Exact solution of a cellular automaton for traffic

TL;DR

This paper provides an exact matrix product solution for a probabilistic cellular automaton modeling traffic flow with open boundaries, revealing new algebraic structures and phase behavior in the system.

Contribution

It introduces a novel quartic algebra for the stationary state of the ASEP with simultaneous updating, extending understanding of traffic models and their phase diagrams.

Findings

Derived the phase diagram for the traffic model.

Computed density profiles and correlation functions.

Mapped results onto other ASEP updating schemes.

Abstract

We present an exact solution of a probabilistic cellular automaton for traffic with open boundary conditions, e.g. cars can enter and leave a part of a highway with certain probabilities. The model studied is the asymmetric exclusion process (ASEP) with {\it simultaneous} updating of all sites. It is equivalent to a special case ($v_{\rm max}=1$) of the Nagel-Schreckenberg model for highway traffic, which has found many applications in real-time traffic simulations. The simultaneous updating induces additional strong short range correlations compared to other updating schemes. The stationary state is written in terms of a matrix product solution. The corresponding algebra, which expresses a system-size recursion relation for the weights of the configurations, is quartic, in contrast to previous cases, in which the algebra is quadratic. We derive the phase diagram and compute various properties such as density profiles, two point functions and the fluctuations in the number of particles (cars) in the system. The current and the density profiles can be mapped onto the ASEP with other time discrete updating procedures. Through use of this mapping, our results also give new results for these models.