Stationary velocity distributions in traffic flows

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, ${R=c}_0{v}_0{t}_0$ 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 $〈m〉\sim R^{\alpha }$ form, and the flux is $J\sim R^{-\gamma }.$ The initial distribution of slow cars governs the statistics. When $P_0\mathrm{}\left(v\right)\mathrm{}\sim v^{\mu }$ as $v\to 0,$ the scaling exponents are $\gamma =1/(\mu +2),$ $\alpha =1/2\mathrm{}$ when $\mu >0,$ and $\alpha =(\mu +1)/(\mu +2)$ when $\mu <0.\mathrm{}$