Non-linear Dynamics, Emergent Behaviors and Controlled Expansions: Towards Effective Modeling of the Congested Traffic

Abstract

We propose a framework for constructing microscopic traffic models from microscopic acceleration patterns that can in principle be experimental measured and proper averaged. The exact model thus obtained can be used to justify the consistency of various popular models in the literature. Assuming analyticity of the exact model, we suggest that a controlled expansion around the constant velocity, uniform headway "ground state" is the proper way of constructing various different effective models. Assuming a unique ground state for any fixed average density, we discuss the universal properties of the resulting effective model, focusing on the emergent quantities of the coupled non-linear ODEs. These include the maximum and minimum headway that give the coexistence curve in the phase diagram, as well as an emergent intrinsic scale that characterizes the strength of interaction between clusters, leading to non-trivial cluster statistics when the unstable ground state is randomly perturbed. Utilizing the universal properties of the emergent quantities, a simple algorithm for constructing an effective traffic model is also presented. The algorithm tunes the model with statistically well-defined quantities extracted from the flow-density plot, and the resulting effective model naturally captures and predicts many quantitative and qualitative empirical features of the highway traffic, especially in the presence of an on-ramp bottleneck. The simplicity of the effective model provides strong evidence that stochasticity, diversity of vehicle types and modeling of complicated individual driving behaviors are \emph{not} fundamental to many observations of the complex spatiotemporal patterns in the real traffic dynamics. We also propose the nature of the congested phase can be well characterized by the long lasting transient states of the effective model, from which the wide moving jams evolve.