Oof. I don’t see how you recover any of thermodynamics. S ≥ k is a really odd axiom to start with. It certainly doesn’t superficially imply the second law. It’s

Oof. I don’t see how you recover any of thermodynamics. S ≥ k is a really odd axiom to start with. It certainly doesn’t superficially imply the second law. It’s also rather at odds with the usual statistical formulation in which the entropy of a system in its unique ground state (or simply a system without meaningful microscopic states) is zero. Imagine trying to describe the behavior of a single (idealized) particle in a box using this law — you would get very confused very quickly. For non-physicists: you can model a gas as a bunch of almost entirely non-interacting particles in a box. You can assume that there is some way for them to exchange energy, but you don’t need any details at all of how that works. With some care (and IMO a really fascinating series of arguments) you can derive the ideal gas law and a whole lot of useful thermodynamics. As a first hack at reducing it to one sentence, I might try “at equilibrium, each unit of phase space (or each discrete microstate in a discrete system) is equally likely.” (I personally find statistical mechanics much more intuitive than classical thermodynamics. I have always felt like E, H, F, and G are mathematically correct quantities that come from manipulating partial derivatives but that they mostly lack intuitive value. The quality of the average thermodynamics textbook doesn’t help.)