Could you pls. point to the said book or some other resource for me to learn about this? As such, I follow what you have said, but would love to see the math too. Thanks.
Boltzmann had originally tried to prove that the second law is a mechanical (statistical) fact and several others tried as well. But Poincaré showed that those systems which, regardless of their initial state, inevitably increase entropy later on inevitably reduce it (his recurrence theorem). There are also in fact reversible processes in nature, so it can't be that mechanics alone implies the procession of time. Something more involved is going on.
Carlo Rovelli has also written a lot about the "thermal time" concept in his books.
My takeaway from them is that you can't really get time out of mechanics by itself (statistical or otherwise). In the same way that you can't get baryon asymmetry. It is intrinsically a selection principle on initial conditions.
I just read about Poincaré's recurrence theorem here: https://en.wikipedia.org/wiki/Poincar%C3%A9_recurrence_theor...
Even with this, would the Poincaré recurrence time not be (probabilistically) be very large for any complex system, and thereby practically imply the second law of thermodynamics?
I also just re-read about Maxwell's demon (https://en.wikipedia.org/wiki/Maxwell%27s_demon), but see that there are already various arguments against it even before arguments about the demon moving the system towards Poincaré recurrence.
Thanks.