Chaotic systems are random

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Chaotic (in the mathematical sense) systems are fundamentally random. Chaos theory explanations generally revolve around the so-called 'Butterfly' effect and world-wide weather - but those explanations don't really do the subject justice. I can explain this but we need a simpler concrete example. The weather is too complicated to discuss - let's talk about a simpler (but still deterministic and still chaotic) system. This is one of my favorites because it's easy to imagine:

The Equipment

Take a couple of small, strong magnets and place them about six inches apart in the middle of a large sheet of paper your desk. Now hang a magnetic pendulum bob a few inches above the magnets on the end of a nice long string suspended from the ceiling. You also need a red and a blue crayon. OK - so here is the experiment:

The Experiment

Hold the pendulum over some point on the paper, release it and after it swings around a bit and if the magnets are strong enough, the pendulum will end up hovering over the center of either one or the other magnet. Colour that 'release point' with a small red dot if the pendulum ended up over the right-hand magnet - colour it blue if it ended up over the left-hand magnet. Repeat this experiment for every point on the paper so it's completely covered in red and blue dots.

So what red/blue pattern results? Well, when you release the pendulum near the right magnet the result is always that the pendulum swings immediately over that magnet - so there is obviously an area around the right magnet that ends up red, and an area around the left magnet that ends up blue. Now, if you hold the magnet a bit further off to the right - beyond the righthand magnet, the pendulum will fly right over the righthand magnet and swing over to the lefthand one - by then it's lost enough energy due to air resistance that it'll stop over the lefthand magnet. But you can imagine that from some places the pendulum loops around one magnet then the other crazy swings until it finally loses energy and winds up over one or the other.

The Results

So you can imagine a fairly complex pattern of red and blue on the paper.

Now this magnetic pendulum setup is 'chaotic' (in the same way that the weather is). If you crunch the math on this, the actual mathematical pattern you wind up with is a fractal - something like the Mandelbrot set. There are regions of our red/blue pattern that are in big solid patches (like immediately around each of the two magnets) - but there are regions where the red and the blue is all mixed up in whorls and patterns of great complexity. Write a computer program to generate this pattern and you can zoom into these patterns and you see more patterns, you can zoom in deeper and deeper into that map - and in some areas you'll keep on getting more and more red/blue patterns no matter how tightly you zoom. The image is infinitely complex...fractal...chaotic.

What is the physical meaning of this infinite complexity of red and blue dots? It means that if you start the pendulum over one of those chaotic regions and release it, the magnet it will end up over is very sensitive to where you started it from. Move a millimeter to one side and the answer may be different. Move a millionth of a millimeter to one side and the result will be different, move the width of a hydrogen atom to one side and you'll get a different answer. In fact, move an INFINITELY SMALL distance to one side or the other and the pendulum may end up over the other magnet. The result is that moving the pendulum by (1/infinity) meters can change the answer...but (1/infinity ) is zero (well, kinda - mathematicians might argue it 'approaches' zero - but the result is the same) if you move the pendulum by zero distance, the answer can change. It's deterministic - in that there is no randomness in the equations - but you need infinite precision to calculate it - and even if you had that, you still wouldn't get the right answer because displacing the initial position of the pendulum by 1/infinity meters changes the answer.

You can see pictures of the results (for three magnets, actually) here: [1].


You don't need quantum effects to get a random answer - you don't even need an inaccurate measurement of the initial position because an error of (1/infinity)% is enough to change the answer. The result is independent of computer power or precision.