Balls-in-Cells Demonstration of a Combinatorial Scheme as an Emergent Statement of the Second Law of Thermodynamics

The article is a continuation of research on the status and formulations of the second law of thermodynamics. In the first paper of this series, it was shown that the Kelvin formulation and the temperature formulation of Clausius are not exhaustive compared to the entropy formulation and the generalized Carnot-type formulation. In the previous paper, it was furthermore shown that the statistical formulation is even more problematic. This has now been supported with quotes from Smoluchowski in contrast to quotes from Eddington. A probabilistic scheme (now refined as a combinatorial one) for the second law of thermodynamics was previously proposed, which is not burdened by a fundamental/non-emergent arrow of time and the Loschmidt reversibility paradox. The present new paper numerically investigates and visualizes the combinatorial scheme for the 12 smallest systems of spheres in cells (including 10 equinumerous cases). Despite the small number of balls (from 2 to 11), the number of macrostates in the last system reaches a value of 356 716, with the number of microstates equal to 285 311 670 611. The combinatorial scheme is additionally expanded by three criteria/properties of emergence: cumulative, cardinal, and probabilistic. The first two properties, cumulative and cardinal, of emergence are confirmed (at least in the equinumerous cases), and halfway probabilistic emergence 50%-50% is postulated as an asymptotic hypothesis for larger systems.