> the classical limit h->0 also exhibits an arrow of time
That is certainly true empirically. But it is much more difficult to explain why this happens in classical terms without begging the question -- you can't just invoke the second law of thermodynamics here. You have to derive the second law from Newtonian mechanics. That is an unsolved problem.
No, deriving the second law from Newton+Liouville is easy (in the form of "mathematically plausible pseudo-theorem", that's late 19th century physics). If you don't adopt Liouville measure as axiom, then it is impossible by (more modern) counter-example: Thermodynamics looks different if you pre-suppose that god cares about weird measures; and it is unavoidable if you pre-suppose that god cares about Liouville measure.
OK... but this seems to me simply like trading one assumption for another. Is there any reason why adopting the Liouville measure as axiom is any less arbitrary than just adopting the second law directly?
Of course it is trading one assumption for another.
It basically comes down to "absent other specifications, one state is as probable as another". As good Bayesians, we should reject this sentence: We always, always need a prior; and if the prior is too crazy (e.g. has zero probability mass on the correct hypothesis), then no amount of observation will ever help us.
Since we deal with a continuous system, this is also a nonsensical sentence on purely mathematical grounds: It could be "absent other specifications, one state has as large probability density as any other, measured relative to Liouville". An easy calculation shows that this state of affair (thermodynamic equilibrium) is preserved by the flow. Furthermore, "If any state has as large probability density as any other, measured to a small (absolutely continuous) distortion of Liouville, and you wait long enough, then it all evens out to Liouville", i.e. small distortions decay (mixing).
Large distortions do not need to decay. For example, you could single out one specific crazy periodic trajectory (there are many crazy periodic trajectories, but the set of crazy trajectories has small ordinary phase-space-volume), and our large distortion is "I am somewhere on this specific periodic trajectory". If we start on the periodic trajectory, then we stay on it; hence, this state of affairs is preserved as well.
In order to separate the two, we need an axiom, like "Liouville is a good prior". The equations of motion don't tell us that the first corresponds better to reality than the second! This is an empirical observation.
Now start with Liouville, but prescribe that our initial data are in a specific low-entropy macro-state. This means that we cut out some non-crazy region of phase space and say "start anywhere here, but then count all states as same probability". Then the second law holds (this is pseudo-theorem).
If we initially started with our crazy measure (concentrated on a single periodic trajectory), then the second law fails / is vacuous. If we permitted to cut out a crazy region of phase space, then it would also fail (crazy region: Take a sane region R0 at time T0 and a sane region R1 at time T1, and our crazy region consists of all points in R0 that will end up in R1 after time T1-T0. All these points will end up in R1, so no second law for you).
This general program is as strong as it gets, and all the "non-crazy" caveats are imo much less arbitrary than just adopting the second law directly. Of course this opens a giant can of worms: What exactly does non-crazy measure mean? What is a non-crazy way of slicing-and-dicing phase space into macro-states? For which systems of physical thermodynamic interest can we formally prove chaoticity / fast mixing?
Ok, I actually know the answer to the last question: Almost none. But I also know the answer to "for which such systems of interest does anyone seriously doubt that we have chaoticity / mixing": Almost none.
> Of course it is trading one assumption for another.
Why "of course"? If would only be "of course" if it were impossible to derive the second law from Newton's laws. AFAIK that hasn't been proven impossible.
But I think you may have misinterpreted my objection. It's not that I object to substituting one axiom for another in general. That can often represent progress. For example: before Einstein, it was observed that inertial mass and gravitational mass were very close to each other, lending considerable weight to the reasonableness of assuming that they were in fact the same. It turns out that you don't have to assume this. There's another set of axioms that allow you to prove this, and these axioms are "better" because they provide a much larger scope of explanatory power for the same axiomatic price.
By way of contrast, Turing machines and the lambda calculus are "the same" in some deep sense that makes it silly to argue about which one is "the right model" of computation.
It's not clear to me whether adopting Liouville really represents progress a la relativity, or whether it's arguing potato-potahto a la Turing vs Church.
BTW, it just occurred to me that there is a fundamental difference between the thermodynamic and quantum arrows of time: the thermodynamic arrow of time is reversible in a non-isolated system. The quantum arrow is not.
That is certainly true empirically. But it is much more difficult to explain why this happens in classical terms without begging the question -- you can't just invoke the second law of thermodynamics here. You have to derive the second law from Newtonian mechanics. That is an unsolved problem.