Since the subject of traffic dynamics has captured the interest of physicists, many surprising effects have been revealed and explained. Some of the questions now understood are the following: Why are vehicles sometimes stopped by ``phantom traffic jams'' even though drivers all like to drive fast? What are the mechanisms behind stop-and-go traffic? Why are there several different kinds of congestion, and how are they related? Why do most traffic jams occur considerably before the road capacity is reached? Can a temporary reduction in the volume of traffic cause a lasting traffic jam? Under which conditions can speed limits speed up traffic? Why do pedestrians moving in opposite directions normally organize into lanes, while similar systems ``freeze by heating''? All of these questions have been answered by applying and extending methods from statistical physics and nonlinear dynamics to self-driven many-particle systems. This article considers the empirical data and then reviews the main approaches to modeling pedestrian and vehicle traffic. These include microscopic (particle-based), mesoscopic (gas-kinetic), and macroscopic (fluid-dynamic) models. Attention is also paid to the formulation of a micro-macro link, to aspects of universality, and to other unifying concepts, such as a general modeling framework for self-driven many-particle systems, including spin systems. While the primary focus is upon vehicle and pedestrian traffic, applications to biological or socio-economic systems such as bacterial colonies, flocks of birds, panics, and stock market dynamics are touched upon as well.

Traffic and related self-driven many-particle systems

to be done in this area. Face recognition is a problem that scarcely clamoured for attention before the computer age but, having surfaced, has involved a wide range of techniques and has attracted the attention of some fine minds (David Mumford was a Fields Medallist in 1974). This singular application of mathematical modelling to a messy applied problem of obvious utility and importance but with no unique solution is a pretty one to share with students: perhaps, returning to the source of our opening quotation, we may invert Duncan's earlier observation, 'There is an art to find the mind's construction in the face!'.

Linear and nonlinear waves, by G. B. Whitham. Pp.636. £50. 1999. ISBN 0 471 35942 4 (Wiley).

Initial-value ordinary differential equation solution via variable order Adams method (SIVA/DIVA) package is collection of subroutines for solution of nonstiff ordinary differential equations. There are versions for single-precision and double-precision arithmetic. Requires fewer evaluations of derivatives than other variable-order Adams predictor/ corrector methods. Option for direct integration of second-order equations makes integration of trajectory problems significantly more efficient. Written in FORTRAN 77.

Solving Ordinary Differential Equations

We introduce a new "second order" model of traffic flow. As noted in [C. Daganzo, Requiem for second-order fluid with approximation to traffic flow, Transportation Res. Part B, 29 (1995), pp. 277--286], the previous "second order" models, i.e., models with two equations (mass and "momentum"), lead to nonphysical effects, probably because they try to mimic the gas dynamics equations, with an unrealistic dependence on the acceleration with respect to the space derivative of the "pressure." We simply replace this space derivative with a convective derivative, and we show that this very simple repair completely resolves the inconsistencies of these models. Moreover, our model nicely predicts instabilities near the vacuum, i.e., for very light traffic.

Resurrection of “second order” models of traffic flow

Traffic flow is a kind of many-body system of strongly interacting vehicles. Traffic jams are a typical signature of the complex behaviour of vehicular traffic. Various models are presented to understand the rich variety of physical phenomena exhibited by traffic. Analytical and numerical techniques are applied to study these models. Particularly, we present detailed results obtained mainly from the microscopic car-following models. A typical phenomenon is the dynamical jamming transition from the free traffic (FT) at low density to the congested traffic at high density. The jamming transition exhibits the phase diagram similar to a conventional gas-liquid phase transition: the FT and congested traffic correspond to the gas and liquid phases, respectively. The dynamical transition is described by the time-dependent Ginzburg-Landau equation for the phase transition. The jamming transition curve is given by the spinodal line. The metastability exists in the region between the spinodal and phase separation lines. The jams in the congested traffic reveal various density waves. Some of these density waves show typical nonlinear waves such as soliton, triangular shock and kink. The density waves are described by the nonlinear wave equations: the Korteweg-de-Vries (KdV) equation, the Burgers equation and the Modified KdV equation. Subjects like the traffic flow such as bus-route system and pedestrian flow are touched as well. The bus-route system with many buses exhibits the bunching transition where buses bunch together with proceeding ahead. Such dynamic models as the car-following model are proposed to investigate the bunching transition and bus delay. A recurrent bus exhibits the dynamical transition between the delay and schedule-time phases. The delay transition is described in terms of the nonlinear map. The pedestrian flow also reveals the jamming transition from the free flow at low density to the clogging at high density. Some models are presented to study the pedestrian flow. When the clogging occurs, the pedestrian flow shows the scaling behaviour.

http://iopscience.iop.org/article/10.1088/0034-4885/65/9/203/pdf

The physics of traffic jams

In the present paper multilane models for vehicular traffic are considered. A microscopic multilane model based on reaction thresholds is developed. Based on this model an Enskog- like kinetic model is developed. In particular, care is taken to incorporate the correlations between the vehicles. From the kinetic model a fluid dynamic model is derived. The macroscopic coefficients are deduced from the underlying kinetic model. Numerical simulations are presented for all three levels of description in [A. Klar and R. Wegener, SIAM J. Appl. Math., 59 (1999), pp. 1002--1011]. Moreover, a comparison of the results is given there.

https://kluedo.ub.uni-kl.de/files/744/itwm_5part1.pdf

A hierarchy of models for multilane vehicular traffic I: modeling

In the present paper a general criticism of kinetic equations for vehicular traffic is given. The necessity of introducing an Enskog-type correction into these equations is shown. An Enskog-like kinetic traffic flow equation is presented and fluid dynamic equations are derived. This derivation yields new coefficients for the standard fluid dynamic equations of vehicular traffic. Numerical simulations for inhomogeneous traffic flow situations are shown together with a comparison between kinetic and fluid dynamic models.

/pdf/enskog-like-kinetic-models-for-vehicular-traffic-sxk7ve10p6.pdf

Enskog-like kinetic models for vehicular traffic

This survey contains a description of different types of mathematical models used for the simulation of vehicular traffic. It includes models based on ordinary differential equations, fluid dynamic equations and on equations of kinetic type. Connections between the different types of models are mentioned. Particular emphasis is put on kinetic models and on simulation methods for these models.

https://kluedo.ub.uni-kl.de/files/591/gruen_158.pdf

Mathematical Models for Vehicular Traffic

This paper has been motivated by the work of Aw and Rascle [SIAM J. Appl. Math., 60 (2000), pp. 916--938]. As noted by Daganzo [Transportation Res. B, 29 (1995), pp. 277--286] some macroscopic models for vehicular traffic encounter difficulties showing nonphysical effects in certain situations. In the paper by Aw and Rascle a new heuristic model has been presented that avoids these difficulties. We present here a kinetic derivation of a class of macroscopic traffic flow equations including the Aw--Rascle equations. The coefficients appearing in the equations are identified from the solutions of the underlying kinetic equations. Moreover, numerical experiments and comparisons between the different models are presented.

Kinetic derivation of macroscopic anticipation models for vehicular traffic

The paper at hand deals with the modeling of turbulence effects on the dynamics of a long slender elastic fiber. Independent of the choice of the drag model, a general aerodynamic force concept is derived on the basis of the velocity field for the randomly fluctuating component of the flow. Its construction as a centered differentiable Gaussian field complies thereby with the requirements of the stochastic k-epsilon turbulence model and Kolmogorov's universal equilibrium theory on local isotropy.

Raimund Wegener

Papers

A hierarchy of models for multilane vehicular traffic I: modeling

Enskog-like kinetic models for vehicular traffic

Mathematical Models for Vehicular Traffic

Kinetic derivation of macroscopic anticipation models for vehicular traffic

Fiber Dynamics in Turbulent Flows: General Modeling Framework