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What is chaos theory? How can something be both deterministic and unpredictable? Is there are hope at all for physics? I discuss these questions and more in today’s Ask a Spaceman!
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EPISODE TRANSCRIPTION (AUTO-GENERATED)
If you listen to this show a lot and if you haven't and it's never too late to change, You've heard me talk about determinism. It's this magical ability in physics to predict the future. If you define the state, this is such a big word in physics, the state of the system, just a list of all the current properties of everything in a system, and that's another word I use all the time in the show is system. Like, a system is the thing you're looking at. The thing you wanna understand in physics is the system.
It could be a ball in a spring. It could be a ball in a box. It could be a ball floating around the universe. Whatever it is, it's a system and you wanna understand. And so a list of all the properties, all the positions of all the particles you're interested in, all the forces involved, energy states, momenta, you know, electric charges, just the state.
And you can even do this in quantum mechanics, but you have to change slightly the definition of state. If you have the state and you know the laws of physics, then you know what the states will be in the future. You can predict the future. This is determinism, and physics is deterministic. If you know the state and you know the laws of physics, you know what the future will be.
There's no question about it. This is all nice and neat and tidy and tied up with a neat little pretty bow. Physics is so awesome except when it isn't, and it isn't most of the time. But here's the crazy part. You can have systems.
You can have situations that are completely and totally deterministic. You can know the state. You can know the laws of physics. Determinism still rules, and yet they are unpredictable. Determinism doesn't mean, after all, that you can predict the future, which seems like a paradox.
Right? Because that's the whole point of determinism is to be able to predict the future. And now you're telling me that you have the ability to predict the future and you can't. What is wrong with that? Let me give you an example.
A pendulum. Right? Everyone loves a pendulum. If you start the pendulum in some position, you know the state of the pendulum and you know the laws of physics. Let's take away friction.
Let's take away air resistance. Any complicating things that just deal with gravity. You know how gravity will act on that pendulum. You can start it from somewhere and it will just go. And you can predict exactly into the future, however far you want, how that pendulum will behave.
Five seconds from now, five years from now, five centuries from now, you know exactly what that pendulum's gonna be doing because we live in a deterministic universe. Everything's cool. Laws of physics. We know the state. Boom.
And if you're to stop it and start it from the exact same position again, you'll get the exact same behavior. Determinism. Now make a double pendulum. That's a pendulum dangling off the end of another pendulum in case you're curious. You started from a position.
Now take away friction, take away air resistance, all that. We're just gonna talk about gravity. Just the two pendulums and gravity and you. That's all there is in the universe. Deterministic.
Right? You know the state. You've you've measured the position, the angles of the starting, like you're good. You know the laws of physics. You know gravity.
You know how gravity will act on those pendulums. You will have no idea how those pendulums will look at a given point in the future. You can track it for a little while. For a little while, you're like, okay. I know gravity.
I know the pendulums. You'll you can make a prediction of how the two pendulums will act in the future. Your prediction will be right for a little while, and then it will go off the rails and you'll have no idea what's going on. And you can say, okay. Okay.
Okay. This makes no sense. So you can stop the pendulum, and you can restart it from the exact same position that you did the first time and let it go, and you will have a completely different result. You'll have completely different behavior of that double pendulum. And you're, what?
This is confusing, so you stop it, restart it from the exact same position. You do it again, you get something new. You do it again, you get something new. You get do it again, you get something new. You cannot get the exact same behavior twice.
It's unpredictable in every sense of the word. You cannot predict what these double pendulums are going to do. What's going wrong? The single pendulum is subject to the laws of physics and it's deterministic and is perfectly predictable. Two pendulums connected together are still deterministic, but are perfectly unpredictable.
It appears totally random. It appears totally chaotic. One more concrete example? Fine. Here's a more concrete example.
Take the solar system, planets, moons, asteroids, gravity. Awesome. If you just have two objects, say, the sun and the Earth orbiting, you can exactly, perfectly, 100% predict what the Earth and the sun will do because of gravity. Deterministic. Predictable.
Add in a third body. Totally unpredictable. Can't do it. Back in the eighteen hundreds, there was a prize offered by the king of Sweden to anyone who could solve this so called three body problem, provide an exact solution to what happens with three gravitating objects. I mean, who does that anymore?
This is a prize from the king of Sweden to solve a physics problem. A French physicist mathematician, Henri Poincare, won the prize, but he didn't solve the problem. Instead, what he did, he wrote a paper, a series of papers outlining the problem, digging deep into it, and he identified a bunch of the same issues that I just talked about with the double pendulums, that there are no firm solutions. Like, you can't just write down an equation to describe how it will behave in the future. You can't make predictions, and that changes the starting points.
If you change the starting point ever so slightly, like, if you start the planets off by by, like, a micrometer, you're gonna get totally different answers. And so he just described this. He's like, well, I can't solve it, but I can talk about it. Classic physicist move. And the king's twins like, well, you know what?
Close enough. I guess. Someone's gotta get the prize. Do you get the prize even though we still can't solve the three body problem. There is no exact description of how three objects in a solar system will behave.
Wanna really bend your mind? Of course, you do. There's a name for a certain mathematical function we call the logistic map. Don't freak out. It's just the name.
The logistic map, it can be used to describe a lot of things because it's just an abstract math concept, but it has lots of cool applications. You can think of the logistic map. If you have, say, a population of buffalo and they eat food and they reproduce and then there's baby buffalo, and then sometimes grown up buffalo die, and sometimes buffalo don't get enough food, and so they starve and they die. And the logistic map can describe that population of buffalo. Like, there's food, they reproduce, the population grows, but then there's not enough food, so some of the population dies, and so there's not as many in the next generation, but then there is enough food for them, and then they're able to reproduce, etcetera, etcetera, etcetera.
What controls this is the population of buffalo is their birthright. You know, if there's only one new buffalo coming online every year, then there's plenty of food to spare. And if there's a couple or or, like, if 50% of the population reproduces, then you got a lot of buffalo. And if every buffalo reproduces, then you got a whole lot of buffalo and you've got a buffalo problem because you don't have enough food for your buffalo. Something really interesting happens with a logistic map.
If the birth rate is low, and I'm just I'm not gonna pin any numbers on this, I'm just gonna speak in general terms. If the population is low, then you can exactly predict what the population of buffalo will be, and it's very, very stable with time. It's just, yep, this is the population of buffalo, some buffalo die, and some buffalo are born, and that's just the way things are. But as you start to increase the birth rate, you start to get more Buffalo, that's fine. So far so good.
Year after year, you know, the same number of Buffalo, some Buffalo die, some are born, everything's cool. But after a certain point, if you increase the birth rate past a certain threshold, then something funny happens. Instead of just having a single stable population of buffalo, you get two populations of buffalo. And they'll go back and forth, like, year after year. Year one, you'll have this many buffalo, and then the next year, you'll have a different number of buffalo.
And then the year after, they'll you'll have the original number and then back and forth, back and forth, back and forth. And then you increase the birth rate again a little bit and passes another threshold. Instead of two levels of buffalo year after year, you have four levels of buffalo year after year. There's level one, then the next year level two, then the next year level three, then the next year level four, and then the next year level one, and then the next year level two, and on and on and on. You increase the birth rate again, and you get eight, and then 16, and then 32.
You have entered what's called in the jargon, in this chaos theory jargon, a period doubling cascade. As the birth rate grows, you pass a point where the population isn't stable anymore. Instead, there are two stable populations and then four stable in eight and sixteen and on and on and on. You have begun to descend into chaos. What does this mean?
It means this is what happens without Patreon. You can prevent chaos by supporting me on Patreon. That's patreon.com/pmsutter. Go there. Figure out how you can keep those shows going, and keep chaos at bay.
It also means that you've entered a really cool realm of mathematics and physics and biology and everything else. You've entered chaos. Because after this period doubling cascade of double and quadruple and October, bah bah bah bah bah, it's just chaotic. Year after year, totally random population, and you have no way of predicting what next year's population will be. Purely unpredictable and even though even though the the equation that describes this behavior, the so called logistic map, hasn't changed at all.
Exact same math going into it, but you start where you start off like totally predictable. I know what the population's gonna be, And then this, like, cycling of populations, like, that's a little weird, but you can deal with it. Now it's just totally unpredictable. It's the exact same equation. All you've changed is the birthright.
Year after year, how many buffalo are we gonna have next year? I have no idea. We're chaotic now. But then it gets even weirder. Because as you start to change this birth rate and, you know, fine tune it, like, okay.
Instead of making big jumps in the birth rate, let's see what happens when the birth rate is just changed by a little bit. You get these so called windows of stability where the chaos disappears and you get back to having a single population. A single predictable population. And then you start to slowly increase the birth rate from there, and then you'll get the two and the four and the 16. You'll get another descent into chaos, and then there'll be a a region of total chaos with a with a associated with a certain range of birth rates.
And then you increase it just a little bit, and then boom, it disappears and you're back to stability. And then you start increasing the birth rate and it descends back into chaos. And if you really fine tune the birth rate, if you're, like, really drilling, like, okay. If the birth rate is 3.6257% versus three point six two five seven eight percent, like, if you if you drill into that level of detail, what you find in the chaotic regions of all these populations year after year after year, you'll find many period doubling cascades. Like, year after year after year, the population is gonna swing wildly.
You have no idea. But if you examine closely, say a few years, you'll see a pattern. Or if you examine closely this range of of birth rates, you'll see this period doubling cascade repeated. What we've discovered is a fractal. The logistic map.
The population of buffalo related to their birth rate is a fractal in this chaotic regime. It's totally and completely unpredictable, and yet there's a pattern there. It seems at first glance that these chaotic systems are unpredictable, and yet there are these little patterns in there. Is there anything we can dig out? Is there anything that we can actually explore and understand and wrap our heads around and make any sort of progress at all?
Or is it just simply that chaotic systems are chaotic and unpredictable, period, end of story? Well, let me talk about the weather, and specifically a guy named Edward Lorenz in the weather. He's a meteorologist, physicist, back in the fifties, sixties, and he was running computer simulations of the weather, some of the simplest and earliest computer simulations. This is very, very simple modeling. This isn't trying to do, like, climate predictions or your weekend outlook forecast or anything like that.
It was just a very, very simple model of the Earth's atmosphere where it was hot on the bottom because of the ground, cold at the top, and it got to mix and there'd be winds. Like, super simple. Just trying to, you know, play around with some simple model of the weather. Edward Lorenz assumed just like everybody else assumed that if you make small changes to the inputs to the system, you'll get small changes to the outputs of the system. You know, like, oh, if I if I change the temperature just a little bit at the beginning, then, you know, there'll just be a little minor change to the temperature at the end.
But he found something pretty strange because he ran a simulation and he wanted to restart it. There was like a bug that cropped up or something. So he started it from, like, a halfway point and he got a totally different answer. So he started from the beginning again, got a totally different answer. So he made some small minor changes like in the sixth decimal point and you got totally different answers.
Like, not even like, a little bit different, wildly different predictions for this weather system. It turned out that for the weather, small, tiny, insignificant changes, seemingly insignificant changes at the beginning, led to wildly different end results. This led to a question of predictability, and there was a conference talk with this title, Does the flap of a butterfly's wing in Brazil set off a tornado in Texas? Something's going on. This system that we call the weather or double pendulums or three gravitational bodies or the logistic map are sensitive to the initial conditions.
You change something, some tiny thing like a butterfly wing, and that is enough to completely and totally alter the future course of the global weather. Think about that. You breathing right now can set off a chain of events that affect, you know, the winter weather in Europe next year, The glacial melt in Antarctica. And it's not just one thing. It's all the countless little changes that are always happening constantly and consistently.
They're all adding up and competing with each other. Is there any hope here? No wonder it's so hard to predict the weather because it's so sensitive. You change one little thing, you change big things. That's why the weather is so hard to predict.
Is there any hope here, or is it just chaos? Well, as I hinted back with the logistic map, there are patterns. Things do appear. And to talk about in the case of weather, I need to talk about phase space. I've talked about states in systems, like the state is the list of all the properties of the system.
In the case of the weather, it could be the current temperature, the current pressure, the current wind speed. That's the current state of the system. And the phase space is a map of the possible states of the system. So as the system changes, as the weather changes, as the pendulum wiggles around, as the planets orbit the sun, they will trace out paths in phase space. You know, like, okay.
One snapshot in time, there's this position, there's this velocity, there's this temperature, there's this pressure, that's a little dot, that's a little point in phase space. And then you wait a while and you take another snapshot and there's a different position, a different velocity, a different temperature, a different pressure. That's another little dot in phase space, and you keep going and going and going and going, and you see phase space allows you to see how the whole system evolves, like all the properties of the system evolves. Within phase space, we have something called attractors. Attractors are where the system likes to be.
What kind of position do planets like to have around a star? What kind of temperatures and pressures does the weather like to have? Like, so you can start off somewhere, like, oh, okay. This is my starting point, and then the system evolves and it just goes to some preferred state. Like, I like to be here with this temperature, this pressure, this position, this speed.
This is this is I'm this is my happy place. This is my attractor. What Lorentz discovered with this simple model of the weather is something called a strange attractor. As this his simplified weather system evolved, it traced out paths in phase space, describing the pressure and the temperature and the wind speed, things like that. There was an attractor there.
It just wasn't all willy nilly. It wasn't totally random. It wasn't just like, okay. It's just gonna be whatever it feels like. No.
There was a structure there. There was a pattern there. There was regularity there, but the same point was never visited twice. The weather never repeated itself. The weather was unpredictable.
Because if you're at one point in face space, you didn't know what the next point was gonna be. And yet there was this larger pattern. And when you zoom in, when you look at this strange attractor, you know, it's a set of curves mapped out. If you drew a little box and zoomed in, you would see the same structure again. Zoomed into a piece of that, you saw the same structure, and the same, it was a fractal.
That's how the weather was never able to repeat itself because it had this fractal pattern in the attractor. So you can't predict the weather, but you can make accurate statements about the likelihood of a future state in a chaotic system. I can't tell you what a chaotic system will be tomorrow, but I can tell you the likelihood of it. Why? Because it has a certain structure in phase space.
I say, I I don't know where that dot's gonna be. I don't know what the state of the weather is gonna be like tomorrow, but I know it will be somewhere on this attractor. It will be somewhere within this part of phase space. It won't be off willy nilly wherever it feels like. No.
It does have a home. It does have a happy place or a range of happy places. There's a lot more to the topic of chaos theory. I just wanted to give you an example. There are applications everywhere in biology, physics, astrophysics.
The big lesson of chaos theory is that order and chaos seem like opposites. Like, you can you can only pick one. You can pick order or you can pick chaos, but you can't have both. Turns out you can have both. You can have orderly equations that lead to chaotic behavior.
The logistic map, Lorenz's model of the weather, the three body gravitational problem that's just Newtonian physics, like, these are very simple, very orderly equations and yet totally chaotic behavior. And even though you can have chaotic behavior, there are some surprising orderly patterns. We don't fully understand chaos theory. We haven't really wrapped our heads around it. We we see these hints.
We see the fractals. We see the patterns. We're still digging out, like, what what's the deeper meaning here? But there is, you know, some deeper orderly meaning and behavior. You can't predict the weather, but you know that the weather follows certain cycles and patterns, and you can predict that.
You know, you generally, the summers will be hot and the winters will be cold. You don't know what the temperature will be on, say, July 18, but you know it will probably be warm. There is a larger pattern there. That larger pattern is not an accident. It's built into the physics.
The main lesson of chaos theory is that randomness, unpredictability, and determinism are actually compatible. You can have a deterministic system that is completely and totally random and unpredictable. And yet out of that randomness and unpredictability, you can still find deeper patterns. You just haven't looked hard enough. Thank you to Carlos t on Patreon, Akansha b on email, at t s fountain works on Twitter, Joyce s on email, and at unplugged wire on Twitter for the questions about chaos theory.
Also, I'm going on an adventure to see the total solar eclipse in November 2021 in the Antarctic Ocean. I am joining Poseidon Expeditions as their token science dude. It's a huge adventure. We're going to, like, Falcon Islands and Georgia Islands and the Antarctic Peninsula itself, and then also enjoying the eclipse. You can email salesusa@poseidonexpeditions.com, or call (347) 801-2610, and tell them you wanna ride the boat with Paul, and they can get you signed up.
Cabins are already selling fast because, of course, they are. It's the eclipse. Thank you to my top Patreon contributors this month, Matthew k, Helghan b, Justin z, Justin g, Kevin o, Duncan m, Corey d, Barbara k, Nuder Dude, Chrissy, Robert m, Nate h, Andrew f, Chris l, John, Elizabeth w, Cameron l, and Nalia. You can go to patreon.com/p. I'm sorry to learn how you can contribute to keep these episodes going.
You can also keep sending me questions, askaspaceman@gmail.com. You can also go to the website, ask a spaceman dot com. Leave a review on iTunes. I always appreciate those. Tell your friends I appreciate that even more and I will see you next time for more complete knowledge of time and space.