### Sync or Swim

In earlier blog posts, I’ve discussed how unconnected oscillators can appear to link up, creating patterns and giving rise to phenomena such as lasers and the spectra of stars. We’ve come a long way from masses on strings: we can understand oscillators as any system that repeats in some manner. This post will look at a particular case of what happens if the oscillators are actually connected, instead of acting independently.

Let’s start by thinking about a peculiar oscillator: a person running on a racetrack at a steady speed. The period of motion for our runner Oscar Later (OK, that joke needs work) is the amount of time he takes to complete one lap. His phase is his position on the track, relative to an imaginary circle like a clock face. Note that if the track isn’t circular, Oscar’s phase won’t change at a constant rate, even if he’s running at a constant speed, but that’s OK: the main thing for now is that he’s repeating his motion in a predictable way.

Two runners (Oscar and Brenda) on a track. They start off together (top frame), but Brenda is a little faster, so she gets ahead (center frame). However, since they are friends and want to run together, she then slows down a bit while Oscar speeds up, so their phases coincide (bottom frame).

If Oscar’s friend Brenda Lum (don’t worry, that’s the last one) runs on the same track when Oscar isn’t there, they are independent oscillators, even if they run at the same pace. However, if they run at the same time, they cease being independent: even if they don’t try to match speeds, each pays attention to the other’s position and how fast the other is running. If they try to run together, then they truly are not independent: if Oscar starts to get behind Brenda, he’ll speed up, while Brenda is likely to slow down a bit to compensate. Even though there is nothing directly connecting them — no bungee cord or anything — their individual natural paces are adjusted to match each other. The end result is that they are running in sync, and if they run side-by-side, their phases will be exactly the same.

Now add more (mercifully nameless) runners to the track. The more runners, the greater the variety of natural speeds, but if they’re running for exercise and not competitively, they might still try to keep together in a pack, achieving some semblance of synchrony. To measure how “together” the pack is, you could take a series of snapshots and see how many people are in each clump (having roughly the same phase) at each time. The fancy physics term for this is a correlation function: if all the people are in the same clump, then the correlation is 1, while if there are no clumps at all, the correlation is 0. For a large enough group of runners, it probably won’t ever be exactly 1 or 0, but if it is stays closer to 1 than to 0 over time, then we say the runners are in phase and synchronized.

The figures show a group of 6 runners as red dots on a circular track. The top sequence of three is the case where the runners are correlated: they stick together in a tight group the whole time. The bottom sequence of three is the case where the runners are uncorrelated: they start off together, but because they aren't linked, they get more spread out over time.

Of course runners, while a realistic example, aren’t particularly illustrative of broader scientific concepts. However, several species of firefly in southeast Asia, eastern Tennessee, and parts of Africa group together in close proximity; the individual fireflies adjust their rate of flashing until they all flash on and off at the same time. They achieve something very close to perfect synchrony! Nothing directly connects them, just as nothing directly connected the runners in the previous example, yet they interact with each other until they match not only rate of flashing, but the particular moment of flashing: they are in phase with one another.

This sequence of three shows the case where the interaction is strong between the "runners": they start off uncorrelated, but the attraction between them is strong, so they quickly group together in one bunch. They will stay clumped up as long as the interaction strength remains high.

Fireflies of this type obviously have a built-in desire to synchronize, but let’s go back to our original two runners for the moment. Let’s say Oscar and Brenda aren’t necessarily friends; they are running on the track, but they might have a variety of responses to each other. If Brenda notices that Oscar has a sexy frequency of oscillation, and Oscar thinks he wants to correlate with Brenda over a cup of coffee, they will quickly adjust their strides until they match both speed and phase. This case represents strong interaction: no matter how far apart their speeds or phases to begin with, they synchronize quickly. Perhaps the attraction isn’t quite so strong, but they still think the other’s phase is worth checking out, they might adjust slightly without ever fully linking up; this would be a weaker correlation. If they don’t particularly notice each other at all, then the interaction is zero: they will individually keep their own pace, and never achieve synchronization, except maybe by momentary accident. (We can also consider revulsion, which itself is a type of correlation: they will match pace of running, but opposite phases to keep as far apart as possible! Another trickier possibility is if (say) Brenda digs Oscar, but Oscar doesn’t reciprocate; this is an asymmetric interaction, and can have lots of variations. I’ll skip over any detailed description of these cases for this post, and focus on positive interaction.)

A real physical system might be analogous to the “multiple runner” case, with a variety of possible interactions between the “runners”. A magnetically-susceptible material (like iron) under ordinary circumstances is not a magnet; inside the material, you have a bunch of electrons whose spinning is uncorrelated. However, if you bring a magnet close to the material, the spins will begin to swing around and produce an internal magnetic field aligned with the magnet, so the iron will stick to the magnet. This is like the runners ignoring each other and keeping their own pace, then suddenly forming a pack when their trainer gives the order. You can also produce a magnet spontaneously in some materials by lowering its temperature; this is like slowing all the runners down to make it easier for them to fall into step. Either way, what is happening is that the effective strength of the interaction between the runners or the electrons goes up, spontaneously changing them from uncorrelated to correlated; for fireflies, it’s like bringing them close enough together so they can see or sense the others’ rhythm. (I should note that electron spins are not oscillators, strictly speaking, but they behave in a similar way when it comes to spontaneous correlations of the type I’m talking about. For the hard-core physicists in the audience, I’m describing the Heisenberg model of magnetism, with the “phase” angle representing spin orientation.)

In fact, a vast number of systems in physics and biology can be treated as interacting oscillators, with various types of behaviors arising based on the strength of coupling (or revulsion). If cell growth and death cycles get out of whack, cancer can be the result; the ordinary interaction is non-functional. Many other systems in the body have their own natural frequencies, and must synchronize to work properly for good health. I admit I don’t know much biology or physiology, so I won’t go into any more detail here, but there is a huge literature on the subject of networks of biochemical oscillators in the body. In physics, besides how magnets work (to answer the Insane Clown Posse’s erudite question), interacting oscillators play a role in superconducting circuits, and many other applications.

Brenda and Oscar are busy little people.

(For more information about synchronization of the type I describe in this post, I highly recommend the book Sync by Steven Strogatz, or if you’re strapped for time, Strogatz’ TED lecture.)