## 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 simultaneous updating of all sites. It is equivalent to a special case (upsilon(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, Wt 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. Thr 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.

Original language | English |
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Pages (from-to) | 45-96 |

Number of pages | 52 |

Journal | Journal of Statistical Physics |

Volume | 95 |

Issue number | 1-2 |

Publication status | Published - Apr 1999 |

## Keywords

- asymmetric exclusion process
- boundary-induced phase transitions
- steady state
- matrix product Ansatz
- cellular automaton
- ASYMMETRIC EXCLUSION MODEL
- OPEN BOUNDARIES
- QUADRATIC ALGEBRAS
- STOCHASTIC-MODELS
- PARALLEL DYNAMICS
- DIFFUSION PROBLEM
- SHOCK PROFILES
- REPRESENTATIONS
- STATES
- UPDATE