Finite-state thermodynamic-limit contract

Use $\mathbb{N}$ from The natural numbers $\mathbb{N}$ (von Neumann), which includes $0$, and define ${\mathbb{N}}_{>0}:=\mathbb{N}\setminus \{0\}$. A finite-state thermodynamic-limit contract specifies a sequence $(X_k,E_k,g_k,V_k,n_k{)}_{k\in {\mathbb{N}}_{>0}}$, where each $X_k$ is a finite nonempty state set, $E_k:X_k\to \mathbb{R}$ assigns finite energies in joules, $g_k:X_k\to (0,\mathrm{\infty })$ assigns positive dimensionless weights, $V_k>0$ is physical volume in ${\mathrm{m}}^3$, and, for one declared species, $n_k>0$ is its amount in moles. A fixed-density sequence has $V_k\to \mathrm{\infty }$ and $n_k\mathrm{/}{V}_k\to \rho \in (0,\mathrm{\infty })$ in $\mathrm{m}\mathrm{o}\mathrm{l}{\mathrm{m}}^{-3}$. Boundary conditions, interaction rules, and the choice of a volume or amount normalization are part of the contract and must be held fixed or explicitly tracked as $k$ varies.

For $T>0$, set $\beta :=(k_{B}T{)}^{-1}$ in ${\mathrm{J}}^{-1}$, with $k_B$ and units from SI quantities and unit conventions. The finite-state canonical partition sum is

$$Z_k(\beta ):=\sum _{x\in X_k}{g}_k(x)e^{-\beta E_k(x)},$$

as the finite case of Countable canonical model. It is a positive finite real number. Define $F_k(\beta ):=-{\beta }^{-1}\log Z_k(\beta )$ in joules and $f_k(\beta ):=F_k(\beta )\mathrm{/}{V}_k$ in $\mathrm{J}{\mathrm{m}}^{-3}$. The specified thermodynamic-limit value is $f(\beta ,\rho ):={\lim }_{k\to \mathrm{\infty }}{f}_k(\beta )$ if this limit exists. A limit per mole or per lattice site is a different normalization and must be stated separately.

This is a contract for what a limit claim must specify, not a theorem that the limit exists. It asserts no uniformity in $\beta$, derivative interchange, ensemble equivalence, convergence rate, or nonanalyticity. A finite model's regularity and any limiting singularity require their own arguments; neither is inferred from the word “thermodynamic limit.”