Stochastic theory of freeway traffic

Extending a new stochastic approach to congestion in traffic flow [R. Mahnke and N. Pieret, Phys. Rev. E 56, 2666 (1997)], the nucleation, growth, and condensation of car clusters in a circular one-lane freeway traffic model is investigated. In analogy to usual aggregation phenomena such as the formation of liquid droplets in supersaturated vapor, the clustering behavior in traffic flow is described by the Master equation. At overcritical densities the transition from the initial free-particle situation (free flow of vehicles) to the final congested cluster state, where one big aggregate of cars has been formed, is shown. In dependence on the concentration of cars on the road, the stationary solution of the Master equation is derived analytically. The obtained fundamental diagram as a flow-density relation indicates clearly the different regimes of traffic flow (free jet of cars, coexisting phase of jams and isolated cars, and highly viscous heavy traffic). In the (thermodynamic) limit of an infinite number of vehicles on an infinitely long road, the analytical solution for the fundamental diagram is in agreement with experimental traffic flow data. As a particular example, we take into account measurements from German highways presented by Kerner and Rehborn.