Canonical sampling assumption for finite and normalized statistical models

Assign a positive temperature $T$ in kelvins and the exact framework constant $k_{\mathrm{B}}$ in $\mathrm{J}{\mathrm{K}}^{-1}$ from SI quantities and unit conventions, and put $\beta =(k_{\mathrm{B}}T{)}^{-1}$ in ${\mathrm{J}}^{-1}$. For a finite microstate model $(\mathrm{\Omega },ϵ)$, adopt the canonical preparation $P_{\beta }(x)=\frac{e^{-\beta E(x)}}{Z(\beta )},Z(\beta )={\sum }_{x\in \mathrm{\Omega }}{e}^{-\beta E(x)}\mathrm{.}$ Here $Z$ is dimensionless, positive, and finite because $\mathrm{\Omega }$ is finite and nonempty.

For a specified nonempty finite or countably infinite state list $I$ with a stated enumeration and finite physical energies $E_i$ in joules, the counted-once branch takes reference weights $g_i=1$. More generally, the model may explicitly assign fixed positive dimensionless reference weights $g_i>0$ and adopts $Z(\beta )={\sum }_{i\in I}{g}_i{e}^{-\beta E_i},P_{\beta }(i)=\frac{g_i{e}^{-\beta E_i}}{Z(\beta )},$ only at parameters with $0<Z(\beta )<\mathrm{\infty }$. Thus the counted-once law is exactly $P_{\beta }(i)=e^{-\beta E_i}\mathrm{/}Z$. To match Countable canonical model, set $E_{\ast }=1\mathrm{J}$, $e_i=E_i\mathrm{/}{E}_{\ast }\in \mathbb{R}$ and $b=E_{\ast }\beta$; its real-valued energy and parameter coordinates are $e_i,b$, and its admissible set is $J_Z$ for the declared parameter interval. Its nonnegative sums are suprema of finite subsums. The weights $g_i$ are declared model data, not inferred degeneracies or measurements.

For a measurable model $(X,\mathrm{\Sigma },\mu ,E)$ with a dimensionless reference measure $\mu$ and measurable finite physical energy $E:X\to \mathbb{R}\mathrm{J}$, adopt the probability density $p_{\beta }=e^{-\beta E}\mathrm{/}Z$ relative to $\mu$, with $Z(\beta )={\int }_X{e}^{-\beta E}d\mu$, only when $0<Z(\beta )<\mathrm{\infty }$. Measurability of $E$ means measurability of its numerical coordinate $E\mathrm{/}{E}_{\ast }$. The integral convention is Measure space and integral at its current draft definition.

This is an adopted statistical-mechanical preparation rule. It is not derived from the macroscopic thermodynamic laws, a finite bath, entropy maximization, or a time-evolution equation. It does not assert that a physical sample has this preparation; that identification is an additional model assumption to be tested against a specified preparation and apparatus.