Single- and multi-population kinetic models for vehicular traffic reproducing fundamental diagrams and with low computational complexity.

In this work, we focus on kinetic theory of vehicular traffic. We introduce (Boltzmann and Fokker-Planck) models having the following properties: they are amenable for computations and analytical investigations, but at the same time they are able to characterize and to explain the features of experimental diagrams. The scattering observed in experimental data is reproduced by a multi-population model. We propose a new interpretation of the dispersion of data since it can be attributed to the heterogeneous composition of the flow. In fact, the scattering is obtained by treating traffic as a mixture of vehicles with different physical and kinematic characteristics. The multi-population model is built as generalization of a new single-population model for which the analytical expression of the steady state can be computed explicitly. This is possible thanks to the particular choice of the microscopic interactions. These models are able to catch the macroscopic properties of the flow at equilibrium, as the phase transition, the capacity drop and the scattering of data. The proposed models are endowed with a robust mathematical structure. We study the mathematical properties which induce the structure of diagrams, the well posedness with the existence and uniqueness proof of the solution of the kinetic equations. A further result of this thesis is the analysis of the effects of the microscopic interactions on the macroscopic dynamics. This purely multiscale issue which is tackled by an asymptotic study of the model in the Fokker-Planck limit.