Microscopic derivation of a traffic flow model with a bifurcation

TL;DR

This paper rigorously derives a macroscopic traffic flow model with bifurcations from a microscopic follow-the-leader model with random parameters, using stochastic homogenization techniques.

Contribution

It introduces the first flux limiter in a stochastic homogenization context for traffic models, connecting microscopic randomness to macroscopic bifurcation behavior.

Findings

Derived a deterministic Hamilton-Jacobi model with flux limiter from stochastic microscopic dynamics.

Established existence of flux limiter using concentration inequalities.

Provided a novel stochastic homogenization approach for traffic flow with bifurcations.

Abstract

The goal of the paper is a rigorous derivation of a macroscopic traffic flow model with a bifurcation or a local perturbation from a microscopic one. The microscopic model is a simple follow-the-leader with random parameters. The random parameters are used as a statistical description of the road taken by a vehicle and its law of motion. The limit model is a deterministic and scalar Hamilton-Jacobi on a network with a flux limiter, the flux-limiter describing how much the bifurcation or the local perturbation slows down the vehicles. The proof of the existence of this flux limiter-the first one in the context of stochastic homogenization-relies on a concentration inequality and on a delicate derivation of a superadditive inequality.