Revisions of the Phenomenological and Statistical Statements of the Second Law of Thermodynamics.

The status of the Second Law of Thermodynamics, even in the 21st century, is not as certain as when Arthur Eddington wrote about it a hundred years ago. It is not only about the truth of this law, but rather about its strict and exhaustive formulation. In the previous article, it was shown that two of the three most famous thermodynamic formulations of the Second Law of Thermodynamics are non-exhaustive. However, the status of the statistical approach, contrary to common and unfounded opinions, is even more difficult. It is known that Boltzmann did not manage to completely and correctly derive the Second Law of Thermodynamics from statistical mechanics, even though he probably did everything he could in this regard. In particular, he introduced molecular chaos into the extension of the Liouville equation, obtaining the Boltzmann equation. By using the theorem, Boltzmann transferred the Second Law of Thermodynamics thesis to the molecular chaos hypothesis, which is not considered to be fully true. Therefore, the authors present a detailed and critical review of the issue of the Second Law of Thermodynamics and entropy from the perspective of phenomenological thermodynamics and statistical mechanics, as well as kinetic theory. On this basis, Propositions 1-3 for the statements of the Second Law of Thermodynamics are formulated in the original part of the article. Proposition 1 is based on resolving the misunderstanding of the of the Second Kind by introducing the of the Third Kind. Proposition 2 specifies the structure of allowed thermodynamic processes by using the Inequality of Heat and Temperature Proportions inspired by Eudoxus of Cnidus's inequalities defining real numbers. Proposition 3 is a Probabilistic Scheme of the Second Law of Thermodynamics that, like a game, shows the statistical tendency for entropy to increase, even though the possibility of it decreasing cannot be completely ruled out. Proposition 3 is, in some sense, free from Loschmidt's irreversibility paradox.