# Traffic & Phase Transitions

It’s 10pm on a Sunday night, and I’m driving west on Interstate 10 right through the middle of downtown Houston. Focused on getting to my destination safely, I obey the traffic laws and proceed through the comparatively deserted interstate at the maximum speed allowed by law. To experimentally do otherwise is a bit hazardous, so instead of trying this in reality I follow Einstein’s lead and imagine an experiment:

I jam on my brakes and swerve to the left across several lanes of traffic. I floor the gas and spin the wheel to the right, overshooting my starting point. Driving like a lunatic with few cars around will have predictable results: the cars within sight of me will avoid me, and I’ll probably eventually get sent to jail for reckless operation. But do I affect anyone other than me? So long as I don’t actually wreck, the most that happens is a small alteration in the course of nearby vehicles. Vehicles farther away won’t even notice. The correlation length – measuring how far the effects of my car’s motions are felt – is small, maybe a few hundred feet. The overall state of the traffic flow remains unaltered no matter what I do.

Now change the picture a bit. It’s now 4:30 pm on a Monday afternoon. Traffic is bumper to bumper and inching along when it’s not actually stopped. I’m not physically able to go swerving wildly around, but I can still try to figure out how far away my movements can affect traffic. In this case too the correlation length is still small. If I tailgate or hold back and try to change lanes or whatever, all that happens is a possible small response by the cars around me. Anything I do gets washed out in the random stop-and-go of all the tens of thousands of cars inching down the road.

Fig 1: Houston traffic, bad by even Houston standards.

From the perspective of my individual car not much is happening in the wide-open-spaces version of traffic or the traffic jam. They have their own statistical properties and characteristics, each interesting and important in their own way. A scientist studying the bulk properties of traffic will have plenty to study even in those cases, just like a scientist can make a career studying the properties of solids or gases. But the transitions between those phases are also certainly key, and as usual they’re harder to deal with. Science prises generalized properties from disparate physical situations all the time however, and transition of traffic flow patters is no exception. It follows some of the same rules as the phase transitions of bulk matter.

As the density of traffic changes from light to heavy, the correlation length between the motions of any one car and its neighbors increases. Very slowly at first, but as the density increases the length begins increasing more rapidly. I brake, and the car behind me brakes, and the car behind him brakes… and suddenly what I do ends up affecting traffic quite far away. Eventually the random motions of the traffic will dissipate my effects, but it takes a greater and greater distance. Suddenly, despite the fact that no obvious single event seems to have happened, the greater traffic density reaches a critical point and the correlation length reaches an “infinite” asymptote. The correlation length becomes infinite, tiny random motions propagate all throughout the traffic, and the traffic changes phase from thick and fast to thick and slow. And the correlation length collapses back down to something small as the individual motions of the slow cars don’t propagate outward very far.

This concept of correlation length becoming infinite is characteristic of almost all phase transitions, from cars to stadium crowds to superfluid helium. Nature is pretty weirdly universal like that.

## Comments

1. #1 NJ
May 26, 2009

I look forward to your exposition on the Landau theory of freeway traffic…

2. #2 Chris P
May 26, 2009

This is clearly not correct. Many times I have worked through a backup on the freeway for several miles only to find ONE vehicle at the front being driven slowly.

I can pretty much predict this now.

Chris P

3. #3 D. C. Sessions
May 26, 2009

The “asymptotic” analysis only applies if the response is proportional to the stimulus. Many traffic flows (like many physical flow systems) are bistable.

4. #4 Chris P
May 26, 2009

From comment #3 – from what I remember the maximum carrying capacity of a road occurs at about 55 mph. If you load beyond that the speed collapses.

5. #5 Bee
May 26, 2009

Hi Matt,

I thought you might find this post interesting: Experimental Traffic Jams. Best,

B.

6. #6 Michael F. Martin
May 26, 2009

This is a nice analogy also for explaining the difference between first and second order phase transitions. There’s no “latent heat of fusion”; the transition is continuous.

By Mermin-Wagner theorem, there can’t be goldstone bosons in a single lane of traffic. Could there be in multiple lanes?

Even more interesting is the implications this has for understanding transition to turbulence…

7. #7 Michael F. Martin
May 26, 2009

This is a nice analogy also for explaining the difference between first and second order phase transitions. There’s no “latent heat of fusion”; the transition is continuous.

By Mermin-Wagner theorem, there can’t be goldstone bosons in a single lane of traffic. Could there be in multiple lanes?

Even more interesting is the implications this has for understanding transition to turbulence…

8. #8 Matt Springer
May 26, 2009

Good point, #2. In fact the reason I picked a large city was so that we were dealing with a large road, with lots of lanes. That way we’re dealing with something closer to a 2-d situation rather than a 1-d chain, which reduces the blockage one car can cause.

9. #9 Nathan
May 26, 2009

I’ve often wondered how a single stationary vehicle on the side of the road can back up four lanes of traffic for miles, or how a single person driving the speed limit can similar delays. It’s amazing, really.

10. #10 Nathan
May 26, 2009

I’ve often wondered how a single stationary vehicle on the side of the road can back up four lanes of traffic for miles, or how a single person driving the speed limit can similar delays. It’s amazing, really.

11. #11 JDHuey
May 26, 2009

Several years ago, I read a newspaper article about a study on traffic patterns. The purpose of the study was to figure out what the underlying causes of traffic jams (other than the obvious causes such as road construction or accidents). The study used some very complex and sophisticated computer modeling. The basic conclusion was that when all traffic traveled in a well behaved fashion that traffic jams didn’t occur (they used the analogy to the laminar flow of a fluid) but when even one car exhibited even slightly irregular behavior a traffic jam would result (turbulence). (Often in the simulations, the offending car didn’t get caught up in the jam but merely left a jam in it’s wake.

12. #12 Eric Lund
May 26, 2009

From comment #3 – from what I remember the maximum carrying capacity of a road occurs at about 55 mph. If you load beyond that the speed collapses.

Safe following distance is (to lowest order) proportional to vehicle speed, so increasing speed leads asymptotically to a fixed capacity. There is a phase transition in the other direction, but in my experience it occurs at a lower speed. Here in New England there are many urban freeways with 50 MPH speed limits, and they flow adequately in moderately heavy traffic.

About a decade ago I had a chance to visit Munich with a small group of colleagues, and we had a rental car. I didn’t drive (Germans prefer stick shift, which I never learned to deal with), but as a passenger I got to observe the dynamics on a stretch of highway (the A9 autobahn north of Munich) with a variable speed limit. When there was very little traffic there would be no speed limit. In moderate traffic the speed limit would be 120 km/h, and as traffic increased the speed limit would be lowered in 20 km/h increments. The highway would still flow at 80 km/h (that’s 50 MPH for you metric-challenged Americans). But if traffic got heavy enough to warrant a 60 km/h limit, the limit would only apply briefly before traffic came to a dead stop.

What often happens is that traffic gets bunched up in waves (which is the result Bee reported in the blog post she linked to). People tend to go as fast as they can when they can go in a traffic jam, which means they catch up to the next backup. I’ve seen this phenomenon in cities on both coasts.

13. #13 Excited State
May 27, 2009

I’m learning more about correlation length (and, dare I say, stat mech in general) from these recent posts than I did from my grad-level stat mech course in the fall. Thanks.

14. #14 manilamac
May 27, 2009

Regarding #s 10 & 11, I trained tractor-trailer drivers for many years on California freeways and also spent many years driving big-rigs where the visibility of traffic patterns was exceptional. Across a wide spectrum of speeds, the wave is definitely the thing; small fluctuations, no matter how meaningless to actual traffic flow, propagate across great distances, even if such is not readily visible from regular passenger cars. I now drive in Asia, where traffic is much different, but one of the rules that seems to apply everywhere is “Try to act normal.” My truck-driving students were capable of doing some rather bizarre things on the public highways and I have seen their antics cause accidents even in the on-coming lanes of divided highways—drivers react very strangely when they witness vehicle behavior they do not understand, even if there is no way it could directly affect them. Stop-and-go waves are often set up by such reactions.

15. Good job! Im glad to read that

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