Traffic and Related SelfDriven ManyParticle Systems
Abstract
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 socalled “phantom traffic jams”, although they all like to drive fast? What are the mechanisms behind stopandgo 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 of the traffic volume cause a lasting traffic jam? Under which conditions can speed limits speed up traffic? Why do pedestrians moving in opposite directions normally organize in lanes, while similar systems are “freezing by heating”? Why do panicking pedestrians produce dangerous deadlocks? All these questions have been answered by applying and extending methods from statistical physics and nonlinear dynamics to selfdriven manyparticle systems. This review article on traffic introduces (i) empirically data, facts, and observations, (ii) the main approaches to pedestrian and vehicle traffic, (iii) microscopic (particlebased), mesoscopic (gaskinetic), and macroscopic (fluiddynamic) models. Attention is also paid to the formulation of a micromacro link, to aspects of universality, and to other unifying concepts like a general modelling framework for selfdriven manyparticle systems, including spin systems. Subjects like the optimization of traffic flows and relations to biological or socioeconomic systems such as bacterial colonies, flocks of birds, panics, and stock market dynamics are touched as well.
Contents
 1 Introduction
 2 Empirical findings for freeway traffic

3 Modelling approaches for vehicle traffic
 3.1 Microscopic followtheleader models
 3.2 Cellular automata (CA)
 3.3 Master equation

3.4 Macroscopic traffic models
 3.4.1 The LighthillWhitham model (LW model)
 3.4.2 The Burgers equation
 3.4.3 Payne’s model and its variants
 3.4.4 The models by Prigogine and Phillips
 3.4.5 The models by Whitham, Kühne, Kerner, Konhäuser, and Lee
 3.4.6 The WeidlichHilliges model
 3.4.7 Common structure of macroscopic traffic models
 3.5 Gaskinetic traffic models and micromacro link

4 Properties of traffic models

4.1 Identical vehicles on homogeneous freeways
 4.1.1 “Phantom traffic jams” and stopandgo traffic
 4.1.2 (Auto)Solitons, KortewegdeVries equation, and GinzburgLandau equation
 4.1.3 Jam formation: Local breakdown and cluster effects, segregation, and selforganized criticality (SOC)
 4.1.4 Instability diagram, stopandgo waves, metastability, and hysteresis
 4.1.5 Selforganized, “natural” constants of traffic flow
 4.1.6 Kerner’s hypotheses
 4.2 Transition to congested traffic at bottlenecks and ramps
 4.3 Heterogeneous traffic
 4.4 Multilane traffic and synchronization
 4.5 Bidirectional and city traffic
 4.6 Effects beyond physics
 4.7 Traffic control and optimization

4.1 Identical vehicles on homogeneous freeways
 5 Pedestrian traffic
 6 Flocking and spin systems
 7 Summary and outlook
1 Introduction
1.1 Motivation: History of traffic modelling and traffic impact on society
The interest in the subject of traffic dynamics is surprisingly old. In 1935, Greenshields has carried out early studies of vehicular traffic, and already in the 1950ies, a considerable publication activity is documented by journals on operations research and engineering. These papers have already addressed topics like the fundamental diagram between traffic flow and vehicle density or the instability of traffic flow, which are still relevant. The reason becomes understandable through the following quotation from H. Greenberg in 1959:
“The volume of vehicular traffic in the past several years has rapidly outstripped the capacities of the nation’s highways. It has become increasingly necessary to understand the dynamics of traffic flow and obtain a mathematical description of the process.”
More than fourty years later, the situation has deteriorated a lot. Some cities like Los Angeles and San Francisco suffer from heavy traffic congestion around the clock. In Europe, as well, the time that drivers spend standing in traffic jams amounts to several days each year. During holiday seasons, jams may grow up to more than 100 kilometers in size. Vehicle emissions like SO, NO, CO, CO, dust particles, smog, and noise have reached or even exceeded a level comparable to those by industrial production and those by private households, being harmful to the environment and human health. On average, every driver is involved in one accident during his lifetime. In Germany alone, the financial damage by traffic due to accidents and environmental impacts is estimated 100 billion dollars each year. The economic loss due to congested traffic is of the same order of magnitude. However, the traffic volume is still growing because of increased demands for mobility and modern logistics, including ecommerce.
Without any doubt, an efficient transportation system is essential for the functioning and success of modern, industrialized societies. But the days, when freeways were free ways, are over. Facing the increasing problems of individual traffic, the following questions came up: Is it still affordable and publicly acceptable to expand the infrastructure? Will drivers still buy cars in view of streets that effectively turn into parking lots? It is for these reasons that automobile companies started to worry about their own future and decided to spend considerable amounts of money for research on traffic dynamics and the question how the available infrastructure could be used more efficiently by new technologies (telematics).
At this time, physicists were motivated to think about what physics could contribute to the field of traffic dynamics. Although, among mathematicians, physicists, and chemists, there were already some early pioneers like Whitham, Prigogine, Montroll, and Kühne, the main activities started in 1992 and 1993 with papers by Biham et al. (1992), Nagel and Schreckenberg (1992), as well as Kerner and Konhäuser (1993). These papers initiated an avalanche of publications in various international physics journals. Meanwhile, it is difficult to keep track of the scientific development and literature. Therefore, it is high time to write a review on traffic, which tries to bring together the different approaches and to show up their interrelations. Doing so, I will take into account many significant contributions by traffic engineers as well. Hopefully, this will stimulate more intense discussions and cooperations among the different disciplines involved.
In the following sections, I will try to give an overview of the field of traffic dynamics from the perspective of manyparticle physics.
1.2 Driven manyparticle systems
A big challenge for physicists are the dynamics and pattern formation in systems far from equilibrium, especially living systems (Nicolis and Prigogine, 1977; Haken, 1977, 1983, 1988; Pasteels and Deneubourg, 1987; Weidlich, 1991; Feistel and Ebeling, 1989; Vicsek, 1992; Kai, 1992; DeAngelis and Gross, 1992; Vallacher and Nowak, 1994; Cladis and PalffyMuhoray, 1995; Helbing, 1995a). Nonequilibrium systems are characterized by not being closed, i.e., by having an exchange of energy, particles, and/or information with their environment. Consequently, they often show complex behavior and, normally, there are no general results such as the laws of thermodynamics and statistical mechanics for closed systems of gases, fluids, or solids in equilibrium. Nevertheless, they can be cast in a general mathematical framework (Helbing, 2001), as I will show here.
1.2.1 Classical mechanics, fluids, and granular media
Let us start with Newton’s equation of motion from classical mechanics, which describes the acceleration of a body of mass subject to pair interactions with other bodies :
(1) 
The interaction forces are mostly dependent on the locations and of the interacting bodies and at time . Often, they depend only on the distance vector , but in special cases, they are also functions of the velocities and . For potential forces, the above manybody system can be characterized by a Hamilton function. A typical example is the description of the motion of celestial bodies.
In driven manybody systems such as fluids under the influence of pressure gradients and boundary forces or vibrated granular media like sand, we have to consider additional interactions with the environment. Therefore, we need to consider additional terms. This concerns (external) driving forces due to boundary interactions and gravitational or electrical fields, (sliding) friction forces with friction coefficient , and individual fluctuations reflecting thermal interactions with the environment (boundaries, air, etc.) or a variation of the surface structure of the particles:
(2)  
From these socalled “microscopic”, moleculardynamics type of equations, one can systematically derive “macroscopic”, fluiddynamic equations for the spatiotemporal evolution of the particle density, the momentum density or average velocity, and the energy density or velocity variance (related to the temperature). The methods for the construction of this micromacro link will be sketched in Sec. 3.5.
In driven systems, the ongoing competition between the driving forces and the dissipative friction forces leads to a spatiotemporal redistribution of energy, which produces a great variety of selforganization phenomena. This results from nonlinearities in the equations of motion, which allow small initial perturbations to be enhanced and nonequilibrium patterns to be dynamically stabilized. In fluids one can find the formation of waves or vortices, bifurcation scenarios like period doubling behavior, the RuelleTakensNewhouse route to chaos, intermittency, or turbulence, depending on the particular boundary conditions (Joseph, 1976; Landau and Lifshits, 1987; Drazin and Reid, 1981; Swinney and Gollub, 1985; Schuster, 1988; Großmann, 2000). In this review, it is important to know that turbulence normally requires three or higherdimensional systems, so it is not expected to appear in one or twodimensional vehicle or pedestrian traffic. However, the instability mechanism for the explanation of the subcritical transition to turbulence in the HagenPoiseuille experiment (Gebhardt and Großmann, 1994; Großmann, 2000) may also be relevant to traffic systems, as pointed out by Krug (cf. the discussion of metastability in Sec. 4.1.4).
In vibrated granular media, one can find emergent convection patterns (Ehrichs et al., 1995; Bourzutschky and Miller, 1995; Pöschel and Herrmann, 1995), collective oscillating states (socalled oscillons, see Umbanhowar et al., 1996), spontaneous segregation of different granular materials (Pöschel and Herrmann, 1995; Santra et al., 1996; Makse et al., 1997), or selforganized criticality with powerlaw distributed avalance sizes in growing sand heaps (Bak et al., 1987, 1988; Bak, 1996) or in the outflow from hoppers (Schick and Verveen, 1974; Peng and Herrmann, 1995).
In spite of many differences between flows of fluids,
granular media, and vehicles or pedestrians due to different conservation laws
and driving terms,
one can apply similar methodological approaches, e.g.
(i) microscopic, molecular dynamic models (see, e.g., Hoover, 1986;
Goldhirsch et al., 1993; Buchholtz and Pöschel, 1993;
Hirshfeld et al., 1997; Sec. 3.1),
(ii) lattice gas automata (Frisch et al., 1986; Chen et al., 1991;
Peng and Herrmann, 1994; Tan et al., 1995)
or cellular automata (see Sec. 3.2),
(iii) gaskinetic (Boltzmann and Enskoglike) models (Boltzmann, 1964; Enskog, 1917;
Chapman and Cowling, 1939; Cohen, 1968, 1969;
Lun et al., 1984; Jenkins and Richman, 1985; Kobryn et al., 1996; Dufty et al., 1996;
Lutsko, 1997;
Cercignani and Lampis, 1988; McNamara and Young, 1993; Sela et al., 1996;
see Sec. 3.5), and
(iv) fluiddynamic models (Haff, 1983; Goldhirsch, 1995; Sela and Goldhirsch, 1995;
Hayakawa et al., 1995; Du et al., 1995; see Sec. 3.4).
1.3 Selfdriven manyparticle systems
In selfdriven manyparticle systems, the driving force is not of external origin (exerted from outside), but associated with each single particle and selfproduced. This requires each particle to have some kind of internal energy reservoir (Schweitzer et al., 1998b; Ebeling et al., 1999).
Selfdriven “particles” are a paradigm for many active or living systems, where they are a simplified and abstract representation of the most important dynamic behavior of cells, animals, or even humans. In order to reflect this, I will generalize Eq. (2) a little, replacing the external driving force by an individual driving force . Moreover, Newton’s third law (actio = reactio) does not necessarily apply anymore to the “selfdriven”, “selfpropelled”, “motorized”, or “active” “particles” we have in mind. I will show that these minor changes will imply various interesting phenomena observed in nature, for example in biological, traffic, or socioeconomic systems. In this context, the masses are sometimes not welldefined, and it is better to rewrite the resulting equation by means of the scaled quantities , , , and , where the accelerations are often loosely called forces as well:
(3) 
From this equation we can see that, with a relaxation time of , the driving term and friction term together lead to an exponentialintime adaptation of the velocity to the desired speed and the desired direction of motion. This is, however, disturbed by fluctuations and interactions with other particles . It is clear that attractive forces will lead to agglomeration effects. Therefore, we will usually investigate systems with vanishing or repulsive interactions, for which one can find various surprising effects.
A further simplification of the model equations can be reached in the overdamped limit of fast (adiabatic) relaxation. In this case and with the abbreviations , , we obtain
(4)  
1.3.1 The concept of social (behavioral) forces
Human behavior often seems to be “chaotic”, irregular, and unpredictable. So, why and under which conditions can we apply the above force equations? First of all, we need to be confronted with a phenomenon of motion in some (quasi)continuous space, which may be also an abstract behavioral space or opinion scale (Helbing, 1992a, 1993b, 1994, 1995a). It is favourable to have a system where the fluctuations due to unknown influences are not large compared to the systematic, deterministic part of motion. This is usually the case in pedestrian and vehicle traffic, where people are confronted with standard situations and react automatically rather than making complicated decisions between various possible alternatives. For example, an experienced driver would not have to think about the detailled actions to be taken when turning, accelerating, or changing lanes.
This automatic behavior can be interpreted as the result of a learning process based on trial and error (Helbing et al., 2001c), which can be simulated with evolutionary algorithms (Klockgether and Schwefel, 1970; Rechenberg, 1973; Schwefel, 1977; Baeck, 1996). For example, pedestrians have a preferred side of walking (Oeding, 1963; Older, 1968; Weidmann, 1993), since an asymmetrical avoidance behavior turns out to be profitable (Bolay, 1998). The related formation of a behavioral convention can be described by means of evolutionary game theory (Helbing, 1990, 1991, 1992a, c, 1993a, 1995a, 1996c).
Another requirement is the vectorial additivity of the separate force terms reflecting different environmental influences and psychological factors. This is probably an approximation, but there is some experimental evidence for it. Based on quantitative measurements for animals and test persons subject to separately or simultaneously applied stimuli of different nature and strength, one could show that the behavior in conflict situations can be described by a superposition of forces (Miller, 1944, 1959; Herkner, 1975). This fits well into a concept by Lewin (1951), according to which behavioral changes are guided by socalled social fields or social forces, which has been put into mathematical terms by Helbing (1991, 1992a, 1993b, 1994, 1995a; see also Helbing and Molnár, 1995). The third Newtonian law, however, is usually not valid.
1.4 What this review is about
In the following sections,
(i) I will focus on the phenomena actually observed in traffic and their
characteristic properties, while I will
discuss models only to the extend, to which they are helpful and necessary for an
interpretation and a better understanding of the observations,
(ii) I will discuss the main methods from statistical physics
relevant for modelling and analyzing traffic dynamics (see the table of contents),
(iii)
I will discuss how far one can get with a
physical, manyparticle description of traffic, neglecting sociopsychological factors
and human behavior.
The limitations of this approach will be shortly discussed in Sec. 4.6.
This review is intended to serve both, experts and newcomers in the field, so some matters will be simplified or explained in more detail for didactical reasons. Moreover, I will try to identify open problems and to shed new light on some controversial topics presently under discussion. The main focus will be on the various kinds of phenomena occuring in selfdriven manyparticle systems and the conditions, under which they appear. I will start with the subject of onedimensional vehicle traffic, continue with twodimensional pedestrian traffic and threedimensional air traffic by birds.
Those physicists who don’t feel comfortable with nonphysical systems may instead imagine special kinds of granular, colloidal, or spin systems driven by gravitational or electrical forces, or imagine particular systems with Brownian motors (Hänggi and Bartussek, 1996; Astumian, 1997; Jülicher et al., Prost, 1997; Reimann, 2000). Moreover, I would like to encourage everyone to perform analogous experiments in these related physical systems.
Despite the complexity of traffic and complications by unknown, latent, or hardly measurable human factors, physical traffic theory is meanwhile a prime example of a highly advanced quantitative description of a living system. An agreement with empirical data has been reached not only on a qualitative, but even on a semiquantitative level (see Sec. 4.2.1). Moreover, there are even “natural constants” of traffic, emerging from the nonlinear vehicle interactions, see Sec. 4.1.5. But, before getting into all this, I would like to mention some other interesting (self)driven manyparticle systems loosely related to traffic.
1.5 Particle hopping models, powerlaw scaling, and SOC
Manyparticle systems in equilibrium can be very well understood with methods from thermodynamics (Keizer, 1987) and statistical physics (Uhlenbeck and Ford, 1963; Landau and Lifshits, 1980; Ma, 1985; Klimontovich, 1986; Huang, 1987). Examples are phase transitions between different aggregate states of matter like vapor, water, and ice, or the magnetization of spin systems composed of many “elementary magnets”. It is known that the phase transition of disordered manyparticle systems with shortrange interactions into states with longrange order is easier in higher dimensions, because of the greater number of neighbors to interact with. Often, there is a certain upper dimension, above which the system can be described by a mean field approach so that the fluctuations can be neglected. For lowerdimensional spaces, one can usually develop approximate theories for the influence of noise by means of suitable expansions and renormalizationgroup treatments based on scaling arguments. These give universal powerlaw scaling exponents for the (“critical”) behaviour of the system close to the phase transition point (critical point) (Stanley, 1971; Domb and Green, 1972–1976; Ma, 1976; Hohenberg and Halperin, 1977; Stinchcombe, 1983; Domb and Lebowitz, 1983–2000; Voss, 1985; Schuster, 1988). In nonequilibrium systems, one can frequently find selforganized criticality (Bak et al., 1987, 1988; Bak, 1996) and fractal properties (Mandelbrot, 1983; Family and Vicsek, 1991; Vicsek, 1992; Barabási and Stanley, 1995). Selforganized criticality (SOC) means that the respective system drives itself to the critical point, which is normally characterized by longrange correlations and scalefree power laws in analogy to thermodynamics.
In the considered manyparticle systems, there is often also a certain lower dimension, below which the system is always disordered, because of the small number of neighbors. For example, it is known that neither ferromagnetic nor antiferromagnetic order is possible in onedimensional equilibrium spin systems (Mermin and Wagner, 1966). The discovery that the situation can be very different in driven, nonequilibrium spin systems, has initiated an intense research activity regarding the phase transitions in onedimensional driven diffusive systems far from equilibrium. It turned out that many properties of equilibrium systems can be generalized to nonequilibrium systems, but others not.
By means of particle hopping models, it was possible to gain a better understanding of directed percolation (Domany and Kinzel, 1984), of spontaneous structure formation (Vilfan et al., 1994), of spontaneous symmetry breaking (Evans et al., 1995), of the roughening transition in certain growth processes (Alon et al., 1996), of the nonequilibrium wetting transition (Hinrichsen et al., 1997), and of phase separation (Evans et al., 1998; Helbing et al., 1999b). Nevertheless, for these nonequilibrium transitions there is still no general theory available which would be of comparable generality as equilibrium thermodynamics or statistical physics. For further reading, I recommend the books by Spohn (1991), Schmittmann and Zia (1995, 1998), Derrida and Evans (1997), Liggett (1999), and Schütz (2000).
Note that the above mentioned models usually assume a random sequential (asynchronous) update, i.e. the state of each particle is randomly updated after an exponentially distributed waiting time . However, it is known that this is not realistic for traffic systems, which rather require a parallel (synchronous) update of the vehicle locations (Schreckenberg et al., 1995), as in other flow problems. To reflect the driver reaction to the corresponding change of the traffic situation, it is common to update the vehicle speeds synchronously as well. However, this is debatable, since some models of spatiotemporal interactions in social systems show artefacts, if updated in parallel (Huberman and Glance, 1993). For example, a parallel update excludes the dynamical reachability of certain states (Schadschneider and Schreckenberg, 1998), which are called paradisical or GardenofEden states (Moore, 1962).
1.5.1 The asymmetric simple exclusion process (ASEP)
The only particle hopping model I will discuss here is the asymmetric simple exclusion process. Particularly relevant contributions to this model go back to Spitzer (1970), Liggett (1975), Domany and Kinzel (1984), Katz et al. (1984), Liggett (1985), Krug (1991), Derrida et al. (1992), Janowski and Lebowitz (1992), Derrida et al. (1993), Ferrari (1993), Schütz and Domany (1993), Stinchcombe and Schütz (1995a, b), Kolomeisky et al.(1998), and Schütz (1998). For overviews see Spohn (1991), Schmittmann and Zia (1995, 1998), Liggett (1999), and Schütz (2000). In the following, I will restrict to the totally asymmetric simple exclusion process (TASEP). This model is defined by sites of a onedimensional lattice, which can be either empty (corresponding to the occupation number ) or occupied by one particle (). The particle locations and occupation numbers are updated every time step . When updated, a particle in cell hops to the right neighboring cell with probability , if this is empty, otherwise it stays in cell . The total rate of moving to the right is, therefore, given by . The boundaries are characterized as follows: A particle enters the system at the leftmost cell with probability , if this is empty. Moreover, a particle in the rightmost cell leaves the system with probability . This corresponds to particle reservoirs at the boundaries which can be described by constant occupation probabilities and . Although it takes enourmous efforts, the stationary states and even the dynamics of the TASEP can be analytically determined. For this, one has to solve the corresponding master equation (see Secs. 3.3.3 and 3.4.1).
1.6 Active Brownian particles
Like usual Brownian particles, active Brownian particles perform a random walk as well. However, they are not only reactive to an external potential , but they are driven by an internal energy reservoir (“pumped particles”) or can actively change the potential while moving (“active walkers”). Therefore, I have recently suggested to call them Brownian agents. For an overview see Schweitzer (2001).
1.6.1 Pumped Brownian particles
Schweitzer et al. (1998b) suggest a model describing how selfdriven particles may take up and consume the internal energy behind their driving force. Their model corresponds to Eq. (2) with the specifications (i.e. no direct interactions), but is replaced by the expression , where the first term is an external potential force and the last term an internal driving force ( being the friction coefficient). The dimensionless (scaled) energy reservoir is assumed to follow the equation
(5) 
Herein, reflects the exploitation rate of energy resources, while the last term describes the consumption rate of energy, which grows quadratically with the speed ( and being suitable parameter values). If the relaxation of this energy equation is fast, we can approximate the driving term by
so that the internal driving and the dissipative friction together can be represented by an active friction coefficient Moreover, the driving direction is given by the normalized actual velocity , and the desired velocity depends on the speed as well: Notice that particle takes up energy only when it moves with some finite speed (i.e. exploits its environment). Therefore, in the absence of a potential , we find the stationary solution . However, this is only stable under the condition . Otherwise, particle will spontaneously start to move with average speed
(6) 
Depending on and , this spontaneous motion displays interesting dynamics such as limit cycles, deterministic chaos, or intermittency (Schweitzer et al., 1998b; see also Chen, 1997). It also allows the particles to climb potential hills, e.g. in a periodic ratchet potential (Schweitzer et al., 2000).
1.6.2 Dissipative Toda and Morse chains
Toda and Morse chains are particles coupled to a heat bath and moving on a ring with particular asymmetrical springs among neighbors, which are described by nonlinear Toda or Morse potentials (Bolterauer and Opper, 1981; Toda, 1983; Jenssen, 1991; Ebeling and Jenssen, 1992; Dunkel et al., 2001). Assuming pumped particles with an active friction coefficient similar to the previous paragraph, Ebeling et al. (2000) have found interesting dynamical patterns for overcritical pumping. This includes uniform rotations, one and multiple solitonlike excitations, and relative oscillations. For Morse potentials, one also observes clustering effects which partly remind of jamming (Dunkel et al., 2001).
1.6.3 Active walker models
By modifying their environment locally, active walkers have indirect interactions with each other, which may lead to the formation of global structures. The versatility of this concept for the description of physical, chemical, biological, and socioeconomic systems is discussed by Schweitzer (2001). For example, a simple model is given by and changes of the environmental potential according to
Herein, and are constants, and denotes Dirac’s delta function, giving contributions only at . Therefore, the first term on the right hand side reflects particles leaving attractive markings at their respective locations . The last term describes a diffusion of the field (e.g., a chemical one), and the previous term its decay.
The result of the dynamics is a spatial agglomeration process. While in the first stage, one finds the emergence of localized clusters at random locations, the clusters eventually merge with each other, resulting in one big cluster. The growth and competition of these clusters can be described by an EigenFisherlike selection equation (Schweitzer and SchimanskyGeier, 1994; Fisher, 1930; Eigen, 1971; Eigen and Schuster, 1979).
1.6.4 Pattern formation of bacterial colonies
Already in 1994, Eshel BenJacob et al. have proposed a socalled communicating walker model for the beautiful pattern formation in certain bacterial colonies (for reviews see BenJacob, 1997; BenJacob et al., 2000). This model takes into account the consumption, diffusion, and depletion of food, the multiplication under good growth conditions, and the transition to an immobile spore state under starvation, as well as the effect of chemotaxis, i.e. the attractive or repulsive reaction to selfproduced chemical fields. The model allows one to reproduce the observed growth patterns of bacterial colonies as a function of the nutrient concentration and the agar concentration determinining how hard the bacteria can move. The same bacterial patterns have recently been reproduced by means of a macroscopic reactiondiffusion model (Golding et al., 1998). The formation of the observed dendritic structures is due to a socalled diffusion instability (BenJacob, 1993, 1997).
1.6.5 Trail formation by animals and pedestrians
Another example is the observed formation of trunk trail systems by certain ant species, which is also based on chemotaxis. It can be modelled by two kinds of chemical markings, one of which is dropped during the search for food, while the other is dropped on the way back to the nest of the ant colony (Schweitzer et al., 1997). Other models for antlike swarm formation have been developed by Deneubourg et al. (1989) and Bonabeau (1996).
Further kinds of active walker models have been proposed for the selforganization of trail systems by pedestrians or animals, where the markings correspond to “footprints” or other kinds of modifications of the ground which make it more comfortable to move (Helbing et al., 1997a, b; Helbing, 1998c). Interestingly enough, the model yields fair solutions and optimal compromises between short and comfortable ways, see Fig. 1 (Helbing, 1998c; Helbing and Vicsek, 1999). The corresponding computer simulations are, therefore, a valuable tool for developing optimized pedestrian facilities and way systems.
1.7 Vehicle and pedestrian traffic
The next sections will mainly focus on unidirectional freeway traffic, but bidirectional traffic and twodimensional city traffic are shortly sketched as well. I will start with an overview over the most important empirical findings. Afterwards, I will concentrate on the different modelling approaches and discuss the properties of these models, starting with onelane traffic of identical vehicles and then adding more and more details, including heterogeneous traffic on multilane roads. Finally, I will discuss twodimensional pedestrian traffic and threedimensional flocks of birds.
2 Empirical findings for freeway traffic
As physics deals with phenomena in our physical world, a physical theory must face the comparison with empirical data. A good theory should not only reproduce all the empirically known properties of a system, but it should also make predictions allowing us to verify or falsify the theory. To gain on overview of data analyses, I recommend to read the review articles and books by Gerlough and Huber (1975), Koshi et al. (1983), the Transportation Research Board (1985), May (1990), Daganzo (1997a, 1999a, b), Bovy (1998), and Kerner (1998b, c, 1999a, b, 2000a, b).
2.1 Measurement techniques
Probably the most refined technique to gain empirical data is based on aerial photography, allowing us to track the trajectories of many interacting vehicles and even their lane changing maneuvers (Treiterer and Taylor, 1966; Treiterer and Myers, 1974). A method suitable for experimental investigations are carfollowing data. Depending on the equipment of the cars, it is possible to determine the location and speed, possibly the acceleration and clearance (netto distance), and sometimes even lane changing maneuvers of the equipped car or of the respective following vehicle (see, for example, Koshi et al., 1983; Bleile, 1997a, b, 1999).
However, most data are obtained by detectors located at certain cross sections of the freeway. For example, single inductionloop detectors measure the numbers of crossing vehicles during a certain sampling interval as well as the times and when a vehicle reaches and leaves the detector. This facilitates to determine the time headways (brutto time separations)
and the time clearances (netto time separations or time gaps) including their respective distributions, as well as the vehicle flow
(7) 
and the time occupancy (where and denotes the last vehicle before the sampling interval begins). The newer double inductionloop detectors additionally measure the vehicle velocities and the vehicle lengths , allowing us to estimate the headways (brutto distances) (assuming constant vehicle speeds) and the clearances (netto distances) The velocity measurements are normally used to obtain the (arithmetic) average velocity
(8) 
but the velocity variance
(9) 
and the local velocity distribution may be determined as well. The vehicle density is often calculated via the fluiddynamic formula (46), i.e. Another method is to relate the density with the time occupancy, as in the formula where is the average vehicle length during the measurement interval and the detector length (i.e. the loop extension) in driving direction (May, 1990).
The problem of the above measurement methods is that the velocity distribution measured over a time interval differs from the one measured on a freeway section of length . In other words, temporal and spatial averaging yield different results, since fast vehicles cross a section of the freeway more frequently than slow ones, which is intuitive for a circular multilane freeway with fast and slow lanes. The problem of determining the empirical density via the formula is that it mixes a temporal average (the flow) with a spatial one (the density). This can be compensated for by the harmonic average velocity defined by
(10) 
giving a greater weigth to small velocities (Gerlough and Huber, 1975; Leutzbach, 1988). Using formula (10) instead of the commonly applied formula (8) together with the relation results in similarly shaped velocitydensity relations, but the density range is much higher (see Fig. 2). Moreover, the velocity and flow values at high densities are somewhat lower. The disadvantage of Eq. (10) is its great sensitivity to errors in the measurement of small velocities .
2.2 Fundamental diagram and hysteresis
Functional relations between the vehicle flow , the average velocity , and the vehicle density or occupancy have been measured for decades, beginning with Greenshields (1935), who found a linear relationship. The name “fundamental diagram” is mostly used for some fit function
(11) 
of the empirical flowdensity relation, where stands for the fitted
empirical velocitydensity relation (see Fig. 2), which is monotonically
falling:
Most measurements confirm the following features:
(i) At low densities , there is a clear and quasi
onedimensional relationship between
the vehicle flow and the density. It starts almost linearly and is bent
downwards. The slope at low densities
corresponds to the average free velocity , which can
be sustained at finite densities as long as there are sufficient
possibilities for overtaking. That is, the velocitydensity relation
of a multilane road starts horizontally, while it tends to have a negative
slope on a onelane road.
(ii) With growing density, the velocity decreases monotonically,
and it vanishes together with the flow
at some jam density , which is
hard to determine because of the above mentioned measurement problems. Estimates by different
researchers from various countries reach from 120 to 200 vehicles per
kilometer and lane, but values between 140 and 160 vehicles per
kilometer are probably most realistic.
(iii) The vehicle flow has one maximum at medium
densities.
(iv) The empirical flowdensity relation is discontinous and looks
comparable to a mirror image of the Greek letter lambda. The two
branches of this reverse lambda are
used to define free lowdensity and congested
highdensity traffic (see, e.g., Koshi et al., 1983;
Hall et al., 1986;
Neubert et al., 1999b; Kerner, 2000a; cf. also Edie and Foote, 1958). In congested
traffic, the average vehicle velocity
is significantly lower than in free traffic. Therefore,
the free and congested part of the flowdensity relation can be
approximately separated by a linear flowdensity relation
, where is the average velocity
below which traffic is characterized as congested. Around this line,
the density of data points is reduced (see Fig. 3a).
The reverselambdalike structure of the flowdensity data implies
several things: First, we have a discontinuity at some
critical density (Edie, 1961; Treiterer and Myers, 1974; Ceder and May, 1976;
Payne, 1984), which has animated researchers (Hall, 1987; Dillon and Hall, 1987;
Persaud and Hall, 1989; Gilchrist and Hall, 1989)
to relate traffic dynamics with catastrophe theory (Thom, 1975; Zeeman, 1977).
Second, there is a certain density
region
with , in which we can have either free or congested traffic, which
points to hysteresis (Treiterer and Myers, 1974). The tip of
the lambda corresponds to highflow states of free traffic (in the passing lanes), which
can last for many minutes (see, e.g., Cassidy and Bertini, 1999). However, these are
not stable, since it is only a matter of time until
a transition to the lower, congested branch of the lambda takes place
(Persaud et al., 1998; see also Elefteriadou et al., 1995). The
transition probability from a free to a congested state
within a long time interval is 0 below the flow
and 1 above the flow . Between and
, it is monotonically increasing, see Fig. 4
(Persaud et al., 1998).
The transition from congested to free traffic does not go back to
highflow states, but to flows belonging to densities , typically around . The highflow states in the density range
are only metastable
(Kerner and Rehborn, 1996b, 1997; Kerner, 1998b, 1999a, b, c, 2000a, b).
(v) The flowdensity data in the congested part are widely scattered
within a twodimensional area (see, e.g., Koshi et al., 1983;
Hall et al., 1986; Kühne, 1991b).
This complexity in traffic flow is usually interpreted as
effect of fluctuations, of jam formation, or of an instability of vehicle
dynamics. The scattering is reduced by increasing
the sampling interval of the data averaging (Leutzbach, 1988).
(vi) By removing the data belonging to wide moving jams (see Fig. 20), Kerner and Rehborn
(1996b) could demonstrate that the remaining data of congested traffic
data still display a wide and twodimensional scattering (see Fig. 3a), thereby
questioning the applicability of a fundamental diagram and
defining the state of “synchronized flow” (“synchronized” because of the typical
synchronization between lanes in congested traffic, see Fig. 15a,
and “flow” because of flowing in contrast to standing traffic in fully developed jams).
Therefore, Kerner and Rehborn classify three traffic phases:
– free flow,
– synchronized flow (see Secs. 2.3, 2.4, and 2.5.2 for details), and
– wide moving jams (i.e. moving jams whose width in longitudinal
direction is considerably higher than the width of the jam fronts).
In contrast to synchronized flow, a wide moving jam propagates either
through free or any kind of sychronized flow and through
bottlenecks (e.g. on and offramps), keeping the propagation
velocity of the jam’s downstream front (Kerner, 2000a, b).
In contrast to wide moving jams, after synchronized flow has occurred
upstream of an onramp, the downstream front of synchronized flow is fixed
at the onramp. In an initially free flow, two types of transitions
are observed: either to synchronized flow or to (a) wide moving jam(s).
Both of them appear to be firstorder phase transitions accompanied by
different hysteresis and nucleation effects (Kerner and Rehborn,
1997; Kerner, 1998a, 1999a, 2000c).
(vii) The slopes
of the lines connecting successive data points are always positive
in free traffic. In synchronized flow, however, they
erratically take on positive and negative values, characterizing
a complex spatiotemporal dynamics (Kerner and Rehborn, 1996b). This
erratic behavior
is quantitatively characterized by a weak crosscorrelation between
flow and density (Neubert et al., 1999b). Banks (1999) showed that
it could be interpreted as a result of random variations in the
time clearances (partly due to acceleration and deceleration maneuvers
in unstable traffic flow). These variations are, in fact, very large
(cf. Fig. 6). Banks points out that, if drivers would keep a save
clearance in congested
traffic, the flow would grow with decreasing (effective)
safe time clearance according to
(12) 
where denotes the minimum frontbumpertorearbumper distance
and
the jam distance with the (average) effective vehicle length .
Simplifying his argument, in the congested regime positive slopes can result from a
reduction in the safe time clearance with growing density,
while negative slopes normally correspond to a
reduction in the speed. Hence, the slopes do not
necessarily have the meaning of a speed of wave propagation.
(viii) The flowdensity data depend on the measurement cross section.
While the congested branch is very pronounced upstream of a bottleneck,
it is virtually not present far downstream of it. However,
immediately downstream of a bottleneck, one finds a positively
sloped congested area slightly below
the free branch (“homogeneousinspeed states”). It
looks similar to the upper part of the free branch, but with a
somewhat lower desired velocity. I suggest that this may indicate
a transition to free traffic in the course of the road
(see Figs. 5 and 43b) and that this “recovering traffic” bears already some signatures of free traffic
(cf. also Persaud and Hurdle, 1988; Hall et al., 1992).
For example, onramp flows just
add to the flows on the freeway, which may produce states with high flows
(Kerner, 2000b). Some observations, however, question the interpretation
of homogeneousinspeed states as recovering traffic, as they can
extend over stretches of more than 3 kilometers (Kerner, 1999a).
2.3 Time headways, headways, and velocities
Time headways show an astonishing individual variation, supporting Banks’ theory discussed above. When distinguished for different density ranges, timeheadway distributions have an interesting property. It turns out that the distribution becomes sharply peaked around approximately 1 s for densities close to congestion (Smulders, 1990; Hoogendoorn and Bovy, 1998c; Neubert et al., 1999b). For lower and higher densities, the distribution is considerably broader, see Fig. 6 (Tilch and Helbing, 2000). This indicates that congestion is an overcritical phenomenon occuring when the maximum efficiency in terms of time headways (related to highflow states) cannot be supported any longer. One may conclude that the effective time clearance has, then, reached the safe time clearance . A further increase in the density forces to reduce the speed, which automatically increases the time headway , even if the effective clearance remains (cf. Banks, 1991b).
The distributions of vehicle headways can be determined as well, but it is always surprisingly broad (see Fig. 7). Therefore, Bleile (1999) suggests to consider this by an individual distance scaling of the velocityclearance relation.
What are the reasons and consequences of the wide scattering of time headways and clearances, apart from driverdependent preferences? By detailed investigations of clearanceoverspeed relations, Koshi et al. (1983) have observed that vehicles keep larger distances in real congested traffic than in free traffic or undisturbed congested traffic under experimental conditions (cf. Fig. 8).
Plotting the speed over the vehicle distance, one can find a densitydependent reduction of the speed (see Fig. 9), which has been called “frustration effect”.
In my opinion, this is partly an effect of a reduction in the vehicle speed with decreasing distance due to safety requirements, combined with the wide distance scattering illustrated in Fig. 7 (Tilch and Helbing, 2000). Another relevant factor is the influence of the relative velocity on driver behavior (Bleile, 1997a, 1999). The average clearances of vehicles which are driving faster or slower than the respective leading vehicles are naturally increased compared to vehicles driving at the same speed, see Fig. 10 (Neubert et al., 1999b; Tilch, 2001). In summary, driver behavior is not only influenced by the clearance to the next car, but also by the relative velocity and the driver velocity (Koshi et al., 1983; Bleile, 1997, 1999; Dijker et al., 1998; Neubert et al., 1999b).
Note that the relative velocity in congested traffic is oscillating because of the instability of traffic flow, see Fig. 11 (Hoefs, 1972; Helbing and Tilch, 1998). As a consequence, the relative velocity variance , as compared to the average velocity , is considerably higher in congested than in free traffic (see Figs. 12 and 17). I suggest that this, together with the factors mentioned in the previous paragraph, may explain the empirical relations depicted in Fig. 9. If this is confirmed, one can drop the assumption of a frustration of drivers in congested traffic.
Measurements of velocity distributions for traffic with small truck fractions are in good agreement with a Gaussian distribution, see Fig. 13 (Pampel, 1955; Leong, 1968; May, 1990). However, if the sampling intervals are taken too large, one may also observe bimodal distributions (Phillips, 1977; Kühne, 1984a, b), reflecting a transition from free to congested traffic. As expected for a Gaussian distribution, 1minute averages of singlevehicle data for the skewness and the kurtosis are scattering around the zero line (Helbing, 1997e, a, c, 1998a).
Like the velocity distribution itself, the higher velocity moments are sensitive to the choice of the sampling interval . Large values of may lead to peaks in the variance when the average velocity changes abruptly. The additional contribution can be estimated as where the overbar indicates a time average over the sampling interval (Helbing, 1997a, b, e). 1minute data of the variance are more or less monotonically decreasing with the vehicle density (see Fig. 14), while 5minute data have maxima in the density range between 30 and 40 vehicles per kilometer and lane, indicating particulary unstable traffic (Helbing, 1997a).
Note that the average of the velocity variances in the different lanes is by an amount of lower than the velocity variance evaluated over all lanes, in particular at low densities . This is due to the different average velocities in the neighboring lanes. For a twolane freeway, the difference in average speeds decreases almost linearly up to a density of 35 to 40 vehicles per kilometer, while it fluctuates around zero at higher densities, see Fig. 15a (Helbing, 1997a, b; Helbing et al., 2001b). This reflects a synchronization of the velocities in neighboring lanes in congested traffic, both in wide moving jams and synchronized flow (Koshi et al., 1983; Kerner and Rehborn, 1996b; for reduced speed differences in congested traffic see also Edie and Foote, 1958; Forbes et al., 1967; Mika et al., 1969). However, according to Figs. 15b, c, there is a difference in the speeds of cars and long vehicles (“trucks”), which indicates that vehicles can still sometimes overtake, apart from a small density range around 25 vehicles per kilometer, where cars move as slow as trucks (Helbing and Huberman, 1998). However, direct measurements of the number of lane changes or of overtaking maneuvers as a function of the macroscopic variables are rare (Sparman, 1978; Hall and Lam, 1988; Chang and Kao, 1991). It would be particularly interesting to look at the dependence of lanechanging rates on the velocitydifference among lanes.
2.4 Correlations
In congested traffic, the average velocities in neighboring lanes are synchronized, see Fig. 16 (Mika et al., 1969; Koshi et al., 1983; cf. also Kerner and Rehborn, 1996b, for the nonlinear features of synchronized flow). The synchronization is probably a result of lane changes to the faster lane, until the speed difference is equilibrated (for recent results see, e.g., Lee et al., 1998; Shvetsov and Helbing, 1999; Nelson, 2000). This equilibration process leads to a higher vehicle density in the socalled “fast” lane, which is used by less trucks (see, for example, Hall and Lam, 1988; Helbing, 1997a, b). Nevertheless, density changes in congested neighboring lanes are correlated as well. Less understood and, therefore, even more surprising is the approximate synchronization of the lanespecific velocity variances over the whole density range (Helbing, 1997a, b, e). It may be a sign of adaptive driver behavior.
Correlations are also found between the density and flow or average velocity. At small densities, there is a strong positive correlation with the flow, which is reflected by the almost linear free branch of the fundamental diagram. In contrast, at large densities, the velocity and density are strongly anticorrelated (see Fig. 16) because of the monotonically decreasing velocitydensity relation (see Fig. 2). The average velocity and variance are positively correlated, since both quantities are related via a positive, but densitydependent prefactor, see Fig. 17 (Helbing, 1996b, 1997a, c, Treiber et al., 1999; Shvetsov and Helbing, 1999; Helbing et al., 2001a, b).
Finally, it is interesting to look at the velocity correlations among successive cars. Neubert et al. (1999b) have found a longrange velocity correlation in synchronized flow, while in free traffic vehicle velocities are almost statistically independent. This finding is complemented by results from Helbing et al. (2001b), see Fig. 18. The observed velocity correlations may be interpreted in terms of vehicle platoons (Wagner and Peinke, 1997). Such vehicle platoons have been tracked by Edie and Foote (1958) and Treiterer and Myers (1974).
2.5 Congested traffic
2.5.1 Jams, stopandgo waves, and power laws
The phenomenon of stopandgo waves (startstop waves) has been empirically studied by a lot of authors, including Edie and Foote (1958), Mika et al. (1969), and Koshi et al. (1983). The latter have found that the parts of the velocityprofile, which belong to the fluent stages of stopandgo waves, do not significantly depend on the flow (regarding their height and length), while the frequency does. Correspondingly, there is no characteristic frequency of stopandgo traffic, which indicates that we are confronted with nonlinear waves. The average duration of one wave period is normally between 4 and 20 minutes for wide traffic jams (see, e.g., Mika et al., 1969; Kühne, 1987; Helbing, 1997a, c, e), and the average wave length between 2.5 and 5 km (see, e.g., Kerner, 1998a). Analyzing the power spectrum (Mika et al., 1969; Koshi et al., 1983) points to white noise at high wave frequencies (Helbing, 1997a, c, e), while a power law with exponent is found at low frequencies (Musha and Higuchi, 1976, 1978). The latter has been interpreted as sign of selforganized criticality (SOC) in the formation of traffic jams (Nagel and Herrmann, 1993; Nagel and Paczuski, 1995). That is, congested traffic would drive itself towards the critical density , reflecting that it tries to reestablish the largest vehicle density associated with free flow (see below regarding the segregation between free and congested traffic and Sec. 4.1.5 regarding the constants of traffic flow).
Based on evaluations of aerial photographs, Treiterer (1966, 1974) has shown the existence of “phantom traffic jams”, i.e. the spontaneous formation of traffic jams with no obvious reason such as an accident or a bottleneck (see Fig. 19). According to Daganzo (1999a), the breakdown of free traffic “can be traced back to a lane change in front of a highly compressed set of cars”, which shows that there is actually a reason for jam formation, but its origin can be a rather small disturbance. However, disturbances do not always lead to traffic jams, as del Castillo (1996a) points out. Even under comparable conditions, some perturbations grow and others fade away, which is in accordance with the metastability of traffic mentioned above (Kerner and Konhäuser, 1994; Kerner and Rehborn, 1996b, 1997; Kerner, 1999c).
While small perturbations in free traffic travel downstream with a densitydependent velocity (Hillegas et al., 1974), large perturbations propagate upstream, i.e., against the direction of the vehicle flow (Edie and Foote, 1958; Mika et al., 1969). The disturbances have been found to propagate without spreading (Cassidy and Windover, 1995; Windower, 1998; Muñoz and Daganzo, 1999; see also Kerner and Rehborn, 1996a, and the flow and speed data reported by Foster, 1962; Cassidy and Bertini, 1999). The propagation velocity in congested traffic seems to be roughly comparable with a “natural constant”. In each country, it has a typical value in the range km/h, depending on the accepted safe time clearance and average vehicle length (see, e.g., Mika et al., 1969; Kerner and Rehborn, 1996a; Cassidy and Mauch, 2000). Therefore, fully developed traffic jams can move in parallel over a long time periods and road sections. Their propagation speed is not even influenced by ramps, intersections, or synchronized flow upstream of bottlenecks, see Fig. 20 (Kerner and Rehborn, 1996a; Kerner, 2000a, b).
Wide moving jams are characterized by stable wave
profiles and by kind of “universal” parameters (Kerner and Rehborn, 1996a).
Apart from
(i) the propagation velocity , these are
(ii) the density inside of jams,
(iii) the average velocity and flow inside of traffic jams,
both of which are approximately zero,
(iv) the outflow from jams (amounting
to about 2/3 of the maximum flow reached in free traffic,
which probably depends on the sampling interval ),
(v) the density downstream of jams, if these are propagating
through free traffic (“segregation effect”
between free and congested traffic). When propagating
through synchronized flow, the outflow of wide moving jams is
given by the density of the surrounding traffic
(Kerner, 1998b).
The concrete values of the characteristic parameters slightly depend
on the accepted safe time clearances, the average vehicle length,
truck fraction, and weather conditions (Kerner and Rehborn, 1998a).
2.5.2 Extended congested traffic
The most common form of congestion is not localized like a wide moving jam, but spatially extended and often persisting over several hours. It is related with a capacity drop to a flow, which is (at least in the United States) typically 10% or less below the “breakdown flow” of the previous highflow states defining the theoretically possible capacity (Banks, 1991a; Kerner and Rehborn, 1996b, 1998b; Persaud et al., 1998; Westland, 1998; Cassidy and Bertini, 1999; see also May, 1964, for an idea how to exploit this phenomenon with ramp metering; Persaud, 1986; Banks, 1989, 1990; AgyemangDuah and Hall, 1991; Daganzo, 1996). For statistical reasons, it is not fully satisfactory to determine capacity drops from maximum flow values, as these depend on the sampling interval . Anyway, more interesting are probably the bottleneck flows after the breakdown of traffic. Because of relation , these may be more than 30% below the maximum free flow (see Fig. 3b). Note that bottleneck flows depend, for example, on ramp flows and may, therefore, vary with the location. The congested flow immediately downstream of a bottleneck defines the discharge flow , which appears to be larger than (or equal to) the characteristic outflow from wide jams: .
In extended congested traffic, the velocity drops much more than the flow, but it stays also finite. The velocity profiles can differ considerably from one cross section to another (see Fig. LABEL:PINCH). In contrast, the temporal profiles of congested traffic flow , when measured at subsequent cross sections of the road, are often just shifted by some time interval which is varying (see Fig. 21). This may be explained with a linear flowdensity relation of the form (12) with (Hall et al., 1986; Ozaki, 1993; Hall et al., 1993; Dijker et al., 1998; Westland, 1998), together with the continuum equation for the conservation of the vehicle number (see the kinematic wave theory in Sec. 3.4.1; Cassidy and Mauch, 2000; Smilowitz and Daganzo, 1999; Daganzo, 1999a). However, a linear flowdensity relation in the congested regime is questioned by the “pinch effect” (see Sec. 2.5.3 and Fig. LABEL:PINCH).
The extended form of congestion is classical and mainly found upstream of bottlenecks, so that it normally has a spatially fixed (“pinned”) downstream front. In contrast, the upstream front is moving against the flow direction, if the (dynamic) capacity of the bottleneck is exceeded, but downstream, if the traffic volume, i.e. the inflow to the freeway section is lower than the capacity. Hence, this spatially extended form of congestion occurs regularly and reproducible during rush hours. Note, however, that bottlenecks may have many different origins: onramps, reductions in the number of lanes, accidents (even in opposite lanes because of rubbernecks), speed limits, road works, gradients, curves, bad road conditions (possibly due to rain, fog, or ice), bad visibility (e.g., because of blinding sun), diverges (due to “negative” perturbations, see Sec. 4.2, weaving flows by vehicles trying to switch to the slow exit lane, or congestion on the offramp, see Daganzo et al., 1999; Lawson et al., 1999; Muñoz and Daganzo, 1999; Daganzo, 1999a). Moving bottlenecks due to slow vehicles are possible as well (Gazis and Herman, 1992), leading to a forward movement of the respective downstream congestion front. Finally, in cases of two subsequent inhomogeneities of the road, there are forms of congested traffic in which both, the upstream and downstream fronts are locally fixed (Treiber et al., 2000; Lee et al., 2000; Kerner, 2000a).
While Kerner calls the above extended forms of congested traffic “synchronized flow” because of the synchronization among lanes (see Secs. 2.3, 2.4), Daganzo (1999b) speaks of 1pipe flow. Kerner and Rehborn (1997) have pointed out that the transition from free to extended congested traffic is of hysteretic nature and looks similar to the firstorder phase transition of supersaturated vapor to water (“nucleation effect”). It is often triggered by a small, but overcritical peak in the traffic flow (playing the role of a nucleation germ). This perturbation travels downstream in the beginning, but it grows eventually and changes its propagation direction and speed, until it travels upstream with velocity . When the perturbation reaches the bottleneck, it triggers a breakdown of traffic flow to the bottleneck flow . In summary, synchronized flow typically starts to form downstream of the bottleneck (cf. Fig. 39b).
Kerner (1997, 1998a) has attributed the capacity drop to increased time headways due to delays in acceleration. This has recently been supported by Neubert et al. (1999b). At a measurement section located directly at the downstream front of a congested region, they found the appearance of a peak at about 1.8 seconds in the timeheadway distribution, when congestion set in, see Fig. 22. The reachable saturation flows of accelerating traffic are, therefore, vehicles per hour. This is compatible with measurements of stopped vehicles accelerating at a traffic light turning green (Androsch, 1978) or after an incident (Raub and Pfefer, 1998).
Kerner (1998b) points out that the above transition to “synchronized” congested flow is,
in principle, also found on onelane roads,
but then it is not anymore of hysteretic nature and not connected with synchronization.
Moreover, he and Rehborn (1996b)
distinguish three kinds of synchronized flow:
(i) Stationary and homogeneous states where both the average
speed and the flow rate are stationary during a relatively long time
interval (see, e.g., also Hall and AgyemangDuah, 1991; Persaud et al.,
1998; Westland, 1998).
I will later call these “homogeneous congested traffic” (HCT).
(ii) States where only the average vehicle speed is stationary,
named “homogeneousinspeed states” (see also Kerner, 1998b; Lee et al., 2000).
I interpret this state as “recovering traffic”,
since it bears several signatures of free traffic
(cf. Secs. 2.2 and 4.3.2).
(iii) Nonstationary and nonhomogeneous states
(see also Kerner, 1998b; Cassidy and Bertini, 1999; Treiber et al., 2000).
For these, I will also use the term “oscillating congested traffic” (OCT).
Not all of these states are longlived, since there are often spatiotemporal sequences of these different types of synchronized flow, which indicates continuous (secondorder) transitions. However, at least states (i) and (iii) are characterized by a wide, twodimensional scattering of the flowdensity data, i.e. an increase in the flow can be either related with an increase or with a decrease in the density, in contrast to free flow (see Sec. 2.2).
2.5.3 Pinch effect
Recently, the spontaneous appearance of stopandgo traffic has been questioned by Kerner (1998a) and Daganzo et al. (1999). In his empirical investigations, Kerner (1998a) finds that jams can be born from extended congested traffic, which presupposes the previous transition from free to synchronized flow. The alternative mechanism for jam formation is as follows (see Fig. LABEL:PINCH): Upstream of a section with homogeneous congested traffic close to a bottleneck, there is a socalled “pinch region” characterized by the spontaneous birth of small narrow density clusters, which are growing while they travel further upstream. Wide moving jams are eventually formed by the merging or disappearance of narrow jams, which are said to move faster upstream than wide jams. Once formed, the wide jams seem to suppress the occurence of new narrow jams in between. Similar findings were reported by Koshi et al. (1983), who observed that “ripples of speed grow larger in terms of both height and length of the waves as they propagate upstream”. Daganzo (1999b) notes as well that “oscillations exist in the 1pipe regime and that these oscillations may grow in amplitude as one moves upstream from an active bottleneck”, which he interprets as a pumping effect based on ramp flows. The original interpretation by Koshi et al. suggests a nonlinear selforganization phenomenon, assuming a concave, nonlinear shape of the congested flowdensity branch (cf. Sec. 4.2.2). Instead of forming wide jams, narrow jams may coexist when their distance is larger than about 2.5 km (Kerner, 1998a; Treiber et al., 2000).
2.6 Cars and trucks
Distinguishing vehicles of different lengths (“cars” and “trucks”), one finds surprisingly strong variations of the truck fraction, see Fig. 24 (Treiber and Helbing, 1999a). This point may be quite relevant for the explanation of some observed phenomena, as quantities characterizing the behavior of cars and trucks are considerably different. For example, this concerns the distribution of desired velocities as well as the distribution of time headways (see Fig. 25).
2.7 Some critical remarks
The collection and evaluation of empirical data is a subject with often
underestimated problems. To make reliable conclusions, in original investigations
one should specify
(i) the measurement site and conditions (including applied control
measures),
(ii) the sampling interval,
(iii) the aggregation method,
(iv) the statistical properties
(variances, frequency distributions, correlations, survival times of traffic states, etc.),
(v) data transformations,
(vi) smoothing procedures,
and the respective dependencies on them.
The measurement conditions include ramps and road sections with their respective in and outflows, speed limits, gradients, and curves with the respectively related capacities, furthermore weather conditions, presence of incidents, and other irregularities. Moreover, one should study the dynamics of the long vehicles separately, which may also have a significant effect (see Sec. 2.6).
A particular attention has also to be paid to the fluctuations of the data, which requires a statistical investigation. For example, it is not obvious whether the scattering of flowdensity data in synchronized flow reflects a complex dynamics due to nonlinear interactions or whether it is just because of random fluctuations in the system. In this connection, I remind of the power laws found in the highfrequency variations of macroscopic quantities (see Sec. 2.5.1).
The statistical variations of traffic flows imply that all measurements of macroscopic quantities should be complemented by error bars (see, e.g., Hall et al., 1986). Due to the relatively small “particle” numbers behind the determination of macroscopic quantities, the error bars are actually quite large. Hence, many temporal variations are within one error bar and, therefore, not significant. As a consequence, the empirical determination of the dynamical properties of traffic flows is not a simple task. Fortunately, many hardtosee effects are in agreement with predictions of plausible traffic models, in particular with deterministic ones (see Secs. 4.2 and 4.3.2).
Nevertheless, I would like to call for more refined measurement techniques, which are required for more reliable data. These must take into account correlations between different quantities, as is pointed out by Banks (1995). Tilch and Helbing (2000) have, therefore, used the following measurement procedures: In order to have comparable sampling sizes, they have averaged over a fixed number of cars, as suggested by Helbing (1997e). Otherwise the statistical error at small traffic flows (i.e. at small and large densities) would be quite large. This is compensated for by a flexible measurement interval . It is favourable that becomes particularly small in the (medium) density range of unstable traffic, so that the method yields a good representation of traffic dynamics. However, choosing small values of does not make sense, since the temporal variation of the aggregate values will mainly reflect statistical variations, then. In order to have a time resolution of about 2 minutes on each lane, one should select , while can be chosen when averaging over both lanes. Aggregate values over both lanes for are comparable with 1minute averages, but show a smaller statistical scattering at low densities (compare the results in Helbing 1997a, c with those in Helbing, 1997e).
Based on the passing times of successive vehicles in the same lane, we are able to calculate the time headways . The (measurement) time interval
(13) 
for the passing of vehicles defines the (inverse of the) traffic flow via which is attributed to the time
Approximating the vehicle headways by , one obtains
where is the covariance between the headways and the inverse velocities . We expect that this covariance is negative and particularly relevant at large vehicle densities, which is confirmed by the empirical data (see Fig. 26). Defining the density by
(14) 
and the average velocity via Eq. (10), we obtain the fluiddynamic flow relation (46) by the conventional assumption . This, however, overestimates the density systematically, since the covariance tends to be negative due to the speeddependent safety distance of vehicles. In contrast, the common method of determining the density via systematically underestimates the density (see Fig. 27). Consequently, errors in the measurement of the flow and the density due to a neglection of correlations partly account for the observed scattering of flowdensity data in the congested regime. However, a considerable amount of scattering is also observed in flowoccupancy data, which avoid the above measurement problem.
3 Modelling approaches for vehicle traffic
In zeroth order approximation, I would tend to say that each decade was
dominated by a certain modelling approach:
(i) In the 50ies, the propagation of shock waves, i.e. of density jumps in
traffic, was described by a fluiddynamic model for kinematic waves.
(ii) The activities in the 60ies concentrated on “microscopic”
carfollowing models, which are often called followtheleader models.
(iii) During the 70ies, gaskinetic, socalled Boltzmannlike models
for the spatiotemporal change of the velocity distribution were florishing.
(iv) The simulation of “macroscopic”, fluiddynamic
models was common in the 80ies.
(v) The 90ies were dominated by discrete cellular automata models of vehicle traffic
and by systematic investigations of the dynamical solutions of the
models developed in the previous 50 years.
(vi) Recently,
the availability of better data brings up more and more experimental studies
and their comparison with traffic models.
Altogether, researchers from engineering, mathematics, operations research, and physics have probably suggested more than 100 different traffic models, which unfortunately cannot all be covered by this review. For further reading, I therefore recommend the books and proceedings by Buckley (1974), Whitham (1974), Gerlough and Huber (1975), Gibson (1981), May (1981), Vumbaco (1981), Hurdle et al.(1983), Volmuller and Hamerslag (1984), Gartner and Wilson (1987), Leutzbach (1988), Brannolte (1991), Daganzo (1993), Pave (1993), Snorek et al. (1995), Wolf et al. (1996), Lesort (1996), the Transportation Research Board (1996), Gartner et al. (1997), Helbing (1997a), Daganzo (1997a), Rysgaard (1998), Schreckenberg and Wolf (1998), Bovy (1998), Brilon et al. (1999), Ceder (1999), Hall (1999), and Helbing et al. (2000c, 2001a, b). Readers interested in an exhaustive discussion of cellular automata should consult the detailed review by Chowdhury et al. (2000b).
In the following subsections, I can only introduce a few representatives for each modelling approach (selected according to didactical reasons) and try to show up the relations among them regarding their instability and other properties (see Secs. 4.1 to 4.4).
Before, I would like to formulate some criteria for good traffic models: For reasons of robustness and calibration, such models should only contain a few variables and parameters which have an intuitive meaning. Moreover, these should be easy to measure, and the corresponding values should be realistic. In addition, it is not satisfactory to selectively reproduce subsets of phenomena by different models. Instead, a good traffic model should at least qualitatively reproduce all known features of traffic flows, including the localized and extended forms of congestion. Furthermore, the observed hysteresis effects, complex dynamics, and the existence of the various selforganized constants like the propagation velocity of stopandgo waves or the outflow from traffic jams should all be reproduced. A good model should also make new predictions allowing us to verify or falsify it. Apart from that, its dynamics should not lead to vehicle collisions or exceed the maximum vehicle density. Finally, the model should allow for a fast numerical simulation.
3.1 Microscopic followtheleader models
The first microscopic traffic models were proposed by Reuschel (1950a, b) and the physicist Pipes (1953). Microscopic traffic models assume that the acceleration of a drivervehicle unit is given by the neighboring vehicles. The dominant influence on driving behavior comes from the next vehicle ahead, called the leading vehicle. Therefore, we obtain the following model of driver behavior from Eq. (3):
(15) 
Herein, describes the repulsive effect of
vehicle , which is generally a function of
(i) the relative velocity
,
(ii) the own velocity due to the velocitydependent safe
distance kept to the vehicle in front,
(iii) of the headway (brutto distance)
or the clearance (netto
distance)
, with
meaning the length of vehicle .
Consequently, for identically behaving vehicles with
, , and , we would have
(16) 
If we neglect fluctuations for the time being and introduce the trafficdependent velocity
(17) 
to which driver tries to adapt, we can considerably simplify the “generalized force model” (15):
(18) 
3.1.1 Noninteger carfollowing model
Models of the type (18) are called followtheleader models (Reuschel 1950a, b; Pipes, 1953; Chandler et al., 1958; Chow, 1958; Kometani and Sasaki, 1958, 1959, 1961; Herman et al., 1959; Gazis et al., 1959; Gazis et al., 1961; Newell, 1961; Herman and Gardels, 1963; Herman and Rothery, 1963; May and Keller, 1967; Fox and Lehmann, 1967; Hoefs, 1972). One of the simplest representatives results from the assumption that the netto distance is given by the velocitydependent safe distance where has the meaning of the (effective) safe time clearance. This implies or, after differentiation with respect to time, Unfortunately, this model does not explain the empirically observed density waves (see Sec. 2.5.1). Therefore, one has to introduce an additional time delay s in adaptation, reflecting the finite reaction time of drivers. This yields the following stimulusresponse model:
(19) 
Herein, is the sensitivity to the stimulus. This equation belongs to the class of delay differential equations, which normally have an unstable solution for sufficiently large delay times . For the above case, Chandler et al. (1958) could show that, under the instability condition a variation of individual vehicle velocities will be amplified. The experimental value is . As a consequence, the nonlinear vehicle dynamics finally gives rise to stopandgo waves, but also to accidents. In order to cure this, to explain the empirically observed fundamental diagrams, and to unify many other model variants, Gazis et al. (1961) have introduced a generalized sensitivity factor with two parameters and :
(20) 
The corresponding noninteger carfollowing model can be rewritten in the form which is solved by with if and otherwise (, being integration constants). For and we will discuss the stationary case related to identical velocities and distances. The vehicle density is then given by the inverse brutto distance , and the corresponding equilibrium velocity is identical with . Therefore, we obtain the velocitydensity relation with the free velocity and the maximum density . Most of the velocitydensity relations that were under discussion at this time are special cases of the latter formula for different values of the model parameters and . Realistic fundamental diagrams result, for example, for the noninteger values and (May and Keller, 1967; Hoefs, 1972) or and (Kühne and Rödiger, 1991; Kühne and Kroen, 1992).
Finally, note that the above noninteger carfollowing model is used in the traffic simulation package MITSIM and has recently been made more realistic with accelerationdependent parameters (Yang and Koutsopoulos, 1996; Yang, 1997). However, other carfollowing models deserve to be mentioned as well (Gipps, 1981; Benekohal and Treiterer, 1988; del Castillo, 1996b; Mason and Woods, 1997). For a historical review see Brackstone and McDonald (2000).
3.1.2 Newell and optimal velocity model
One of the deficiencies of the noninteger carfollowing models is that it cannot describe the driving behavior of a single vehicle. Without a leading vehicle, i.e. for , vehicle would not accelerate at all. Instead, it should approach its desired velocity in free traffic. Therefore, other carfollowing models do not assume an adaptation to the velocity of the leading vehicle, but an adaptation to a distancedependent velocity which should reflect the safety requirements and is sometimes called the “optimal velocity”. While Newell (1961) assumes a delayed adaptation of the form
(21) 
Bando et al. (1994, 1995a, b) suggest to use the relation
(22) 
with constants and , together with the optimal velocity model
(23) 
which may be considered as a firstorder Taylor approximation of Eq. (21) with . For the optimal velocity model one can show that small perturbations are eventually amplified to traffic jams, if the instability condition
(24) 
is satisfied (Bando et al., 1995a), i.e., if we have large relaxation times or big changes of the velocity with the clearance (see Fig. 28). Similar investigations have been carried out for analogous models with an additional explicit delay (Bando et al., 1998; Wang et al., 1998b).
3.1.3 Intelligent driver model (IDM)
As the optimal velocity model does not contain a driver response to the relative velocity with respect to the leading vehicle, it is very sensitive to the concrete choice of the function and produces accidents, when fast cars approach standing ones (Helbing and Tilch, 1998). To avoid this, one has to assume particular velocitydistance relations and choose a very small value of , which gives unrealistically large accelerations (Bleile, 1997b, 1999). In reality, however, the acceleration times are about five to ten times larger than the braking times. Moreover, drivers keep a larger safe distance and decelerate earlier, when the relative velocity is high. These aspects have, for example, been taken into account in models by Gipps (1981), Krauß et al. (1996, 1997), Helbing (1997a), Bleile (1997b, 1999), Helbing and Tilch (1998), Wolf (1999), or Tomer et al. (2000).
For illustrative reasons, I will introduce the socalled intelligent driver model (Treiber and Helbing, 1999b; Treiber et al., 2000). It is easy to calibrate, robust, accidentfree, and numerically efficient, yields realistic acceleration and braking behavior and reproduces the empirically observed phenomena. Moreover, for a certain specification of the model parameters, its fundamental diagram is related to that of the gaskineticbased, nonlocal traffic (GKT) model, which is relevant for the micromacro link we have in mind (see Sec. 3.5). For other specifications, the IDM is related with the Newell model or the NagelSchreckenberg model (see Sec. 3.2.1).
The acceleration assumed in the IDM is a continuous function of the velocity , the clearance , and the velocity difference (approaching rate) of vehicle to the leading vehicle:
(25) 
This expression is a superposition of the acceleration tendency on a free road, and the deceleration tendency describing the interactions with other vehicles. The parameter allows us to fit the acceleration behavior. While corresponds to an exponentialintime acceleration on a free road, as assumed by most other models, in the limit , we can describe a constant acceleration with , until the desired velocity is reached. The deceleration term depends on the ratio between the “desired clearance” and the actual clearance , where the desired clearance
(26) 
is dynamically varying with the velocity and the approaching rate , reflecting an intelligent driver behavior. The IDM parameters can be chosen individually for each vehicle , but for the moment, we will assume identical vehicle parameters and drop the index for readability. These are the desired velocity , the safe time clearance , the maximum acceleration , the comfortable deceleration , the acceleration exponent , the jam distances and , and the vehicle length , which has no dynamical influence. To reduce the number of parameters, one can assume , , and , which still yields good results.
In equilibrium traffic with and , drivers tend to keep a velocitydependent equilibrium clearance to the front vehicle given by Solving this for the equilibrium velocity leads to simple expressions only for and , , or . In particular, the equilibium velocity for the special case and is From this equation and the micromacro relation between clearance and density follows the corresponding equilibrium traffic flow as a function of the traffic density . The acceleration coefficient influences the transition region between the free and congested regimes. For and