The Secret of Synchronization

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The second law of thermodynamics tells us that everything in the universe tends towards disorder. And in complex systems, chaos is the norm. So you’d naturally expect the universe to be messy. And yet, we can observe occasions of spontaneous order, the synchronization of metronomes, the perfectly timed orbits of moons, the simultaneous flashes of fireflies, and even the regular beating of your heart. What puts these things in order in spite of nature’s tendency for disorder.

On June 10 2000, the Millennium Bridge, a new footbridge Across the River Thames in London was opened to much excitement. But as crowds filled the bridge, it began to wobble back and forth. Police started restricting access to the bridge, but that only resulted in long lines to get on the wobble was unaffected. Two days later, the bridge which had cost 18 million pounds, was fully closed, and it wouldn’t reopen for another two years.

So what went wrong? Well, it’s long been known that armies should break step when crossing bridges. This dates back to an accident in 1831. When 74 men from the 60 of Rifle Corps were marching across the Broughton suspension bridge in northern England. It collapsed under their synchronized footsteps.

60 men fell into the river 20 of whom suffered injuries like broken bones or concussions. Luckily, no one was killed. But after this, the British Army ordered all troops to break step when crossing bridges. Now look at the people walking across the Millennium Bridge. Most of them are walking in step with each other.

But they are not part of an army. They’re random members of the public so why are they walking together? And why couldn’t a modern bridge designed for heavy pedestrian traffic handle this? Well, to understand it, we have to go back 350 years. In 1656, famous Dutch physicist Christian Huygens created the first working pendulum clock.

The goal was to help sailors figure out where they were on the globe. Latitude can be judged by measuring the position of the sun or stars. But for longitude you also need to know the time at some fixed location, say your home port. But clocks at the time were routinely out by around 15 minutes a day. So they were effectively useless.

wagons pendulum clocks by contrast, were accurate to around 10 to 15 seconds per day.Huygensplan was to attach his clocks to a heavy hanging mass on the ship, so they wouldn’t get tossed around by the rolling seas. His plan called for two clocks in case one stopped or was damaged. But testing out this arrangement while at home sick in February 1665. He made a remarkable discovery.

To have his clocks hung from a wood beam across some chairs.

Watching the pendulums sway back and forth for hours. He noticed after half an hour or so they would spontaneously synchronize. As one clock swung one way, the second would swing the other way. As one would tick, the other would talk.

So he tried disturbing the clocks. He set them ticking out of sync but again, within 30 minutes or so they were back to the same lockstep. Wiggins thought this strange sympathy of clocks must have been caused by air currents between the pendulums, so he placed a large board in between them, but their clocks continued to sync up. It wasn’t the air currents. When he separated the clocks, the synchrony would disappear, their times drifting apart.

But when he brought them back together, the synchrony returned. Wiggins realized the two clocks were synchronizing because they were hung from the same wood beam. He transferred mechanical vibrations from one clock to the other, making the two oscillators coupled. wegens was the first to observe this kind of spontaneous synchronization in inanimate objects.

And although he qualitatively described what was happening, he was only a few decades ago that scientists started fleshing out a rigorous theory of synchronization.

You may have seen this demo where you put several metronomes on a light wobbly platform and start them out of sync. It’s trickier than people make it look. When you do get it to work, though. It’s kind of magical.

These metronomes don’t have exactly the same natural frequency, and yet they still beat in time.

To understand how this works, it’s easiest to first consider a couple metronomes oscillating in sync with each other. When the large masses accelerate to the left, they push the platform to the right. And when they accelerate to the right, they push the platform to the left. So the center of mass of the system always stays roughly in the same spot. Now, if you start another metronome completely out of sync with the first two, the motion of the platform gives it a kick every half swing, speeding it up until it’s in time with the first two.

This works, regardless of the number of metronomes, you have, the platform just goes whichever way the majority of metronomes are pushing it.

We can represent the position of a metronome pendulum or any other oscillator as a point on a circle. This shows its phase that is what part of the cycle it’s in. So you could call the rightmost point of the pendulum zero degrees. And then the leftmost point is 180 degrees.

And as the pendulum oscillates back and forth, the point goes around the circle, the higher the frequency of the oscillator, the faster that point goes around. So this represents two metronomes with different frequencies. And this represents two metronomes with the same frequency, but completely out of phase. When the metronomes are synchronized in phase, their dots go around the circle together, we can use this depiction to illustrate a mathematical model for the synchronizing behavior we’ve been looking at. It’s called the Kuramoto model.

It says the rate each dot goes around the circle equals its natural frequency, plus some amount related to how far it is from all the other dots.

And the size of this term is determined by the coupling strength. I like to think of it actually visually by thinking about people that are running around a track, like suppose you’re running with your friend, and maybe your friend is faster than you, your friend says, you know, come on, move it or hurry it up. Because you’re dawdling, you’re slow, you’re falling behind. So if you have enough fortitude, and you know, you try hard enough, and if the friend is sympathetic enough to slow down, then the coupling between you is strong enough to overcome that inherent difference in your natural running speeds.

But if you’re not very good friends, or, you know, if you can’t quite suck it up to move yourself faster, then the coupling will not be strong enough to overcome that difference in one person will start lapping the other, the fireflies of Southeast Asia are apparently good enough friends, because they synchronize their flashes. Even though each one has its own particular frequency at which it likes to flash, they coupled to each other strongly enough so that hundreds, even 1000s can flash together in the same split second. There’s a great simulation of this by Nicky Case, you start with individual fireflies just doing their thing. And then you can turn on the interaction between them. Now in the Kuramoto model, this would mean every Firefly has an effect on every other one.

But in this simulation, a firefly is only affected by its neighbors. If it sees a flash close by it nudges its internal clock forward a little bit, so it’ll flash sooner than it would have otherwise.

Now, what’s remarkable about this is even though the interactions are small and close range, over time, you can see waves traveling through all the fireflies, and eventually, they’re all flashing at once. Like you might think if you increase the coupling, you just sort of gradually get a system more and more synchronized. That’s not what happens.

It’s sort of like the way water doesn’t gradually freeze as you lower the temperature, its water, water water as you’re lowering the temperature. And then at a critical temperature, the molecules suddenly start to change their state and become solid instead of liquid.

And this is a sort of time rather than space version of the same thing. They sort of lock their phases in time, once you pass a critical level of coupling. And at that point, the sort of crystallization in time is the phenomenon that we call synchronization.

This is an audience in Budapest applauding after a performance. But what happens next is completely spontaneous. They’re not being instructed by anyone to see if you can spot the phase transition. this phenomenon of synchronization that we’ve been talking about one of the things that i find most appealing about it is how universal it is that it occurs at every scale of nature from subatomic to cosmic it uses every communication channel that nature has ever devised from gravitational interactions electrical interactions chemical mechanical i mean you name it anyway the two things can influence each other nature uses that to get things in sync take our own moon for example we only ever see one side of it because it rotates on its axis exactly once for every time it goes around the earth we say it is tidally locked to the earth and this is a common effect in our solar system there are 34 moons that are tidally locked to their planet the way this happens goes something like this a moon starts out with its own rotational frequency but the gravitational attraction to the planet is stronger on the side closer to the planet and so it distorts the moon into an egg shape which is greatly exaggerated here as the moon continues to orbit and rotate on its axis those bulges swing out of alignment with the planet and so the gravitational force on them is constantly pulling them back into alignment and this slows the rotation of the moon until it is locked to the planet if the moon is initially rotating too slowly this same mechanism can speed it up until it’s locked there are all kinds of other beautiful synchronization phenomena in our solar system the three innermost moons of jupiter io europa and ganymede are not only tidally locked to the planet they’re also in a one to four orbital resonance with each other for every time ganymede goes around jupiter europa goes around twice and io four times in the 1950s some russian chemists went looking for a chemical reaction that would oscillate like a chemical analog of a pendulum like could you get something going back and forth say between blue and orange over and over again and naively you might say that’s impossible because there’s principles of thermodynamics which say that closed systems just increase their entropy over time that they’re just going to come to equilibrium but there’s no principle in chemistry or thermodynamics that says you have to go monotonically to equilibrium you are allowed to oscillate and damp out to equilibrium in an facilitatory way this is exactly what Boris Belousov off and later Anatol Zhabotinsky discovered so this reaction is known as the Belousov Zhabotinsky or BZ reaction i’ve sped it up because it can continue for half an hour or more oscillating between these colors now it spends more time on the burnt orange color so i sped up those sections more it’s very spectacular and it’s kind of shocking to see a chemical reaction doing these periodic changes in color like chemicals acting like a clock like a pendulum so the stirred reaction has the advantage that you you really get a sense of the collectivity of you know i don’t know quadrillions of molecules avogadro’s number of molecules all doing the same thing at the same time on the other hand if you don’t stir if you just put like a petri dish of the bz reaction you can see something even more amazing i think which is that you can see spiral waves of color or target patterns expanding circles of color moving through the liquid maybe i should emphasize the liquid itself is not moving it’s not like we’re seeing ripples on a pond but what’s not still is chemical concentrations you can see these blue waves in the bz reaction that are chemical waves not not water waves and that will just propagate and they move at a constant speed and or they can look like a spiral that just grows and grows and spins around and what’s really spooky and uncanny about this is that the same phenomenon is seen in the heart you can see spiral waves of electrical excitation in a heart that look exactly like the spiral waves in chemical oscillations and chemical waves in the bz reaction and this was the sort of thing that inspired my mentor a guy named art winfrey who used chemical reaction waves to give himself insight into cardiac arrhythmias you know you may have heard the most deadly kind of a arrhythmia that the kind that will kill you it really in a matter of minutes ventricular arrhythmias ventricular fibrillation in particular winfrey’s work seeing these rotating spirals on hearts as well as in chemistry led him to a theory about what’s really causing ventricular fibrillation and how could we design for example better defibrillators That are gentler, that could be a good outcome of this theory.

You know, the lack of synchronization in a fibrillation heart is what causes no blood to be pumped, and then sudden death ensues.

So too little synchronization is obviously a problem. But too much synchronization can also cause trouble. Remember, the wobbly Millennium Bridge, it was all apparently down to something called crowd synchrony Was it the people walking in step that caused it to oscillate? Actually, kind of the opposite. The Millennium Bridge was designed to look like a ribbon of light.

So its construction is unique. Unlike a typical suspension bridge, its supporting cables run alongside it stretched taut, like guitar strings. In the civil engineering literature, all designers know that you do not build a footbridge with a resonant frequency equal to the frequency of human walking.

So we take about two strides per second one with your left foot one with your right foot. So everybody who takes civil engineering knows if people are going to walk on the bridge, It better not have a resonant frequency in the vertical direction of two hertz.

Okay, everybody knows that, including the people who, who built the Millennium Bridge. But what they didn’t know. And what was new that day is that half the frequency is also important, a frequency of one cycle a second, which is the frequency with which you put down, say, your left foot, half the time you’re doing your left foot.

So why does that matter? Because when you’re walking across a bridge, and you put your left foot down, you put a tiny force sideways on the bridge.

And normally, that wouldn’t matter because people are all walking at their own pace, they’re not synchronized, so their sideways forces which are only about a 10th, as big as their downward forces that they impart on the bridge, that would be negligible, and it wouldn’t do anything to the bridge. But if the bridge happens to have a sideways frequency of one cycle a second, which the Millennium Bridge had happened did, then people can actually start to get the bridge moving a little bit.

After the bridge was closed, engineers got their colleagues to walk across it in increasing numbers, while they measured its acceleration. With 50 people on the bridge, there was very little motion. At 100, the vibrations had barely increased at 156, there was still no wobble, but with just 10 more people 166 the acceleration grew dramatically.

The bridge suede, just like it had on opening day, the system had undergone a phase transition. If people can get the bridge moving a little. It turns out, people don’t like to walk on a platform that’s moving a little bit sideways. If you’ve ever been in a train, that’s kind of going faster, if you stand up in a rowboat, and it starts moving sideways, people spread their legs apart, to try to stabilize themselves.

And they will actually start to walk in step.

With the sideways motion of the bridge, you can see footage from the BBC, of people doing that. It’s spectacular and crazy. So it wasn’t people walking in sync that got the bridge to wobble. It was the wobbling bridge that got people to walk in sync. And so as the people got in step with the motion of the bridge, by adopting this weird kind of penguin gate, they ended up inadvertently pumping more energy into the bridge and making its motion worse.

And so this was this positive feedback loop between the motion of the crowd, causing the bridge to move more, which causes more people to get in step with the bridge, which made more people you know, drive the bridge. Once the problem was identified, they could solve it by decreasing the coupling strength.

They installed energy dissipating dampers all along the bridge. It was a tremendous embarrassment, and it cost several million pounds to repair the bridge. In science, we do reductionism, all of our science courses tell us the way to solve a problem is to break it into smaller parts and analyze the parts.

And that has been phenomenally successful for every branch of science.

But the great frontier in science today is what happens when you try to go back to put the parts together to understand the whole. That’s the field of complex systems. That’s why we don’t understand the immune system very well. We don’t understand consciousness very well or the economy.

It seems like the whole is more than the sum of the parts. That’s the cliche that has entranced me for my whole research career. I want to understand how can you figure out the properties of the whole given the properties of the parts? Hey, this video is sponsored by Kiwico, which makes awesome hands on projects for kids.

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