A Stochastic Multi—Cluster Model of Freeway Traffic

Abstract

A stochastic approach based on the Master equation is proposed to describe the process of formation and growth of car clusters in traffic flow in analogy to usual aggregation phenomena such as the formation of liquid droplets in supersaturated vapour. We have extended the stochastic theory of freeway traffic allowing a coexistence of many clusters on a one-lane circular road. Analytical equations have been derived for calculation of the stationary cluster distribution and related physical quantities of an infinitely large system of interacting cars. If the probability per time to decelarate a car without an obvious reason tends to zero in an infinitely large system, our multi-cluster model behaves essentially in the same way as the one-cluster model presented at Traffic and Granular Flow ’97. In particular, there are three different regimes of traffic flow (free jet of cars, coexisting phase of jams and isolated cars, highly viscous heavy traffic) and two phase transitions between them. In a general case, some qualitative differences in the behaviour of these two models have been observed.