On Cellular Automata Models of Single Lane Traffic

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 $v_{max}=\infty$.