Thought Toys · Emergence · Exhibit 01

Phantom traffic jams

No crash. No bottleneck. No bad driver. Just a loop of cars, each following one simple rule — and a traffic jam appears out of nothing and crawls backward forever. Slow the drivers down to summon it; speed them up to dissolve it.

0.00Avg speed
—Flow

A ring of cars, each following one rule; below a critical reaction speed the smooth flow breaks into jams that travel backward.

position around the loop →time ↓
stopped crawling free flow

What you're seeing

Every car obeys a single instruction: ease onto the gas when the gap ahead opens up, ease off when it closes. That's the whole brain of every driver here — no phones, no rubbernecking, no merging trucks.

Yet when their reactions are a touch too slow, a tiny wobble doesn't smooth out — it amplifies. One driver brakes a hair too hard, the next brakes harder, and a wall of stopped cars assembles itself and rolls backward through the traffic while every car keeps trying to go forward. The cars flow through the jam; the jam never leaves. Nudge "reaction speed" to the right and watch the same crowd suddenly glide. The space-time strip records the history: those backward-leaning red stripes are the ghosts.

This is why a highway can stop dead and then, a mile later, open up with nothing to show for it. The jam had no cause. It was just traffic being traffic.

The rule, exactly. Each car steers toward a desired speed set only by its gap b: V(b) = tanh(b−2) + tanh(2), accelerating toward it at a rate you control with the reaction slider. Below a critical reaction speed the uniform flow becomes unstable and stop-and-go waves emerge on their own. (The Bando "optimal velocity" model, 1995.) Counter-example, verified in node: above the critical reaction (a=2) the uniform flow is stable — a fully stopped car's disturbance decays (speed spread → 0) and no jam forms.

Also in Emergence: Schelling's Segregation →

All 14 in Emergence
  1. 01Phantom Traffic Jams — you are here
  2. 02Schelling's Segregation
  3. 07Conway's Game of Life
  4. 11Percolation
  5. 15Reaction & diffusion
  6. 18Sandpiles & self-organized criticality
  7. 22Diffusion-limited aggregation
  8. 29The simplest rule that makes complexity
  9. 32Fireflies that sync
  10. 41Small-world networks
  11. 43The Ising model
  12. 81Nobody panicked, and the factory went haywire
  13. 84The tragedy of the shared field
  14. 88Nobody's leading. The flock turns as one.

Thought Toys — a cabinet of explorable explanations. Exhibit 01.