Finite Size Effects in the Nagel-Schreckenberg Traffic Model

TL;DR

This paper investigates finite-size effects in the Nagel-Schreckenberg traffic model across different maximum speeds, revealing universal scaling behavior near the jamming transition and how it depends on maximum speed.

Contribution

It provides a finite-size scaling analysis of the jamming transition in the Nagel-Schreckenberg model, highlighting the dependence of scaling exponents on maximum speed.

Findings

Finite-size effects are significant at the transition for large maximum speeds.

Universal scaling behavior is observed with exponents depending on $V_{max}$.

Jamming nucleation occurs at large $V_{max}$, but not at small $V_{max}$.

Abstract

We examine the Nagel-Schreckenberg traffic model for a variety of maximum speeds. We show that the low density limit can be described as a dilute gas of vehicles with a repulsive core. At the transition to jamming, we observe finite-size effects in a variety of quantities describing the flow and the density correlations, but only if the maximum speed $V_{\text{max}}$ is larger than a certain value. A finite-size scaling analysis of several order parameters shows universal behavior, with scaling exponents that depend on $V_{\text{max}}$. The jamming transition at large $V_{\text{max}}$ can be viewed as the nucleation of jams in a background of freely flowing vehicles. For small $V_{\text{max}}$ no such clean separation into jammed and free vehicles is possible.