## Phases and phase transitions

Try to tell somebody that you are studying the transitions of matter from one phase to another and they will immediately think of the way water turns into ice. Yes, this is a classical example, something that we have learned when we were little kids: ice and snow melt in your hands and turn into water and when water gets frozen, it becomes ice. Actually there are still a lot of questions about these processes that are not well understood, such as snowflakes patterns, the dependence of the freezing temperature on various factors, etc. However, the physics of phase transitions would be very boring if it was only limited to what happens to water.

Fortunately, there are many other substances in the world. Some are solids, some are liquids and some are gases, some are magnetic and some are not, some can easily conduct electrical current and some are insulating. One of the most interesting kind of phenomena is when a certain material suddenly qualitatively changes its properties when we only slightly vary the external conditions, such as temperature. Just like water turns into ice at 0^{o} C (32^{o} F), iron can become magnetic and mercury superconducting at their own *critical* temperatures.

However, the concept of phase transitions goes even beyond that. Every complex system has some unique properties that specify the current phase. And when the external conditions change smoothly, this system can suddenly begin to develop qualitatively new features. In fact, some of these complex systems have nothing to do with physics! One of the excellent examples is something that most of us are familiar with in our everyday lives.

## Traffic

Imagine a highway with lots of rapidly moving cars. What is the probability that a given car at given instant has exactly the speed at 51.918654 miles per hour? Of course, zero. It is hard even to measure the velocity with such precision. What is the probability that two cars are moving at exactly the same speed? Also zero, unless one car is towing another. In real life you can not make two numbers precisely equal and when we say "their speeds are equal," we mean that they are approximately the same, that is, they will become equal if you round off each number a bit.

But there is a special condition on the road when some of the cars have the speeds known exactly, with arbitrary precision. This special condition is also characterized by a remarkable property that some of the cars have precisely the same speed! Because this speed is *zero*. The cars are not moving at all! We are all familiar with such a state of highway traffic: it is called *traffic jam*.

Traffic jam does not mean that each car has permanently stopped. From the point of view of each individual driver, he moves for a while, then stops, then moves again, then stops again, and so on. From the point of view of the helicopter hovering over the highway, some of the cars are stopped at any given moment, and not just one car or two -- there is some nearly constant macroscopic *fraction* of all the cars on the highway (say, 10% or 30%), whose speed is zero.

This fraction is clearly a new measure characterizing the traffic flow. In the "normal" state of the highway this measure is zero. All the cars have different speeds (the system is *disordered*) and none have stopped (if one or two cars have stopped for some reason, this does not really count, because we are interested in macroscopic phenomena that affect all cars).

In the case of the traffic jam, this fraction is nonzero and each car on the highway will stop at some moment. Ironically, it is conventional to call such a state as *ordered* one, because the state when some cars are not moving is substantially less random than when all cars move with different velocities. This description agrees with our experience that traffic jam is a state of the flow on the highway that is qualitatively different from free traffic. Can physics tell us some nontrivial facts about it?

## Order and symmetry

In the physics of phase transitions, ordered phases are usually characterized by an *order parameter*, a quantity that is zero in the normal state and takes a nonzero value in the ordered one. It appears that the fraction of the stopped cars at the given moment has this property, indeed. However, it actually does not give a complete description. Namely, it does not reflect the fact that the formation of traffic jams is associated with breaking of a certain *symmetry*.

What is a symmetry? The symmetry is a transformation of the system that leaves it invariant. The most obvious symmetries appear in elementary geometry. After you rotate a circle about its center by any angle, you can superpose it with the original circle. If you reflect a square about any line of symmetry (there are four of them and they all pass through the center), it will match the original square.

The free traffic has a symmetry, too! It is nearly impossible to keep the speed of the car perfectly constant, therefore, each car either accelerates or slows down, at least slightly. If you videotape the highway and then replay the tape in the opposite direction (that is, effectively, reverse time), the accelerating cars will replace the decelerating ones and *vice versa*. However, even if you do not reverse time, such a replacement does not change the flow substantially. As long as each driver has a choice -- either to accelerate or to slow down -- he will help the flow to be as close to optimal one as possible, from any initial state of the highway -- as long as the traffic is free. In mathematics, such symmetry is denoted as _{ 2}.

The situation qualitatively changes when a driver absolutely **can not** accelerate and **must** slow down (or stay still, if he has already stopped). This may happen due to an accident ahead, due to the local jam, due to traffic signals or even because of another irresponsible driver. Of course, in practice, this means that the driver no longer controls the speed with the pressure on the gas pedal and moves his foot over the brakes pedal. The fact that the driver no longer has a choice, that it is impossible to cause him to accelerate even a tiny bit, means that the symmetry is *broken*.

Consequently, the state of the traffic jam has an order parameter that reflects the breaking of this symmetry. This order parameter is the number of the cars that can not accelerate (divided by the total number of the cars). Although the situations in which a driver has no choice but to slam on the breaks may occur spontaneously, the important thing is the average picture. If the dangerous situation has disappeared a few seconds later, then it has been merely a fluctuation. However, if the relative number of the cars that are unable to accelerate remains constant -- one car this moment, another car next moment -- then the order parameter is nonzero and the highway is in a separate, ordered state.

## Jam

The fact that we have identified the traffic jam as an ordered phase, a special state of the traffic flow with broken symmetry and an order parameter, is very important, because now we can proceed to the further analysis of its unique properties and of what happens when the jam just begins.

First of all, it is quite obvious that the transition of the flow into the jammed state can be triggered by such factors as the density of cars on the highway (as viewed from the helicopter) or the condition of the road, or even the slow reaction of the drivers. For low density of cars, the traffic is free. The more cars on highway, the more cars pass through it. We can define traffic *current* as the number of cars that passes by a certain location on the highway per second. This will be our measure of the traffic flow. Then for free traffic, when the average speed of the cars does not substantially depend on their number, this current is simply proportional to the density of cars.

What happens as the density increases? The studies show that at some moment the current saturates. The density of cars increases, but the traffic current does not. And then the current suddenly drops. That is right, there is almost discontinuous drop in current as the density increases slightly above the critical value! After that the dependence of the traffic current on the density of cars becomes reverse: as the density increases, the overall flow decreases.

This is, of course, due to the jam. Normally, each driver is acting in a way to keep traffic smooth. If somebody slows down a bit, others quickly adjust their speeds, some people change lanes, and as a result, the flow returns to the initial state. This means that when the traffic flow is "optimal," small fluctuations in the speeds of individual cars will rapidly decay.

However, the jammed cars can not adjust their speeds, because they can not accelerate. As a result, the fluctuations in the velocities never decay. Instead, they propagate as waves (*collective modes*). What the drivers typically see is that the cars ahead start to move, then accelerate, then slow down, and then stop. Each driver knows that he will follow precisely the same pattern when the wave reaches him. The fact that traffic can not restore its optimal state has catastrophic consequences for the flow, which leads to the abrupt decrease of the current mentioned above.

Maybe it is possible to drive without fluctuations? The answer is "No." As I have already mentioned, you can not maintain a perfectly constant speed even if you are alone on the highway. As a result, some fluctuations occur all the time.

## People and their will

At this moment somebody could interrupt me: "Wait a minute! People do not want to sit in traffic jams. People are actually doing their best to keep the flow smooth!"

That is exactly what happens... but it can still be described perfectly well in the presented picture!

Indeed, people fight the traffic jams as much as they can. When the density of cars approaches critical, the drivers begin to coordinate their efforts, which leads to a *synchronized* flow. The synchronized flow is still a free flow (that is, without jams), however, the average speed of the cars begins to decrease with the density so that the overall traffic current is nearly constant. Although all cars, strictly speaking, move with different speeds, their velocities become very close to each other. For example, the drivers try not to speed up when a gap opens in front of them, avoid changing lanes, and so on.

The net result of these coordinated efforts is that some of the fluctuations simply disappear, because people do not let them propagate. In order for this to happen, these fluctuations have to be characterized by a time scale that is less than human reaction time, that is, they have to be rather slow. However, near the point of the transition from free traffic into the traffic jam, the most "dangerous" fluctuations are precisely the slow ones. Essentially, the transition can not take place without them at all! Therefore, when the density of cars is near critical one, the suppression of slow fluctuations makes the transition into the jammed state impossible and the traffic remains synchronized.

As the density of cars continues to increase, the time scale for the "dangerous" fluctuations decreases. At some point people can no longer react fast enough to suppress them, the local jams emerge, which rapidly grow, and soon the entire highway slides into the jammed state. At the moment of the transition, the order parameter, which was introduced above, jumps from zero to a finite value, which is a signature of so called *first-order* phase transitions in physics.

I would like to emphasize that in the state of the synchronized flow, the exact density of cars, at which the jam develops, substantially depends on the drivers. The better is their reaction and the higher is their culture, the longer they are able to postpone the transition. A single irresponsible or inexperienced driver can ruin the efforts of the rest and can create the traffic jam.

## Summary

Thus, what happens with traffic when the number of cars on the highway increases?

Initially, the traffic is free and the number of cars on the highway does not substantially affect their average speed. Therefore the flow linearly increases with the density of cars. The drivers accelerate or slow down a bit in order to keep the flow close to the optimal one.

Then people begin to notice more and more dangerous fluctuations in the distance and the velocities of the cars ahead and around them. Thanks to their appropriate reaction, they can suppress the slow fluctuations, but the fast fluctuations survive and propagate as density waves in the opposite direction of the flow of traffic. The people begin to drive more slowly and to adjust their speeds close to each other, leading to a synchronized flow. The entire highway traffic balances at the edge of falling into the traffic jam.

However, as the density of cars further increases, the fast fluctuations begin to grow, the synchronized flow breaks down and the flow becomes jammed. Some of the cars can not accelerate and stop, while the rest can temporarily enjoy "free traffic." In fact, every car spends some time in the jam and at any given moment the number of stopped cars is macroscopically large and is almost constant. The fraction of the cars that are temporarily unable to accelerate is the order parameter, which is a quantitative measure of the traffic jam. The stopped cars often group together, in which case such local jams appear to propagate slowly at nearly constant speed, corresponding to the velocity of the collective modes.

Once the traffic is in the jammed state, increasing the density of cars will cause the order to enhance, too. There will be fewer and fewer cars able to move fast, which will decrease the traffic flow.

This is an example of how the modern vision of the physics of phase transitions allows one to understand conceptually how the traffic jams occur and what characterizes them. Of course, this post is only an overview, since the research on the traffic flow has been active for the past 20 years.

If you sit in a some gridlock this summer, you might come to the end of the line of cars and realize, hey, there's nothing there. No accident, no police on the shoulder, just a bunch of cars that aren't getting where they want to go. Over at the Universities of Exeter (in England), Bristol and Budapest, mathematicians now think they've figured out why this happens (and wastes lots of gasoline in the process).

The short answer: braking and full roads. When there are between 10 and 15 vehicles on a one-kilometer stretch of highway and the front one hits the brakes, a "backward travelling wave" is created that can sometimes lead to traffic jams. As Dr. Gábor Orosz of the University of Exeter's School of Engineering, Computing and Mathematics, said in a statement: "As many of us prepare to travel long distances to see family and friends over Christmas, we're likely to experience the frustration of getting stuck in a traffic jam that seems to have no cause. Our model shows that overreaction of a single driver can have enormous impact on the rest of the traffic, leading to massive delays." He continued: "When you tap your brake, the traffic may come to a full stand-still several miles behind you. It really matters how hard you brake - a slight braking from a driver who has identified a problem early will allow the traffic flow to remain smooth. Heavier braking, usually caused by a driver reacting late to a problem, can affect traffic flow for many miles."

This seems like a problem with no solution. Not braking could lead to accidents, which certainly don't make the highways easier to travel on. And removing cars from the road would be appreciated by many, until public transportation becomes a better option, it ain't gonna happen. So, if you get stuck on the way to or from grandmother's house this year, at least you now kind of know why.

The DOT may not do much (they never look like they're DOING anything), but at least I can say I'm having a productive day and put it in terms everyone can relate to, if not completely understand. Don't you just love math and science now?

Walter

## No comments:

Post a Comment