Simulation four of five

The jam

On a circular road with no junction, no traffic light, no accident and no bottleneck, touch one brake pedal for a moment and a standing jam appears, travels backwards through the traffic, and is still there long after the driver who braked has gone round again.

Most explanations of a traffic jam are hunts for a cause: an accident, a bad merge, a narrowing up ahead. Sometimes there is one and sometimes there is not, and the jams with no cause at all are the ones worth understanding, because they are the ones the causal habit cannot see.

The road is a loop with nothing on it but cars, each obeying one rule: aim for the speed the gap in front can support, and change speed towards that aim at the rate your reaction allows, with no aggression and no distraction anywhere in the model.

01The road and the trace

The right panel is the same road with time running downwards, so a jam shows as a dark band and the slope of that band is the speed of the jam itself.

Figure one run it, then tap one brake

The loop from above

The same road with time running downwards

02The rule, written out

A driver has a target speed set by the gap ahead and a rate at which it moves 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 you set, six metres is a car plus a little, and one point two seconds is the headway a driver keeps.

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 already at the cap does not respond to the gap at all, so nothing can propagate and nothing can grow: that is free flow, 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, so a disturbance now has something to travel through. Each driver amplifies what it receives by a factor that grows with its reaction time, and when that factor passes one a disturbance that arrives small leaves larger, and 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. Raise the density one step at a time, tapping the brake each time, until it does not: the line is fairly sharp, and either side of it the same tap on the same brake produces two completely different futures.

Now hold the density where the jam persists and lower the reaction time instead: attentive drivers absorb the same disturbance that inattentive ones amplify, with nothing about the road or the number of cars changed.

Cars travel forwards through a jam the whole time: one reaches the back, crawls through and accelerates away at the front, while the jam itself walks in the opposite direction. The jam is made of no 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 with no representation anywhere inside any agent: no driver knows there is a jam, and none is doing anything but keeping a reasonable distance. The jam is real, stable, moving at a definite speed and measurable, and it exists at a level of description none of the agents that constitute it can reach.

The programme's interest here is that an outcome can be produced entirely by the interaction of ordinary local rules, which makes looking for a cause among the agents looking in the wrong place. When the collective object is a jam that 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, junctions, entries, exits or other roads; the loop keeps the number of cars 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, headway and target speed, while real variation between drivers would change the numbers without removing the effect.
A crude ruleThe target speed is a straight line in the gap with a cap on it, while real car following is more complicated and the literature has many rules. The instability shown here is a general property of delayed following, though the precise threshold is an artefact of this particular line.
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, the only thing in the model outside the formula above: no car may come closer to the one ahead than the six metres at which the rule already puts its target speed at zero.
No noiseA 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 need nobody 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, a reading of this model rather than 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, 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 has been done with real cars and real drivers, rather than claimed from a simulation.

Vehicles were driven around a circuit, evenly spaced and moving uniformly with no bottleneck present, and 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 at 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.