On Cellular Automata Models of Single Lane Traffic

TL;DR

This paper analyzes the jamming transition in cellular automaton traffic models, revealing different behaviors depending on speed limits and connecting these to surface growth and phase transition theories.

Contribution

It provides a detailed analysis of the jamming transition in cellular automaton traffic models, linking it to surface growth classes and phase transition phenomena.

Findings

For v_max=1, the model relates to KPZ surface growth.

At 1<v_max<∞, the relaxation time peaks near the jamming density.

In the v_max=∞ limit, the model exhibits a first order transition.

Abstract

The jamming transition in the stochastic cellular automaton model (Nagel-Schreckenberg model) of highway traffic is analyzed in detail, by studying the relaxation time, a mapping to surface growth problems and the investigation of correlation functions. Three different classes of behavior can be distinguished depending on the speed limit $v_{max}$. For $v_{max} = 1$ the model is closely related to KPZ class of surface growth. For $1<v_{max} < \infty$ the relaxation time has a well defined peak at a density of cars $\rho$ somewhat lower than position of the maximum in the fundamental diagram: This density can be identified with the jamming point. At the jamming point the properties of the correlations also change significantly. In the $v_{max}=\infty $ limit the model undergoes a first order transition at $\rho \to 0$. It seems that in the relevant cases $1<v_{max} < \infty$ the jamming transition is under the influence of second order phase transition in the deterministic model and of the first order transition at $v_{max}=\infty $.