Boltzmann-type description with cutoff of Follow-the-Leader traffic models

TL;DR

This paper develops a Boltzmann-type kinetic model for Follow-the-Leader traffic flow with a cutoff collision kernel, deriving explicit stationary distributions via a Fokker-Planck approximation that align with empirical traffic data.

Contribution

It introduces a Boltzmann-type model with a cutoff for traffic dynamics and derives explicit stationary solutions through a Fokker-Planck approximation, connecting theory with empirical observations.

Findings

Explicit stationary distributions derived from the model.

The Fokker-Planck approximation justifies empirical traffic data.

The cutoff kernel excludes unphysical interactions.

Abstract

In this paper we consider a Boltzmann-type kinetic description of Follow-the-Leader traffic dynamics and we study the resulting asymptotic distributions, namely the counterpart of the Maxwellian distribution of the classical kinetic theory. In the Boltzmann-type equation we include a non-constant collision kernel, in the form of a cutoff, in order to exclude from the statistical model possibly unphysical interactions. In spite of the increased analytical difficulty caused by this further non-linearity, we show that a careful application of the quasi-invariant limit (an asymptotic procedure reminiscent of the grazing collision limit) successfully leads to a Fokker-Planck approximation of the original Boltzmann-type equation, whence stationary distributions can be explicitly computed. Our analytical results justify, from a genuinely model-based point of view, some empirical results found in the literature by interpolation of experimental data.