The jam
A circular road with no junction, no traffic light, no accident and no bottleneck of any kind. Every driver follows the same undramatic rule. Touch one brake pedal for a moment and a standing jam appears and starts travelling backwards through the traffic, and it will still be there long after the driver who braked has gone round again.
Most explanations of a traffic jam are hunts for a cause. There must have been an accident. Someone must have merged badly. The road must narrow up ahead. Often there is such a thing and often there is not, and the jams that have no cause at all are the ones worth understanding, because they are the ones the causal habit cannot see.
The road below is a loop. Nothing is on it except cars, and every car obeys one rule: aim for the speed that the gap in front of you can support, and change your speed towards that aim at the rate your reaction allows. There is no aggression in the model and no distraction. Every driver is equally competent and equally patient.
01The road and the trace
On the left is the road seen from above, with the cars on it shaded by speed: pale is moving freely, dark is barely moving. On the right is the same road with time added. Position runs left to right around the loop, time runs downwards, and the shade is the local speed. Free flow is a pale field. A jam is a dark band, and the slope of that band is how fast the jam itself is moving.
The loop from above
The same road with time running downwards
02The rule, written out
There is only one, and it has two parts. A driver has a target speed set by the gap ahead, and a rate at which it can move its actual speed towards that target. The reaction time does both jobs: it is how stale the gap you are responding to is, and it is how long you take to respond to it.
target speed for a gap g: V(g) = ( g − 6 metres ) / 1.2 seconds, capped at 30 metres per second actual speed: dv/dt = ( V( gap as it was τ seconds ago ) − v ) / τ τ is the reaction time the reader sets. Six metres is a car plus a little, and one point two seconds is the headway a driver keeps. Nothing else is in the model.
When the road is empty the gap is large, the target speed is at its cap, and the derivative of the target with respect to the gap is zero. A driver who is already at the cap does not respond to the gap at all, so nothing can propagate and nothing can grow. That is free flow, and it is stable for any reaction time whatsoever.
Add cars. Below a certain gap every driver is on the sloping part of the rule, where a change in the gap ahead changes the target speed, and now a disturbance has something to travel through. Each driver amplifies what it receives by a factor that grows with its reaction time. When that factor passes one, a disturbance that arrives small leaves larger, and it comes back round the loop larger still.
03Finding the edge
Set the density low and the reaction time high, and tap a brake. The disturbance spreads backwards a little, thins out and dies. Now raise the density one step at a time, tapping the brake each time, until it does not die. There is a fairly sharp line, and either side of it the same tap on the same brake produces two completely different futures.
Then do it the other way. Hold the density where the jam persists and lower the reaction time instead. Attentive drivers absorb the same disturbance that inattentive ones amplify. Nothing about the road changed and nothing about how many cars are on it changed.
Once a jam has formed, watch the trace on the right rather than the road on the left. The dark band leans backwards. Cars are travelling forwards through it the whole time: a car reaches the back of the jam, crawls through, and accelerates away at the front, while the jam itself walks in the opposite direction. The jam is not made of any particular cars, and it outlives every car's membership of it.
04Why this belongs to SIMEA
Because it is the cleanest case I know of a collective object that has no representation anywhere inside any agent. No driver on the road knows there is a jam. No driver is doing anything other than keeping a reasonable distance. The jam is real, it is stable, it moves at a definite speed and it can be measured, and it exists at a level of description that none of the agents that constitute it has access to.
The programme's interest in this is not traffic. It is the reminder that an outcome can be produced entirely by the interaction of ordinary local rules, so that looking for a cause among the agents is looking in the wrong place. When the collective object is a jam this is a curiosity. When it is a panic, a run on a bank, or a population's opinion, the same mistake is expensive.
05Where the model stops
Assumptions and limits
- One lane, one loopNo overtaking, no junctions, no entries, no exits, no other roads. The loop is there so that the number of cars stays fixed, which is what makes the experiment clean and also what makes it unlike any road you have driven on.
- Identical driversEvery driver has the same reaction time, the same headway and the same target speed. Real variation between drivers matters and would change the numbers, though it does not remove the effect.
- A crude ruleThe target speed is a straight line in the gap with a cap on it. Real car following is more complicated and the literature has many rules. The instability shown here is a general property of delayed following and not an artefact of this particular line, but the precise threshold is an artefact of it.
- One rule added by handAt a long reaction time the following rule on its own will let a car drive through the car in front, because the gap it is responding to is out of date. A hard constraint stops that: no car may come closer to the one ahead than the six metres at which the rule already puts its target speed at zero. It is a physical fact rather than a driving behaviour, and it is the only thing in the model that is not in the formula above.
- No noiseNothing is random. A perfectly even line of cars stays perfectly even for ever, which is why the brake has to be tapped by hand. Real traffic is never even, so real jams do not need anybody to start them.
- The jam speed is an estimateThe figure reported is fitted to the movement of the slowest point on the loop over the last stretch of the run. It is a reading of this model, not a measurement of anything on a road.
- Not calibratedThe six metres, the one point two seconds and the thirty metres per second are plausible round numbers chosen so the mechanism is visible. They are not measured values and no number on this page describes real traffic.
06Source
That a jam can appear on a circular road with no bottleneck at all is not a claim I am making from a simulation. It has been done with real cars and real drivers.
- Vehicles were driven around a circuit, starting evenly spaced and moving uniformly, with no bottleneck present. Below a critical density the uniform flow held. Above it the flow became unstable, a tiny fluctuation grew, and a jam cluster appeared and propagated backward like a solitary wave with the same speed as a jam cluster on a highway. The authors conclude that a bottleneck is only a trigger and not the essential origin of a traffic jam, and present the work as the first experimental evidence that a jam is a collective phenomenon. Yuki Sugiyama, Minoru Fukui, Macoto Kikuchi, Katsuya Hasebe, Akihiro Nakayama, Katsuhiro Nishinari, Shin-ichi Tadaki and Satoshi Yukawa, Traffic jams without bottlenecks: experimental evidence for the physical mechanism of the formation of a jam, New Journal of Physics 10, 2008, article 033001. iopscience.iop.org/article/10.1088/1367-2630/10/3/033001 Checked 2026 08 22.