Kinetic models for traffic flow resulting in a reduced space of microscopic velocities

TL;DR

This paper introduces a kinetic traffic flow model with a reduced set of microscopic velocities, providing explicit steady-state solutions and enabling derivation of macroscopic equations, thus simplifying analysis while maintaining model richness.

Contribution

It presents a novel kinetic model with a quantized velocity space, offering explicit steady-state solutions and a framework for deriving macroscopic traffic equations.

Findings

Explicit steady-state distributions supported on quantized velocities.

The number of velocities is linked to vehicle acceleration parameters.

Numerical investigations support the uniqueness of solutions.

Abstract

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.