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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