Finite statistical microstate model

A finite statistical microstate model is a pair $(\mathrm{\Omega },ϵ)$ where $\mathrm{\Omega }$ is a finite nonempty set and $ϵ:\mathrm{\Omega }\to \mathbb{R}$ assigns a finite numerical energy to each microstate. The physical energy of $x\in \mathrm{\Omega }$ is $E(x)=ϵ(x)\mathrm{J}$, with its zero and any retained internal offsets fixed by the declared material model. For a physical internal-energy interpretation, the declared companion mechanical model supplies, for every admitted microstate preparation $x$, a positive total mass and a center-of-mass position map with a defined componentwise velocity at that preparation, obeying the Galilean transformation conditions of Galilean inertial charts and the center-of-mass rest frame. That velocity selects the statewise instantaneous rest chart in which $E(x)$ is recorded; a finite set and energy map alone do not construct these mechanical data or a common rest chart for a time interval. Bulk translational kinetic energy and separately tracked external-field energy are excluded from these internal-energy values. The energy zero and any internal offsets are fixed consistently across admitted states. If the statistical model also assigns a single equilibrium macrostate to a preparation, $U$ is its internal-energy coordinate of Equilibrium state domains; the pair $(\mathrm{\Omega },ϵ)$ alone does not assert that macroscopic identification. Equal energy values at distinct elements of $\mathrm{\Omega }$ count as distinct microstates.

When two such models are declared independent, their joint microstate set is the Cartesian product ${\mathrm{\Omega }}_A\times {\mathrm{\Omega }}_B$, and additivity of the microscopic energy is the explicit condition ${ϵ}_{AB}(x,y)={ϵ}_A(x)+{ϵ}_B(y)$. Independence and this energy equation are model data; they do not follow merely from the two systems being separated. The definition itself assigns no weights, temperature, probability law, or dynamics.