Stationary Velocity Distributions in Traffic Flows,
E. Ben-Naim and P.L. Krapivsky
We introduce a traffic flow model that incorporates clustering and
passing. We obtain analytically the steady state characteristics of
the flow from a Boltzmann-like equation. A single dimensionless
parameter, $R=c_0v_0t_0$ with $c_0$ the concentration, $v_0$ the
velocity range, and $t_0^{-1}$ the passing rate, determines the nature
of the steady state. When $R\ll 1$, uninterrupted flow with single
cars occurs. When $R\gg 1$, large clusters with average mass $\langle
m\rangle\sim R^{\alpha}$ form, and the flux is $J\sim R^{-\gamma}$.
The initial distribution of slow cars governs the statistics. When
$P_0(v)\sim v^{\mu}$ as $v\to 0$, the scaling exponents are
$\gamma=1/(\mu+2)$, $\alpha=1/2$ when $\mu>0$, and
$\alpha=(\mu+1)/(\mu+2)$ when $\mu<0$.
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