I will be a bit sloppy with my description here: There are two ways to "prove" thermodinamics. You can start from the axioms of probability and statistics *together* with an assumption that matter is made out of atoms or similar constituents, and you can derive, through statistical physics, most of thermodynamics. That is a perfectly reasonable and practical way to do it.

However, it is extremely valuable to also understand that the vast majority of thermodynamics (including our understanding of entropy) can be derived from completely independent axioms that have no relationship to information theory or the knowledge of discreteness of matter. You need to define temperature as an associative observable (zeroth law of thermodynamics), you need to define heat as a type of energy (first law of thermodynamics), and you need some version of the second law (there are a few equivalent ones, but the easiest is that "heat flows only from high temperature to low temperature"). Lastly, you need equations of state as a starting point (instead of assumptions about the microscoping constituents of matter). Tadaaa, you have all that is necessary to derive all our knowledge of thermodynamics without knowing statistics or atoms.

Aesthetically this is incredibly pleasing and intellectually it is quite surprising that these two sets of axioms that look nothing alike produce the same self-consistent physics. To be productive in cutting edge research you usually need to be comfortable with both.

> it is quite surprising that these two sets of axioms that look nothing alike produce the same self-consistent physics

Why is that surprising? This sort of thing seems to be ubiquitous in our universe. F=ma is equivalent to the principle of least action. Turing machines are equivalent to the lambda calculus. Electrodynamics is equivalent to the existence of a finite reference velocity.

Also:

> You need to define temperature as an associative observable

Did you mean transitive? My understanding is that the required axiom is that if system A is in thermal equilibrium with system B, and B is in equilibrium with C, then A will be in equilibrium with C. That's transitivity, not associativity.

Indeed, this type of "surprises" is how you know you are probably on the right track with your choice of abstraction. But I still find the word "surprise" reasonable for the emotion I feel when I observe something like that. Very pleasing reassuring intellectually satisfying surprise. A "sufficiently smart armchair theorist" might discover these "surprises" in advance simply from consistency and beauty and simplicity considerations, but these armchair theorists are usually just a made up thing we imagine when teaching.

I corrected the wrong use of the word associative, thanks!

Sure enough, the principles of Carnot's thermodynamics and the premises of statistical mechanisc look quite differently. The thing is: since both form the same thermodynamics there must be a connection.

I submit: the qualification 'completely independent axioms' is incorrect. There is the observation: temperature is transitive. This transitive property is a statement of conservation (In terms of Carnot's thermodynamic it used to be thought of as conservation of Caloric.) The concept of Conservation of a quantity correlates with information.

We have that statistical mechanics subsumed Carnot's thermodynamics.

The laws of Carnot's thermodynamics are theorems of statistical mechanics. (Those theorems weren't necessarily stated explicitly. I'm saying the principles of statistical mechanics are sufficient to imply the laws of Carnot's thermodynamics.)

Statistical mechanics needs to assume the existence of atoms to give you the results of Carnot's. Carnot's thermodynamics does not. That makes it a bit less straightforward to say one strictly follows from the other. Otherwise the gist of what you say seems reasonable.

Here is one way of looking at it: statistical mechanics introduced the concept of entropy.

Years ago, in school, the physics teacher gave the following vivid demonstration:

The demonstration involved two beakers, stacked, the openings facing each other, initially a sheet of thin cardboard separated the two.

In the bottom beaker a quantity of Nitrogen dioxide gas had been had been added. The brown color of the gas was clearly visible. The top beaker was filled with plain air, so it was colorless.

Nitrogen dioxide is denser than air. If the gases would not mix then all of the Nitrogen dioxide would stay in the bottom beaker. But of course the two do mix.

When the separator was removed we saw the brown color of the Nitrogen dioxide rise to the top. In less than half a minute the combined space was an even brown color.

And then the teacher explained the significance: in the process of filling the entire space the heavier Nitrogen dioxide molecules had displaced lighter molecules. That is: a significant part of the population of Nitrogen dioxide had moved against the pull of gravity. This move against gravity is probability driven.

Statistical mechanics provides the means to treat this process quantitatively. You quantify by counting numbers of states. Mixed states outnumber separated states - by far.

The climbing of the Nitrogen dioxide molecules goes at the expense of the temperature of the combined gases. That is, if you make sure that in the initial state the temperature in the two compartments is the same then you can compare the final temperature with that. The temperature of the final mixture will be a bit lower than the starting temperature. That is, some kinetic energy has been converted to gravitational potential energy.

So in this particular demonstration probability was acting in a direction opposite to gravity, and overall probability had the upper hand.

Probability effects fall in the category of emergent phenomena. An emergent phenomenon is somewhat of an in-between category. Not quite as fundamental as the law of gravity, but there is no denying that it has an existence of its own.