The Physics of Traffic and Regional Development
Abstract
This contribution summarizes and explains various principles from physics which are used for the simulation of traffic flows in large street networks, the modeling of destination, transport mode, and route choice, or the simulation of urban growth and regional development. The methods stem from manyparticle physics, from kinetic gas theory, or fluiddynamics. They involve energy and entropy considerations, transfer the law of gravity, apply cellular automata and require methods from evolutionary game theory. In this way, one can determine interaction forces among drivervehicle units, reproduce breakdowns of traffic including features of synchronized congested flow, or understand changing usage patterns of alternative roads. One can also describe daily activity patterns based on decision models, simulate migration streams, and model urban growth as a particular kind of aggregation process.
I Introduction
From the point of view of a physicist, cars on a highway or pedestrians in a mall are many particle systems. Yet, the particles of those systems are more complicated than typical particles in physical systems. A question therefore is in how far the methods from physics can be applied to systems with these more complicated particles.
It turns out that in fact quite a lot can be explained with the methods from physics, which combine simple microscopic modeling assumptions with advanced mathematics and/or advanced computational science. For example, those systems display a surprising variety of states, such as jammed, laminar, or oscillating. Those states can be explained using several concepts, from microscopic to fluiddynamical. Physics methods have contributed significantly to better understanding those states, and to alleviate them where possible.
Traffic usually unfolds on networks, rather than on flat twodimensional space. And indeed, traffic has some similarity to the steadystate flow of electrons in a fuse network with nonlinear resistors. Besides the fact that the traffic dynamics can be more complicated than this steadystate behavior, as pointed out in the last paragraph, the main difference between human travelers and electrons in a network is that human travelers typically have an individual destination. This introduces an important additional nonlinearity, and also brings game theory into the picture, in particular the concept of a Nash Equilibrium (NE). Game theory states that a system is at a NE when no player can gain by unilaterally switching strategies. What is more of an interest to physicists is that certain types of learning systems can have a NE as fixed point attractor – they can, however, also have other types of attractors, such as periodic or chaotic ones. And indeed, route choice experiments with human subjects confirm that in many cases the players approach a NE, but also that this NE does not have to be stable and displays behavior reminiscent of intermittency and volatility clustering.
This and other evidence demonstrates that the route choice behavior of real world people can be approximated by a fastest path algorithm, which is relatively cheap in terms of computational complexity. From a practical point of view, the next problem then becomes to generate the destinations of the travelers. Two different approaches are discussed in the paper: (i) gravity models, which make the probability to select a destination dependent on the distance; and (ii) activitybased demand generation, which attempts to generate complete daily plans for all travelers in a system. And once more the situation is similar to physics: While it may be hopeless to realistically describe the behavior of individual people in the system, there is some hope that the macroscopic (“emergent”) behavior of many travelers together can be derived from simple microscopic rules, akin to the derivation of the ideal gas equation. Also, behavioral invariances such as the energy consumption of the human – not the vehicle – can sometimes be found and used.
Intuitively, the next important question is then how the city itself develops – where humans find residences, where retail and other industries find locations, and where the city grows when the space inside its current boundaries is no longer sufficient. Once more, models from physics, this time growth models based on the cellular automata paradigm, can offer important insight. The use of such models is related to the fact that some quantities of realworld cities, such as the area, the perimeter, or the size distribtion, display fractal properties. An alternative approach is to use models from Synergetics, with its associated master equation, as the starting point to describe migration dynamics.
Similar to the molecular dynamics approach in physics, it is now possible to build large scale “agentbased” simulation systems of real world transport systems which are based on the “first principles” introduced above. Since a typical metropolitan area has about inhabitants, this is well within the range of computational feasibility, although some computational efficiency is lost because of the more complicated particles. The main challenge currently is to integrate the different levels, once more a challenge that is similar to multiscale modeling in physics.
Results achieved so far indicate that simulation models based on those principles are, albeit just at the beginning of their development, at least as good as established methods. In consequence, one can start to move those models into the realm of realworld infrastructure planning. In addition, one would expect that additional scientific insights, better implementation techniques allowing faster turnaround, and the increasingly better availability of electronic data will make them considerably better in the future.
Ii FluidDynamic and GasKinetic Traffic Models
Every driver has probably encountered the widespread phenomenon of socalled “phantom traffic jams”, for which there is no visible reason such as an accident or a bottleneck Review . So, why are vehicles sometimes stopped although everyone likes to drive fast? Ency
The first wellknown approach to describe density waves in traffic flows goes back to Lighthill and Whitham LiWi in 1955. They start with the typical continuity equation,
(1) 
where denotes density and denotes flow or throughput. The equation reflects that vehicles are not generated or lost in the absence of ramps, intersections, or accidents. As usual, one also has the relation , where is the velocity at place and time .
Lighthill and Whitham postulated that the traffic flow could be specified as a function of the density only. This assumes an instantaneous adaptation of to some equilibrium flowdensity relation . The corresponding curve is often called the “fundamental diagram” and obtained as a fit to empirical data.
Because of , one obtains
(2) 
which is a nonlinear wave equation where the propagation velocity
(3) 
of socalled kinematic waves depends on the vehicle density only. Kinematic waves have the property that they keep their amplitude, while their shape changes until shock waves (i.e. discontinuous changes in the density) have developed. The densities and immediately upstream and downstream of a shock front determine its propagation speed
(4) 
Experimental observations of traffic patterns show additional features which cannot be reproduced by the above traffic model. While traffic flow appears to be stable with respect to perturbations at small and large densities, there is a linearly unstable range at medium densities, where already small disturbances of uniform traffic flow give rise to traffic jams. Between the stable and linearly unstable density ranges, one finds meta or multistable ranges and , since there exists a densitydependent, critical amplitude , so that the resulting traffic pattern is path or historydependent KerKon94 ; KerKoSchi95 ; KerKoSc96 ; Ker97 (see Fig. 1). While subcritical perturbations fade away, supercritical perturbations cause a breakdown of traffic flow (“nucleation effect”). Consequently, traffic displays critical points, nonequilibrium phase transitions, noiseinduced transitions, and fluctuationinduced ordering phenomena. One may view this as nonequilibrium analogue of the phase transitions between vapour (free flow), water (“synchronized”, queued traffic flow above freeway capacity), and ice (jams with standing vehicles and zero flow). Note that noiselike effects enter the above deterministic models only via the boundary conditions, but some models take care of them by additional fluctuation terms or by distinguishing different drivervehicle types, i.e. partial flows with nonlinear interactions.
According to Krauß Krauss , traffic models show the observed hysteretic phase transition related with metastable traffic and high flows, if the typical maximal acceleration is not too large and the deceleration strength is moderate. In such kinds of models, is the density where the flowdensity relation
(5) 
of traffic with fully developed traffic jams (the socalled “jam line”) intersects with the free branch of the flowdensity diagram . denotes the net time gap (time clearance) in congested traffic and the density inside of traffic jams. The intuitive interpretation of Eq. (5) is that traffic in the congested regime is composed of moving areas and of areas where vehicles are stopped. The fraction of the moving areas is given by .
Moreover, the outflow from traffic jams is a selforganized constant KerKon94 ; Ker97 ; KerReh96a , which lies between and HelHenTre99 . The jam line corresponds to the flowdensity relation for traffic patterns with a selforganized, stationary profile KerKon94 . These propagate with the velocity km/h, which is another traffic constant. The explanation of this is the following: Once a traffic jam is fully developed, vehicles leave the downstream jam front at a constant rate, while new ones join it at the upstream front. This makes the jam move upstream with constant velocity.
In order to reproduce such emergent traffic jams and metastable density regimes, the LighthillWhitham model from Eqs. (1) to (4) needs to be generalized. For this, it is helpful that fluiddynamic models can be derived from gaskinetic models, which relate to models of interactive driver behavior HelHeShTr01a ; HelHeShTr01b . The gaskinetic approach has been introduced by Prigogine et al. PriAnd ; Prig ; PriHer and is inspired by kinetic gas theory. It describes the spatiotemporal change of the phase space density (= vehicle density velocity distribution). The related equations are either of Boltzmannlike type (for pointlike vehicles or low densities) or of Enskoglike type, if vehicular space requirements at moderate and high densities are taken into account (see HelHeShTr01a and references therein). They allow the systematic derivation of macroscopic equations for the vehicle density , the average velocity , the velocity variance , etc. This hierarchy of equations is usually closed after the velocity or variance equation, although the separation of time scales assumed by the underlying approximations is weak. Nevertheless, the observed traffic dynamics is rather well reproduced by the resulting coupled partial differential equations. The density equation is just the continuity equation
(6) 
where and denote on and offramp flows, respectively. The velocity equation can be cast into the form
(7) 
In theoretically consistent macroscopic traffic models such as the gaskineticbased traffic model, the “traffic pressure” and the velocity are nonlocal functions of the density , the average velocity , and the variance HeTre98 ; ShvHe99 . Several wellknown models are special cases of Eq. (7):

Kerner’s and Konhäuser’s model KerKon93 ; KerKon94 is a variant of Kühne’s model Kuehne and corresponds to the specifications and , where and are positive constants. The corresponding equation is a NavierStokes equation with a viscosity term and an additional relaxation term describing the delayed adaptation to the velocitydensity relation .
Both for the Payne model and for the KernerKonhäuser model, the linearly unstable regime is determined by the densities fulfilling the condition
(8) 
In consequence, both the Payne model and the KernerKonhäuser model have linearly unstable ranges if is large, while the LighthillWhitham model is marginally stable.
Beyond linear stability analysis, in some cases it is also possible to find analytical expressions for the large amplitude instability in the metastable ranges and (Fig. 1). For these more complicated aspects, the interested reader is referred to Kerner and Konhäuser (1994).
In simple terms, the reason for emergent traffic jams is as follows: Due to the finite adaptation time (= reaction + acceleration time), a small disturbance in the traffic flow can cause an overreaction (overbraking) of a driver, if the safe vehicle speed (which satisfies ) drops too rapidly with increasing vehicle density . At high enough densities , this will give rise to a chain reaction of the followers, as other vehicles will have approached before the original speed can be regained. This feedback can eventually cause the unexpected standstill of vehicles known as traffic jam.
Iii Congested Traffic States in Theory and Empirical Data
With this knowledge one can understand the various congested traffic states observed on freeway sections with bottlenecks HelHenTre99 ; Lee3LeKi99 ; Lee3LeKi00 ; inTGF01 ; TS . Given that the flow and density are measured per lane, a bottleneck due to ramp flows is determined by , where is the used length of the onramp and the number of freeway lanes. The corresponding bottleneck strength is
(9) 
iii.1 Theoretical Phase Diagram of Traffic States
Let denote the traffic flow per lane upstream of the bottleneck and
(10) 
the total capacity required downstream of the ramp. Then, we expect to always observe free traffic (FT) below the first instability threshold at density , i.e. if
(11) 
Traffic flow will always be congested, if the maximum flow (the capacity) is exceeded, i.e.
(12) 
At least if the density related with the maximum flow lies between and , traffic states between the two diagonal lines and in the vs phase space can be either congested or free, depending on the initial and boundary conditions TS (see Fig. 2). While homogeneous free flow may persist over long time periods, large perturbations tend to produce congested states. Extended congested traffic can emerge above the line
(13) 
in the phase diagram, i.e. if is greater than the dynamic capacity . This line does not have to be parallel to the previously mentioned phase boundaries, as may depend on the bottleneck strength TreHeHe00 ; Review . For , congested traffic states are always localized, i.e. they never grow over long sections of the freeway, as the outflow can cope with the overall traffic volume.
The traffic flow resulting in the congested area plus the inflow or bottleneck strength are normally given by the outflow , i.e.
(14) 
(if vehicles cannot enter the freeway downstream of the congestion front). One can distinguish the following cases: Homogeneous congested traffic (HCT) such as typical traffic jams during holiday seasons can occur, if the density associated with the congested flow
(15) 
lies in the stable or metastable range
(16) 
Oscillating forms of congested traffic can emerge, if
(17) 
That is, lower bottleneck strengths tend to produce less serious congestion, namely oscillating rather than homogeneous congested flow. We either find oscillating congested traffic (OCT), triggered stopandgo traffic (TSG), or moving localized clusters (MLC) TS . In contrast to OCT, stopandgo traffic is characterized by a sequence of moving jams, between which traffic flows freely. This state can either emerge from a spatial sequence of homogeneous and oscillating congested traffic Koshi , which is also called the “pinch effect” pinch , or it can be caused by the inhomogeneity at the ramp. In the latter case, each traffic jam triggers another one by inducing a small perturbation in the inhomogeneous freeway section (see Fig. 2), which propagates downstream as long as it is small, but turns back when it has grown large enough (“boomerang effect”). This growth requires the downstream traffic flow to be linearly unstable. If it is metastable (when the traffic volume is sufficiently small), small perturbations will fade away. Therefore, if
(18) 
one expects to find localized traffic states, either a single moving localized cluster (MLC), or a pinned localized cluster (PLC) at the location of the ramp. The latter requires the traffic flow in the upstream section to be stable, i.e.
(19) 
so that no traffic jam can survive there. In contrast, moving localized clusters and triggered stopandgo waves require TS
(20) 
The simulation results displayed in Figure 2 summarize, for freeways with a single bottleneck, which states are typically found in different areas of the phase diagram. Apart from the detailed shape and exact location of the phase boundaries, this phase diagram is expected to be universal for all microscopic and macroscopic, stochastic and deterministic traffic models with the same instability diagram (with stable, metastable, and unstable density ranges) Review . Results for more complex freeway geometries are available as well.
iii.2 Wide Scattering of Traffic Data in Heterogeneous Traffic
In the congested traffic regime, flowdensity data are characterized by a surprisingly wide scattering KerReh96b (see Fig. 3a). This has led people to question the applicability of the jam line (5) critic1 ; critic2 ; critic3 . However, it can be shown that, taking into account the variation of the average net time gap (time clearance) , the variations in the data can be reproduced with a correlation coefficient of 0.92 (compared to 0.35 when only the density is varied).
A closer analysis reveals a large variation of time gaps between vehicles. The time gaps of trucks are particularly large. Simulations of heterogeneous traffic with different kinds of vehicles (i.e. different parameter values) suggest that at least part of the scattering of flowdensity data may be explained by the mixture of cars and trucks Scattering and of different driving behaviors (see Fig. 4). Although heterogeneous traffic can be treated by macroscopic models (see Fig. 4), a simulation of individual vehicles is much more easy, flexible, and efficient, in particular if one likes to simulate network traffic of drivers with different origins and destinations. Therefore, we will focus on some “microscopic” traffic models in the following.
Iv Driven ManyParticle Systems and Microscopic Traffic Simulation
The observations in freeway traffic can not only be described by fluiddynamic or “macroscopic” traffic models. They are also well reproduced by “microscopic” models TreHeHe00 . Note that a theoretical connection between both approaches exists. It is called the micromacro link HelHeShTr01b .
iv.1 CarFollowing Models
Microscopic models are often followtheleader models specifying the acceleration of each single vehicle as a function of their speed , their distance headway with respect to the leading vehicle , and/or their relative velocity :
(21) 
are random variations in the acceleration behavior, while may be interpreted as average situationdependent acceleration force. A typical example is the noninteger carfollowing model
(22) 
with the reaction time s and the parameters , , and Gazis . It has a linearly unstable range for . (The time delay on both sides facilitates to determine analytical solutions.) A simpler model is the optimal velocity model
(23) 
where is the “optimal” velocitydistance relation and the adaptation time Bando1 ; Bando2 . This model has an unstable range for . The respective nonlinearly coupled differential equations (or stochastic differential equations, if fluctuations are taken into account) are numerically solved as in molecular dynamics.
Let us now define the maximum, desired, or free velocity by . The expression
(24) 
reflects the interaction force as a function of the net distance (clearance) between two successive vehicles, where denotes the length of vehicle . With this, Eq. (23) can be reformulated in terms of the “social force model”
(25) 
iv.2 Interaction Forces Among Vehicles
The difficulty is now to specify the interaction forces among vehicles in an appropriate way. Their exact specification is essential for realistic traffic simulations, which are required for the design of efficient and reliable traffic optimization measures such as intelligent onramp controls, driver assistance systems, lanechanging assistants, etc. If the fluctuation term was zero (i.e. ), the interaction force could simply be determined from the velocitydensity relation. Since the average of the headways determines the inverse density , we would simply find the relation .
However, things are considerably more difficult, when random fluctuations are not negligible. One may think of applying scattering theory to determine the interaction potential from statistical distributions, but learning about human interactions requires a somewhat different approach. Progress has recently been made by means of random matrix theory, a powerful method from quantum and statistical physics. It allows one to determine the interaction potentials via net distance distributions of vehicles.
Random matrix theory Mehta has been developed for manyparticle systems exposed to a ”thermal bath” of a given reciprocal temperature , i.e. to random forces of a certain variance and statistics. The resulting velocity and net distance distributions allow one to draw conclusions about the interaction potential , as this determines their shapes. Although a generalization to arbitrary potentials is possible, here we will focus our attention on powerlaw potentials
(26) 
for simplicity. The exponent is a fit parameter and is the net distance (clearance) between two successive cars. Such a potential describes the repulsive tendency of drivers to keep a safe distance to the respective car ahead. The related net distance distributions can be calculated to be
(27) 
where is the vehicle density and , represent normalization constants determined by requiring and
Originally, the applied random matrix method is an equilibrium concept assuming the conservation of energy , i.e. a transformation of potential energy into kinetic energy with , , , and being the location, length, mass, and velocity of the th vehicle. It can however be shown fpg that this is a reasonable approximation for an ensemble of vehicles in a stationary state at a given density and inverse “temperature” (velocity variance) , while the stopandgo regime between 20 and 40 vehicles per kilometer and lane must be excluded from an investigation of this kind.
Milan Krbalek and one of us have separately analyzed eight density intervals of Dutch single vehicle data in the free lowdensity regime and eight density intervals in the congested traffic regime. In accordance with the theoretical predictions, the velocity distributions fit Gaussian distributions very well milan . Moreover, the best fit of the net distance distributions is obtained for the integer parameter in the free traffic regime, while we find an excellent agreement with in the congested regime (see Figs. 5, 6). This is not only a strong support of studies questioning a uniform behavior of drivers in all traffic regimes Neubert . It also offers an interpretation of the mysterious fractional distancescaling exponent in classical followtheleader models MayKe67 , which average over the driver behavior in the free and congested regimes.
Note that the net distance distribution (27) agrees with the Pearson III type of distribution May which has been suggested to fit empirical time headway distributions already before our theoretical explanation had been found. Therefore, the determination of interaction potentials in freeway traffic contributes to the challenging problem of understanding time headway or distance distributions of cars May ; Hoog . In addition, the identification of behavioral regimes points to adaptive driver behavior.
V Simulation of Large Traffic Networks with Cellular Automata
v.1 Cellular Automata Rules for Links
An alternative approach to carfollowing models are rulebased cellular automata (CA). For the CA technique, road links are divided into cells, say of length 7.5 m, which can contain at most one car each, and there is one array of cells for each lane, see Fig. 7. Movement is performed by jumping from one cell to another, where the new cell is determined by a set of “driving rules.” A good update time step is 1 sec (justified by reaction time arguments). Taking this together means, for example, that a jump of cells in a time step models a speed of m/sec = 135 km/h.
Driving rules of traffic on a link consist of two parts: car following, and lane changing. Typical rules for car following are: deterministic speed calculation, randomization, and movement. The deterministic speed calculation rule first computes a new speed for each car based on its current speed and closeness to the car in front of it. An example for such a rule is
where models acceleration, is a speed limit, and is the gap, i.e. the number of empty spaces to the vehicle ahead. See Fig. 7 for an illustration of gap. Next, the randomization rule introduces a probability for each car so that instead of following the above speed deterministically, it may instead drive one velocity level more slowly:
In the movement rule, each particle is moved forward:
To illustrate the above rules, Fig. 7 shows traffic moving to the right. Using a value of 0.2, the leftmost vehicle accelerates to velocity 2 with probability 0.8 and stays at velocity 1 with probability 0.2. The middle vehicle slows down to velocity 1 with probability 0.8 and to velocity 0 with probability 0.2. The rightmost vehicle accelerates to velocity 3 with probability 0.8 and stays at velocity 2 with probability 0.2. Velocities are in “cells per time step.” All vehicles are then moved according to their velocities using the movement rule described above.
Fig. 8 shows, in a socalled spacetime diagram for traffic, the emergence of a traffic jam. Lines show configurations of a segment of road in secondbysecond time steps, with time increasing in the downward direction; cars drive from left to right. Integer numbers denote the velocities of the cars. For example, a vehicle with speed “3” will move three sites (dots) forward. Via this mechanism, one can follow the movement of vehicles from left to right, as indicated by an example trajectory.
v.2 Lane Changing
Typical rules for lane changing (see Fig. 7) include a reason to change lanes and a safety criterion for changing lanes. First, there needs to be a reason why a vehicle wants to change lanes, for example that the other lane is faster, or that it needs to get into a certain lane in order to make an intended turn at the end of the link. A possible rule to model the first is “check if the (forward) gap in the other lane is larger than the gap in the current lane.” Second, the vehicle needs to check that there is really enough space in the destination lane. A possible rule is that the forward gap in the other lane needs to be larger than , and the backward gap in the other lane needs to be larger than the velocity of the oncoming vehicle.
Figure 7 illustrates the lane changing rules. Only lane changes to the left lane are considered. In situation I, the leftmost vehicle on the bottom lane will change to the left because the forward gap on its own lane, 1, is smaller than its velocity, 3; the forward gap in the other lane, 10, is larger than the gap on its own lane, 1; the forward gap in the target lane is large enough; and the backward gap is large enough. In situation II, the second vehicle from the right on the right lane will not perform a lane change because the gap backwards on the target lane is not sufficient.
Due to the complexity of the dynamics, it is inconvenient to do both lane changing and car following in one parallel update. It is however possible, and convenient for parallel computing, to do the update in two completely parallel subtimesteps: First, all lane changes are executed in parallel based on information from time , resulting in intermediate information, at time . , which results in information for time . Then, all car following rules are executed in parallel, based on information from time
v.3 Intersections
The typical modeling substrate for transportation is the network, which is composed of links and nodes that model roads and intersections, respectively. Since the full simulation of an intersection is rather complicated, for large scale applications one attempts to reduce the complexity by resolving all conflicts at the entering of the intersection. Under such a simplification, it is still possible to model things like left turns against oncoming traffic, but the decision is made before the vehicle enters the intersection, not inside the intersection. In consequence, complications inside the intersection, such as the interesting dynamics of a large roundabout in Paris, are not modeled by this approach. Nevertheless, the simplified approach is quite useful for large scale applications.
Given this, there are only two remaining cases for traffic: Protected and unprotected turns. A protected turn means that a signal schedule takes care of possible conflicts, and thus the simulation does not have to worry about them. In this case, it is enough if the simulation makes vehicles stop when a signal is red, and lets them go when a signal is green.
In unprotected turns, conflicts are resolved by the legal rules, not by signals. For example, a vehicle making a left turn needs to give priority to all oncoming vehicles (Fig. 7). This entails that one needs, for each turning direction, to encode which other oncoming lanes have the priority. Once this is done, a vehicle that intends to do a certain movement only needs to check if there is a vehicle approaching on any of these interfering lanes. If so, the vehicle under consideration has to stop, otherwise it can proceed. As said before, once the vehicle has entered the intersection, other vehicles are no longer regarded. Figure 7 shows an example of a left turn against oncoming traffic. The turn is accepted because on all three oncoming lanes, the gap is larger or equal to three times the first oncoming vehicle’s velocity.
v.4 Modeling Queues
Sometimes, even the relatively simple CA rules are computationally too slow. Another factor of ten in the execution speed can often be gained by simplifying the dynamics even more. In the socalled queue model (Gazis:queue, ; Gawron:queue, ), links have no internal structure any more, they can be considered as vehicle reservoirs. These reservoirs have a storage constraint, , which varies from link to link, and is based on the physical attributes of the link, such as length and number of lanes. Once the storage constraint is exhausted, no more vehicles can enter the link until some other vehicles have left the link.
Vehicles can only leave the link if their destination link has space, if they have spent the time it takes to traverse the link, and if the flow capacity constraint is fulfilled. The flow capacity constraint is a maximum rate with which vehicles can leave the link.
Except for the storage constraint and its consequences, this is just a regular M/M/1 queuing model. The storage constraint does, however, necessitate one adaptation of the model: Since destination links can be full, there is now competition for space on the destination link. A good option is to give that space randomly to incoming links, with a bias toward incoming links with higher capacity (Cetin and Nagel, 2003).
With the queue model and using parallel computing, it is possible to simulate a full day of all car traffic of all of Switzerland (approx. 7 mio inhabitants) in less than 5 minutes Cetin and Nagel (2003). Although data movement is still a challenge, this makes it possible to run completely microscopic (agentbased) studies of large metropolitan areas Raney et al. (2003).
Vi Entropy Laws and Destination Choice
The simulation of traffic in street networks requires data on the number of trips between origins and destinations as a function of the time of the day, the weekday etc. These origindestination data are relatively scarce and expensive. However, the number of trips starting in a city or quarter of a city and the number of trips ending in is relatively wellknown, as this requires only instead of data, where denotes the number of dinstinguished origins/destinations. Apart from this, and can be automatically measured by detectors, while the determination of the entries of the traffic flow matrix requires to obtain origindestination pairs, i.e. to ask drivers.
Therefore, the relative frequencies from the origins to the destinations , where , are often determined via the “most likely” distribution under the constraints
(28) 
This is obtained by minimizing the “information gain”
(29) 
compared to some “natural distribution” . This is analogous to maximizing entropy in cases where is an equidistribution. The only difference is that (29) takes into account a “resistance function” , which reflects the “natural distribution” as a function of the distance and other behaviorally relevant variables in the absence of constraints like (28). The appropriate specification of will be discussed in Sec. VI.1.
Instead of minimizing (29) under the constraints (28), it is easier to minimize
(30)  
where the additional terms vanish when the constraints (28) are fulfilled. Differentiation with respect to gives
(31) 
or
(32) 
The Lagrange parameters and are now determined in a way that the constraints (28) are satisfied. This requires the use of numerical iteration procedures. For an overview of numerical solution algorithms see Refs. Lohse1 ; Lohse2 ; Lohse3 . These references also suggest ways to treat the choice of destinations and transport modes simultaneously.
vi.1 Energy Laws and Resistance Function
According to the last paragraphs, the distribution of traffic significantly depends on the “natural distribution” . Robert Kölbl and one of us have investigated the electronically available empirical data of the UK National Travel Survey during the years 1972–1998 eins . When we distinguished between different daily modes of transport , we found that the average modal travel time per day and person remained almost constant over the 27 years of observation, despite variations by a factor 3.8 between different modes. More specifically, the average travel times were min during a day with walking (), without any usage of other means of transport, min for cyclists (), min for stage bus users (), min for car drivers (), min for car passengers (), and min for train passengers ().
When we scaled the axis of the modal traveltime distributions by the average travel times , we discovered that, within the statistical variation, the resulting distributions collapsed onto one single curve
(33) 
with , two fit parameters and , and the normalisation constant (see Fig. 9a). This implies a universal law of human travel behaviour and the resistance function or natural distribution
(34) 
where is the travel time from to divided by the average travel time for transport mode .
In the semilogarithmic representation of (33), the term is relevant only for short time scales up to , while the linear relationship dominates clearly over a wide range of scaled travel times (see Fig. 9b). Therefore, we suggest the following interpretation: The dominating term corresponds to the canonical energy distribution
(35) 
where with stands for the energy consumed by the physical activity of travelling with transport mode for a time period . The term possibly represents the additional amount of energy required for the mobilization of short trips, i.e. the Simonson effect drei . With an average travel energy budget of kJ, this interpretation is consistent with values of the energy consumption per unit time for certain activities obtained by ergonomic measurements of the related Oconsumption vier1 ; vier2 ; vier3 .
Based on entropy maximization, the canonical distribution can be shown to be the most likely distribution, given that the average energy consumption per day by an ensemble of travellers is fixed for the area of investigation. This agrees well with the investigated data and generalizes the concept of a constant traveltime budget f1 ; f2 ; f3 ; f4 ; f5 ; f6 ; f7 ; f8 . In addition, the above theory can be combined with trip distribution models MNL ; s1 ; s2 ; s3 ; s4 ; s5 , e.g. the multinomial logit model
(36) 
where is a parameter (see Sec. VII.2) and the modedependent energy consumption enters the utility function as a (negative) cost term, which was shown to be a significant variable of human travel behaviour.
In summary, when daily travel time distributions by different modes of transport such as walking, cycling, bus or car travel are appropriately scaled, they turn out to have a universal functional relationship. A closer investigation reveals a law of constant average energy consumption by the physical activity of daily travelling. The advantage of the behavioral law (33) is its expected longterm validity over more than 25 years even under changing conditions. It will, therefore, prove to be important for urban, transport, and production planning: Compared to previous models, it facilitates improved quantitative conclusions about trip distributions, modal splits and induced traffic after the more reliable determination of fewer parameters, which are constant over typical planning horizons. It also helps to assess the increase in the acceptability of public transport, when the comfort of travel is improved, to forecast the change in travel behavior in an aging society, to predict the usage of new modes of transport, and to estimate potential market penetrations of new travelrelated products.
Vii Activity Patterns
Sometimes, the models outlined in the previous sections are not sufficient to model demand generation for traffic. The first obvious shortfall is that they do not say anything about a timeofday dynamics. This is sometimes introduced by assuming that all trips from residential to commercial locations represent morning traffic, and the afternoon/evening peak is then obtained by transposing the trip matrix.
Often, even that is not enough. This happens typically when there are significant correlations between travelers’ attributes and their decisions. For example, the number of trips starting in a zone in general does not just depend on the number of residents, but also on their income or age; destination choice in general depends on vehicle availablity; the acceptance of a new mode of transport in general depends on the question if people can reorganize their entire day plans so that the new facility can be efficiently integrated.
A possible way out of this is to make the entire approach completely microscopic, or agentbased. This means that throughout the whole simulation, all travelers are maintained as individuals, and individual behavioral models are applied at all steps of the decisionmaking. For example, synthetic ( simulated) travelers may plan their activity patterns (e.g. home – work – shop – home – leisure – home), then the corresponding locations, then the modes of transportation, and finally the precise timing of their activities. The advantage of this approach over the ones discussed so far is that, at each step of the decisionmaking, all “internal” information of the agent is available.
Such an approach draws from many areas of science. Clearly, psychology is involved because it describes the behavior of human individuals. Yet, since the goal is not to describe each individual traveler correctly but to get correct aggregate results, economics and the social sciences are involved as well, and there is in fact a community of “travel behavior research” (iatbr00book, ) which applies these principles to traffic. However, the approach to go to microscopic principles when an analysis on the macroscopic level fails is also deeply rooted in physics, and in fact, both the computing methods to largescale problems and the general intuition that simple (and therefore individually wrong) microscopic models may lead to correct macroscopic results come to a large extent from physics.
Given its highly interdiscplinary nature plus the fact that only recently data and computing methods became available to do activitybased demand generation for large urban areas, it is understandable that the area is in rapid flux. There is one method, discrete choice theory (BenAkiva:book, ), that is arguably better established than others, and it will be described in a bit more detail in the following. However, there are some drawbacks to that theory, and it is questionable whether it will be possible to correct them within the framework of that theory. For that reason, alternative, less well established approaches will also be discussed.
vii.1 Discrete Choice Theory
Assume, for simplicity, a situation where an agent has a choice between several alternatives, , for example between several locations for the same activity. Also assume that there are utilities associated with those alternatives. Discrete choice theory starts from two assumptions:

It is possible for the observer/modeler to measure/predict some part of the utilities. These measurable parts are called “systematic utilities” .

The differences between the and the are random and statistically independent. These differences are denoted as .
With these assumptions, one obtains essentially the following algorithm:

Compute the based on your observations. For example, one might have observed that one location is accessible only via some steep stairs, and therefore avoided by physically less fit persons.

Add the random components: .

Select the alternative that now has the highest utility. Note that, because of the random component, it can well be that the alternative with a smaller in the end “wins”.
There are at least two different interpretations where the random parts come from: Economic theory states that economic agents always select the option with the largest utility, and the difference between the observed/predicted utilities comes from socalled unobserved attributes of the agent which are thus unknown to the observer/analysis. Psychology, in contrast, might say that there is just a random component to people’s behavior.
vii.2 Multinomial Logit Model (MNL)
The question now is how to obtain realworld numbers for the and the . The typical way to progress is in two steps: (i) make some assumptions about the ; (ii) assume that the are additive in their contributions and do a maximum likelihood estimation of the parameters.
The random components.
A conceptually easy way to proceed is to assume that the are normally distributed. The result of this is called a (multinomial) probit model. Its solution, however, cannot be written down in terms of simple functions. Another path is to assume that the follow socalled Gumbeldistributions with zero mean, i.e. their generating function is , where is a width parameter. In this case, the result is that the probability for agent to select option follows the Boltzmann distribution
(37) 
This is known as the (multinomial) logit model, cf. Eq. (36). results from the “widths” of the distributions. It models how “rational” the agents behave with respect to what is observed: a small makes the agents nearly random with respect to ; for one obtains an agent that always takes the option with the best .
The systematic components.
In order to make further progress, one has to make an assumption about the functional form of the , and its free parameters. The most typical assumption is that the are linear in their components. For example, the utility for agent to go swimming at location might be
(38) 
where is the travel time to get there, is the monetary cost, and is ’s gender (e.g. 0 for male, and 1 for female). Normally, and will be negative. A positive would express that, all other things being equal, females like to go swimming more than males. Note that attributes of the alternative as well as attributes of the person can occur, and that attributes can be continuous or discrete.
One now proceeds by asking a large number of people about their individual , , , plus the decision that they actually made (revealed preference) or would make (stated preference) in the given situation. Maximum likelihood estimation then finds coefficients such that the probability to obtain the decisions from the survey is maximized ( stands for the different possible indices). More precisely, it finds products , meaning that the from Eq. (37) is already included. Note that it is not necessary for Eq. (38) to be linear; something like or is possible (where is ’s income). In the past, the utility function had to be linear in the (because of the need for a partial derivative for the maximum likelihood estimation), but that requirement is now lifted BIOGEME www page (accessed 2004).
vii.3 Extensions
Multinomial logit models (MNLs) have a property called independence from irrelevant alternatives (IIA). It means that taking an alternative out of the choice set does not change the relative probabilities: does not depend on a third alternative . This has sometimes odd consequences, typically explained in terms of the socalled “red bus blue bus” paradoxon. Assume that a traveler has the option to take a car (option 1) or to take the bus, and that she takes the bus with 50% probability. Now assume that the bus route is served by two buses, one of them red (option 2) and one of them blue (option 3), and both of them depart at exactly the same time, they have exactly the same service quality, and they are always nearly empty. Estimation of an MNL would result in a 25% probability for the red bus and a 25% probability for the blue bus.
So far, this is all fine. However, now assume that the blue bus gets taken out of service. The model for our traveler now predicts that she uses the car with probability , and the red bus with probability . This is due to the fact that the ratio of the probabilities of option 1 and option 2 has to remain unchanged.
The problem sits in the fact that for this example, the random components and are not independent. One way to deal with this problem is to make the model hierarchical (“nested”), where first the choice is between car and bus, and then about the type of bus. This reduces computational complexity, but demands that the analyst specifies the hierarchical structure. Another option is to use models in which correlations between the random variables are included as covariances. The results are more sophisticated than for the MNL model, but with increasingly better computers they enter the domain of being useful for practical cases.
Another extension concerns the parameters , where again stands for the different possible indices. For example, in Eq. (38), it would make sense to assume that the ratio , related to the “value of time”, depended on income. The two options in this situation are (i) to assume an explicit incomedependent model such as (where once more and would be expected to be negative) or (ii) to assume random taste variations across the population, which means that the coefficients become random variables, with accordingly more free parameters to be estimated.
Software for the estimation of the models described in this section and pointers to relevant literature can be found at BIOGEME www page (accessed 2004).
vii.4 Discrete Choice Theory for Activities
Existing discrete choice models for the generation of daily activities use the nested MNL approach, which translates into a hierarchical decision tree: First a traveler decides on the daily pattern, then on activity locations, then on modes of transportation, and then on exact times Bowman:thesis ; Bowman:etc:Portland:acts . A problem is that decisions on the higher level depend on decisions on the lower levels; for example, the decision about the inclusion of an activity depends on how close its location is to the locations of other possible activities. The practical approach to that problem is to first compute the lower level options and then pass them on as “expected” utilities to the higher levels. For example, the utility of doing a trip both by car and by bus is computed, and (roughly) the maximum of both is passed on to the location choice module.
vii.5 Alternative Methods for the Generation of Activity Patterns
In spite of the random term, discrete choice theory assumes extremely rational agents. In fact, a traveler modeled by a discrete choice model for activity planning will consider the utilities of all different options before making a decision. A consequence of this is that the computation is in fact much more expensive than finding the best option, since for a best option, suboptimal branches in the search tree can be pruned, while for discrete choice theory, all branches need to be followed to the leaves. In addition, discrete choice models for human activity planning need of the order of several hundred free parameters, and it is questionable how well so many parameters can be estimated from surveys.
For those reasons, it makes sense to look at alternative models. For example, one can make models where travelers do not look forward on the time axis Kitamura:CO2 . Such a model is much easier to calibrate, but it is not sensitive to changes in the time structure: For example, the model will not make people get up later in the morning if opening times of shops are extended in the evening. An improvement of this is the use of Qlearning, where agents learn, by doing the same day over and over again, to backpropagate the temporal effects to the decision points q4acts . Again, other models are entirely based on rules ALBATROSS or on genetic algorithms ga4acts . A newer development is the use of mental maps, where the agent remembers which parts of the system it knows Unger:phd ; arentze:mentalmap:iatbr03 ; Kistler (2004).
Viii Learning, Feedback, and Evolutionary Game Theory
Neither entropybased destination choice (Sec. VI) nor complete activity patterns (Sec. VII) nor route choice (Sec. IX) can be sensibly modeled without including the effect of learning or adaptation. The simple reason for this is that there are nonlinear circular dependencies: Human plans depend on (expected) congestion, but congestion is the result of (the execution of) plans.
The obvious solution for this is any kind of relaxation method, i.e. some version of “make initial plans – compute congestion – adjust the plans – recompute congestion” etc. Another interpretation of the same approach is that it models human learning from one day to the next.
It is interesting to note that there is actually some theory available to describe this process. For example, if all agents always play “best reply” (i.e. the best answer to the last iteration), and if the iterative dynamics converges to a fixed point attractor, than that fixed point is also a Nash Equilibrium (NE) Hofbauer and Sigmund (1998). For the traditional method of transportation planning, called static assignment (similar to flows in electric networks except that particles know their destinations), one can actually show that under some assumptions the NE is unique (in the link flows), and therefore the iterative process does not have to be explicitly modeled Sheffi:book . For the more complicated models discussed in this paper, no such mathematical statement is available, and therefore multiple fixed point attractors could be possible, or the iterative process could converge to a periodic or chaotic attractor. If the models are stochastic, then under mild assumptions they converge to a fixed steadystate statespace density Cascetta:Cantarella:day2day . In practice, only a small number of iterations is feasible, and effects such as broken ergodicity Palmer:broken:ergodicity need to be taken into account. A very illustrative example of these different types of possible learning dynamics will be provided in Sec. IX.
Real world scenarios are even more complicated. Decisions of different agents may happen on different time scales, which depending on their hierarchical order leads to different solutions Zuylen:Taale:tgf03 ; Nagel:etc:sqrtutl . This is related to the issues of sequential games in game theory. Agents may have limited information, or they may not even use “best reply” at all, but rather some constraint satisficing method, in which they improve until they are satisfied. Technically, this means that some important real world results may lie in the transients rather than in the steady state behavior. A research agenda for the future needs to clarify these questions, and needs to identify the limits of predictability for such systems.
Ix Route Choice
Once the activity pattern of an individual is selected (including the destination and the mode of transport Lohse1 ; Lohse2 ; Lohse3 ), one of the alternative routes to the destination needs to be chosen. This is typically done by maximizing the utility , which in most cases comes down to minimizing the overall travel time. As a consequence, the fastest connection is filled with vehicles, until the traffic density has increased so much that their travel times become comparable with the travel time on an alternative road with greater length or smaller speed limit. In this way, a distribution of traffic is produced, in which the travel times of all alternative routes are the same. This distribution is called the Wardrop equilibrium Wardrop . However, does this distribution really describe the route choice behavior of drivers? The following paragraphs will show that the Wardrop equilibrium systematically underestimates travel times, as drivers do not match the ideal traffic distribution due to coordination problems.
ix.1 Decision Experiments
The efficient distribution of road capacities among users based on individual decisions is still a fundamental problem. As individuals normally have aggregate information (such as radio news) rather than complete information, one frequently observes farfromoptimal travel times. In order to learn more about this decision behavior, we have carried out a variety of route choice experiments Control ; Bonn (see Figs. 10 to 13).
In these experiments, test persons had to repeatedly decide between two alternative routes and should try to maximize their resulting payoffs , which were chosen proportionally to the inverse travel times. If the average vehicle speed on route is approximated by the linear relationship Gree235
(39) 
the inverse travel times obey the payoff relations with
(40) 
Herein, is the maximum velocity (speed limit) and the number of drivers on route , its length, and its capacity, i.e. the maximum possible number of vehicles on route . For an improved approach to determine the travel times in road networks see Ref. traveltime . Note that alternative routes can reach comparable payoffs and travel times only when the total number of vehicles is large enough to fulfil the relations and . Our route choice experiment has focussed on this traffic regime. Furthermore, we have the capacity restriction , as is connected with a complete gridlock.
The Wardrop or user equilibrium corresponds to equal travel times and payoffs for both alternative decisions. It is found for a fraction
(41) 
of persons choosing alternative 1 and agrees with the system optimum only in the limit of many participants. Small groups were chosen to investigate the fluctuations in the system. It was found that the test groups managed to adapt to the user equilibrium on average (see Fig. 12a). However, although it appears reasonable to stick to the same decision once the equilibrium is reached, the standard deviation stayed at a finite level. This was not only observed in “treatment” 1, where all players knew only their own (previously experienced) payoff, but also in treatment 2, where the payoffs and for both, 1 and 2decisions, were transmitted to all players (analogous to radio news). Moreover, treatment 2 could increase the average payoffs, as the additional information allowed for a better adaptation without having to change decisions to explore the better performing alternative (see Ref. Selten and Fig. 12).
With treatment 3, we managed to reveal the mysterious persistence in the changing behavior and to achieve more than three times higher payoff increases. Every test person was informed about the own payoff [or ] and the potential payoff [or ] he or she would have obtained, if a fraction of persons had additionally chosen the other alternative (here: ). Consequently, in the user equilibrium with every player knew that he or she would not get the same, but a reduced payoff, if he or she would change the decision. That explains why treatment 3 could reach a better adaptation performance, reflected by a low standard deviation and closetooptimal average payoffs. Moreover, even the smallest individual cumulative payoff exceeded the highest one in treatment 1. Therefore, treatment 3’s way of information presentation is much superior to the ones used today.
Figure 13 clears up why players changed their decision in the user equilibrium at all. We discovered intermittent behavior, i.e. quiescent periods without changes, followed by turbulent periods with many changes. This is reminiscent of volatility clustering in stock market indices, where individuals also react to aggregate information summarizing all decisions (the trading transactions). Single individuals seem to change their decision speculating for aboveaverage payoffs. In fact, although the cumulative individual payoff is anticorrelated with the average changing rate, some individuals receive higher payoffs with larger changing rates than others, i.e. they profit from the overreaction in the system: Once the system is out of equilibrium, all individuals respond in one way or another. Typically, there are too many decision changes.
In treatment 4, we have tested the group performance under changing environmental conditions, when the participants got the same information as in treatment 3. That is, some of the payoff parameters were varied in time, implying a timedependent user equilibrium. Even without recommendations, the group managed to adapt surprisingly well to the variable conditions, but the standard deviation and changing rate were approximately as high as in treatment 2 (see Fig. 12). This adaptability (the collective ‘group intelligence’) is based on complementary responses Selten ; Bonn . That is, if some players do not react to the changing conditions, others will take the chance to earn additional payoff. This experimentally supports the behavior assumed in the theory of efficient markets, but here the efficiency is limited by overreaction.
To avoid overreaction, in treatment 5 we have recommended a number of players to change their decision and the other ones to keep it. These userspecific recommendations helped the players to reach the smallest overreaction of all treatments and a very low standard deviation, although the payoffs were changing in time as in treatment 4 (see Fig. 12).
X Migration and the Law of Gravity
Urban and regional evolution, in particular the change of street infrastructures, housing, industrial and business quarters, has a natural feedback on the spatial distribution of people. The temporal development of the population distribution in space can be described by migration models. Pioneers in this field have been Weidlich and Haag WeidlHa88 . In the following, we will discuss a variant of their master equation model WeidlHa83 ; Weidl91 ; Weidl00
(42) 
According to this equation, the stream of immigrants into place is given by the fraction of the population at place which could potentially move to place . The proportionality factor is the transition rate from to , i.e. the migration probability to per unit time, given one lives at place before. Analogously, the stream of emigrants leaving place is proportional to the fraction of the population living in , and the proportionality factor is the migration rate from to . This migration rate is proportional to the fraction of people living at place , as the number of acquaintances with friends, family members, or former collegues increases linearly with the number of inhabitants of a town . However, the migration rate decreases significantly with the distance between two places and (and/or with the transaction costs). Accordingly, we assume Hel95a
(43) 
where is a monotonously growing function in . The parameter reflects the mobility of the population and the utility/attractiveness of living at place . Note that formula (43) reminds of the gravity law in physics Zip46 , where and have the meaning of the masses of two celestial bodies and and . Moreover, Eq. (43) is analogous to Eq. (32) with , , and . That is, the gravity law can also be motivated by entropy considerations, but this time is adapted rather than and . A great advantage of Eq. (42) is that it can describe disequilibrium situations such as resulting migration streams when borders between countries are opened.
Xi Urban and Regional Evolution
Weidlich and Haag have modeled urban and regional evolution with a master equation approach similar to Eq. (42), but with additional birth and deathterms describing the generation or the removal of entities from the system. Moreover, in their simulations, they distinguished different kinds of entities (“agents”): inhabitants, who may migrate, streets which may be expanded, industrial areas which may grow or decay, building blocks in which people may live, and leisure areas such as parks. Positive and negative feedbacks between these different entities were represented by interaction terms in the utility functions determining the corresponding transition rates . In this way, they could describe the empirically observed differentiation in spatial usage patterns and interdependencies between them. Moreover, they have applied their approach to the evolution of cities in China WeidCity .
An advantage of models based on the master equation is their huge flexibility. A disadvantage is that there is a large number of coefficients that need to be defined. Therefore, there is also a search for models that naturally include (usually geometric) constraints and by doing so can explain some aspects of urban dynamics with a much smaller number of free parameters. These models originate from the observation that some properties of cities are fractal. For example, the fractal dimension of the bulk of cities seems to be around 1.9, and the fractal dimension of the (outer) perimeter of cities seems to be around 1.3 Frankhauser:fractalesurbaines ; Batty:book . This implies that one might attempt to describe city growth with local growth models. There are both attempts to explain the fractal dimension by variations of models originally from physics Berlinselforg ; Makse:Batty:Stanley and attempts to recreate (and thus predict) the growth of realworld cities White:CA ; Rabino:urbanca . Yet, not all aspects of urban growth can be explained by entirely local rules. For example, construction of a large mall or a golf course certainly depends on the availability of a similar facility somewhere in the city. Unfortunately, this leads to models and simulations where everything depends on everything, which means complicated models and long simulation times. Yet, plausibly, this is also not correct: Although large and important facilities have a farreaching impact, the effect still decays with distance. In consequence, hierarchical models, as they are known from particle dynamics with long range interactions, are now also transfered to urban dynamics steen:white:urbangrowth .
As was the case at other places in this paper, the higher level dynamics of urban growth is not independent from lower level dynamics such as traffic congestion. In consequence, the ultimate challenge of this area may be to present a uniform view that bridges all scales from car driving via human activity planning to urban evolution. In such a view, the models presented so far would be limiting cases for the situation where a separation of (time) scales is possible. There are three approaches to achieve this bridging of the scales:

Parameterization: If a separation of (time) scales is not possible, then lower level processes need to be taken into account at least via parameterizations, i.e. simplified aggregated models of the lower level processes. As is well known, parameterization needs to be preceeded by scientific understanding.

Model coupling: Instead of using a parameterization, i.e. a macroscopic description of the lower level dynamics, a (computational) model of the lower level dynamics could be used directly. Such model coupling is another big challenge, not only for computational reasons, but also because the combined model systems are once more dynamical systems with a behavior that is usually not well understood.

Microscopic across all scales: A further alternative is to construct a model that is microscopic or agentbased across all scales, that is, at all levels all decisions are made by synthetic versions of the respective actors, usually humans. This is similar to a molecular dynamics simulation in physics. It also faces the same challenge: It is difficult to simulate large scale systems in spite of all the computational progress we had in recent times Nagel and Marchal (2003). Nevertheless, the first partial systems of this approach are emerging TRANSIMS www page (accessed 2004); MATSIM www page (accessed 2004); Salvini and Miller (2003), and validations of partial aspects look rather promising Esser and Nagel (2001); Raney et al. (2003).
Once such models are available and operational, they could be applied to a rather wide range of questions, ranging from zoning via infrastructure planning to the deliberate attraction of commercial companies and the effects of migration. And although such models cannot take away the responsibility for decisions from politicians or from society, they might help that such decisions are based on a more informed understanding of the system than today.
Xii Summary and Outlook
Physics is a science that has a tradition in deriving large scale emergent phenomena from microscopic rules. Sometimes, these derivations can be done analytically, and in such cases the thermodynamic limit is useful. In other situations, the microscopic rules are too complicated to make analytical progress, and the computer is used. However, even with today’s computational power, only up to about particles can be simulated directly.
The situation in traffic and regional development is similar in the sense that most effects are also caused by the combined behavior of many particles. In contrast to materials science, the number of particles in regional simulations, typically several millions, is well within the range that is computationally accessible. On the other hand, the rules that each individual particle follows are considerably more complicated.
In this situation, it makes sense to look at different scenarios one by one. The first question is to identify those areas where very simple models already lead to good levels of understanding and prediction. This refers, for example, to the area of traffic flow, where already models which just include limits on acceleration, braking, and excluded volume generate a wide range of dynamical phenomena that can all be identified in the real world. Similar models are tried in the areas of route choice, activity generation, migration, or urban growth.
If these simple models fail, then more complicated models are in order. These models typically contain more behavioral parameters, and because the behavior of humans is difficult to predict, these models are more difficult to calibrate or validate. The challenge is, as so often, to keep the models as simple as possible, but no simpler. Also, one has to keep in mind that we are interested in macroscopic, “emergent” phenomena such as traffic jams, or general urban patterns, and those should be easier to predict than the behavior of individual humans. Nevertheless, one has to get used to validation error bars that are much larger than in physics.
With respect to realworld prediction, models that cover a large range of scales would be useful. Most notably, there is a rather old dream for integrated land use transport models, in which the location choices of inhabitants and commercial entities lead to traffic, which in turn triggers revised location choices, etc. Timmermans (2003) It has, however, been difficult to put that dream into practice Timmermans (2003), and one could argue that the main obstacle has been that the interactions between the scales were not well understood but rather included in an adhoc manner. Simultaneously, computing power in the past has not really been sufficient for sound solutions.
However, in our view the situation is now changing. Based on the combination of computational, analytical, and experimental work, there has been considerable progress with respect to understanding aspects at the different levels, leading to better and faster submodels in the future. At the same time, it is now possible to build computational models that bridge large ranges of scales while remaining microscopic, which, although computationally demanding, again solves a large variety of conceptual problems.
Finally, one should also mention that data availability in the field of geoscience has made a quantum jump forward. Satellitederived high resolution spatial data is now available for nearly everywhere; decent census data is available in nearly every industrialized country; vector data for the infrastructure is increasingly becoming available; and a whole range of telematics devices is capable of collecting important data about spatial behavior, although aspects of privacy need to be valued against scientific interest. This also means that the traditional approach of physics, where theory, computation, and realworld measurement enhance each other, can now be applied to traffic and regional systems.
Overall, it is our impression that the new computing and data collection technologies have had a huge impact in the area of traffic and regional systems, and that both the computational and the theoretical work are struggling to keep up with the new possibilities. This, combined with the obvious societal relevance of the area, makes this a very exciting field.
Xiii Authors’ Biographies
Dirk Helbing (*19/Jan/1965) is the Managing Director of the Institute for Economics and Traffic at Dresden University of Technology. Originally, he studied Physics and Mathematics in Göttingen, Germany, but soon he got fascinated in interdisciplinary problems. Therefore, his master thesis dealt with physical models of pedestrian dynamics, while his Ph.D. thesis at the University of Stuttgart focussed on modeling interactive decisions and behaviors with methods from statistical physics and the theory of complex systems. After his habilitation on the physics of traffic flows, he received a Heisenberg scholarship and worked at Xerox PARC in Silicon Valley, the Weizmann Institute in Israel, and the Collegium Budapest—Institute for Advanced Study in Hungary. He gets excited when physics meets traffic or social science, economics, or biology, and if the results are potentially relevant for everyday life.
Kai Nagel (*17/Sep/1965) is Professor for Transport Systems Analysis and Transport Telematics at the Institute of Land and Sea Transport Systems at the Technical University Berlin in Germany. He studied physics and meteorology at the University of Cologne and the University of Paris, with one master’s thesis in the area of cellular automata models for cloud formation and another one in the area of large scale climate simulations. His Ph.D., in computer science at the University of Cologne, was about cellular automata models for large scale traffic simulations. He then was postdoc, staff member, and research team leader at Los Alamos National Laboratory, working on the TRANSIMS (Transportation ANalysis and SIMulation System) project. In 1999–2004, he was assistant professor for computer science at ETH Zurich in Switzerland. His interests are in large scale simulation and in the simulation and modeling of socioeconomic systems.
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