Dyck paths, Motzkin paths and traffic jams

Richard Blythe, W Janke, D A Johnston, R Kenna

Research output: Contribution to journalArticlepeer-review

Abstract

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.

Original languageEnglish
Article numberP10007
Pages (from-to)-
Number of pages22
Journal Journal of Statistical Mechanics: Theory and Experiment
DOIs
Publication statusPublished - Oct 2004

Keywords

  • driven diffusive systems (theory)
  • ASYMMETRIC EXCLUSION MODEL
  • YANG-LEE THEORY
  • NONEQUILIBRIUM PHASE-TRANSITION
  • DRIVEN DIFFUSIVE SYSTEMS
  • OPEN BOUNDARIES
  • CELLULAR-AUTOMATON
  • LATTICE
  • DYNAMICS
  • STATES

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