Self-organized systems PACS 89.75.Kd -Patterns in complex systems PACS 45.70.Vn -Granular models of complex systems; traffic flowAbstract -We study a lattice model of two perpendicular intersecting flows of pedestrians represented by hard core particles of two types, eastbound ('E ') and northbound ('N '). Each flow takes place on a strip of width M so that the intersection is an M ×M square lattice. In experiment and simulation there occurs on this square spontaneous formation of a diagonal pattern of alternating E and N particles. We show that this pattern formation may be understood in terms of a linear instability of the corresponding mean field equations. A refined investigation reveals that the pattern actually consists of chevrons rather than straight diagonals. We explain this effect as the consequence of the existence of a nonlinear mode sustained by the interaction between the two types of particles.
We introduce a new update algorithm for exclusion processes, more suitable for the modeling of pedestrian traffic. Pedestrians are modeled as hard-core particles hopping on a discrete lattice, and are updated in a fixed order, determined by a phase attached to each pedestrian. While the case of periodic boundary conditions was studied in a companion paper, we consider here the case of open boundary conditions. The full phase diagram is predicted analytically and exhibits a transition between a free flow phase and a jammed phase. The density profile is predicted in the frame of a domain wall theory, and compared to Monte Carlo simulations, in particular in the vicinity of the transition.
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