert-Rolland and H.J. Hilhorst for their collaboration in this work.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
Cividini, J. (2015). Generic Instability at the Crossing of Pedestrian Flows. In: Chraibi, M., Boltes, M., Schadschneider, A., Seyfried, A. (eds) Traffic and Granular Flow '13. Springer, Cham. https://doi.org/10.1007/978-3-319-10629-8_2
Download citation
DOI: https://doi.org/10.1007/978-3-319-10629-8_2
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-10628-1
Online ISBN: 978-3-319-10629-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)