Kinetics of Clustering in Traffic Flows,
E. Ben-Naim, P.L. Krapivsky and S. Redner
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,
$\pvim$ as $v\to 0$, a simple scaling argument shows that the average
cluster size grows as $n \sim t^{\va}$ and that the average velocity
decays as $v \sim t^{-\vb}$ as $t\to \infty$. 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.
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