# American Institute of Mathematical Sciences

June  2014, 7(3): 395-409. doi: 10.3934/dcdss.2014.7.395

## Pattern formation in 2D traffic flows

 1 Laboratoire de Physique Théorique, bâtiment 210, Université Paris-Sud, 91405 Orsay Cedex, France

Received  May 2013 Revised  September 2013 Published  January 2014

We review numerical and analytical results that have been obtained for a general model of intersecting flows of two types of particles propagating towards east ($\varepsilon$) and north ($\mathcal{N}$) on a bidimensional $M \times M$ square lattice. The behaviour of this model can also be reproduced by a system of mean field equations. The low density behaviour of both models is studied, with a focus on pattern formation. Using periodic boundary conditions, particles self-organize into a pattern of alternating diagonal stripes, which corresponds to an instability in the mean-field equations. 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 differs from $45^\circ$, both for the particle system and the equations. The angle of inclination of the stripes is measured using two different numerical methods. Finally, simplified theoretical arguments based on a particular mode of propagation give a quantitative estimate for the angle of the stripes. A complementary understanding of the phenomenon in terms of effective interactions between particles is presented.
Citation: Julien Cividini. Pattern formation in 2D traffic flows. Discrete and Continuous Dynamical Systems - S, 2014, 7 (3) : 395-409. doi: 10.3934/dcdss.2014.7.395
##### References:
 [1] C. Appert-Rolland, J. Cividini and H. J. Hilhorst, Frozen shuffle update for an asymmetric exclusion process with open boundary conditions, J. Stat. Mech., (2011), P10013. [2] O. Biham, A. A. Middleton and D. Levine, Self-organization and a dynamic transition in traffic-flow models, Phys. Rev. A, 46 (1992), R6124-R6127. doi: 10.1103/PhysRevA.46.R6124. [3] D. Chowdhury, L. Santen and A. Schadschneider, Statistical physics of vehicular traffic and some related systems, Phys. Rep., 329 (2000), 199-329. doi: 10.1016/S0370-1573(99)00117-9. [4] J. Cividini and C. Appert-Rolland, Wake-mediated interaction between driven particles crossing a perpendicular flow, J. Stat. Mech., 2013 (2013), P07015. doi: 10.1088/1742-5468/2013/07/P07015. [5] J. Cividini, C. Appert-Rolland and H. J. Hilhorst, Diagonal patterns and chevron effect in intersecting traffic flows, Europhys. Lett., 102 (2013), 20002. doi: 10.1209/0295-5075/102/20002. [6] J. Cividini, H. J. Hilhorst and C. Appert-Rolland, Crossing pedestrian traffic flows,diagonal stripe pattern, and chevron effect, J. Phys. A: Math. Theor., 46 (2013), 29 pp. [7] Z.-J. Ding, R. Jiang and B.-H. Wang, Traffic flow in the Biham-Middleton-Levine model with random update rule, Phys. Rev. E, 83 (2011), 047101. doi: 10.1103/PhysRevE.83.047101. [8] R. M. D'Souza, Coexisting phases and lattice dependence of a cellular automaton model for traffic flow, Phys. Rev. E, 71 (2005), 066112. doi: 10.1103/PhysRevE.71.066112. [9] H. J. Hilhorst and C. Appert-Rolland, A multi-lane TASEP model for crossing pedestrian traffic flows, J. Stat. Mech., 2012 (2012), P06009. doi: 10.1088/1742-5468/2012/06/P06009. [10] J. M. Molera, F. C. Martinez, J. A. Cuesta and R. Britot, Theoretical approach to two-dimensional trafIic flow models, Phys. Rev. E, 51 (1995), 175-187. doi: 10.1103/PhysRevE.51.175. [11] N. H. Packard and S. Wolfram, Two-dimensional cellular automata, J. Stat. Phys., 38 (1985), 901-946. doi: 10.1007/BF01010423. [12] S.-I. Tadaki, Two-dimensional cellular automaton model of traffic flow with open boundaries, Phys. Rev. E, 54 (1996), 2409-2413. doi: 10.1103/PhysRevE.54.2409. [13] S. Wolfram, Statistical mechanics of cellular automata, Rev. Mod. Phys., 55 (1983), 601-644. doi: 10.1103/RevModPhys.55.601.

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##### References:
 [1] C. Appert-Rolland, J. Cividini and H. J. Hilhorst, Frozen shuffle update for an asymmetric exclusion process with open boundary conditions, J. Stat. Mech., (2011), P10013. [2] O. Biham, A. A. Middleton and D. Levine, Self-organization and a dynamic transition in traffic-flow models, Phys. Rev. A, 46 (1992), R6124-R6127. doi: 10.1103/PhysRevA.46.R6124. [3] D. Chowdhury, L. Santen and A. Schadschneider, Statistical physics of vehicular traffic and some related systems, Phys. Rep., 329 (2000), 199-329. doi: 10.1016/S0370-1573(99)00117-9. [4] J. Cividini and C. Appert-Rolland, Wake-mediated interaction between driven particles crossing a perpendicular flow, J. Stat. Mech., 2013 (2013), P07015. doi: 10.1088/1742-5468/2013/07/P07015. [5] J. Cividini, C. Appert-Rolland and H. J. Hilhorst, Diagonal patterns and chevron effect in intersecting traffic flows, Europhys. Lett., 102 (2013), 20002. doi: 10.1209/0295-5075/102/20002. [6] J. Cividini, H. J. Hilhorst and C. Appert-Rolland, Crossing pedestrian traffic flows,diagonal stripe pattern, and chevron effect, J. Phys. A: Math. Theor., 46 (2013), 29 pp. [7] Z.-J. Ding, R. Jiang and B.-H. Wang, Traffic flow in the Biham-Middleton-Levine model with random update rule, Phys. Rev. E, 83 (2011), 047101. doi: 10.1103/PhysRevE.83.047101. [8] R. M. D'Souza, Coexisting phases and lattice dependence of a cellular automaton model for traffic flow, Phys. Rev. E, 71 (2005), 066112. doi: 10.1103/PhysRevE.71.066112. [9] H. J. Hilhorst and C. Appert-Rolland, A multi-lane TASEP model for crossing pedestrian traffic flows, J. Stat. Mech., 2012 (2012), P06009. doi: 10.1088/1742-5468/2012/06/P06009. [10] J. M. Molera, F. C. Martinez, J. A. Cuesta and R. Britot, Theoretical approach to two-dimensional trafIic flow models, Phys. Rev. E, 51 (1995), 175-187. doi: 10.1103/PhysRevE.51.175. [11] N. H. Packard and S. Wolfram, Two-dimensional cellular automata, J. Stat. Phys., 38 (1985), 901-946. doi: 10.1007/BF01010423. [12] S.-I. Tadaki, Two-dimensional cellular automaton model of traffic flow with open boundaries, Phys. Rev. E, 54 (1996), 2409-2413. doi: 10.1103/PhysRevE.54.2409. [13] S. Wolfram, Statistical mechanics of cellular automata, Rev. Mod. Phys., 55 (1983), 601-644. doi: 10.1103/RevModPhys.55.601.
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