A kinetic model of vehicular traffic and its associated bimodal equilibrium solutions

Traffic flow in a single lane (no passing) with all vehicles having the same desired speed is considered, with a view toward developing a kinetic equation in which speeding-up interactions are treated in the same fundamental manner as traditionally has been the case for slowing-down interactions. Such a kinetic equation is developed, based upon a specific correlation model expressing the leading-vehicle distribution function in terms of the single-vehicle distribution function and a specific mechanical model describing the circumstances and manner in which drivers change speeds in response to their situation vis-à-vis the vehicle immediately ahead of them. The mechanical model employs an extension (the generalized vehicular chaos hypothesis) of the limited form of the vehicular chaos hypothesis that classically has been employed for the term representing slowing-down interactions. The distributions that represent (local) equilibrium solutions of this equation on the time scale of interactions of individual vehicles are shown to comprise a family of bimodal distributions, with peaks at zero speed and at the desired speed. Interpretations and consequences of such equilibria are discussed from the view of traffic flow theory, especially in regard to the corresponding traffic stream model (flow-density relation) and to stop-and-go traffic patterns.