Clocks falling into step
How a crowd of different clocks can end up beating together, with no leader.
What you are looking at
Thousands of dots each swing around a circle at their own speed. At first they are out of step. Slowly, more and more of them move together.
The idea
Think about a crowd clapping after a concert. At first everyone claps at their own speed. But each person hears the people near them and nudges their own clapping a little toward theirs. After a while, the whole hall is clapping in time.
No one is in charge. Each person only listens to the others. And it does not work if the nudges are too weak or if everyone's natural speeds are too different. Once the nudging is strong enough, the group suddenly snaps into step.
This is how many fireflies come to flash together, and how the cells that set your heartbeat keep the same time.
Where else you see it
- Fireflies in some places flash in unison.
- Pacemaker cells in the heart.
- Two pendulum clocks on the same wall slowly swing together (first noticed by Huygens in 1665).
Go deeper: the proper names and the math
This is the Kuramoto model: dθi/dt = ωi + (K/N) Σj sin(θj − θi). Each oscillator i has its own natural frequency ωi drawn from a spread of values, and K is the coupling strength.
The amount of synchrony is measured by the order parameter r eiψ = (1/N) Σ eiθj. For a symmetric single-peaked frequency distribution, r stays near zero until K reaches a critical value Kc = 2 / (π g(0)), and then it rises sharply: a phase transition, like water freezing, but for rhythms.