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 11 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

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