Jamming transition in a homogeneous one-dimensional system: the Bus Route Model

TL;DR

This paper introduces the Bus Route Model, a one-dimensional driven diffusive system demonstrating a transition from a jammed to a homogeneous phase, with the transition becoming a sharp crossover as passenger arrival rate approaches zero.

Contribution

The study provides a new model combining simulation, heuristic, and mean-field approaches to analyze jamming transitions in a one-dimensional system with conserved and non-conserved variables.

Findings

Evidence of a phase transition between jammed and homogeneous states.

Transition becomes a sharp crossover as passenger arrival rate approaches zero.

Coarsening dynamics of gaps in the jammed phase studied.

Abstract

We present a driven diffusive model which we call the Bus Route Model. The model is defined on a one-dimensional lattice, with each lattice site having two binary variables, one of which is conserved (``buses'') and one of which is non-conserved (``passengers''). The buses are driven in a preferred direction and are slowed down by the presence of passengers who arrive with rate lambda. We study the model by simulation, heuristic argument and a mean-field theory. All these approaches provide strong evidence of a transition between an inhomogeneous ``jammed'' phase (where the buses bunch together) and a homogeneous phase as the bus density is increased. However, we argue that a strict phase transition is present only in the limit lambda -> 0. For small lambda, we argue that the transition is replaced by an abrupt crossover which is exponentially sharp in 1/lambda. We also study the coarsening of gaps between buses in the jammed regime. An alternative interpretation of the model is given in which the spaces between ``buses'' and the buses themselves are interchanged. This describes a system of particles whose mobility decreases the longer they have been stationary and could provide a model for, say, the flow of a gelling or sticky material along a pipe.