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.
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.
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.
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.
"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).
"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.
There is a tension between them. If you take the second law as 'rigorous', the entropy must always increase. But Poincare recurrence says you will inevitably return to the lower entropy state, which of course is a violation.
One way around this is to relax the definition of the second law to be something like "almost always": the set of states with higher entropy are incredibly unlikely. And when you compute the recurrence time, it's astoundingly large; like trillions upon trillions of ages of the universe.
Cosmologically, there is some escape in that the expansion of the universe breaks the assumption of finite accessible phase space, but there is an underlying difficulty of what it means for the universe to be truly infinite or divergent in size. Not just really big, but truly infinite: where did it all come from?
My personal take is that the second law is about "macroscopic" experiments. You are given a box and you can only do things like push the side of box or put it on a hot stove or in a magnetic field or whatever, and the laws are about what you can or cannot do with those operations. The microscopic system might decide to do something incredibly unlikely, but you can't control or even anticipate it, and that is just the extremely long tail of the probability distribution that is a thermodynamic state.
I don't claim my view is rigorous; people like Boltzmann and Ehrenfest were much smarter than me and struggled with what irreversibilty means, and you can kind of go crazy worrying about this (likewise with the 'measurement problem' of quantum mechanics). In the end, whether a situation can be mapped to the exact axioms of a model like QM or thermodynamics is very tricky and perhaps unknowable.
Your personal one is the correct one from my perspective (as someone that knows a lot of math and is familiar with statistical mechanics as math and not physics). The ODEs that we get from classical mechanics are typically reversible: we can write down an ODE that does the same thing but backwards.
You cannot do that for the PDEs that arise in statistical mechanics and the result is the second law. These PDEs arise from approximating many copies of deterministic systems as continuous distributions of states. Entropy is not a concept that makes sense when discussing single trajectories of systems — only the macroscopic view of many copies of that system evolving according to the same dynamics.
