3 ms·

I wrote about this at https://r6.ca/blog/20100802T005835Z.html https://r6.ca/blog/20100802T005835Z.html. Basically entropy is a function of descriptions of sta

I wrote about this at https://r6.ca/blog/20100802T005835Z.html https://r6.ca/blog/20100802T005835Z.html. Basically entropy is a function of descriptions of state rather than states themselves. So when you talk about jar of gas having a certain volume and pressure, there is a set of states that satisfy that description and the logarithm of the volume of that set of states (in phase space) is the entropy of that description. Liouville’s theorem says that the volume of a subset of phase space is preserved under time evolution of the system. So if you want to have a machine that reliably transform from one description to another description, then the volume of the latter description must be at least as large as the initial description. That is the second law of thermodynamics. (This are a bit more complicated than I state above because you have to work with differentials, but the basic idea is there). The Poincaré recurrence theorem says individual states eventually almost reoccur, but this doesn't apply to a volume of states. The volume of states usually gets smeared out throughout phase space. Like if you start the Lorenz equations on a square of points instead of an individual point, it gets smeared out over the entire Lorenz attractor. And while individual points in that square nearly reoccur, the entire square does not. And thus you cannot build a machine to reliably reduce entropy because you cannot predict when your reoccurance will reoccur (even if you can compute an upper bound on how long it will take) since you never know precisely the initial conditions to begin with (you start somewhere in that initial square).

sickofthisshit 3y ago
Except the recurrence theorem applies to subsets of the space, not just elements. Your "doesn’t apply to a volume of states" isn't really an exception, apart from technicalities about subsets of measure zero.
- rssoconnor 3y ago
  Do you have a reference on this? I don't think the space of subsets is going to be bounded, which means the Poincaré recurrence theorem won't apply to it. That coupled with the fact that starting our Lorenz equation on a box shaped subset will cause that box to stretch and distort, asymptotically approaching the Lorenz attractor. This implies that it can never become box shaped again. The set just gets closer and closer to the shape of the attractor as it evolves. Otherwise it wouldn't really be an attractor now would it? P.S. If you take a discrete subset, like pixels on the screen, then I agree those sets of pixels will reoccur, making it look like the box has reappeared. But in reality that is just no longer a representative sample of the twisted form of the actual set wrapped around the attractor.
  - sickofthisshit 3y ago
    I'm not sure exactly what you mean by "space of subsets". All I was trying to point out was that the argument that proves the recurrence theorem itself uses a volume of space around the initial state, and how "preimages" of that volume work. So it applies to volumes of states, too. The Lorenz attractor generally avoids the recurrence because its dynamics are dissipative: nothing drives points near the attractor to points far from the attractor. But once you are on the attractor, you can't just stay on the attractor forever getting "smeared out" without recurrence: you can only get at most smeared out over the finite area of the attractor, and eventually the smearing reaches your initial location on the attractor again.
    - rssoconnor 3y ago
      You are correct about the Lorenz attractor being dissipative. I just assumed the chaos of the Lorenz attractor and, say a frictionless double pendulum, were the same phenomenon. However it seems they are, in fact, quite different. Even still > So it applies to volumes of states, too. I'm pretty sure this is false. As I mention in another comment, http://philsci-archive.pitt.edu/9838/1/recurrence.pdf http://philsci-archive.pitt.edu/9838/1/recurrence.pdf says "In [the case of the space of probability distributions over phase space], the reason that classical dynamics fails to abide by the linear recurrence theorem is that the distribution space is infinite-dimensional, even if phase space has finite volume: distributions can have structure on arbitrarily short scales. Yes the proof of the recurrence theorem uses volumes; but the result still only applies to individual points within the volume (or maybe sets of measure 0 at best).
  - rssoconnor 3y ago
    Replying to myself: Apparently strange attractors can only occur in dampened systems: https://physics.stackexchange.com/questions/125931/does-a-simple-double-pendulum-have-transients https://physics.stackexchange.com/questions/125931/does-a-si... So my analogy between the Lorenz attractor and phase-space chaos (like that of a double pendulum) is false. However, I think my point is still true: http://philsci-archive.pitt.edu/9838/1/recurrence.pdf http://philsci-archive.pitt.edu/9838/1/recurrence.pdf says "In [the case of the space of probability distributions over phase space], the reason that classical dynamics fails to abide by the linear recurrence theorem is that the distribution space is infinite-dimensional, even if phase space has finite volume: distributions can have structure on arbitrarily short scales.

sickofthisshit 3y ago

Except the recurrence theorem applies to subsets of the space, not just elements. Your "doesn’t apply to a volume of states" isn't really an exception, apart from technicalities about subsets of measure zero.

rssoconnor 3y ago
Do you have a reference on this? I don't think the space of subsets is going to be bounded, which means the Poincaré recurrence theorem won't apply to it. That coupled with the fact that starting our Lorenz equation on a box shaped subset will cause that box to stretch and distort, asymptotically approaching the Lorenz attractor. This implies that it can never become box shaped again. The set just gets closer and closer to the shape of the attractor as it evolves. Otherwise it wouldn't really be an attractor now would it? P.S. If you take a discrete subset, like pixels on the screen, then I agree those sets of pixels will reoccur, making it look like the box has reappeared. But in reality that is just no longer a representative sample of the twisted form of the actual set wrapped around the attractor.
- sickofthisshit 3y ago
  I'm not sure exactly what you mean by "space of subsets". All I was trying to point out was that the argument that proves the recurrence theorem itself uses a volume of space around the initial state, and how "preimages" of that volume work. So it applies to volumes of states, too. The Lorenz attractor generally avoids the recurrence because its dynamics are dissipative: nothing drives points near the attractor to points far from the attractor. But once you are on the attractor, you can't just stay on the attractor forever getting "smeared out" without recurrence: you can only get at most smeared out over the finite area of the attractor, and eventually the smearing reaches your initial location on the attractor again.
  - rssoconnor 3y ago
    You are correct about the Lorenz attractor being dissipative. I just assumed the chaos of the Lorenz attractor and, say a frictionless double pendulum, were the same phenomenon. However it seems they are, in fact, quite different. Even still > So it applies to volumes of states, too. I'm pretty sure this is false. As I mention in another comment, http://philsci-archive.pitt.edu/9838/1/recurrence.pdf http://philsci-archive.pitt.edu/9838/1/recurrence.pdf says "In [the case of the space of probability distributions over phase space], the reason that classical dynamics fails to abide by the linear recurrence theorem is that the distribution space is infinite-dimensional, even if phase space has finite volume: distributions can have structure on arbitrarily short scales. Yes the proof of the recurrence theorem uses volumes; but the result still only applies to individual points within the volume (or maybe sets of measure 0 at best).
- rssoconnor 3y ago
  Replying to myself: Apparently strange attractors can only occur in dampened systems: https://physics.stackexchange.com/questions/125931/does-a-simple-double-pendulum-have-transients https://physics.stackexchange.com/questions/125931/does-a-si... So my analogy between the Lorenz attractor and phase-space chaos (like that of a double pendulum) is false. However, I think my point is still true: http://philsci-archive.pitt.edu/9838/1/recurrence.pdf http://philsci-archive.pitt.edu/9838/1/recurrence.pdf says "In [the case of the space of probability distributions over phase space], the reason that classical dynamics fails to abide by the linear recurrence theorem is that the distribution space is infinite-dimensional, even if phase space has finite volume: distributions can have structure on arbitrarily short scales.