Selected grand-variational phase uniqueness

Fix the selected stable regularly tempered pair model with Borel-measurable $\widehat{\phi }=\phi \mathrm{/}{E}_0$ and activity $z>0$, with the SI conventions of Stable regularly tempered classical positional preparation. Select exactly one parameter branch: either a finite temperature $T>0$ in kelvins with $\beta =(k_{\mathrm{B}}T{)}^{-1}>0$ in ${\mathrm{J}}^{-1}$, or the formal retained-hard-admissibility parameter $\beta =0$, to which no finite temperature is assigned. In the mathematical numerical-energy convention use $\widehat{\beta }=E_0\beta$; the two branches correspond to $\widehat{\beta }>0$ and $\widehat{\beta }=0$, respectively. For the superstable branch, require measurable even pair potentials ${\phi }_s,{\phi }_r$ with $\phi ={\phi }_s+{\phi }_r$, where ${\phi }_s$ is stable in joules and ${\phi }_r(x)\ge 0$ joules for every $x$. Require constants $ϵ>0$ joules and $r_0>0$ in dimensionless coordinates such that ${\phi }_r(x)\ge ϵ$ whenever $\mathrm{\parallel }x{\mathrm{\parallel }}_{\mathrm{\infty }}\le r_0$, where $\mathrm{\parallel }x{\mathrm{\parallel }}_{\mathrm{\infty }}={\max }_j\mathrm{\mid }{x}_j\mathrm{\mid }$. After energy conversion, ${\widehat{\phi }}_s={\phi }_s\mathrm{/}{E}_0$ is stable and ${\widehat{\phi }}_r={\phi }_r\mathrm{/}{E}_0\ge 0$ with core lower bound $ϵ\mathrm{/}{E}_0$, exactly as in Superstable regular pair interactions and configurational ensemble notation. This is an additional model hypothesis, not a consequence of stability alone.

Under the Axiom of Choice, use the global measurable configuration space $\mathrm{\Omega }$ of Poisson Configuration Reference, consisting of locally finite simple configurations with sigma-algebra generated by bounded-Borel count maps. For $a\in {\mathbb{R}}^d$, let $T_a\omega (A)=\omega (A-a)$ be its measurable translation. A probability law $P$ on $\mathrm{\Omega }$ is stationary exactly when $P\circ T_a^{-1}=P$ for every $a\in {\mathbb{R}}^d$. For a bounded Borel region $A$, equip its finite simple configuration set ${\mathcal{C}}_A$ with ${\mathcal{F}}_A^{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{t}}=\sigma (N_B:B\subseteq A\text{ Borel})$, without completing it. On this space $r_A(\omega )=\omega \cap A$ is measurable: each generating count satisfies $N_B\circ r_A=N_B$, a bounded-Borel global count. Write $P_A=P\circ r_A^{-1}$ on this exact count sigma-algebra for every stationary $P$. This assertion does not extend $P_A$ to the completed quotient sigma-algebra. The reference $Q_A=Q\circ r_A^{-1}$ is the restriction of the same unit-intensity law $Q$; by Poisson Reference Counts and Translation Invariance and the count-generated restriction sigma-algebra, it equals the restriction to ${\mathcal{F}}_A^{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{t}}$ of the finite-sector law $e^{-\mathrm{\mid }A\mathrm{\mid }}{\sum }_{N\ge 0}{\mu }_{A,N}$ of Indistinguishable-particle positional measure. This count sigma-algebra is contained in the completed quotient sigma-algebra: in each sector, the pullback of $N_B$ is ${\sum }_{j=1}^N{\mathbf{1}}_B(x_j)$, a Borel function, so each count generator is quotient measurable. Thus the stated restriction of the positional measure is defined. Relative entropy and the restrictions below always use ${\mathcal{F}}_A^{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{t}}$, exactly as in the mathematical global supplier, rather than an unproved completion for arbitrary $P$. Indeed, a finite Borel partition of $A$ has multinomial counts under each fixed-$N$ Lebesgue sector, and summing these sector measures with the factor $e^{-\mathrm{\mid }A\mathrm{\mid }}$ gives independent Poisson counts of the corresponding volumes. For each bounded Borel $A$, write $D_A(P)=D(P_A\mathrm{\parallel }{Q}_A)$ using Relative Entropy of Probability Measures: if $P_A\ll Q_A$ and $h_A=d{P}_A\mathrm{/}d{Q}_A$, then $D_A(P)=\int h_A\log h_{A}d{Q}_A$ with $0\log 0=0$; otherwise $D_A(P)=+\mathrm{\infty }$. Let $C_0=[0,1{)}^d$ and ${\mathrm{\Lambda }}_L=[0,L{)}^d$. Define $\rho (P)=E_P{N}_{C_0}$ and let ${\mathcal{P}}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}}^1$ be the stationary laws with finite intensity $\rho (P)<\mathrm{\infty }$. For $P\in {\mathcal{P}}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}}^1$, define each of the following quantities when its own limit exists and is finite; existence or finiteness of $I(P)$ does not require existence or finiteness of $\mathrm{\Phi }(P)$:

$$I(P)=\underset{L\to \mathrm{\infty }}{\lim }\frac{D_{{\mathrm{\Lambda }}_L}(P)}{\mathrm{\mid }{\mathrm{\Lambda }}_L\mathrm{\mid }},\mathrm{\Phi }(P)=\underset{L\to \mathrm{\infty }}{\lim }\frac{E_P[H_{{\mathrm{\Lambda }}_L}^{\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}}]}{\mathrm{\mid }{\mathrm{\Lambda }}_L\mathrm{\mid }}\mathrm{.}$$

Here $I(P)$ and $\rho (P)$ are dimensionless per dimensionless reference volume; the physical number density corresponding to $\rho (P)$ is $\rho (P)\mathrm{/}{\mathrm{\ell }}_0^d$ in ${\mathrm{m}}^{-d}$. The energy density $\mathrm{\Phi }(P)$ is in joules per dimensionless reference volume; the corresponding energy density per physical $d$-volume is $\mathrm{\Phi }(P)\mathrm{/}{\mathrm{\ell }}_0^d$ in $\mathrm{J}{\mathrm{m}}^{-\mathrm{d}}$. For $\beta >0$, use the laws $P\in {\mathcal{P}}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}}^1$ with finite $I(P)$ and $\mathrm{\Phi }(P)$ and adopt, at the selected $(\phi ,\beta ,z)$, that ${\mathrm{arg\hspace{0.17em}min}}_P\{I(P)+\beta \mathrm{\Phi }(P)-\rho (P)\log z\}=\{P^{\ast }\}\mathrm{.}$

For the $\beta =0$ branch, require a nonempty hard-core displacement set $B=\{x:\phi (x)=+\mathrm{\infty }\}$ and use Retained hard-core continuation at zero inverse temperature. A global configuration is hard-admissible when no pair of its distinct points has displacement in $B$. Restrict to $P\in {\mathcal{P}}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}}^1$ assigning probability one to hard-admissible configurations and having finite $I(P)$; adopt, at the selected $z$, that ${\mathrm{arg\hspace{0.17em}min}}_P\{I(P)-\rho (P)\log z\}=\{P^{\ast }\}\mathrm{.}$ These are two explicit parameter-specific physical model restrictions. They do not assert uniqueness for every activity or temperature, do not apply at coexistence when the minimizer set has several elements, and do not infer uniqueness from thermodynamic smoothness.