I’ve noticed you using the word “chaos” a few times across your posts. I think you’re using it colloquially to mean something like “rapidly unpredictable”, but it does have a technical meaning that doesn’t always line up with how you use it, so it might be useful to distinguish it from a couple other things. Here’s my current understanding of what some things mean. (All of these definitions and implications depend on a pile of finicky math and tend to have surprising counter-example if you didn’t define things just right, and definitions vary across sources.)
Sensitive to initial conditions. A system is sensitive to initial conditions if two points in its phase space will eventually diverge exponentially (at least) over time. This is one way to say that you’ll rapidly lose information about a system, but it doesn’t have to look chaotic. For example, say you have a system whose phase space is just the real line, and its dynamics over time is just that points get 10x farther from the origin every time step. Then, if you know the value of a point to ten decimal places of precision, after ten time steps you only know one decimal place of precision. (Although there are regions of the real line where you’re still sure it doesn’t reside, for example you’re sure it’s not closer to the origin.)
Ergodic. A system is ergodic if (almost) every point in phase space will trace out a trajectory that gets arbitrarily close to every other point. This means that each point is some kind of chaotically unpredictable, because if it’s been going for a while and you’re not tracking it, you’ll eventually end up with maximum uncertainty about where it is. But this doesn’t imply sensitivity to initial conditions; there are systems that are ergodic, but where any pair of points will stay the same distance from each other. A simple example is where phase space is a circle, and the dynamics are that on each time step, you rotate each point around the circle by an irrational angle.
Chaos. The formal characterization that people assign to this word was an active research topic for decades, but I think it’s mostly settled now. My understanding is that it essentially means this;
Your system has at least one point whose trajectory is ergodic, that is, it will get arbitrarily close to every other point in the phase space
For every natural number n, there is a point in the phase space whose trajectory is periodic with period n. That is, after n time steps (and not before), it will return back exactly where it started. (Further, these periodic points are “dense”, that is, every point in phase space has periodic points arbitrarily close to it).
The reason these two criteria yield (colloquially) chaotic behavior is, I think, reasonably intuitively understandable. Take a random point in its phase space. Assume it isn’t one with a periodic trajectory (which will be true with “probability 1”). Instead it will be ergodic. That means it will eventually get arbitrarily close to all other points. But consider what happens when it gets close to one of the periodic trajectories; it will, at least for a while, act almost as though it has that period, until it drifts sufficiently far away. (This is using an unstated assumption that the dynamics of the systems have a property where nearby points act similarly.) But it will eventually do this for every periodic trajectory. Therefore, there will be times when it’s periodic very briefly, and times when it’s periodic for a long time, et cetera. This makes it pretty unpredictable.
There are also connections between the above. You might have noticed that my example of a system that was sensitive to initial conditions but not ergodic or chaotic relied on having an unbounded phase space, where the two points both shot off to infinity. I think that if you have sensitivity to initial conditions and a bounded phase space, then you generally also have ergodic and chaotic behavior.
Anyway, I think “chaos” is a sexy/popular term to use to describe vaguely unpredictable systems, but almost all of the time you don’t actually need to rely on the full technical criteria of it. I think this could be important for not leading readers into red-herring trails of investigation. For example, all of standard statistical mechanics only needs ergodicity.
I’ve noticed you using the word “chaos” a few times across your posts. I think you’re using it colloquially to mean something like “rapidly unpredictable”, but it does have a technical meaning that doesn’t always line up with how you use it, so it might be useful to distinguish it from a couple other things. Here’s my current understanding of what some things mean. (All of these definitions and implications depend on a pile of finicky math and tend to have surprising counter-example if you didn’t define things just right, and definitions vary across sources.)
Sensitive to initial conditions. A system is sensitive to initial conditions if two points in its phase space will eventually diverge exponentially (at least) over time. This is one way to say that you’ll rapidly lose information about a system, but it doesn’t have to look chaotic. For example, say you have a system whose phase space is just the real line, and its dynamics over time is just that points get 10x farther from the origin every time step. Then, if you know the value of a point to ten decimal places of precision, after ten time steps you only know one decimal place of precision. (Although there are regions of the real line where you’re still sure it doesn’t reside, for example you’re sure it’s not closer to the origin.)
Ergodic. A system is ergodic if (almost) every point in phase space will trace out a trajectory that gets arbitrarily close to every other point. This means that each point is some kind of chaotically unpredictable, because if it’s been going for a while and you’re not tracking it, you’ll eventually end up with maximum uncertainty about where it is. But this doesn’t imply sensitivity to initial conditions; there are systems that are ergodic, but where any pair of points will stay the same distance from each other. A simple example is where phase space is a circle, and the dynamics are that on each time step, you rotate each point around the circle by an irrational angle.
Chaos. The formal characterization that people assign to this word was an active research topic for decades, but I think it’s mostly settled now. My understanding is that it essentially means this;
Your system has at least one point whose trajectory is ergodic, that is, it will get arbitrarily close to every other point in the phase space
For every natural number n, there is a point in the phase space whose trajectory is periodic with period n. That is, after n time steps (and not before), it will return back exactly where it started. (Further, these periodic points are “dense”, that is, every point in phase space has periodic points arbitrarily close to it).
The reason these two criteria yield (colloquially) chaotic behavior is, I think, reasonably intuitively understandable. Take a random point in its phase space. Assume it isn’t one with a periodic trajectory (which will be true with “probability 1”). Instead it will be ergodic. That means it will eventually get arbitrarily close to all other points. But consider what happens when it gets close to one of the periodic trajectories; it will, at least for a while, act almost as though it has that period, until it drifts sufficiently far away. (This is using an unstated assumption that the dynamics of the systems have a property where nearby points act similarly.) But it will eventually do this for every periodic trajectory. Therefore, there will be times when it’s periodic very briefly, and times when it’s periodic for a long time, et cetera. This makes it pretty unpredictable.
There are also connections between the above. You might have noticed that my example of a system that was sensitive to initial conditions but not ergodic or chaotic relied on having an unbounded phase space, where the two points both shot off to infinity. I think that if you have sensitivity to initial conditions and a bounded phase space, then you generally also have ergodic and chaotic behavior.
Anyway, I think “chaos” is a sexy/popular term to use to describe vaguely unpredictable systems, but almost all of the time you don’t actually need to rely on the full technical criteria of it. I think this could be important for not leading readers into red-herring trails of investigation. For example, all of standard statistical mechanics only needs ergodicity.