in Math, physics

Chaos theory — the mysterious staple of pop science is a fascinating topic. It has the ability to both enlighten and bewilder. Its one in a long set of human endeavors that show us, truly, our limitation in the face of the ever-mystery that is nature.

I found out about chaos theory in any detail two years ago contemplating the limitations of mathematics. Most aspects of different phenomena can be reduced to a mathematical model which can be reasoned about. The model can help us answer questions and make predication about how the system would evolve. Questions like where will the ball land if we throw it with a certain velocity. Or if and when the moon might disappear from the sky. But others have a measure of “difficulty” to them that so far have defied all mathematical reductions. And it appears, though unproven in most  cases, that this is a fundamental limitation, not just a simple result of our ignorance. I’m not convinced of that, but it’s an open questions where most informed  opinion is that some questions are just not amenable to the simplifications we can get for the ball moving through the air.

And that’s where chaos theory comes in. What makes predicting the weather harder than predicting the change in velocity of a thrown ball? What makes the economy intractable even for people who claim it as an expertise, while even a kid can program a computer to “solve” a tic-tac-toe game? Those questions and others led to the eventual development of field of mathematics called chaos theory.

Chaos theory in short is the the study of dynamical systems which are sensitive to initial conditions. This definition might seem like a mouthful but the basic insight that led to the development of the field is the realization that in most* examples where predictability was lost it was due to small changes in the way the system was set up leading to vastly different results after some time.

* By most I allude to the fact that in some instances it can be easy to solve a problem in theory, but not in practice. Consider for example multiplication, it’s an easy problem for both humans and computers but if the numbers involved are large enough even the most powerful computers couldn’t multiply them in any reasonable time or even at all due to physical limitations.

To illustrate this point we can use a simplified version of the weather prediction example from earlier. In our example lets imagine a big box with a 10 balloons filled with lighter than air gas. Let’s put all the balloons  in one corner of the box evenly spaced one centimeter from each other.

When we begin the system, the balloons would want to move up, but due to temperature variations in the box and their shape they would have a small amount of horizontal motion as well. They would start colliding and after 10 seconds each balloon would find itself in a different location in the box. If we run the experiment again, changing nothing* the outcome would be the same.

If we change the distance between one of the balloons to its neighbor by one tenth of a centimeter while keeping everything else the same, after 10 seconds the locations of the balloons would be very different than in our first experiment. This seems obvious for most of our intuitions. But it’s important to emphasize, there’s nothing random about this system. The chaos or lose of predictability was not due to not having enough information. It’s due to the nature of the system. More specifically the balloons interacted with each other, and each interactions was dependent on the previous interactions, and small changes build upon each other, until after some time the system is in a very different state.

I choose this example for a reason. It might seem reasonable to assume that we can’t predict what the atmosphere would do since it’s made up of countless particles but that’s not the case, like the balloon example shows. In a sense the complexity is built in due to the dynamics of the system.

* In the real world changing nothing is rarely possible in systems like the balloon example. But fortunately mathematics does not face this limitation.

Chaotic systems have some very interesting properties one of those is attractors which I hope to tackle in the next installment of this series.

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