Kinetic models for traffic flow resulting in a reduced space of microscopic velocities
For researchers in traffic flow modeling, this provides a theoretical bridge between kinetic and macroscopic models, but the result is incremental as it extends existing Boltzmann-type approaches.
The paper shows that kinetic traffic flow models with continuous velocity space can yield steady-state distributions supported on a quantized, small number of speeds, enabling simpler macroscopic equations. Numerical evidence supports uniqueness of these solutions.
The purpose of this paper is to study the properties of kinetic models for traffic flow described by a Boltzmann-type approach and based on a continuous space of microscopic velocities. In our models, the particular structure of the collision kernel allows one to find the analytical expression of a class of steady-state distributions, which are characterized by being supported on a quantized space of microscopic speeds. The number of these velocities is determined by a physical parameter describing the typical acceleration of a vehicle and the uniqueness of this class of solutions is supported by numerical investigations. This shows that it is possible to have the full richness of a kinetic approach with the simplicity of a space of microscopic velocities characterized by a small number of modes. Moreover, the explicit expression of the asymptotic distribution paves the way to deriving new macroscopic equations using the closure provided by the kinetic model.