The Physics of Traffic

Abstract

Springer Complexity is a publication program, cutting across all traditional disciplines of sciences as well as engineering, economics, medicine, psychology and computer sciences, which is aimed at researchers, students and practitioners working in the field of complex systems. Complex Systems are systems that comprise many interacting parts with the ability to generate a new quality of macroscopic collective behavior through self-organization, e.g., the spontaneous formation of temporal, spatial or functional structures. This recognition, that the collective behavior of the whole system cannot be simply inferred from the understanding of the behavior of the individual components, has led to various new concepts and sophisticated tools of complexity. The main concepts and tools-with sometimes overlapping contents and methodologies-are the theories of self-organization, complex systems, synergetics, dynamical systems, turbulence, catastrophes, instabilities, nonlinearity, stochastic processes, chaos, neural networks, cellular automata, adaptive systems, and genetic algorithms. The topics treated within Springer Complexity are as diverse as lasers or fluids in physics, machine cutting phenomena of workpieces or electric circuits with feedback in engineering, growth of crystals or pattern formation in chemistry, morphogenesis in biology, brain function in neurology, behavior of stock exchange rates in economics, or the formation of public opinion in sociology. All these seemingly quite different kinds of structure formation have a number of important features and underlying structures in common. These deep structural similarities can be exploited to transfer analytical methods and understanding from one field to another. The Springer Complexity program therefore seeks to foster crossfertilization between the disciplines and a dialogue between theoreticians and experimentalists for a deeper understanding of the general structure and behavior of complex systems. The program consists of individual books, books series such as "Springer Series in Synergetics", "Institute of Nonlinear Science", "Physics of Neural Networks", and "Understanding Complex Systems", as well as various journals.

Fully adaptive multiresolution schemes for strongly degenerate parabolic equations with discontinuous flux

“…Numerous mathematical models have been proposed for one-directional flows of vehicular traffic; reviews of this topic are given in the monographs by Helbing [62], Kerner [63] and Garavello and Piccoli [64], as well as in the articles by Bellomo et al [65,66,67]. These and other works vividly illustrate that the number of balance equations (for the car density, velocity, and possibly other flow variables) that form a time-dependent model based on partial differential equations, as well as the algebraic structure of these equations, is a topic of current research.…”

Fluid-dynamical and microscopic description of traffic flow: a data-driven comparison

“…This time, the slope of the decline is the speed of the backwardtravelling jam wave. Note, however, that more general fundamental diagrams may exist, or even none at all (Kerner 2004). The model in equation (2.9) is already an acceptable description of traffic flow from a macroscopic point of view.…”

“…In the more physics-minded part of the traffic community, there is also the further subdivision into one-phase, two-phase or three-phase models. They differ in the details of the car-following dynamics: onephase models always have a stable fixed point in car following, in two-phase models, there exist some values of the density where this fixed point becomes unstable, creating start-and-stop jam waves, while three-phase models (see Kerner 2004) have many marginally stable fixed points in car following, even for the same speed.…”

Nonlinear dynamics of autonomous vehicles with limits on acceleration

“…The reader is referred to several good references on traffic and ACC [12][13][14][15][16]. However, I will describe the salient developments as they relate to the present work.…”