Traffic and Granular Flow ’99

The problem of the relation between recurrence and irreversibility is an old and universal one: it has been discussed by philosophers, physicists, historians, and social scientists. After briefly mentioning philosophical formulations of the problem, the controversy, being deeply inherent in the notions of Statistical Physics, is discussed in terms of Poincare’ recurrence theorem versus irreversible equations such as the Boltzmann equation. The conclusion is that the neglection of certain correlations, i.e., an approximation, leads from recurrence to irreversibility. Thereupon the inverse problem is considered, in which manner recurrent, in particular periodic or quasiperiodic sub-processes can appear to be embedded in a globally irreversible process. Some approaches how to trace and recognize embedded recurrent (and even periodic) sub-processes are discussed. Finally, selected examples of model-based (quasi-) periodic processes in social systems are presented. They belong to the sectors demography (migration) and sociology (group dynamics).

In recent years, a considerable number of physicists have started applying physics concepts and methods to understand economic phenomena. The term “Econophysics” is sometimes used to describe this work. Economic fluctuations can have many repercussions, and understanding fluctuations is a topic that many physicists have contributed to in recent years. Further, economic systems are examples of complex interacting systems for which a huge amount of data exist and it is possible that the experience gained by physicists in studying fluctuations in physical systems might yield new results in economics. Much recent work in econophysics is focused on understanding the peculiar statistical properties of price fluctuations in financial time series. In this talk, we discuss three recent results. The first result concerns the probability distribution of stock price fluctuations. This distribution decreases with increasing fluctuations with a power-law tail well outside the Lévy stable regime and describes fluctuations that differ by as much as 8 orders of magnitude. Further, this non stable distribution preserves its functional form for fluctuations on time scales that differ by 3 orders of magnitude, from 1 min up to approximately 10 days. The second result concerns the accurate quantification of volatility correlations in financial time series. While price fluctuations themselves have rapidly decaying correlations, the volatility estimated by using either the absolute value or the square of the price fluctuations has correlations that decay as a power-law and persist for several months. The third result bears on the application of random matrix theory to understand the correlations among price fluctuations of any two different stocks. We compare the statistics of the cross-correlation matrix constructed from price fluctuations of the leading 1000 stocks and a matrix with independent random elements, i.e., a random matrix. Contrary to first expectations, we find little or no deviation from the universal predictions of random matrix theory for all but a few of the largest eigenvalues of the cross-correlation matrix.

This paper deals with description of all possible minimal cost extensions of transportation flows in the developing urban system. The enumeration of such extensions is based on competitive exclusion behavioral rules for suppliers and demanders connected by means of minimal cost solution of the classical Linear Programming Transportation Problem: in such a way the complete set of all topological structures for all possible optimal extensions of transportation network is constructed in the form of all maximal tries without cycles with exactly m + n — 1 basic sells (where m is a number of suppliers and n is a number of demanders). It is important to underline that there exist only finite number of such trees. The catastrophe effect analysis in this enumeration set is based on the polyhedral form of general sensitivity analysis for classical minimal cost transportation problem: for each preset topological structure of the minimal cost flow there is a polyhedral cone K in the space of supply-demand and the polyhedral wedge W in the space of transportation costs, such that the choice of arbitrary supply-demand within the cone K and the choice of arbitrary set of transportation costs within the wedge W will give the existing minimal cost flow with a preset topological structure. The finite numbers of Cartesian products K xW represents the domains of structural stability of minimal cost flows with preset topological structures. Eventually the vector method of potentials is elaborated for the construction of the domains of structural stability K xW. As a practical numeric example the expanding set of bounded Christaller-Loesh and Beckman-McPherson Central Place System are considered in detail.

We investigate the dynamics of social processes and, in particular, processes of innovation in socio-economical evolution, technological change, and scientific research in discrete and continuous state spaces. The discrete description is based on the occupation number formalism and transition probabilities (Master equation formalism). Our special attention is devoted to the creation and survival of the NEW (i.e., new behavior, new technologies, new ideas etc.). The second (continuous) model is based on the idea that evolution is hill-climbing in an adaptive landscape over a continuous characteristics space. The behavior of an individual, a technological product, or a scientific problem is described by a large number of characteristics covering behavioral aspects, technology-inherent and economic parameters, or thematic dimensions. Further, we define a real-valued multi-modal fitness function/functional and a population density over the characteristics space. The evolutionary dynamics including competition and innovations is modeled by reaction-diffusion equations of Fisher-Eigen or Lotka-Volterra type.

Group decision making is assumed to obey some universal features which are independent of both the social nature of the group making the decision and the nature of the decision itself. On this basis a simple magnetic like model is built. Pair interactions are introduced to measure the degree of exchange among individuals while discussing. An external uniform field is included to account for a possible pressure from outside. Individual biases with respect to the issue at stake are also included using local random fields. A unique postulate of minimum conflict is assumed. The model is then solved with emphasize on its psycho-sociological implications. Counter-intuitive results are obtained. At this stage no new physical technicality is involved. Instead, the full psycho-sociological implications of the model are drawn. Few cases are then detailed to enlight them.

We study a model of opinion formation based on the theory of social impact and the concept of cellular automata. The case is considered when two strong agents influence the group: a strong leader and an external social impact acting uniformly on every individual. There are two basic stationary states of the system: cluster of the leader’s adherents and unification of opinions. In the deterministic limit the variation of parameters like the leader’s strength or external impact can change the size of the cluster or, when they reach some critical values, make the system jump to another phase. In the presence of noise (social temperature) the rapid changes can be regarded as the first order phase transitions. When both agents are in a kind of balance, a second order transition and critical behaviour can be observed. Analytical results obtained within a mean field approximation are well reproduced in computer simulations.

In this work we analyze an apparently simple question: Is there any sign of intelligence in human motion? Traditional comparative ethology has shown that individual and most collective motions are common in many species of very different levels of intelligence. We show here that an almost unambiguous identification of intellect is feasible by detecting some form of absolutely senseless collective motion of large masses.

In this contribution, intelligent transportation systems (ITS) and their impact on traffic systems are discussed. Although traffic forecast offers the possibility to rearrange the temporal distribution of traffic patterns, it suffers from a fundamental problem because the reaction of the driver to the forecast is a priori unknown. On the other hand the behaviour of drivers can have a serious impact on the quality of a traffic forecast since it can result in a feedback - an anticipatory forecast is needed. To include such effects we propose a two-layered agent architecture for modelling drivers’ behaviour in more detail. The layers distinguish different tasks of road users.

Traffic systems in large cities might be viewed as interdependent decision situations. Natural N-person extensions of familiar games may serve as templates to cover some system relations. An agent based simulation model is proposed. Several decision levels are considered in the model in order to cover aspects of institutional framework.

The model is an attempt to synthesise the Weidlich-Haag social synergetics probabilistic approach with Brillouin’s information-thermodynamics method of reasoning. It proposes mathematical modelling and physical explanation of one of the basic human and social phenomena: The need of change — change for the sake of change (without visible external motivations and reasons; reasons and motivations which in the Weidlich-Haag model are expressed by the traditional concept of utility widely used in economics). The computations make use of the Monte-Carlo method, in which the histories of each individual are followed. The results are discussed in terms of really observed social phenomena. The model can also be regarded as an attempt to “motorise” the elements of a stochastically behaving complex system composed of independent elements with free energy depots. This would extend the approach of other contributors to the TGF 99 conference extending their models of stochastic movement in the geometrical space into motion in the axiological space.

The power of Darwinian evolution is based on the dichotomy of genotype and phenotype with the former being the object under variation and the latter constituting the target of selection. Only the simplest case of an evolutionary process, the optimization of RNA molecules in vitro, where phenotypes axe understood as RNA structures, can be handled explicitly. We derive a model based on differential equations with stochastic terms which includes unfolding of genotypes to yield phenotypes as well as the evaluation of the latter. The relations between genotypes and phenotypes are understood as mappings from sequence space into shape space, the space of molecular structures. Generic properties of this map are derived and analyzed for RNA secondary structures as an example. The optimization of molecular properties in populations is modeled in silico through replication and mutation in a flow reactor. The approach towards a predefined structure is monitored and reconstructed in terms of a relay series being an uninterrupted sequence of phenotypes from initial structure to target. Analysis of the molecular shapes in the relay series provides the basis for a novel definition of continuity in evolution. Discontinuities can be identified as major changes in molecular structures.

Is it possible to extract energy from random fluctuations and put it to beneficial use? This challenging question has provoked discussions ever since the early days of Brownian motion studies. Generally, noise in dynamical systems is considered a nuisance. But in certain nonlinear systems the presence of noise can in fact enhance the detection of weak signals. This phenomenon, called Stochastic Resonance (SR) does find useful applications in physical, technological and biomedical contexts [1]. In a second class of systems that are periodic - but which lack reflection symmetry - directed, noise induced transport can take place. The directed motion of particles in periodic potentials requires at least one source of non-equilibrium which must inherit an explicit or inherent statistical asymmetry. Such non-equilibrium systems have become known in the literature under the label of “ratchets” [2]. In both situations the importance of fluctuations is elevated to a level where noise must be viewed as source of order and complexity in its own right.

During the course of evolution, bacteria have developed sophisticated cooperative behavior and intricate communication capabilities [1-3]. Utilizing these capabilities, bacterial colonies develop complex spatio-temporal patterns in response to adverse growth conditions. It is now understood that the study of cooperative self-organization of bacterial colonies is an exciting new multidisciplinary field of research, necessitating the merger of biological information with the physics of non-equilibrium processes and the mathematics of non-linear dynamics. At this stage, several experimental systems have been identified, and preliminary modeling efforts are making significant progress in providing a framework for the understanding of experimental observations [4-12]. This endeavour is not limited to bacteria alone. Studies have been performed of other types of microorganisms as well, such as amoeba [13] and yeast [14]

We discuss simulations of flocking and the principle of optimal self-organisation during self-driven motion of many similar objects. In addition to driving and interaction the role of fluctuations is taken into account as well. In our models, the particles corresponding to organisms locally interact with their neighbours by choosing at each time step a velocity depending on the directions of motion of them. Our numerical studies of flocking indicate the existence of new types of transitions. As a function of the control parameters both disordered and long-range ordered phases can be observed, and the corresponding phase space domains are separated by singular “critical lines”. In particular, we demonstrate both numerically and analytically that there is a disordered-to-ordered-motion transition at a finite noise level even in one dimension. We also present computational and analytical results indicating that driven systems with repulsive interactions tend to reach an optimal state corresponding to minimal interaction. This extremal principle is expected to be relevant for a class of biological and social systems involving driven interacting entities.

Active motion relies on the supply of energy. In order to turn passive into active motion, we need to consider mechanisms of energy take-up, storage and conversion. A suitable approach which considers both the energetic and stochastic aspects of active motion is provided by the model of active Brownian particles. For a supercritical supply of energy these particles are able to move in a “high velocity” or active mode, which results in deviation from the Maxwellian velocity distribution. We investigate different types of complex motion of active Brownian particles moving in external potentials. Among the examples are the occurrence of stochastic limit cycles, transitions between Brownian and directed motion, the “uphill” motion against the direction of an external force, or the establishment of positive or negative net currents in a ratchet potential, dependent on energy supply and stochastic influences.

We start from the basic equations of a pulse-coupled neural network with arbitrary couplings (“synaptic strengths”) between its elements. The axonal pulses are described by means of a phase, whose rotation speed depends on the dendritic inputs (“lighthouse model”). We include the effects of noise by means of fluctuating forces. We also allow for delays between the neurons. The introduction of time-averaged axonal pulse rates ω_ℓ allows us to convert the original, highly nonlinear and stochastic equations into rather simple equations for ui that can be solved directly. The solutions can be interpreted as action of an associate memory.

For a company to stay competitive, providing “intelligent” application solutions, services and products to its customers is indispensable. For example, forthcoming telematics applications require technologies like adaptive control, model building by learning, and focusing attention on relevant features. Neural Networks offer appropriate architectures for these purposes. This contribution gives an overview on activities at Siemens Corporate Research using Neural Networks for prediction and control purposes.

We show how optimization methods from economics known as portfolio strategies can be used for minimizing down-load times in the Internet and travel times in freeway traffic. While for Internet traffic, there is an optimal restart frequency for requesting data, freeway traffic can be optimized by a small percentage of vehicles coming from on-ramps. Interestingly, the portfolio strategies can decrease the average waiting or travel times, respectively, as well as their standard deviation (“risk”). In general, portfolio strategies are applicable to systems, in which the distribution of the quantity to be optimized is broad.

Boltzmann’s equation describes the dynamics of the one particle phase space distribution density including two particle interactions. These drive the system towards equilibrium. The physical properties of this equilibrium turn out to be those of an ideal gas, giving the leading order of the equation of state, the pressure, etc, only. No viral corrections originating from the two particle interactions appear, as one finds them in equilibrium statistical mechanics. This means the Boltzmann dynamics is insufficient to imply the proper equilibrium. It has to be upgraded, clearly, in its so called flow terms, which contribute only in order n (particle density) corresponding to a free motion between collisions. If properly derived, the equation of motion contains also order n contributions in the flow terms, weighted by the real part of the forward scattering amplitude. Then the equilibrium limit of the dynamics coincides with the findings of equilibrium statistical physics. Various implications can be identified. It is pointed out, that the corresponding upgrade has to be checked carefully in interdisciplinary modeling, as is used, e.g., in traffic flow or granular systems descriptions.

The paper reviews a general theory predicting the general importance of fractional evolution equations. Fractional time evolutions are shown to arise from a microscopic time evolution in a certain long time scaling limit. Fractional time evolutions are generally irreversible. The infinitesimal generators of fractional time evolutions are fractional time derivatives. Evolution equations containing fractional time derivatives are proposed for physical, economical and traffic applications. Regular non-fractional time evolutions emerge as special cases from the results. Also for these regular time evolutions it is found that macroscopic irreversibility arises in the scaling limit.

The stationary states of one-dimensional driven diffusive systems, connected to boundary reservoirs with fixed particle density are shown to be selected by an extremal principle for the macroscopic current. Given the current one obtains the exact first- and second-order non-equilibrium phase transition lines for the bulk density as a function of the boundary densities. The basic dynamical mechanism behind the extremal principle is an intriguing generic interplay between the motion of shocks and localized perturbations.

A recently developed accelerated deterministic method of approximation of the Boltzmann collision operator is applied in numerical modelling of the process of shock wave focusing in a rarefied noble gas. The results are compared with the results obtained previously for the same problem with the Boltzmann collision operator evaluated by the Monte Carlo quadrature.

The dynamics of a ring of masses including dissipative forces (passive and active friction) and Toda interactions between the masses are investigated. The characteristic attractor structure and the influence of noise by coupling to a heat bath are studied. The system may be driven from the thermodynamic equilibrium to far from equilibrium states by including negative friction. We show, that over-critical pumping with free energy may lead to a partition of the phase space into attractor regions corresponding to several types of collective motions including uniform rotations, one- and multiple soliton excitations and relative oscillations. With Lennard-Jones like interaction potentials the particles form clusters moving along the ring.

We investigate a simple model corresponding to pedestrians walking in opposite directions and interacting via a repulsive potential. The pedestrians move off-lattice in a periodic corridor and are subject to random forces as well. We show that this model - which can be considered as a continuum version of some driven diffusive systems - exhibits a paradoxical, new kind of transition called here “freezing by heating”. The most interesting feature of the transition is that a crystallized state with a higher total energy is obtained from a fluid state by increasing the amount of fluctuations

A review of an experimental study of phase transitions in traffic flow is presented. A critical comparison of real features of phase transitions with recent numerical results is given. A qualitative theory of congested traffic flow recently developed is discussed.

By the year 2000, more than half of the world’s population will live in cities. This means that not only an extensive exchange of population, goods and information in a globalised world can be expected but also that the transport of goods, population and information must be effectively managed. A city which wants to represent an important node in this network must provide beside appropriate economic, social and cultural conditions and political stability, qualified labour, and an urban as well as an internationally operating interurban system of transport. Moreover, all cities face a common problem: they must possess the capacity to sustain unprecedented numbers of citizens within limited budgets and severe environmental constraints.

Single-vehicle data of freeway traffic as well as selected Floating-Car (FC) data are analyzed in detail. Traffic states are distinguished by means of aggregated data. We propose a method for a quantitative classification of these states. The data of individual vehicles allow for insights into the interaction of vehicles. The time-headway distribution reveals a characteristic structure dominated by peaks and controlled by the underlying traffic states. The tendency to reach a pleasant level of “driving comfort” gives rise to new v(Δx)-diagrams, known as Optimal-Velocity (OV) curves. The insights found at locally fixed detectors can be confirmed by FC data. In fundamental diagrams derived from local measurements new high-flow states can be observed.

In highway traffic one observes metastabiltiy in a certain range of the traffic density. Perturbations of a large enough amplitude develop into jams or stop-andgo waves. Observations and simulations indicate that the latter behave like solitary waves. Current traffic control systems based on varying speed limits cannot always prevent the creation of stop-and-go waves nor can they damp them. We have designed a nonlinear controller which is able to extinguish stop-and-go waves; its performance is demonstrated within simulations using a realistic traffic model. The construction is based on a transformation to a coordinate system moving with the propagation speed of the stop-and-go waves.

For advanced traffic information systems and traffic control systems, the detailed and accurate knowledge of the current state of a road network is of great importance for their efficiency. In real traffic, measurements are mostly collected at local points, from sensors like loops, infrared sensors, or video cameras. To get the overall state of a traffic system, online traffic simulations or traffic state estimations are used. This paper focuses on the main differences between these two approaches. An example with real traffic data shows, that the state estimation can cope better with inaccurate initial conditions and measurement errors.

A necessary prerequisite for numerous services in traffic telematics is a good knowledge of the current traffic situation by location and time. An interdisciplinary solution is presented, which provides a location-time-dynamical description of the traffic situation by revealing traffic domains with uniform traffic states. For this pattern recognition task in a dynamical process with even chaotic traffic state transitions, the underlying measurements are classified into different traffic states. Some specific problems arising in this context are solved: 1) Data from different sources and of different units of measurement, e g., velocities, flows and densities, which are only sparsely available with respect to location and time are combined (“data fusion”). 2) “Floating Car Data (FCD)” is integrated using morphological filters and recognizing its location-time relation compared to other data. 3) The stochastic data are filtered without suppressing significant state transitions. 4) The subjective feeling of roadusers, that it is difficult to distinguish between different traffic states, is taken into account by fuzzy-classification. 5) Using a region growing method the segmentation problem of traffic domains is solved without being restricted to a predefined coarse road segmentation. 6) Good stability is obtained despite contradictory demands for a high resolution, a short reaction time and the differentiation of more than two traffic states. The chosen approach is confirmed by results with actual traffic data. The author knows of no other procedure with comparable capabilities.

In this paper we study dependencies of fundamental diagrams, time gap distributions, and velocity-distance relations on vehicle types, lanes and/or measurement sites. We also propose measurement and aggregation methods that have more favourable statistical properties than conventional methods.

Results of investigations of a recent method for the automatic tracing of moving traffic jams and of the prediction of time-dependent vehicle trip times are presented using different levels of data inputs. The method is based on the previous findings that moving jams possess some characteristic parameters, i e., the parameters are unique, coherent, predictable and reproducible. Based on available data it is found that the method, which performs without any validation of the parameters of a model under different infrastructures of a highway, weather, etc., can be applied for a reliable forecasting of traffic congestions on a highway.

Multiple congested traffic states are found from empirical data. A preliminary empirical phase diagram of congested traffic flow is constructed. This paper also discusses connections between observations and theoretical analysis.

Recently, an on-line simulation for traffic flow in urban areas has been presented [1,2]. In this contribution, a framework for on-line simulations of freeways is proposed. It is based on cellular automaton models which are supplemented by real world traffic data stemming from about 2,500 inductive loops distributed over the freeway network of North Rhine-Westphalia (NRW). We propose and analyse different methods to tune the simulation with regard to the real world measurements. The possibility of a short-term traffic forecast is discussed.

The primary purpose of this project1 is to collect data required to adjust the parameters of a traffic model, allowing realistic predictions of the traffic in Atlanta. This project will provide information with a completeness and accuracy which has not been collected elsewhere. The acquired data will serve the traffic simulation as well as providing an empirical basis for future development in traffic modeling. The verified and calibrated traffic flow model will be implemented in a computer simulation program, which then can be used to focus on the development of strategies for improvement of the traffic flow, optimal positioning for driver-information displays, and evaluation of the significance of the drivers’ response rate for the success of control measures.

We present simulations of congested traffic in open systems with a new carfollowing model. The model parameters are all intuitive and can be easily calibrated. Microsimulations with identical vehicles on a single lane produce the same traffic states as recent macrosimulations of open systems with on-ramps, which also qualitatively agree with real traffic data. The phase diagram in the phase space spanned by the traffic flow and the bottleneck strength is nearly equivalent to the macroscopic phase diagram. In agreement with macroscopic models, we found hysteresis, coexistent states, and a small region of tristability. We simulated the process of obtaining time-averaged traffic data by “virtual detectors”. While for identical vehicles, the resulting flow-density data do not look very realistic, microsimulations of heterogeneous (multi-species) traffic offer a natural explanation of the observed wide scattering of congested traffic data.

We discuss a phenomenological approach to the description of unstable vehicle motion on multilane highways that could explain in a simple way such observed self-organizing phenomena as the sequence of the phase transitions“free flow → synchronized motion → jam” and the hysteresis in them. We introduce a new variable called order parameter that accounts for possible correlations in the vehicle motion at different lanes. So, it is principally due to “many-body” effects in the car interaction in contrast to such variables as the mean car density and velocity being actually the zeroth and first moments of the “one-particle” distribution function. Therefore, we regard the order parameter as an additional independent state variable of traffic flow and formulate the corresponding evolution equation governing the lane changing rate. In this context we analyze the instability of homogeneous traffic flow manifesting itself in both of these phase transitions and endowing them with the hysteresis. Besides, the jam state is characterized by the vehicle flows at different lanes being independent of one another.

We discuss the numerical treatment of boundary conditions for a non-local macroscopic model of uni-directional traffic. Furthermore, we propose a general scheme to derive non-local macroscopic models from given microscopic car-following models. Assuming identical (macroscopic) initial and boundary conditions, we show that there are microscopic models for which the corresponding macroscopic version displays a very similar dynamics. This enables us to combine micro- and macrosimulations of road sections by simple algorithms and even to simulate them simultaneously.

We derive a transformation that relates car-following models to their analogous continuum counterpart by using an integral representation of the density. A Taylor expansion of the latter yields an ordinary differential equation in terms of the density and the headway, hence a relation between the two characteristic variables in the corresponding models. This formal approach is supported by similar numerical solutions of the optimal-velocity (OV) model [1] and its continuum analogue. Moreover, the same stability criterion holds in both models. It can be seen that dispersive and anticipation terms of continuum models are a result of applying our transformation to car-following models.

The “fundamental diagram”of macroscopic traffic flow modelling is related to the nearest-neighbour interaction of an exemplary follow-the-leader model, crucially taking into account microscopic fluctuations.

A Lagrangian particle scheme for the solution of the macroscopic traffic flow equations is presented. The scheme transforms the partial Navier-Stokes-like differential equations into a set of ordinary differential equations that axe given as sums over particle contributions. The continuity equation is fulfilled automatically by construction. The method is applied to the ’standard scenario’ of a road loop where the traffic flow is governed by the macroscopic equations of Kerner and Konhäuser.

Within the formalism of the matrix product ansatz, we study a two-species asymmetric exclusion process with backward and forward site-ordered sequential updates. This model describes a two-way traffic flow with a dynamic impurity and shows a phase transition between the free flow and the traffic jam. We investigate characteristics of this jamming and examine similarities and differences between our results and those with the random sequential update [1].

Exact solutions of the first order differential equation with delay are derived. The equation has been introduced as a model of traffic flow. The solution describes the traveling cluster of jam, which is characterized by Jacobi’s elliptic function. The system is related to some soliton systems.

A new single lane inertial car following model of traffic flow is presented. The model demonstrates the presence of three phases in traffic flow: free flow, nonhomogeneous congested flow and homogeneous congested flow. In the non-homogeneous congested flow we find many periodic stable states with different values of flux and wavelength. We also find that states with relatively low and relatively high values of wavelength are metastable.

A simulation to model traffic on a multilane freeway is introduced starting from microscopic driving rules. The model takes each individual car into account with its individual features and actual situations so that a distribution of parameters as well as different behavior can easily be analyzed. Therefore, a detailed study of certain situations, driving tactics, vehicle properties, and their influence on the global traffic flow can be performed. The model and first results are discussed, namely, the influence of the driver behavior on the fundamental diagram.

Although traffic simulations with cellular-automata models give meaningful results compared with empirical data, highway traffic requires a more detailed description of the elementary dynamics. Based on recent empirical studies we present a modified Nagel-Schreckenberg cellular automaton model which incorporates both a slow-to-start and an anticipation rule, which takes into account especially brake lights. The focus in this article lies on the comparison with empirical single-vehicle data.

We present a new cellular automaton model of vehicular traffic in cities by combining ideas borrowed from the Biham-Middleton-Levine (BML) model of city traffic and the Nagel-Schreckenberg (NaSch) model of highway traffic. The model exhibits a dynamical phase transition to a completely jammed phase a critical density which depends on the time periods of the synchronized signals.

We consider the generalization of the asymmetric exclusion model (ASEP) with parallel update where cars can move with velocities v ≤ w_max and v_max > 1. For stochastic open boundary conditions we find a line of a first-order transition separating the free flow phase from the jammed phase. For maximum velocities V_max ≥ 3 so-called “buffers” develop due to the hindrance an injected car feels from the front car at the beginning of the system. As a consequence, the phase diagram qualitatively differs from that for V_max ≤ 2.

A stochastic approach based on the Master equation is proposed to describe the process of formation and growth of car clusters in traffic flow in analogy to usual aggregation phenomena such as the formation of liquid droplets in supersaturated vapour. We have extended the stochastic theory of freeway traffic allowing a coexistence of many clusters on a one-lane circular road. Analytical equations have been derived for calculation of the stationary cluster distribution and related physical quantities of an infinitely large system of interacting cars. If the probability per time to decelarate a car without an obvious reason tends to zero in an infinitely large system, our multi-cluster model behaves essentially in the same way as the one-cluster model presented at Traffic and Granular Flow ’97. In particular, there are three different regimes of traffic flow (free jet of cars, coexisting phase of jams and isolated cars, highly viscous heavy traffic) and two phase transitions between them. In a general case, some qualitative differences in the behaviour of these two models have been observed.

We report experimental results on two model granular media submitted to gravity flows, i.e., a bidimensional hopper and a vertical column. For the hopper, the statistics of the flow blockade is monitored. For the vertical column, we study the flowing dynamics and we show, for narrow openings, regimes of density waves and full blockade depending on the dissipative character of the grains. For the least dissipative grains the density, velocity and temperature fields are measured and compared with the prediction of a granular hydrodynamics theory.

It is shown how the Boltzmann equation corresponding to a model granular gas can be systematically analyzed in order to obtain constitutive relations as well as boundary conditions. It appears that the formulation leading to the establishment of boundary conditions is the first systematic approach to the problem even in the realm of molecular gases. Limitations on the results, in particular the restriction of their validity to near elastic systems are explained and shown to follow from the mesoscopic nature of granular gases.

We outline the development of a kinetic theory for particle segregation in collisional flows of a binary mixture of nearly elastic spheres. We take care to place the segregation mechanism in the context of an appropriately weighted difference of the momentum balances for the two species.

The influence of the granular packing length on the avalanche parameters, such as its mass and the critical angles at which it begins and stops, is studied for packings of mono size glass beads under a controlled humidity environment. In order to understand the dynamics of this kind of systems, experiments are performed in boxes of two different dimensions to see the influence on the parameters. For both boxes, the critical angles show the same qualitative behavior. While the number of layers involved in the avalanche are determined by the box dimensions.

A stochastic model is proposed to describe the ripple formation. Saltation and avalanches are the unique ingredients of the model. The dynamics of ripple formation is studied using a cellular automata. The “ripple state” turns out to be metastable. An extension of the model to the case of binary mixtures is also discussed.

The interface dynamics in rotating cylinders of an initially well-segregated binary particle configuration is studied numerically. The process is characterized by calculating diffusion coefficients for different friction and density ratios. A transient segregation wave is observed in the mixing regime.

The experimental motivation for this study are recent publications on cohesive granular materials [2–4,10]. Our central question is, in which regime and by which mechanism the the movement of grains changes from movement of independent particles to a movement of small clusters with increasing cohesion. Cohesion introduces an additional length scale, so that the effects become size-dependent. The cohesive force acting on a volume element of size I x I x I is proportional to its surface, or ∝ I2. The repulsive force generated by the mass of the volume element is ∝ I3. The strength of the cohesion and the density of the particles determine the size for which repulsion and cohesion are in equilibrium for a certain characteristic length d.

A system of two coupled partial differential equations is proposed as a model for the standing and rolling layers of slowly moving granular matter with external sources. The model describes the accumulation of matter on tables and in silos with prescribed areas and cross sections. For these boundary conditions stationary and similarity solutions are characterized and explicitly presented.

Using the discrete element method (DEM), I have constructed a numerical code to investigate the static properties of granular particles. In this model, only the gravitational and the contact forces are considered. I focus on piles made up of grains in the two-dimensional geometry. After depositing grains, a static configuration of the grains is obtained, with a reasonable angle of repose. In addition, the network structure of the pile is visualized by connecting centers of grains in contact, showing an interesting dispersion of densities. Furthermore, the horizontal and vertical pressure components along the bottom of the pile are measured. The existence of a dip, still a disputed issue, is inconclusive in this study.