Kinetics of clustering in traffic flows

We study a simple aggregation model that mimics the clustering of traffic on a one-lane roadway. In this model, each ‘‘car’’ moves ballistically at its initial velocity until it overtakes the preceding car or cluster. After this encounter, the incident car assumes the velocity of the cluster which it has just joined. The properties of the initial distribution of velocities in the small-velocity limit control the long-time properties of the aggregation process. For an initial velocity distribution with a power-law tail at small velocities, ${\mathit{P}}_0$(v)∼${\mathit{v}}^{\mathrm{\mu }}$ as v→0, a simple scaling argument shows that the average cluster size grows as m∼${\mathit{t}}^{(\mathrm{\mu }+1)/(\mathrm{\mu }+2)}$ and that the average velocity decays as v∼${\mathit{t}}^{\mathrm{-}1/(\mathrm{\mu }+2)}$ as t→∞. We derive an analytical solution for the survival probability of a single car and an asymptotically exact expression for the joint mass-velocity distribution function. We also consider the properties of spatially heterogeneous traffic and the kinetics of traffic clustering in the presence of an input of cars.