# Why traffic suddenly turns into a traffic jam

Original author: Benjamin Seibold
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One of the most obscure phenomena in a car trip is suddenly arising phantom traffic jams. Most of us have come across this: the car in front of you suddenly slows down, causing you to brake, which forces the driver to slow down behind you. But soon you and the cars around you will again accelerate to the original speed, and it becomes obvious that there are no visible obstacles on the road, nor noticeable reasons for the slowdown.

Since the movement quickly restores its original speed, phantom plugs usually do not cause serious delays. But they are not just insignificant annoying interference. These are foci of accidents, because they cause an unexpected brake. And the jerking movement to which they lead damages the car, reduces the resource and increases fuel consumption.

So what is going on? To answer this question, mathematicians, physicists, and transport engineers have developed many different kinds of traffic patterns. For example, microscopic models calculate the paths of individual cars and are well suited to describe the interaction of individual cars. Macroscopic models describe traffic in the form of liquid, and the machines in it are interpreted as particles of liquid. They are effective in the study of large-scale phenomena involving many 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 reasons for the formation of phantom traffic jams, we first need to learn about the many effects present in real traffic, which can probably contribute to traffic jams: different types of vehicles and drivers, unpredictable behavior, entry and exit from the highway , change of lanes, and so on. It can be assumed that some combination of these effects is necessary to create a phantom plug. One of the great benefits of learning mathematical models is that all of these different effects in theoretical analysis or computer simulation can be turned off. So we can create a group of identical predictable drivers traveling along a single-lane highway without any exits from it. In other words, the perfect road to home.

Surprisingly, when you turn off all these effects, phantom plugs still occur! This observation tells us that phantom congestion is not the fault of individual drivers, but the result of the collective behavior of all drivers on the road. It works like that. Imagine a uniform flow of transport: all cars are evenly distributed on the highway and drive at the same speed. In perfect conditions, such ideal traffic can last forever. However, in reality, the movement is constantly subjected to small fluctuations: imperfection of the asphalt surface, small problems with the engines, fractions of a second, to which the driver weakens his attention, and so on. To predict the evolution of such a traffic flow, an important question must be answered: are all these small fluctuations damped or amplified?

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

In macroscopic (hydrodynamic) models, each driver, interpreted as a fluid particle of a traffic stream, observes a local traffic density around itself at any moment of time and accordingly chooses the speed to be kept: high if there are few cars nearby or low when there is a lot of traffic jam. Then it accelerates or decelerates to this target speed. In addition, he suggests that traffic will do next. This forecasting motion effect is modeled by “traffic pressure”, which in many ways behaves similar to the pressure in a real fluid.

A 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 traffic pressure dampens the oscillations. The state of a homogeneous flow is stable if the prediction effect dominates, and this happens at a low flux density. The delay effect dominates at a high traffic density, which causes destabilization and ultimately phantom traffic jams.

The transition from a uniform flow to a flow in which jamiton dominates is similar to how water passes 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 delay effect when adjusting the speed. The most amazing 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 emergence of jamiton. The flow, flowing from left to right, leads to the spread of jamiton from right to left. The vertical axis indicates the density of cars on the road. A sharp transition from low to high density (and from high to low speed) is a characteristic of all jamitons.

Consequently, the occurrence of traffic waves (jamitons) can be explained by the behavior during a phase transition. But in order to understand how to prevent phantom traffic jams, you also need to understand the details of the structure of a fully established jamiton. In macroscopic traffic models, jamitons are a 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. Cars that collide with a shock wave are forced to brake sharply. After the strike, 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 point of view of drivers, there is a “point of the line of transition through the speed of sound”.

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

Therefore, the behavior of the machine must be influenced before it enters the jamiton. One way to achieve this 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 an event of sudden braking, immediately followed by acceleration, it can broadcast a “jamiton warning” to cars moving after it within one mile. The drivers of these cars can at least prepare for unexpected braking; or, which is also good, increase the interval to contribute to the scattering of the traffic wave.

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

Besides the fact that phantom plugs are an important example for mathematical study, they are probably also an interesting and visual social system. In places where jamitons originate, they are caused by the collective behavior of all drivers, and not by a few “black sheep”. Those who act in the lead can disperse the jamitons and help all the drivers following them. This is a classic example of the effectiveness of the golden rule of morality.

Therefore, the next time you find yourself in a gratuitous, meaningless and spontaneous traffic jam, then remember how much harder it seems.

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