In statistical physics, is there anything more general than Liouville's equation?

Question

If I understand correctly, in statistical physics there is a hierarchy of equations, each of which can be derived from the previous one. Currently my mental model is the following:

• The more general is Liouville's equation
• Then there is BBGKY hierarchy (being equivalent to Liouville's when the hierarchy is infinite if I understand correctly)
• Then there is Boltzmann's equation, which correspond to the first level of the BBGKY hierarchy using molecular chaos to "close" the system
• Then there are approximations of Boltzmann's equation in different regimes like Navier-Stokes assuming continuity, or Vlasov in the collisionless regime

QUESTIONS:

1. Is this mental model correct, and if not how to correct it?

2. Is there anything more "general" than Liouville's equation or Liouville's equation is exact and work for any system we can think of?

3. In case Liouville's equation is exact, I guess it assume classical physics. What are Liouville's equation in the context of: a) Quantum mechanics b) Special relativity c) Quantum mechanics + special relativity d) General relativity (and what are the associated equations).

Answer 1

One very important thing about statistical physics is its background assumptions that are not codified in equations. These are usually covered in the introductory chapters of statistical physics textbooks and therefore easily overlooked... yet, they answer many questions that arise when dealing with the equations and laws.

Ergodicity
One important assumption is one formulated as "ergodicity" (although other approaches are possible) - the one that says that following a trajectory of a system in a phase space and averaging its state over time is equivalent to averaging over an ensemble of systems. This actually underlies the very application of the Liouville theorem.

"Residual interactions"
Another assumption is neglecting the "residual interactions" that are supposedly responsible for the evolution of the system towards thermal equilibrium and/or the establishment of equilibrium between the system and the thermostat.

Thermodynamic limit
Another point is the thermodynamic limit - that is taking the size of the system and the number of particle to infinity, while keeping its density constant (or some other appropriate limit taking the number of particles to infinity.)

The results like the second law of thermodynamics come as handwaving/dancing around this assumptions. Indeed, it is a trivial exercise to show that a system performing Hamiltonian evolution (i.e., described by the Liouville equation) would never reach a thermodynamic equilibrium: see What maximizes entropy?

There are exist other approaches to thermodynamics, which bypass these assumptions (and avoid the Liouville theorem altogether)... but at the expense of making other ones. E.g., Jaynes maximum entropy approach allows deriving all the thermodynamic distributiosn by simply maximizing the entropy with appropriate constraints:
Microcanonical ensemble through Maximum Entropy method
Notion of Entropy as a functional - is it neccessary?
What is the differentiating factor between work and heat?'

Bonus: Six kinds of entropy