Multivalued fundamental diagrams of traffic flow in the kinetic Fokker-Planck limit

TL;DR

This paper investigates how microscopic stochastic interaction rules influence macroscopic traffic flow diagrams, revealing multivalued flux-density and speed-density relationships through a Fokker-Planck approximation.

Contribution

It introduces a novel analytical approach linking microscopic stochastic dynamics to multivalued macroscopic traffic flow diagrams using a Fokker-Planck model.

Findings

Multivalued flux-density diagrams are derived analytically.

Existence of a one-parameter family of stationary distributions.

Fokker-Planck approximation effectively models microscopic influences.

Abstract

Starting from interaction rules based on two levels of stochasticity we study the influence of the microscopic dynamics on the macroscopic properties of vehicular flow. In particular, we study the qualitative structure of the resulting flux-density and speed-density diagrams for different choices of the desired speeds. We are able to recover multivalued diagrams as a result of the existence of a one-parameter family of stationary distributions, whose expression is analytically found by means of a Fokker-Planck approximation of the initial Boltzmann-type model.