I think that there are some things that are sensitively dependant on other parts of the system, and we usually just call those bits random.
Suppose I had a magic device that returned the exact number of photons in its past lightcone. The answer from this box would be sensitively dependant on the internal workings of all sorts of things, but we can call the output random, and predict the rest of the world.
The flap of a butterflies wing might effect the weather in a months time. The weather is chaotic and sensitively dependant on a lot of things, but whatever the weather, the earths orbit will be unaffected (for a while, orbits are chaotic too on million year timescales)
We can make useful predictions (like planetary orbits, and how hot the planets will get) based just on the surface level abstractions like the brightness and mass of a star, but a more detailed models containing more internal workings would let us predict solar flares and supernova.
I think that there are some things that are sensitively dependant on other parts of the system, and we usually just call those bits random.
One key piece missing here: the parts we call “random” are not just sensitively dependent on other parts of the system, they’re sensitively dependent on many other parts of the system. E.g. predicting the long-run trajectories of billiard balls bouncing off each other requires very precise knowledge of the initial conditions of every billiard ball in the system. If we have no knowledge of even just one ball, then we have to treat all the long-run trajectories at random.
That’s why sensitive dependence on many variables matters: lack of knowledge of just one of them wipes out all of our signal. If there’s a large number of such variables, then we’ll always be missing knowledge of at least one, so we call the whole system random.
To expand on the billiard ball example, the degree of sensitivity is not always realised. Suppose that the conditions around the billiard table are changed by having a player stand on one side of it rather than the other. The difference in gravitational field is sufficient that after a ball has undergone about 7 collisions, its trajectory will have deviated too far for further extrapolation to be possible — the ball will hit balls it would have missed or vice versa. Because of exponential divergence, if the change were to move just the cue chalk from one edge of the table to another, the prediction horizon would be not much increased.
Do you have a source on that? My back-of-the-envelope says 7 is not enough. For 1 kg located 1 m away from 60 kg, gravitional force is on the order of 10^-9 N. IIRC, angular uncertainty in a billiards system roughly doubles with each head-on collision, so the exponential growth would only kick in around a factor of 128, which still wouldn’t be large enough to notice most of the time. (Though I believe the uncertainty grows faster for off-center hits, so that might be what I’m missing.)
I heard it a long long time ago in a physics lecture, but I since verified it. The variation in where a ball is struck is magnified by the ratio of (distance to the next collision) / (radius of a ball), which could be a factor of 30. Seven collisions gives you a factor of about 22 billion.
I also tried the same calculation with the motion of gas molecules. If the ambient gravitational field is varied by an amount corresponding to the displacement of one electron by one Planck length at a distance equal to the radius of the observable universe, I think I got about 30 or 40 collisions before the extrapolation breaks down.
I think that there are some things that are sensitively dependant on other parts of the system, and we usually just call those bits random.
Suppose I had a magic device that returned the exact number of photons in its past lightcone. The answer from this box would be sensitively dependant on the internal workings of all sorts of things, but we can call the output random, and predict the rest of the world.
The flap of a butterflies wing might effect the weather in a months time. The weather is chaotic and sensitively dependant on a lot of things, but whatever the weather, the earths orbit will be unaffected (for a while, orbits are chaotic too on million year timescales)
We can make useful predictions (like planetary orbits, and how hot the planets will get) based just on the surface level abstractions like the brightness and mass of a star, but a more detailed models containing more internal workings would let us predict solar flares and supernova.
One key piece missing here: the parts we call “random” are not just sensitively dependent on other parts of the system, they’re sensitively dependent on many other parts of the system. E.g. predicting the long-run trajectories of billiard balls bouncing off each other requires very precise knowledge of the initial conditions of every billiard ball in the system. If we have no knowledge of even just one ball, then we have to treat all the long-run trajectories at random.
That’s why sensitive dependence on many variables matters: lack of knowledge of just one of them wipes out all of our signal. If there’s a large number of such variables, then we’ll always be missing knowledge of at least one, so we call the whole system random.
To expand on the billiard ball example, the degree of sensitivity is not always realised. Suppose that the conditions around the billiard table are changed by having a player stand on one side of it rather than the other. The difference in gravitational field is sufficient that after a ball has undergone about 7 collisions, its trajectory will have deviated too far for further extrapolation to be possible — the ball will hit balls it would have missed or vice versa. Because of exponential divergence, if the change were to move just the cue chalk from one edge of the table to another, the prediction horizon would be not much increased.
Do you have a source on that? My back-of-the-envelope says 7 is not enough. For 1 kg located 1 m away from 60 kg, gravitional force is on the order of 10^-9 N. IIRC, angular uncertainty in a billiards system roughly doubles with each head-on collision, so the exponential growth would only kick in around a factor of 128, which still wouldn’t be large enough to notice most of the time. (Though I believe the uncertainty grows faster for off-center hits, so that might be what I’m missing.)
I heard it a long long time ago in a physics lecture, but I since verified it. The variation in where a ball is struck is magnified by the ratio of (distance to the next collision) / (radius of a ball), which could be a factor of 30. Seven collisions gives you a factor of about 22 billion.
I also tried the same calculation with the motion of gas molecules. If the ambient gravitational field is varied by an amount corresponding to the displacement of one electron by one Planck length at a distance equal to the radius of the observable universe, I think I got about 30 or 40 collisions before the extrapolation breaks down.
Awesome. Thanks for the spot-check, I’ll probably use this as a dramatic example going forward.