A statistical mechanics approach to macroscopic limits of car-following traffic dynamics

TL;DR

This paper derives macroscopic traffic flow models from microscopic car-following dynamics using a kinetic approach, revealing how different microscopic interaction rates influence the resulting macroscopic equations.

Contribution

It introduces a physical derivation of macroscopic traffic models from particle dynamics, highlighting the impact of microscopic process rates on model form and stability.

Findings

Inhomogeneous Aw-Rascle-Zhang model arises when FTL interactions dominate.

Lighthill-Whitham-Richards model emerges when OV relaxation rate is comparable to FTL interactions.

Numerical simulations confirm the theoretical derivations.

Abstract

We study the derivation of macroscopic traffic models from car-following vehicle dynamics by means of hydrodynamic limits of an Enskog-type kinetic description. We consider the superposition of Follow-the-Leader (FTL) interactions and relaxation towards a traffic-dependent Optimal Velocity (OV) and we show that the resulting macroscopic models depend on the relative frequency between these two microscopic processes. If FTL interactions dominate then one gets an inhomogeneous Aw-Rascle-Zhang model, whose (pseudo) pressure and stability of the uniform flow are precisely defined by some features of the microscopic FTL and OV dynamics. Conversely, if the rate of OV relaxation is comparable to that of FTL interactions then one gets a Lighthill-Whitham-Richards model ruled only by the OV function. We further confirm these findings by means of numerical simulations of the particle system and the macroscopic models. Unlike other formally analogous results, our approach builds the macroscopic models as physical limits of particle dynamics rather than assessing the convergence of microscopic to macroscopic solutions under suitable numerical discretisations.