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I think I did a decent job in part 7 of getting across the main paradox in physics that has confused me over the years. And there's a very similar paradox that I found on the wandering sets Wikipedia page (http://en.wikipedia.org/wiki/Wandering_set), so now seems like a good time to return to that.

They define a wandering point as a point which has a neighborhood in phase space which, after some time in the future, never gets back to where it intersects itself again. Similarly, they define a wandering set as a set whose points never intersect each other again after a certain time in the future. One seemingly minor caveat, which may be important, is that the intersection doesn't have to be exactly zero (no points in common), just so long as it has measure zero in the entire space. Measure is sort of like volume (but more mathematically rigorous). So for example, the set of points in a 2D plane has zero volume in a 3D space. So if two 3D objects intersect only in a 2D plane, it doesn't count as a true intersection, since the volume of that intersection is zero. Same goes for higher dimensional spaces, except the definition of volume is different.

Going hand in hand with their definition of a wandering set is their definition of a "dissipative action". The action is the specific rules that time-evolve the system from the past into the future. It's defined as dissipative if and only if there is some wandering set in the space under that action. If there is no wandering set in the space, then it's a conservative action.

But now for the contradiction I thought I saw on the page (after re-reading it many more times, I've figured out why it's not actually a contradiction). They define one more thing, a "completely dissipative action", whose definition seemed to me at first to be completely incompatible with their definition of a dissipative action. They define a completely dissipative action as an action that time-evolves a wandering set of positive measure into the future in such a way that the path it sweeps out through the space (it's "orbit") ends up taking up the entire space--or more precisely, the measure of its orbit is the same as the entire space. The reason this seemed to be a contradiction to me is that I was picturing a case such as mixing, where you start out with an initial condition that takes up some limited subspace of the entire phase space (like one cube of the Rubik's cube we talked about earlier), and then after you time-evolve it forward it ends up expanding to fill the whole space. But if it expands to fill the whole space, then it can't be a wandering set, because the intersection between the whole space and the original set is non-zero (it's the original set)!

So how does one resolve this paradox? Well, the main mistake I was making was confusing the final image of the set after it gets time-evolved with its orbit. The final image is where it is in a single snapshot in time, while the orbit is like the image you get when you leave a camera lens open for a long time (called "time exposure" in photography, I believe). Basically, it's the union of all of the different images it takes up as it progresses in time, not just a single snapshot. If it were just the final image, then their definition of a completely dissipative action is indeed contradictory, and can't coexist with their definition of a wandering set.

Ok, so it's the orbit, not the final image. Even then, it's a bit hard to imagine a scenario that would be "completely dissipative". The reason I was hoping it would be easier to imagine this scenario is because I was hoping that maybe the simplest kind of dissipation would be complete dissipation. And maybe understanding that would be a big step in the process towards understanding any kind of dissipation. In order to imagine what kind of a scenario would work, we need to find a case where the original set never wanders back to it's starting point but ends up sweeping out a path that fills all of space. To do that, it's best to think about what the reason would be that a set might never wander back to its original starting point. In most normal situations in physics, if you've got things moving around according to nice simple laws of physics, and you didn't start at any special starting point, you'd expect the motion to fill the whole space and eventually wander back an infinite number of times. The only case where it wouldn't get back, is if somehow it gets trapped in some subspace. This could be a single point that it ends up approaching, or a line, or a plane, or even a circle for instance. For example, if you had a planet that goes near a solar system and gets sucked into the orbit of that solar system, it would end up getting trapped in an ellipse, and never continue its nice straight motion, never getting back to the original starting point, even if the galaxy were inside a giant box. There's a name for this kind of occurrence in physics, it's called an "attractor". Basically, it seems that in order to have dissipation in the sense describe on the wandering set Wikipedia page, you would need some form of attractor. The ellipse I described would be a regular attractor, but in chaos theory you also have weirder more fractal patterns called "strange attractors". Chaos theory (also known as non-linear dynamics) seems intimately connected with the topic of dissipation, as many of the dissipative systems I mentioned in the beginning (such as hurricanes) are chaotic systems. I wasn't kidding when I said the question I'm wondering about here involves "all areas of physics"! :-)

So now unlike the mixing case, where you're going from a small dimensionality to a higher dimensionality, we've got the opposite happening. You start the motion in a space of higher dimensionality, and then get trapped in an attractor of lower dimensionality. Because no matter where you start in the higher dimensional space, you tend to end up in the lower dimensional attractor, you've got a many-to-1 mapping from initial to final states. In other words, you've got irreversibility and hence dissipation! And this higher to lower dimensional transition seems much more similar to the collapse of the wavefunction in quantum mechanics, which also goes from higher to lower. As opposed to the mixing case which seems to go from lower to higher. So this is not just an issue of quantum mechanics working one way, and thermodynamics working the other--now we have the same paradox appearing solely within regular classical thermodynamics, (which hearkens back to my earlier point that this issue has been around longer than quantum mechanics).

For this case of moving from somewhere in the bulk space into a smaller attractor, the definition of a "completely dissipative action" makes more sense. If you pick as your starting set, the entire space except the attractor, and then all of those points move into the attractor, you have exactly satisfied the definition. The orbit includes both everything outside of the attractor (which is what you started with) as well as everything in the attractor (what you end up with, or near enough to count as the same measure). So the orbit does indeed take up the entire space. But the set is still wandering, since there is no intersection between itself and the final attractor. Presumably, an action that is only partially dissipative (as opposed to completely dissipative) would include an attractor which captures starting points in some of the rest of the space, but not all.

We're getting closer and closer, so hopefully in the next part I'll be able to resolve this higher/lower dimensionality paradox in both thermodynamics and quantum mechanics (or if not both, at least one of them).

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