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_0v_0t_0 with c_0 the concentration, v_0 the velocity range, and 1/t_0 the passing rate, determines the nature of the steady state. When R<<1, uninterrupted flow with single cars occurs. When R>>1, large clusters with average mass <m> ~ R^{\alpha} form, and the flux is J ~ R^{-\gamma}. The initial distribution of slow cars governs the statistics. When P_0(v) ~ v^{\mu} as v->0, the scaling exponents are \gamma=1/(\mu+2), \alpha=1/2 when \mu>0, and \alpha=(\mu+1)/(\mu+2) when \mu<0.