Exact Steady States of Disordered Hopping Particle Models with Parallel and Ordered Sequential Dynamics

TL;DR

This paper provides exact solutions for the steady states of disordered one-dimensional driven lattice gases under various updating schemes, revealing persistent phase transitions and linking steady state velocity to Bose system fugacity, with implications for traffic flow modeling.

Contribution

It presents the first exact solutions for steady states in disordered hopping particle models with parallel and ordered sequential dynamics, extending previous work on random sequential updates.

Findings

Phase transition persists across different updating schemes.

Critical density varies with the type of dynamics, higher in ordered updates.

Steady state velocity relates to Bose system fugacity, indicating a velocity minimization principle.

Abstract

A one-dimensional driven lattice gas with disorder in the particle hopping probabilities is considered. It has previously been shown that in the version of the model with random sequential updating, a phase transition occurs from a low density inhomogeneous phase to a high density congested phase. Here the steady states for both parallel (fully synchronous) updating and ordered sequential updating are solved exactly and the phase transition shown to persist in both cases. For parallel dynamics and forward ordered sequential dynamics the phase transition occurs at the same density but for backward ordered sequential dynamics it occurs at a higher density. In both cases the critical density is higher than that for random sequential dynamics. In all the models studied the steady state velocity is related to the fugacity of a Bose system suggesting a principle of minimisation of velocity. A generalisation of the dynamics where the hopping probabilities depend on the number of empty sites in front of the particles, is also solved exactly in the case of parallel updating. The models have natural interpretations as simplistic descriptions of traffic flow. The relation to more sophisticated traffic flow models is discussed.