Burgers equation for kinetic clustering in traffic flow

A kinetic clustering of cars is analyzed using a limiting procedure and a reductive perturbation method. By using the limiting procedure, the difference–difference equation to describe the clustering is obtained. We derive the coarse-grained equation describing the hydrodynamic mode, using the reductive perturbation method. It is shown that this hydrodynamic equation is given by the Burgers equation and that the typical headway and velocity scale as t1/2 and t−1/2 in time for the initial random velocity distribution.