sickofthisshit

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.

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.

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.