Exact steady states of disordered hopping particle models with parallel and ordered sequential dynamics

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. The phase transition is shown to persist in both cases with the critical densities being higher than that for random sequential dynamics. The steady-state velocities are related to the fugacity of a Bose system suggesting a principle of minimization of velocity. A generalization of the dynamics, to the case 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 Bow models is discussed.