### Abstract

Original language | English |
---|---|

Article number | P10007 |

Journal | Journal of Statistical Mechanics: Theory and Experiment |

Volume | 2004 |

DOIs | |

Publication status | Published - 21 Oct 2004 |

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### Bibliographical note

The full text is also available from: http://de.arxiv.org/abs/cond-mat/0405314This is an author-created, un-copyedited version of an article accepted for publication/published in Journal of Statistical Mechanics: Theory and Experiment. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record is available online at http://dx.doi.org/10.1088/1742-5468/2004/10/P10007.

### Cite this

*Journal of Statistical Mechanics: Theory and Experiment*,

*2004*, [P10007]. https://doi.org/10.1088/1742-5468/2004/10/P10007

**Dyck paths, Motzkin paths and traffic jams.** / Blythe, R. A.; Janke, W.; Johnston, D. A.; Kenna, Ralph.

Research output: Contribution to journal › Article

*Journal of Statistical Mechanics: Theory and Experiment*, vol. 2004, P10007. https://doi.org/10.1088/1742-5468/2004/10/P10007

}

TY - JOUR

T1 - Dyck paths, Motzkin paths and traffic jams

AU - Blythe, R. A.

AU - Janke, W.

AU - Johnston, D. A.

AU - Kenna, Ralph

N1 - The full text is also available from: http://de.arxiv.org/abs/cond-mat/0405314 This is an author-created, un-copyedited version of an article accepted for publication/published in Journal of Statistical Mechanics: Theory and Experiment. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record is available online at http://dx.doi.org/10.1088/1742-5468/2004/10/P10007.

PY - 2004/10/21

Y1 - 2004/10/21

N2 - It has recently been observed that the normalization of a one-dimensional out-of-equilibrium model, the asymmetric exclusion process (ASEP) with random sequential dynamics, is exactly equivalent to the partition function of a two-dimensional lattice path model of one-transit walks, or equivalently Dyck paths. This explains the applicability of the Lee–Yang theory of partition function zeros to the ASEP normalization. In this paper we consider the exact solution of the parallel-update ASEP, a special case of the Nagel–Schreckenberg model for traffic flow, in which the ASEP phase transitions can be interpreted as jamming transitions, and find that Lee–Yang theory still applies. We show that the parallel-update ASEP normalization can be expressed as one of several equivalent two-dimensional lattice path problems involving weighted Dyck or Motzkin paths. We introduce the notion of thermodynamic equivalence for such paths and show that the robustness of the general form of the ASEP phase diagram under various update dynamics is a consequence of this thermodynamic equivalence.

AB - It has recently been observed that the normalization of a one-dimensional out-of-equilibrium model, the asymmetric exclusion process (ASEP) with random sequential dynamics, is exactly equivalent to the partition function of a two-dimensional lattice path model of one-transit walks, or equivalently Dyck paths. This explains the applicability of the Lee–Yang theory of partition function zeros to the ASEP normalization. In this paper we consider the exact solution of the parallel-update ASEP, a special case of the Nagel–Schreckenberg model for traffic flow, in which the ASEP phase transitions can be interpreted as jamming transitions, and find that Lee–Yang theory still applies. We show that the parallel-update ASEP normalization can be expressed as one of several equivalent two-dimensional lattice path problems involving weighted Dyck or Motzkin paths. We introduce the notion of thermodynamic equivalence for such paths and show that the robustness of the general form of the ASEP phase diagram under various update dynamics is a consequence of this thermodynamic equivalence.

U2 - 10.1088/1742-5468/2004/10/P10007

DO - 10.1088/1742-5468/2004/10/P10007

M3 - Article

VL - 2004

JO - Journal of Statistical Mechanics: Theory and Experiment

JF - Journal of Statistical Mechanics: Theory and Experiment

SN - 1742-5468

M1 - P10007

ER -