Why traffic suddenly turns into traffic jam

One of the most incomprehensible phenomena on a car trip is suddenly arising phantom traffic jams. Most of us faced this: the car in front of you suddenly slows down, forcing you to brake, which causes the driver to brake on you. But soon you and the cars around you are accelerating again to their original speed, and it becomes obvious that there are no visible obstacles on the road or any noticeable reasons for slowing down.

Since the movement quickly restores the original speed, phantom traffic jams usually do not cause serious delays. But they are not just insignificant annoying interference. These are the centers of accidents, because they make suddenly slow down. And the jerky movement to which they lead, harms the car, reduces resource and increases fuel consumption.

So what happens? To answer this question, mathematicians, physicists, and transport engineers have developed many different types of traffic patterns. For example, microscopic models calculate the paths of individual cars and are well suited for describing the interaction of individual cars. Macroscopic models describe traffic as a liquid, and machines in it are interpreted as fluid particles. They are effective in studying large-scale phenomena involving a variety of cars. Finally, cellular models divide the road into segments and prescribe the rules by which cars move from cell to cell, creating a structure for describing the uncertainty inherent in real road traffic.

In order to begin to understand the causes of phantom traffic jams, we first need to learn about the many effects present in real traffic that can probably contribute to traffic jams: different types of transport and drivers, unpredictable behavior, entry and exit from the highway , changing lanes, and so on. It can be assumed that some combination of these effects is necessary to create a phantom cork. One of the great advantages of studying mathematical models is that all these various effects can be turned off in a theoretical analysis or computer simulation. So we can create a group of identical predictable drivers traveling along a single-lane highway without any congresses from it. In other words, the perfect way to go home.
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Surprisingly, if you turn off all these effects phantom traffic jams still arise! This observation tells us that phantom traffic jams are not the fault of individual drivers, but the result of the collective behavior of all drivers on the road. It works like this. Imagine a uniform traffic flow: all cars are evenly distributed along the highway and travel at the same speed. In perfect conditions, such perfect traffic can last forever. However, in reality, the movement is constantly subject to slight fluctuations: the imperfection of asphalt pavement, minor problems with engines, a fraction of a second, to which the driver weakens attention, and so on. To predict the evolution of such a traffic flow, you need to answer an important question: do all these small fluctuations fade out or amplify?

If they fade out, the flow is stable and there are no traffic jams. But if they increase, the uniform flow becomes unstable, and small fluctuations grow into waves moving back, called “jamitons” (jamitons, from jam - traffic jam). Such waves can be observed in reality, they are noticeable in various types of models and computer simulations, and also were recreated in carefully controlled experiments.

In macroscopic (hydrodynamic) models, each driver, interpreted as a particle of transport flow fluid, observes the local traffic density around itself at any given time and appropriately selects the speed to follow: high if there are few cars nearby, or low if there is a large traffic jam. It then accelerates or decelerates to this target speed. In addition, he assumes that traffic will do next. This motion effect with prediction is modeled by “traffic pressure”, which behaves in many ways like pressure in a real fluid.

Mathematical analysis of traffic patterns shows that these two effects compete. The delay before reaching the desired speed leads to an increase in fluctuations, and the pressure of traffic dampens vibrations. The state of a uniform flow is stable if the prediction effect dominates, and this happens at a low flux density. The delay effect dominates at high traffic density, which causes destabilization and eventually phantom traffic jams.

The transition from a uniform stream to a stream dominated by a jamiton is similar to how water changes from a liquid to a gaseous state. In the flow of cars, this phase transition occurs when the flux density reaches a certain critical threshold at which drivers' expectations are balanced by the effect of a delay in speed regulation. The most surprising aspect of this phase transition is that the nature of the movement changes dramatically, although individual drivers do not change their behavior at all.


Video of the appearance of jamiton. The stream flowing from left to right leads to the spread of the jamiton from right to left. The vertical axis indicates the density of cars on the road. The abrupt transition from low to high density (and from high to low speed) is characteristic of all jamitons.

Consequently, the occurrence of traffic waves (jamitons) can be explained by the behavior of the phase transition. But in order to understand how to prevent phantom traffic jams, it is also necessary to understand the details of the structure of a fully established jamiton. In macroscopic traffic patterns, jamitons are the mathematical analogue of detonation waves that occur in the real world during explosions. All jamitons have a localized area of ​​high traffic density and low speed. The transition from high to low speed is extremely sharp - like a shock wave in a liquid. Machines colliding with a shock wave are forced to brake sharply. After the impact, there is a “reaction zone” in which drivers try to accelerate again to their original speed. Finally, at the end of the phantom plug, from the drivers' point of view, there is a “point of the transition line through the speed of sound”.

The name “point of the transition line through the speed of sound” (sonic point) arose from an analogy with detonation waves. In an explosion, this is the point at which a liquid turns from supersonic to subsonic. This has important implications for the flow of information in both the detonation wave and jamiton. The transition point creates an information boundary similar to the black hole event horizon: no information downstream can affect the jamiton on the other side of the transition point. Because of this, it is quite difficult to disperse the jamitons - after passing through the transition point the car cannot affect the jamiton.

Therefore, the behavior of the machine must be influenced before it gets into the jamiton. One way to achieve this goal is wireless communication between cars, and modern mathematical models allow us to develop suitable ways to use the technology of the future . For example, when a car detects a sudden braking event, immediately followed by acceleration, it can broadcast a “jamiton warning” to vehicles moving within one mile. Drivers of these cars can at least prepare for unexpected braking; or, which is also good, increase the interval to contribute to the dispersion of the traffic wave.

The results obtained by observing hydrodynamic models of traffic flows can help in solving many other problems of the real world. For example, logistics chains exhibit behavior similar to traffic jams. Phenomena of traffic jams, queues and waves can also be observed in gas pipelines, information networks and biological network flows - all of them can be considered as analogs of fluid flows.

Besides the fact that phantom traffic jams are an important example for mathematical study, they are probably also an interesting and visual social system. In the places of the origin of the jamitons, they are caused by the collective behavior of all drivers, and not by several “black sheep”. Those who act proactively can dispel the jamitons and help all drivers behind them. This is a classic example of the effectiveness of the golden rule of morality.

So the next time you get into a causeless, meaningless and spontaneous traffic jam, remember how difficult it is than it seems.

About the author: Benjamin Saybold is a professor of mathematics at Temple University.