Ok then~
I have a lot of random thoughts so I'll just first post something rough otherwise I'll never post anything if I'm trying to be precise haha.
There's a lot in foundations of SM that interests me, but the concept of entropy has always been a bit of a mystery. Mostly, I think, for a few reasons:
1. There's several definitions of entropy, the Claussius ##dS=\frac{\delta Q}{T}##, Boltzmann ##S=k\text{ln}\Omega##, Gibbs ##S=-k\sum \rho\text{ln}\rho##, Von Neumann ##S=\text{Tr}(\rho\text{ln}\rho)##, and Shannon entropy to name a few. I was basically taught (all those many years ago) these are all the same thing or maybe they're just for slightly different circumstances (but that they measure the same "physical thing" in different circumstances). But they really seem not, despite formal similarities with each other. Especially the Shannon one seems very different than the rest.
2. So many definitions "under different circumstances" it is hard to keep them straight in my head, and I really can't see how they define "the same thing".
3. If they're not all the same thing then it would seem the second law must be shown to be true in each of their cases. And if they are all the same thing then the equivalence would have to be shown case by case.
Furthermore, the ontological status of entropy is unclear and seems to differ for each definition. What I mean by "ontological status" is vague, but roughly speaking it's the question "is entropy a physical quantity like mass/distance/time/energy (it certainly has units Joule/Kelvin) or is it a consequence of our ignorance of the microphysics like not knowing how a dice will land after being thrown?" I.e. if we "knew everything there is to know" would entropy disappear?
Even within one definition, the ontological status seems murky. Take the Boltzmann entropy as an example. We could partition the phase space up into separate chunks based on "our knowledge of the macro states", course grain it (this concept is also fuzzy to me, but I see it very roughly as some way to define a "unit volume" in phase space, correct me if I'm wrong) and count ##\Omega##. The entropy then seems to be a property of the partition (essentially, the log of the quantized phase space volume) and not the actual trajectory (in Classical mechanics this is simply one point in phase space). Each time I think on this I come to a different conclusion on whether entropy is a physical thing (my trajectory is in this particular partition, so at least it makes some certain sense to attribute all physical configurations in here to have some certain value of entropy) or not (the partitions seem arbitrary based on what integrals of the motion I have knowledge of).
Other views might also bolster a physical interpretation -- e.g. that the arrow of time is somehow defined using the second law. But then you get things like the Poincare recurrence which seems to just massively violate the second law given long enough time scales.
Anyways I'll stop here before I ramble on too long. I'll just say I am heavily influenced by the thinking of David Albert and Tim Maudlin, but even their expositions here are not entirely convincing. I can't seem to form a concrete opinion on the matter.