Dyck paths, Motzkin paths and traffic jams
Dyck paths, Motzkin paths and traffic jams
复制标题
Dyck 路径、Motzkin 路径和交通堵塞
DOI：
--
复制
发表时间：
2004
期刊：
影响因子：
--
通讯作者：
R. Kenna
中科院分区：
文献类型：
--
作者：
R. Blythe;W. Janke;D. Johnston;R. Kenna
最近观察到，一维不平衡模型的归一化，即具有随机序列动力学的非对称排斥过程(ASEP)，恰好等价于一次过境行走的二维格子路径模型的配分函数，或者等价于Dyck路径。这解释了配分函数零点的Lee-Yang理论对ASEP归一化的适用性。在本文中，我们考虑了并行更新的ASEP的精确解，这是交通流的Nagel-Schrecenberg模型的一个特例，其中ASEP的相变可以解释为阻塞相变，并且发现Lee-Yang理论仍然适用。我们证明了并行更新的ASEP正规化可以表示为包含加权Dyck或Motzkin路径的几个等价的二维格子路径问题之一。我们引入了这种路径的热力学等价性的概念，并证明了在不同的更新动力学下ASEP相图的一般形式的稳健性是这种热力学等价性的结果。
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.