Unraveling the puzzling intermediate states in the Biham-Middleton-Levine traffic model

Abstract

The Biham-Middleton-Levine (BML) traffic model, a cellular automaton with eastbound and northbound cars moving by turns on a square lattice, has been an underpinning model in the study of collective behavior by cars, pedestrians, and even internet packages. Contrary to initial beliefs that the model exhibits a sharp phase transition from freely flowing to fully jammed, it has been reported that it shows intermediate stable phases, where jams and freely flowing traffic coexist, but there is no clear understanding of their origin. Here, we analyze the model as an anisotropic system with a preferred fluid direction (northeast) and find that it exhibits two differentiated phase transitions: the system is either longer in the flow direction (longitudinal) or perpendicular to it (transversal). The critical densities where these transitions occur enclose the density interval of intermediate states and can be approximated by mean-field analysis, all derived from the anisotropic exponent relating the longitudinal and transversal correlation lengths. Thus, we arrive at the interesting result that the puzzling intermediate states in the original model are just a superposition of these two different behaviors of the phase transition, solving by the way most mysteries behind the BML model, which turns out to be a paradigmatic example of such anisotropic critical systems.

Traffic gridlock on a honeycomb city

“…A comparison with the square lattice. The critical density ρ c =0.244(3) for the BML model on a honeycomb is lower that the value of 0.283(2) for the lowest transition on a square lattice [28]. However, this order is reversed in at least two cases.…”

“…Driven by car density, the system falls into three different phases: free flow (all vehicles move), jammed phase (all vehicles are stuck) and intermediate states where jams and free flow coexist on a wide density range [25][26][27]. A recent study have shown that such intermediate states are a consequence of the anisotropy inherent to the model [28], which produces two different phase transitions: one if the system is longer in the flow direction (longitudinal) and other if the system is longer in the perpendicular one (transversal). It has also been reported that this intermediate phase disappears when some kind of randomization is introduced [26,29,30], or the traffic periods for the two cars are increased [31].…”

“…If the density is large enough, the system reaches a jamming state after a transient period. Following the methods applied in [28], we define the parallel (perpendicular) spatial correlation function [18] as…”

The impact of intelligent vehicles on a two-route system with a work zone

“…This problem has attracted the interest of scientists and engineers to research the dynamic characteristics of tra±c°ow, and a lot of models have been proposed, such as°uid *Corresponding author. models, [1][2][3][4] car-following models, 5-10 cellular automata models (CA model) [11][12][13][14][15][16][17] and so on. By those models, some basic characteristics of tra±c°ow, i.e.…”