Phantom traffic jams

Why does a jam form and then clear on an open motorway with no crash or obstacle anywhere?

You crawl for four minutes, inch forward, stop again — and then, without warning, the road ahead is empty and you are back at seventy. You pass no crash, no breakdown, no lane closure, nothing. The jam had no cause you can point at.

The tempting conclusion is that the cause was there and you simply missed it, or that it cleared just before you arrived. Both are sometimes true. But there is a stranger possibility worth testing: that a jam can exist with nothing causing it at all, sustained only by the cars in it. What would have to be true of driving for that to happen?

Start with one car and one small event. Someone ahead of you eases off — a glance at a phone, a merging car, a slight rise in the road. Their speed drops by, say, five kilometres per hour. What do you do?

Two things, and both matter. You react late, because perceiving the change and moving your foot takes something like a second. And in that second the gap has closed further than you would like, so you do not simply match their five — you take off rather more, to rebuild the space you have lost. The driver behind you sees a bigger disturbance than you saw, reacts late to it in turn, and takes off more again.

Follow that down the line. Each car receives the disturbance slightly amplified and passes it on more amplified still. Twenty cars back, "five kilometres per hour slower" has grown into a full stop. Nobody did anything unusual; the growth is in the chain, not in any driver.

Now notice where the jam is. Each car begins braking before it reaches the spot where the car ahead began braking — it has to, because the gap is closing. So each successive driver's stopping point lies a little further back up the road, and the place where cars stop drifts steadily upstream, against the flow, at a speed commonly measured in the region of fifteen to twenty kilometres per hour. The cars go forwards; the jam goes backwards through them. That is precisely why you emerged with nothing to see: you did not reach the end of the jam, the jam passed over you and travelled away behind you toward the drivers who have yet to meet it.

But if amplification is inherent to driving, why is the motorway not permanently jammed? Because there is a second variable: spacing. With large gaps, you have room to absorb a five-kilometre-per-hour drop by coasting; the disturbance shrinks as it travels and dies out a few cars back. Squeeze the gaps and the same disturbance grows instead. There is a density threshold, and the same drivers behaving in exactly the same way produce smooth flow below it and stop-and-go waves above it. The instability is a property of the traffic, not of any individual in it.

That is a satisfying story, but is it more than a story? The decisive test removes every possible obstacle. In 2008 Sugiyama and colleagues put twenty-two cars on a circular track about 230 metres round and asked the drivers to hold a steady speed at even spacing. There is no junction on a circle, no lane drop, no incident, no lead car to blame. For a while the ring held. Then the spacing wobbled, the wobble grew, and within minutes a genuine stop-and-go wave was circulating backwards around the track while the cars went forwards. A jam had been produced out of nothing but drivers and density.

Three clarifications, one of which is a boundary marker.

First, the two ingredients are separable. Reaction delay alone would give you a wave that travels; asymmetric response — braking harder than the car ahead, then rebuilding speed slowly — is what makes it grow. Remove either and you get disturbances that fade. This is why smoothing behaviour helps so much out of proportion to the number of drivers doing it: experiments on ring tracks have shown that a single vehicle holding a deliberately steady, generous gap can damp the wave for everyone behind it.

Second, this is not the same phenomenon as induced traffic, which this collection treats separately. Induced traffic is about demand — more road capacity eventually calling forth more trips. Phantom jams are about flow at a fixed demand: the same cars, the same road, the same hour, arranged either stably or unstably depending on how closely packed they are. One is an economic response over months; the other is a physical instability over seconds. They are often mentioned together and they are not the same claim.

Third, "no cause" is loose talk. There is always a triggering perturbation — a lane change, a gradient, a driver reaching for a coffee. What is missing is a sustaining cause. The trigger is trivial and long gone; what you sat in was the amplified descendant of it, kept alive by density.

• What sets the critical density, and why weather and lane discipline shift it.
• Why variable speed limits upstream of a wave can dissolve it, when slowing traffic down seems like the wrong move.

Every term the collection defines is gathered in the glossary.