Countable canonical model

Let $I$ be a nonempty finite or countably infinite set equipped with a specified enumeration, let $E_i\in \mathbb{R}$ be finite energies, let $g_i>0$ be weights, and let $J\subseteq \mathbb{R}$ be an open parameter interval. For a nonnegative family $(a_i{)}_{i\in I}$ define ${\sum }_{i\in I}{a}_i:={\sup }_{F\subset I,F\text{ finite}}{\sum }_{i\in F}{a}_i$, with finite sums as in Finite sums and finite products, by recursion and the supremum as in Complete ordered field (least-upper-bound property) when the finite subsums are bounded; set the value to $+\mathrm{\infty }$ when they are unbounded. This definition is independent of the specified enumeration. For $\beta \in J$, define the nonnegative partition sum $Z(\beta ):={\sum }_{i\in I}{g}_i\exp (-\beta E_i)$ by this rule. The admissible parameter set is $J_Z:=\{\beta \in J:0<Z(\beta )<\mathrm{\infty }\}$. For $\beta \in J_Z$, define $p_i(\beta ):=g_i\exp (-\beta E_i)\mathrm{/}Z(\beta )$. A countable probability law on $I$ means a nonnegative family with total sum $1$ under the finite-subsum convention above; these weights form such a law because ${\sum }_i{p}_i(\beta )=Z(\beta )\mathrm{/}Z(\beta )=1$. For any real family $X_i$ with ${\sum }_i{p}_i\mathrm{\mid }{X}_i\mathrm{\mid }<\mathrm{\infty }$, define $⟨X{⟩}_{\beta }:={\sum }_i{p}_i{X}_i$. If ${\sum }_i\mathrm{\mid }{p}_i\log (p_i\mathrm{/}{g}_i)\mathrm{\mid }<\mathrm{\infty }$, define the reference-relative entropy $H_g(p):=-{\sum }_i{p}_i\log (p_i\mathrm{/}{g}_i)$. When $g_i=1$, this is the Shannon entropy with natural logarithms. A lower bound on the energies alone is not part of this definition and does not imply $Z(\beta )<\mathrm{\infty }$. For countably infinite $I$, any specified enumeration is used only to write a list; the nonnegative sum itself is the supremum over finite subsums above.

The sum is interpreted using the series convention of Series, partial sums, convergence and the sum, divergence, and the tail series. The model is purely mathematical; any physical interpretation of $E_i$, $g_i$, or $\beta$ requires separate premises.