Unravelling the puzzling intermediate states in the Biham-Middleton-Levine traffic mode

TL;DR

This paper analyzes the Biham-Middleton-Levine traffic model as an anisotropic system, revealing that its intermediate states are due to two distinct phase transitions related to system dimensions, clarifying longstanding mysteries.

Contribution

It identifies two separate phase transitions in the BML model based on anisotropic properties, explaining the origin of intermediate states and advancing understanding of anisotropic critical systems.

Findings

The model exhibits two differentiated phase transitions based on system orientation.

Intermediate states result from the superposition of two phase transition behaviors.

Mean-field analysis approximates critical densities for phase transitions.

Abstract

The Biham-Middleton-Levine (BML) traffic model, a cellular automaton with east-bound and north-bound cars moving by turns on a square lattice, has been an underpinning model in the study of collective behaviour 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 (north-east) and find that it exhibits two differentiated phase transitions: either if the system is 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 to the interesting result that the puzzling intermediate states in the original model are just a superposition of these two different behaviours of the phase transition, solving by the way most mysteries behind the BML model, which turns to be a paradigmatic example of such anisotropic critical systems.