Maxwell model of Traffic Flows,

E. Ben-Naim and P.L. Krapivsky

We investigate traffic flows using the kinetic Boltzmann equations with a Maxwell collision integral. This approach allows analytical determination of the transient behavior and the size distributions. The relaxation of the car and cluster velocity distributions towards steady state is characterized by a wide range of velocity dependent relaxation scales. Furthermore, these relaxation time scales decrease with the velocity, with the smallest scale corresponding to the decay of the overall density. The steady state cluster size distribution follows an unusual scaling form $P_m \sim \langle m\rangle^{-4} \Psi(m/\langle m\rangle^2)$. This distribution is primarily algebraic, $P_m\sim m^{-3/2}$, for $m\ll \langle m\rangle^2$, and is exponential otherwise.


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