R A Blythe et al J. Stat. Mech. (2004) P10007 doi:10.1088/1742-5468/2004/10/P10007
Dyck paths, Motzkin paths and traffic jams
R A Blythe1, W Janke2, D A Johnston3,5 and R Kenna4
Show affiliationsIt 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.
- Keywords
- PACS
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05.50.+q Lattice theory and statistics (Ising, Potts, etc.)
- MSC
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82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs
82B26 Phase transitions (general)
82B41 Random walks, random surfaces, lattice animals, etc. (See also 60G50, 82C41)
- Subjects
- Dates
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Issue 10 (October 2004)
Received 25 May 2004, accepted for publication 11 October 2004
Published 21 October 2004
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Dyck paths, Motzkin paths and traffic jams
R A Blythe et al J. Stat. Mech. (2004) P10007
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