Kinetics of clustering in traffic flows.

A variety of approaches have been applied to describe the collective properties of traffic flows [1]. For example, to mimic congested traffic flow in two dimensions, cellular automation models have been proposed [2,3]. Asymmetric hopping processes have also been applied to model traffic flow on a one-dimensional road [4-6]. When the number of cars is large, traffic flows can be modelled phenomenologically in terms of a one-dimensional compressible gas [7-9]. Such an approach predicts the appearance of shock waves, where hydrodynamic quantities, such as the average density and velocity, become discontinuous. However, the hydrodynamic approach does not naturally describe the behavior of traffic flows in the low-density limit where there are large heterogeneities in traffic density. For this situation, a microscopic model may provide a more appropriate description. In this article, we introduce a ballistic aggregation process to model the kinetics of clustering in one-dimensional traffic flows. Our approach is inspired, in part, by the recent interesting results that have been obtained for a variety of reaction processes which involve ballistic particles including: ballistic agglomeration, Ai+Aj → Ai+j , with momentum conserving collisions [10,11]; ballistic annihilation, A + A → 0, [12,13]; and several nucleation and ballistic growth processes [14-16]. In our model, cars move ballistically in a one direction, say to the right, according to an initial velocity distribution. Clusters form whenever a faster car overtakes a slower car or cluster. The overtaking car then assumes the velocity of the lead car in the cluster. This model is an idealized description for one-lane traffic flow. While there are obvious shortcomings in our model, it is exactly soluble and permits a thorough understanding of the kinetics of the aggregation process. This paper is organized as follows. In section II, we present the model and postulate the scaling behavior for the velocity and the concentration of the clusters. This approach makes use of the statistical properties of the minimal random variable within a large sample. In section III, we investigate the distribution of cluster velocities. For this distribution, the cluster size is irrelevant and this feature allows us to consider a simpler “coalescence only” model. For this reduced problem, the velocity distribution is obtained exactly in terms of the initial distribution of car velocities and then evaluated for general continuous distributions. Building on these results, the general clustering process is solved in section IV and an asymptotically exact expression for the joint cluster mass-velocity distribution is obtained. In section V, we present a formal solution for the velocity distribution function for an inhomogeneous initial distribution of particles. We examine the temporal behavior that arises for a simple step function initial spatial distribution. In section VI, we investigate another generalization of the model to the situation with a spatially and temporally homogeneous input of cars. Depending on the functional form of the input velocity distribution in the low-velocity limit, the input can give rise to a steady state or to a system which continues to evolve indefinitely. We give our conclusions in section VII. The details of specific calculations are given in the Appendices.