The physics of gridlock
By: Stephen Budiansky
Date: 2000-12-08
Illustration by Maris Bishofs.
Copyright � 2000 by The Atlantic Monthly Company.
Courtesy of The Atlantic Monthly; December
2000; Volume 286, No. 6; page 20-24.
BERTRAND
Russell once observed that animal behaviorists studying the problem-solving
abilities of chimpanzees consistently seemed to detect in their experimental
subjects the "national characteristics" of the scientists themselves. A divergence
in the findings of the practical-minded Americans and the theoretically inclined
Germans was particularly apparent. Animals studied by Americans rush about
frantically, with an incredible display of hustle and pep, and at last achieve
the desired result by chance. Animals observed by Germans sit still and think,
and at last evolve the solution out of their inner consciousness.
In science, Germans tend to come up with things like the uncertainty principle. Americans tend to come up with things like the atomic bomb.
The latest field to host this conflict of national styles is one that seems at first glance to offer little prospect of a sporting contest. Bigger and better highways are as American as fast-food restaurants and sport utility vehicles, and when it comes to making the crooked straight and the rough places plain, the practicality of American traffic engineers is hard to argue with. As an American academic discipline, traffic engineering is centered in civil-engineering departments, and civil engineers tend to believe in solving problems by going at them head on. A recent study funded by nine state departments of transportation to examine the doubling in congestion on urban highways and primary roads that has occurred over the past two decades listed in its final report various ways that traffic engineers have tried to alleviate the problem. These included "add road space" and "lower the number of vehicles." This would not, as the saying goes, appear to be rocket science.
Even when American traffic engineers have ventured closer to rocket science, with computer simulations of traffic flow on multi-lane highways, the results have tended to reinforce the American reputation for practicality and level-headedness. The mathematical and computer models indicate that when traffic jams occur, they are the result of bottlenecks (merging lanes, bad curves, accidents), which constrict flow. Find a way to eliminate the bottlenecks and flow will be restored.
SUCH was the happy, practical, and deterministic state of affairs up until a few years ago, when several German theoretical physicists began publishing papers on traffic flow in Physical Review Letters, Journal of Physics, Nature, and other publications not normally read by civil engineers. The Germans had noticed that if one simulated the movement of cars and trucks on a highway using the well-established equations that describe how the molecules of a gas move, some distinctly eerie results emerged. Cars do not behave exactly like gas molecules, to be sure: for example, drivers try to avoid collisions by slowing down when they get too near another car, whereas gas molecules have no such aversion. But the physicists added some terms to the equations to take the differences into account, and the overall description of traffic as a flowing gas has proved to be a very good one. The moving-gas model of traffic reproduces many phenomena seen in real-world traffic. When a flowing gas encounters a bottleneck, for example, it becomes compressed as the molecules suddenly crowd together--and that compression travels back through the stream of oncoming gas as a shock wave. That is precisely analogous to the well-known slowing and queuing of cars behind a traffic bottleneck: as cars slow at the obstruction, cars behind them slow too, which causes a wave of stop-and-go movement to be transmitted "upstream" along the highway.
The eeriest thing that came out of these equations, however, was the implication that traffic congestion can arise completely spontaneously under certain circumstances. No bottlenecks or other external causes are necessary. Traffic can be flowing freely along, at a density still well below what the road can handle, and then suddenly gel into a slow-moving ooze. Under the right conditions a small, brief, and local fluctuation in the speed or spacing of cars--the sort of fluctuation that happens all the time just by chance on a busy highway--is all it takes to trigger a system-wide breakdown that persists for hours after the blip that triggered it is gone. In fact, the Germans' analysis suggested, such spontaneous breakdowns in traffic flow probably occur quite frequently on highways.
THOUGH a decidedly unnerving discovery, this was very much of a piece with the results of mathematical models of many physical and biological systems that exhibit the phenomena popularized under the heading "chaos theory." In any complex interacting system with many parts, each of which affects the others, tiny fluctuations can grow in huge but unpredictable ways. Scientists refer to these as nonlinear phenomena--phenomena in which seemingly negligible changes in one variable can have disproportionately great consequences. Nonlinear properties have been discovered in the mathematical equations that describe weather, chemical reactions, and populations of biological organisms. Some combinations of variables for these equations give rise to sudden "phase shifts," in which the solution to the equation jumps abruptly from one value to another; others set off truly chaotic situations in which for a time the solution to the equation fluctuates wildly and without any seeming pattern, and then suddenly calms down.
Such mathematical discoveries do seem to be borne out in the real world. Biological populations often exhibit erratic booms and busts that cannot be explained by any external cause. Long-term weather patterns defy prediction by the most powerful supercomputers. And a whole class of chemical reactions has been discovered in which the chemicals do not merely react and create a product, as they did in high school chemistry class, but oscillate back and forth between reactants and products. (Some especially nice ones cause color changes in the solution, so you can sit there and watch the stuff in the beaker go back and forth every few seconds.) The consistent story in all these discoveries is that the components of the system and their interactions themselves--rather than any external cause--give rise to the nonlinear behavior of the system as a whole. A rough analogy is a dozen dogs standing on a water bed. If one dog moves, he starts the bed sloshing around, which causes another dog to lose his balance and shift his weight, which sets up another wave of disturbance, until true chaos is reached.
IN the case of traffic, the German physicists--principally Dirk Helbing and Boris Kerner, of Stuttgart--found that given a certain combination of vehicle density and vehicle flow rate along a highway, the solution to their equations undergoes a sudden phase shift from freely moving traffic to what they call "synchronized traffic." Cars in all lanes abruptly slow down and start moving at the same speed as the cars in adjacent lanes, which makes passing impossible and can cause the whole system to jam up for hours.
In the traditional picture of traffic flow and congestion, the number of cars per minute that pass a given point on the highway at first steadily increases as the density of cars on the highway increases. (As long as everything keeps moving freely, the more cars there are on a mile of a highway, the more flow by per minute.) Eventually, however, further increases in density will cause a decrease in flow, as drivers begin braking to maintain a safe distance from the cars in front of them. A graph of flow versus density thus forms an inverted V shape. The uphill side corresponds to free flow, the downhill to congested flow. The Germans found, in effect, that under the right (or, rather, wrong) circumstances the solution to the equations can tunnel right through this hill without ever reaching the top, jumping from a state of (submaximal) free flow straight to congestion.
Such a leap from one state to another is like what happens when a chemical substance changes phase from vapor to liquid. It often happens that water in a cloud remains in the gas phase even after temperature and density have reached the point where it could condense into water droplets. Only when a speck of dust happens along, providing a surface on which condensation can take place (a "condensation nucleus"), does the transition finally occur. Helbing and Kerner basically found that free flow and synchronized flow can occur under the same conditions, and that under such "metastable" conditions a small fluctuation in traffic density can act as the speck of dust causing the shift from one to the other.
Worse, they found that it is easier to start a traffic jam than to stop one. The phase shifts they discovered exhibit what is known in the terminology of nonlinear phenomena as hysteresis. That is, a small and transient increase in, say, the number of cars entering a highway from a ramp can trigger a breakdown in flow, but even after the on-ramp traffic drops to its original level (in fact, even after it drops well below its original level), the traffic jam persists. Looking at actual data recorded by sensors on Dutch and German highways, the physicists found apparent examples of this phenomenon in action, in which a sluggish synchronized flow came on suddenly and persisted for hours, even after the density of traffic had dropped.
If breakdowns in flow can result from such small and random fluctuations, then the world is a very different place from the one that most traffic engineers are accustomed to. The very notion of maximum capacity for a highway is called into question, because even at traffic densities well below what a highway is designed to handle, jams can spontaneously arise. "If this flow breakdown can take place just anywhere," says James Banks, a professor of civil and environmental engineering at San Diego State University, "then we're in trouble, because there's a lot more potential for congested traffic than we thought was the case. And it makes a control strategy much more difficult."
For example, it may not be enough simply to limit the rate at which cars are allowed to enter a highway, as is now done on some congested freeways; rather, it may be necessary to time each car's entry precisely to coincide with a transient drop in density along the main road, thus aiming to smooth out the fluctuations that can trigger a phase shift. There may even be situations in which widening roads or "metering" on-ramp flow could backfire, making flow breakdowns more likely. Preventing flow breakdowns in a nonlinear, chaotic world could ultimately require realizing an Orwellian idea that has been suggested from time to time: directly controlling the speed and spacing of individual cars along a highway with central computers and sensors that communicate with each car's engine and brake controls.
TO say that not all American traffic engineers like these discoveries in chaos theory and their implications for traffic is an understatement. Banks acknowledges that there has been a strong, almost visceral, reaction against the Germans' conclusion, because of its assault on rational determinism and common sense, and also on what might be termed culture-of-science grounds. "Scientists and engineers are human beings,"he says, "and the first reaction is, These guys are not only physicists--they also have a knack for getting themselves in the press. So right away there's an envy factor: Who do these guys think they are?" It doesn't help that the German theoreticians' papers are very difficult to understand. "They're written in such a way that those of us who aren't physicists never know if it's their English, or whether they're using physics jargon, or whether they just don't make sense," Banks says.
The Americans also question how well the Germans' theoretical results relate to traffic in the real world. All mathematical models involve assumptions, and just because a model re-creates certain real-world phenomena doesn't mean it accurately reflects reality in toto; there is always the possibility that the weird properties of the equations are artifacts of the model itself and its assumptions. The Germans' theory "is one plausible description," says Carroll Messer, a research engineer at Texas A&M University, using words that are obviously chosen carefully, "but that's not saying it's been verified." Indeed, some American researchers have questioned whether elaborate chaos-theory interpretations are needed at all, since at least some of the traffic phenomena the Germans' theories predict seem to be much like things that have been appearing in the traffic-engineering literature under other names for years, and these have straightforward cause-and-effect explanations. Banks published a paper in 1999 pointing out that data from monitors that record how many cars pass a fixed point--the sort of data the Germans obtained from Dutch and German highways, which they say verify their predictions--often fail to capture the complete picture of what is happening on the road. He suggests that the behavior of drivers may in fact offer a simpler explanation for the phase shifts and other nonlinear features of the Germans' theoretical models. A sudden slowdown in traffic may have less to do with chaos theory and self-organizing phenomena of systems than with driver psychology. Synchronized flow, for example, has appeared in American traffic literature for decades, under the name "speed sympathy," and Banks says it often happens as traffic gets heavier simply because of the way individual drivers react to changing conditions. As the passing lane gets more crowded, aggressive drivers move to other lanes to try to pass, which also tends to homogenize the speed between lanes. Another leveling force is that when a driver in a fast lane brakes a bit to maintain a safe distance, the shock wave travels back much more rapidly than it would in other lanes, because each following driver has to react more quickly. So as a road becomes congested, the faster-moving traffic is the first to slow down.
Thus many American traffic engineers insist that when breakdowns in flow occur for no apparent reason, it is only because no one has looked hard enough to find the reason, which could be anything from a bad stretch of pavement to a deer running across the road. Much work is now under way on both sides of the Atlantic on a "theory of bottlenecks" that may help to settle the matter.
Even if traffic engineers manage to slay the mathematical bogeyman that theoretical physics and chaos theory have unleashed, another bogeyman may be lurking nearby. It turns out that the properties collectively exhibited by large numbers of cars moving over a network of roadways have many mathematical features in common with the behavior of other things that flow over networks, such as data carried by telephone lines and the Internet. The mathematics of networks is a well-studied topic in communications research, and a recent paper draws on this body of theory to establish an interesting paradox about the flow of vehicular traffic: adding a new road segment to an existing network of roadways can under certain circumstances reduce the car-carrying capacity of the network as a whole. The safest advice for budding engineers may be, If you want determinacy, stick to something simple--like rockets or atomic bombs.
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