The Aw–Rascle Traffic Model: Enskog-Type Kinetic Derivation and Generalisations

Abstract

We study the derivation of second order macroscopic traffic models from kinetic descriptions. In particular, we recover the celebrated Aw-Rascle model as the hydrodynamic limit of an Enskog-type kinetic equation out of a precise characterisation of the microscopic binary interactions among the vehicles. Unlike other derivations available in the literature, our approach unveils the multiscale physics behind the Aw-Rascle model. This further allows us to generalise it to a new class of second order macroscopic models complying with the Aw-Rascle consistency condition, namely the fact that no wave should travel faster than the mean traffic flow.

Model-based assessment of the impact of driver-assist vehicles using kinetic theory

“…The first term on the right-hand side is analogous to the one found in [25]. Owing to the convexity of Φ, it is non-positive, thus it confers a non-increasing trend on the time derivative of H along the solutions to (23). Conversely, the second term is computed as:…”

“…In this paper, we propose to ground such a multiscale analysis on the sound mathematical-physical framework of the kinetic theory, which has now a quite consolidated tradition also in the modelling of vehicular traffic. Far from claiming to be exhaustive, we recall here the pioneering works [41,43] together with modern kinetic methods [17,29,33,34] and recent advancements [23,30,50,51,55]. Taking advantage of a statistical approach, kinetic-type equations naturally link the microscopic dynamics of few interacting agents representative of a much larger group to the observable macroscopic trends of the system as a whole.…”

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

“…the speed update rules based on FTL interactions and OV relaxation, respectively. Two remarks are in order about the first term on the right-hand side of (12). First, this term is written under the (usual) assumption of propagation of chaos, which states that any two randomly chosen vehicles are statistically independent at the moment of their interaction.…”

“…The description of traffic flow by appealing to principles of statistical mechanics and kinetic theory dates back to the pioneering works [27,28] and in successive times has been the object of a renewed interest. Without intending to review all the pertinent literature, we refer the interested reader to e.g., [12,18,19,21,22] and references therein. In particular, in [21,22] the authors were the first to show that the method of moment equations applied to an Enskog-type kinetic description of generic traffic dynamics may formally lead to the (homogeneous) ARZ model.…”

“…In particular, in [21,22] the authors were the first to show that the method of moment equations applied to an Enskog-type kinetic description of generic traffic dynamics may formally lead to the (homogeneous) ARZ model. Their approach will be very much inspiring for us but at the same time we will improve it in the spirit of [12]. In particular, we will focus on a rigorous upscaling of microscopic interaction rules specifically motivated by the superposition of FTL and OV dynamics, which will allow us to obtain explicit closure relationships at the kinetic and macroscopic levels and to avoid partly heuristic ones like in [21,22].…”