  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.
