Ludger Santen

Statistical physics of vehicular traffic and some related systems

Abstract In the so-called “microscopic” models of vehicular traffic, attention is paid explicitly to each individual vehicle each of which is represented by a “particle”; the nature of the “interactions” among these particles is determined by the way the vehicles influence each others’ movement. Therefore, vehicular traffic, modeled as a system of interacting “particles” driven far from equilibrium, offers the possibility to study various fundamental aspects of truly nonequilibrium systems which are of current interest in statistical physics. Analytical as well as numerical techniques of statistical physics are being used to study these models to understand rich variety of physical phenomena exhibited by vehicular traffic. Some of these phenomena, observed in vehicular traffic under different circumstances, include transitions from one dynamical phase to another, criticality and self-organized criticality, metastability and hysteresis, phase-segregation, etc. In this critical review, written from the perspective of statistical physics, we explain the guiding principles behind all the main theoretical approaches. But we present detailed discussions on the results obtained mainly from the so-called “particle-hopping” models, particularly emphasizing those which have been formulated in recent years using the language of cellular automata.

Metastable states in cellular automata for traffic flow

Abstract:Measurements on real traffic have revealed the existence of metastable states with very high flow. Such states have not been observed in the Nagel-Schreckenberg (NaSch) model which is the basic cellular automaton for the description of traffic. Here we propose a simple generalization of the NaSch model by introducing a velocity-dependent randomization. We investigate a special case which belongs to the so-called slow-to-start rules. It is shown that this model exhibits metastable states, thus sheding some light on the prerequisites for the occurance of hysteresis effects in the flow-density relation.

Towards a realistic microscopic description of highway traffic

Simple cellular automata models are able to reproduce the basic properties of highway traffic. The comparison with empirical data for microscopic quantities requires a more detailed description of the elementary dynamics. Based on existing cellular automata models, we propose an improved discrete model incorporating anticipation effects, reduced acceleration capabilities and an enhanced interaction horizon for braking. The modified model is able to reproduce the three phases (free-flow, synchronized, and stop-and-go) observed in real traffic. Furthermore we find a good agreement with detailed empirical single-vehicle data in all phases.

Single-vehicle data of highway traffic: a statistical analysis

In the present paper, single-vehicle data of highway traffic are analyzed in great detail. By using the single-vehicle data directly, empirical time headway distributions and speed-distance relations can be established. Both quantities yield relevant information about the microscopic states. Several fundamental diagrams are also presented, which are based on time-averaged quantities and compared with earlier empirical investigations. In the remaining part, time-series analyses of the averaged as well as the single-vehicle data are carried out. The results will be used in order to propose objective criteria for an identification of the different traffic states, e.g., synchronized traffic.

The Asymmetric Exclusion Process: Comparison of Update Procedures

The asymmetric exclusion process (ASEP) has attracted a lot of interest not only because of its many applications, e.g., in the context of the kinetics of biopolymerization and traffic flow theory, but also because it is a paradigmatic model for nonequilibrium systems. Here we study the ASEP for different types of updates, namely random-sequential, sequential, sublattice-parallel, and parallel. In order to compare the effects of the different update procedures on the properties of the stationary state, we use large-scale Monte Carlo simulations and analytical methods, especially the so-called matrix-product Ansatz (MPA). We present in detail the exact solution for the model with sublattice-parallel and sequential updates using the MPA. For the case of parallel update, which is important for applications like traffic flow theory, we determine the phase diagram, the current, and density profiles based on Monte Carlo simulations. We furthermore suggest an MPA for that case and derive the corresponding matrix algebra.

Disorder effects in cellular automata for two-lane traffic

For single-lane traffic models it is well known that particle disorder leads to platoon formation at low densities. Here we discuss the effect of slow cars in two-lane systems. Surprisingly, even a small number of slow cars can initiate the formation of platoons at low densities. The robustness of this phenomenon is investigated for different variants of the lane-changing rules as well as for different variants on the single-lane dynamics. It is shown that anticipation of drivers reduces the influence of slow cars drastically.

SINGLE-VEHICLE DATA OF HIGHWAY TRAFFIC: MICROSCOPIC DESCRIPTION OF TRAFFIC PHASES

We present a detailed analysis of single-vehicle data, which sheds some light on the microscopic interaction of the vehicles. Besides the analysis of free flow and synchronized traffic the data sets especially provide information about wide jams that persist for a long time. The data have been collected at a location far away from ramps and in the absence of speed limits, which allows a comparison with idealized traffic simulations. We also resolve some open questions concerning the time-headway distribution.

A realistic two-lane traffic model for highway traffic

A two-lane extension of a recently proposed cellular automaton model for traffic flow is discussed. The analysis focuses on the reproduction of the lane usage inversion and the density dependence of the number of lane changes. It is shown that the single-lane dynamics can be extended to the two-lane case without changing the basic properties of the model, which are known to be in good agreement with empirical single-vehicle data. Therefore it is possible to reproduce various empirically observed two-lane phenomena, like the synchronization of the lanes, without fine tuning of the model parameters.

Shock formation in an exclusion process with creation and annihilation

We investigate shock formation in an asymmetric exclusion process with creation and annihilation of particles in the bulk. We show how the continuum mean-field equations can be studied analytically and hence derive the phase diagrams of the model. In the large system-size limit direct simulations of the model show that the stationary state is correctly described by the mean-field equations, thus the predicted mean-field phase diagrams are expected to be exact. The emergence of shocks and the structure of the phase diagram are discussed. We also analyze the fluctuations of the shock position by using a phenomenological random walk picture of the shock dynamics. The stationary distribution of shock positions is calculated, by virtue of which the numerically determined finite-size scaling behavior of the shock width is explained.

Absence of thermodynamic phase transition in a model glass former

The glass transition can be viewed simply as the point at which the viscosity of a structurally disordered liquid reaches a universal threshold value. But this is an operational definition that circumvents fundamental issues, such as whether the glass transition is a purely dynamical phenomenon. If so, ergodicity gets broken (the system becomes confined to some part of its phase space), but the thermodynamic properties of the liquid remain unchanged across the transition, provided they are determined as thermodynamic equilibrium averages over the whole phase space. The opposite view claims that an underlying thermodynamic phase transition is responsible for the pronounced slow-down in the dynamics at the liquid–glass boundary. Such a phase transition would trigger the dynamic standstill, and then be masked by it. Here we perform Monte Carlo simulations of a two-dimensional system of polydisperse hard disks far within its glassy phase. The approach allows for non-local moves in a way that preserves micro-reversibility. We find no evidence for a thermodynamic phase transition up to very high densities; the glass is thus indistinguishable from the liquid on purely thermodynamic grounds.