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PATTERN FORMATION IN 2D TRAFFIC FLOWS

机译：二维交通流中的花样形成

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摘要

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 (ε) and north (N) on a bidimensional M × 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°, 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.
机译：我们回顾了在二维M×M方格上，两种类型的粒子向东（ε）和北（N）传播的相交流的一般模型的数值模型和分析结果。该模型的行为也可以通过平均场方程组来重现。研究了两个模型的低密度行为，重点是图案形成。使用周期性边界条件，粒子会自组织成交替的对角线条纹图案，这对应于均场方程中的不稳定性。在开放边界条件下，平移对称性被破坏。然后，人们观察到两种类型粒子的组织之间的不对称性，导致对粒子系统和方程式均产生倾斜的对角线，其倾斜角与45°不同。使用两种不同的数值方法来测量条纹的倾斜角度。最后，基于特定传播模式的简化理论论证给出了条纹角度的定量估计。提出了关于颗粒之间有效相互作用的现象的补充理解。

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