STATISTICAL HYDRODYNAMICS OF TRAFFIC FLOW. IN VEHICULAR TRAFFIC SCIENCE

In a prior paper, the authors studied the relation between the macroscopic hydrodynamic description of traffic flow and the statistical approach as based on a generalized kinetic equation. It was shown that a local steady-state theory (SST) leads to the undamped kinematic waves of the Lighthill-Whitham macroscopic hydrodynamic description. This steady-state approach can usually only be valid in the limit of perturbations characterized by a sufficiently large characteristic length. The local SST may then be considered as a long wavelength theory. This paper deals with the space-time behavior of traffic flow for the entire relevant range of wavelengths from the average distance between cars to infinity. The dispersion and absorption of the perturbation waves are discussed by analyzing the behavior of traffic flow under the influence of small perturbations. This is possible only on the basis of a statistical model. Closely related to the problem of the absorption of perturbation waves is the question of stability of traffic flow. Indeed, a positive absorption coefficient means the amplification of initially present inhomogeneities. This paper then, presents also a new alternate approach to the problem of stability of traffic flow under circumstances in which an approach based on the "follow the leader" type theory is not applicable since in the latter theory passing is not allowed, while in this approach it is the dominant relaxation mechanism.