Interacting Equilibrium Preparations

Fix an integer spatial dimension $d\ge 1$, a reference length ${\mathrm{\ell }}_0>0$ measured in metres, and a bounded Lebesgue-measurable region $\mathrm{\Lambda }\subset {\mathbb{R}}^d$ with dimensionless volume $V=\mathrm{\mid }\mathrm{\Lambda }\mathrm{\mid }<\mathrm{\infty }$. Its physical region is ${\mathrm{\ell }}_0\mathrm{\Lambda }$. The coordinate $x\in \mathrm{\Lambda }$ is dimensionless and represents physical position ${\mathrm{\ell }}_{0}x$. Use the Euclidean norm $\mathrm{\mid }x{\mathrm{\mid }}_2=({\sum }_{j=1}^d{x}_j^2{)}^{1\mathrm{/}2}$ and the maximum norm $\mathrm{\mid }x{\mathrm{\mid }}_{\mathrm{\infty }}={\max }_{1\le j\le d}\mathrm{\mid }{x}_j\mathrm{\mid }$. A finite positional configuration of $N$ classical point particles in $\mathrm{\Lambda }$ is an unordered simple set $\omega =\{x_1,\dots ,x_N\}\subset \mathrm{\Lambda }$; simple means that no point is repeated. Particle number $N$ is a dimensionless nonnegative integer. When $V>0$, the physical number density corresponding to the dimensionless density $N\mathrm{/}V$ is $N\mathrm{/}({\mathrm{\ell }}_0^{d}V)$, in ${\mathrm{m}}^{-d}$; no density $N\mathrm{/}V$ is assigned when $V=0$.

A pair potential is an even measurable function $\phi :{\mathbb{R}}^d\to \mathbb{R}\cup \{+\mathrm{\infty }\}$, where $\phi (x)$ is the interaction energy in joules at physical displacement ${\mathrm{\ell }}_{0}x$ and evenness means $\phi (x)=\phi (-x)$. Set $E_0=1\mathrm{J}$ and $\widehat{\phi }=\phi \mathrm{/}{E}_0$. Whenever $\widehat{\phi }$ is Borel measurable, it is the dimensionless numerical potential used by Superstable regular pair interactions and configurational ensemble notation. For a labelled representative of $\omega$, define the positional pair energy $H_N(\omega )={\sum }_{1\le i<j\le N}\phi (x_i-x_j),H_0=H_1=0.$ If any pair is forbidden by the potential, its energy and hence $H_N$ are $+\mathrm{\infty }$. This model contains no kinetic energy, internal molecular coordinates, quantum exchange terms, or external field.

The interaction is stable if there is a constant $B\ge 0$, measured in joules, such that $H_N(\omega )\ge -BN$ for every finite simple configuration. It has a regular integrable tail if there are $R>0$ and a nonnegative nonincreasing measurable function $\psi :[R,\mathrm{\infty })\to [0,\mathrm{\infty })$, measured in joules, such that $\mathrm{\mid }\phi (x)\mathrm{\mid }\le \psi (\mathrm{\mid }x{\mathrm{\mid }}_2)$ for $\mathrm{\mid }x{\mathrm{\mid }}_2\ge R$ and ${\int }_R^{\mathrm{\infty }}{r}^{d-1}\psi (r)dr<\mathrm{\infty }\mathrm{.}$ A stable potential with this tail condition is the stable regularly tempered class used for the free-cube results on this page. Superstability and a positive divergent core are stronger, separate hypotheses and are not implied by stability.

At temperature $T>0$ measured in kelvins, use the exact SI Boltzmann constant $k_{\mathrm{B}}=1.380649\times {10}^{-23}\mathrm{J}{\mathrm{K}}^{-1}$ of SI quantities and unit conventions and write $\beta =(k_{\mathrm{B}}T{)}^{-1}$ in joules inverse. Thus $\beta H_N$ is dimensionless. Pass the dimensionless numerical inverse temperature $\widehat{\beta }=E_0\beta =E_0\mathrm{/}(k_{\mathrm{B}}T)$ and numerical energy ${\widehat{H}}_N=H_N\mathrm{/}{E}_0$ to the mathematical partition definitions of Superstable regular pair interactions and configurational ensemble notation; for finite energies, $\widehat{\beta }{\widehat{H}}_N=\beta H_N$. For the periodic energy defined below, use the same conversion ${\widehat{H}}_{L,\mathrm{p}\mathrm{e}\mathrm{r}}=H_{L,\mathrm{p}\mathrm{e}\mathrm{r}}\mathrm{/}{E}_0$. Infinite-energy configurations receive zero positive-temperature weight without evaluating $\beta (+\mathrm{\infty })$. The activity $z>0$ is dimensionless relative to the reference intensity ${\mathrm{\ell }}_0^{-d}$. A free-wall finite-volume model contains only positions in $\mathrm{\Lambda }$ and uses $H_N$ above.

For periodic boundary conditions, identify a cubic cell of side $L$ in these dimensionless coordinates with the torus ${\mathbb{R}}^d\mathrm{/}(L{\mathbb{Z}}^d)$. For its half-open representative ${\mathrm{\Lambda }}_L=[0,L{)}^d$ and a finite simple configuration $\omega \subset {\mathrm{\Lambda }}_L$, define the periodic continuation ${\omega }^{\mathrm{p}\mathrm{e}\mathrm{r}}=\{x+Lk:x\in \omega ,k\in {\mathbb{Z}}^d\}$ and $H_{L,\mathrm{p}\mathrm{e}\mathrm{r}}(\omega )=\frac{1}{2}{\sum }_{x\in \omega }{\sum }_{\begin{array}{c}y\in {\omega }^{\mathrm{p}\mathrm{e}\mathrm{r}}\\ y\mathrm{\ne }x\end{array}}\phi (y-x)\mathrm{.}$ Here the inner sum omits only the identical point $y=x$; it includes images of the same particle in other cells. The factor $1\mathrm{/}2$ removes double counting of ordered representatives of each periodic pair. For any fixed $L>0$ and finite $\omega$, the $\mathrm{\mid }k{\mathrm{\mid }}_{\mathrm{\infty }}=m$ image shell has $O_d(m^{d-1})$ terms and, outside finitely many $m$, $\mathrm{\mid }y-x+Lk{\mathrm{\mid }}_2\ge (L\mathrm{/}2)m$. Monotonicity and the finite weighted tail integral imply ${\sum }_m{m}^{d-1}\psi (Lm\mathrm{/}2)<\mathrm{\infty }$, so the far-image tail is absolutely convergent. The remaining near-image set is finite; a term $+\mathrm{\infty }$ makes $H_{L,\mathrm{p}\mathrm{e}\mathrm{r}}=+\mathrm{\infty }$, and otherwise the finite near terms plus the absolutely convergent tail define the sum. Translating a centered half-open cell to $[0,L{)}^d$ preserves every displacement and this energy. Free-wall and periodic preparations are distinct boundary conditions. All limits on this page concern positional configurations and do not by themselves include a momentum integral or a full phase-space energy shell.

Use the dimensionless coordinates, simple configurations, and pair potential of Stable regularly tempered classical positional preparation, with $d\ge 2$ as stated in this physical scope. The boundary $\mathrm{\partial }\mathrm{\Lambda }$ is the topological boundary in the Euclidean topology of ${\mathbb{R}}^d$, and $\mathrm{dist}(x,\mathrm{\partial }\mathrm{\Lambda })={\inf }_{y\in \mathrm{\partial }\mathrm{\Lambda }}\mathrm{\mid }x-y{\mathrm{\mid }}_2$. For a bounded measurable region $\mathrm{\Lambda }$, define its two-sided layer of width $s>0$ by ${\mathrm{\partial }}_s\mathrm{\Lambda }=\{x\in {\mathbb{R}}^d:\mathrm{dist}(x,\mathrm{\partial }\mathrm{\Lambda })\le s\}\mathrm{.}$ A sequence $({\mathrm{\Lambda }}_k{)}_{k\ge 1}$ of bounded measurable regions is van Hove if $0<\mathrm{\mid }{\mathrm{\Lambda }}_k\mathrm{\mid }<\mathrm{\infty }$, $\mathrm{\mid }{\mathrm{\Lambda }}_k\mathrm{\mid }\to \mathrm{\infty }$, and, for every fixed $s>0$, $\frac{\mathrm{\mid }{\mathrm{\partial }}_s{\mathrm{\Lambda }}_k\mathrm{\mid }}{\mathrm{\mid }{\mathrm{\Lambda }}_k\mathrm{\mid }}⟶0.$ This is the two-sided Euclidean boundary-layer condition used by the mathematical shape suppliers; it controls points on both sides of $\mathrm{\partial }{\mathrm{\Lambda }}_k$. Increasing volume alone does not imply it. A free wall permits particles only in ${\mathrm{\Lambda }}_k$ and includes no exterior particles. A periodic wall uses the specified periodic cell and its image convention.

For the deterministic exterior class, let $C=[-1\mathrm{/}2,1\mathrm{/}2{)}^d$ and ${\mathrm{\Lambda }}_n=[-n-1\mathrm{/}2,n+1\mathrm{/}2{)}^d$, with dimensionless volume $V_n=\mathrm{\mid }{\mathrm{\Lambda }}_n\mathrm{\mid }$. A configuration $\zeta \subset {\mathbb{R}}^d$ is locally finite if $\mathrm{\mid }\zeta \cap K\mathrm{\mid }<\mathrm{\infty }$ for every bounded set $K\subset {\mathbb{R}}^d$. For such $\zeta$, write $N_{C+i}(\zeta )=\mathrm{\mid }\zeta \cap (C+i)\mathrm{\mid }$. The parameter $t$ is dimensionless. Given $0\le t<\mathrm{\infty }$, define $\mathrm{\Omega }(t)=\{\zeta :{\sum }_{i\in {\mathrm{\Lambda }}_n\cap {\mathbb{Z}}^d}{N}_{C+i}(\zeta {)}^2\le t{V}_n\text{ for every integer }n\ge 0\}\mathrm{.}$ For an interior finite configuration $\omega \subset {\mathrm{\Lambda }}_n$ and $\zeta \in \mathrm{\Omega }(t)$, its exterior interaction is $W_n(\omega ,\zeta )={\sum }_{x\in \omega }{\sum }_{y\in \zeta \setminus {\mathrm{\Lambda }}_n}\phi (x-y)\mathrm{.}$ This sum is well-defined in $\mathbb{R}\cup \{+\mathrm{\infty }\}$. Indeed, Cauchy--Schwarz and the defining bound for $\mathrm{\Omega }(t)$ give $\mathrm{\mid }\zeta \cap {\mathrm{\Lambda }}_m\mathrm{\mid }\le \sqrt{t}{V}_m=O_t(m^d)$. Since $\omega$ is finite and bounded, for each fixed $x\in \omega$ and all sufficiently large dyadic shells $2^k\le \mathrm{\mid }y{\mathrm{\mid }}_2<2^{k+1}$, one has $\mathrm{\mid }x-y{\mathrm{\mid }}_2\ge c{2}^k$ for some $c>0$. The absolute interaction on that shell is at most a constant times $2^{kd}\psi (c{2}^k)$. These bounds are summable: monotonicity gives ${\int }_{c{2}^{k-1}}^{c{2}^k}{r}^{d-1}\psi (r)dr\ge C_d{2}^{kd}\psi (c{2}^k)$ for a constant $C_d>0$, and the tail integral is finite. Only finitely many near exterior points remain; if one contributes $+\mathrm{\infty }$, set $W_n=+\mathrm{\infty }$, and otherwise the near sum and absolutely convergent tail define $W_n$. A positive infinite value has zero Boltzmann weight. The class $\mathrm{\Omega }(t)$ is a fixed uniform bound on exterior cell counts, not permission for an arbitrary volume-filling external potential. The van Hove and uniformly tempered exterior conditions are separate boundary regimes.

Let $\mathrm{\Lambda }\subset {\mathbb{R}}^d$ be a bounded Lebesgue-measurable region in the dimensionless coordinates of Stable regularly tempered classical positional preparation. Use the completed product-Lebesgue sigma-algebra on ${\mathrm{\Lambda }}^N$. For $N\ge 1$, let $D_{\mathrm{\Lambda },N}=\{(x_1,\dots ,x_N)\in {\mathrm{\Lambda }}^N:x_i\mathrm{\ne }{x}_j\text{ whenever }i\mathrm{\ne }j\},{\mathcal{C}}_{\mathrm{\Lambda },N}=D_{\mathrm{\Lambda },N}\mathrm{/}{S}_N,$ where the permutation group $S_N$ acts by relabelling coordinates. Give ${\mathcal{C}}_{\mathrm{\Lambda },N}$ the quotient sigma-algebra: $A\subseteq {\mathcal{C}}_{\mathrm{\Lambda },N}$ is measurable exactly when its preimage under the quotient map $q_N:D_{\mathrm{\Lambda },N}\to {\mathcal{C}}_{\mathrm{\Lambda },N}$ is measurable in the trace of the completed product-Lebesgue sigma-algebra. Identify each orbit with its unordered simple configuration. For $N=0$, set ${\mathcal{C}}_{\mathrm{\Lambda },0}=\{\mathrm{\varnothing }\}$. If ${\lambda }_{\mathrm{\Lambda }}^{\otimes N}$ is product Lebesgue measure restricted to $D_{\mathrm{\Lambda },N}$, define the sector reference measure by ${\mu }_{\mathrm{\Lambda },N}=\frac{1}{N!}(q_N{)}_{\mathrm{\#}}{\lambda }_{\mathrm{\Lambda }}^{\otimes N}(N\ge 1),{\mu }_{\mathrm{\Lambda },0}(\{\mathrm{\varnothing }\})=1.$ Equivalently, for every nonnegative measurable function $f$ on ${\mathcal{C}}_{\mathrm{\Lambda },N}$, $\int f(\omega ){\mu }_{\mathrm{\Lambda },N}(d\omega )=\frac{1}{N!}{\int }_{D_{\mathrm{\Lambda },N}}f(q_N(x_1,\dots ,x_N))d^{dN}x\mathrm{.}$ Collision diagonals have product Lebesgue measure zero, so ${\mu }_{\mathrm{\Lambda },N}({\mathcal{C}}_{\mathrm{\Lambda },N})=\mathrm{\mid }\mathrm{\Lambda }{\mathrm{\mid }}^N\mathrm{/}N!$; in particular, symmetric integrands may use the equivalent integral over all of ${\mathrm{\Lambda }}^N$. Thus ${\mu }_{\mathrm{\Lambda },N}$ is a finite measure, not generally a probability. Two labelled tuples represent the same state exactly when they are in one $S_N$ orbit. The factor $1\mathrm{/}N!$ is an adopted indistinguishable-particle positional convention, not a theorem about classical trajectories or every classical description.

Let ${\mathcal{C}}_{\mathrm{\Lambda }}={⨆}_{N\ge 0}{\mathcal{C}}_{\mathrm{\Lambda },N}$ with the disjoint-union sigma-algebra, and regard each ${\mu }_{\mathrm{\Lambda },N}$ as a measure on this union supported on its $N$-point sector. Define the unit-intensity finite-region Poisson reference by $Q_{\mathrm{\Lambda }}=e^{-\mathrm{\mid }\mathrm{\Lambda }\mathrm{\mid }}{\sum }_{N=0}^{\mathrm{\infty }}{\mu }_{\mathrm{\Lambda },N}\mathrm{.}$ Its total mass is one because ${\mu }_{\mathrm{\Lambda },N}({\mathcal{C}}_{\mathrm{\Lambda },N})=\mathrm{\mid }\mathrm{\Lambda }{\mathrm{\mid }}^N\mathrm{/}N!$ and $e^{-\mathrm{\mid }\mathrm{\Lambda }\mathrm{\mid }}{\sum }_{N\ge 0}\mathrm{\mid }\mathrm{\Lambda }{\mathrm{\mid }}^N\mathrm{/}N!=1$. This is a positional reference measure in the chosen dimensionless coordinates; it supplies no empirical claim about particle trajectories.

Use ${\mathrm{\Lambda }}_n=[-n-\frac{1}{2},n+\frac{1}{2}{)}^d$, its side length $L_n=2n+1$ and dimensionless volume $V_n=L_n^d$, the periodic positional energy $H_{L_n,\mathrm{p}\mathrm{e}\mathrm{r}}$ of Stable regularly tempered classical positional preparation, and the exact-$N$ measure ${\mu }_{{\mathrm{\Lambda }}_n,N}$ of Indistinguishable-particle positional measure. Given $N_n\in {\mathbb{N}}_0$, a reduced configuration-energy center $e\in \mathbb{R}$ joules per dimensionless reference volume, and a width ${\delta }_n>0$ in the same units, define $S_n(N_n,e,{\delta }_n)=\{\omega \subset {\mathrm{\Lambda }}_n:\mathrm{\mid }\omega \mathrm{\mid }=N_n,\mathrm{\mid }{H}_{L_n,\mathrm{p}\mathrm{e}\mathrm{r}}(\omega )\mathrm{/}{V}_n-e\mathrm{\mid }\le {\delta }_n\}\mathrm{.}$ When $0<{\mu }_{{\mathrm{\Lambda }}_n,N_n}(S_n)<\mathrm{\infty }$, the uniform positional shell law is $M_n(A)=\frac{{\mu }_{{\mathrm{\Lambda }}_n,N_n}(A\cap S_n)}{{\mu }_{{\mathrm{\Lambda }}_n,N_n}(S_n)}\mathrm{.}$ A shrinking energy-density shell uses some specified sequence ${\delta }_n\downarrow 0$. Its total energy width is $2{\delta }_n{V}_n$, which is $o(V_n)$ but need not be microscopic or bounded. Existence of a normalizing shell and the existence of a suitable shrinking sequence are separate theorem obligations. This is not a surface measure on an exact-energy hypersurface and not a phase-space shell for $K(p)+H_N^{\mathrm{p}\mathrm{e}\mathrm{r}}$.

For the pair potential $\phi$ of Stable regularly tempered classical positional preparation, let $B=\{x\in {\mathbb{R}}^d:\phi (x)=+\mathrm{\infty }\}$. A finite configuration is hard-admissible if every pair displacement avoids $B$. For $z>0$, define the retained-admissibility zero-inverse-temperature free-wall weight of $\omega$ to be $w_0(\omega )=z^{N(\omega )}{\mathbf{1}}_{\{H_{N(\omega )}(\omega )<\mathrm{\infty }\}}\mathrm{.}$ Its finite-volume partition sum, when finite, is ${\mathrm{\Xi }}_{\mathrm{\Lambda }}^{\mathrm{r}\mathrm{e}\mathrm{t},0}(z)={\sum }_{N\ge 0}\int w_0(\omega ){\mu }_{\mathrm{\Lambda },N}(d\omega )$, using Indistinguishable-particle positional measure. This definition sets the weight of a forbidden configuration to zero; it does not assign a value to the undefined product $0\cdot (+\mathrm{\infty })$. When $B=\mathrm{\varnothing }$ the measure is the unrestricted Poisson activity model; for a nonempty hard core it is generally not Poisson. The convention is explicit model data and carries no statement about an arbitrary zero-width energy shell.

For a periodic cell of side $L$ in dimensionless coordinates, use the exact image energy $H_{L,\mathrm{p}\mathrm{e}\mathrm{r}}$ of Stable regularly tempered classical positional preparation. Its periodic retained-admissibility weight is $w_{L,0}^{\mathrm{p}\mathrm{e}\mathrm{r}}(\omega )=z^{N(\omega )}{\mathbf{1}}_{\{H_{L,\mathrm{p}\mathrm{e}\mathrm{r}}(\omega )<+\mathrm{\infty }\}}=z^{N(\omega )}{\mathbf{1}}_{\{\phi (y-x)<+\mathrm{\infty }\text{ for every }x\in \omega ,y\in {\omega }^{\mathrm{p}\mathrm{e}\mathrm{r}},y\mathrm{\ne }x\}}\mathrm{.}$ Thus every forbidden pair between distinct particles and every forbidden periodic-image pair, including a particle with one of its own nontrivial images, receives zero weight. When finite, the corresponding partition sum is ${\mathrm{\Xi }}_L^{\mathrm{r}\mathrm{e}\mathrm{t},0,\mathrm{p}\mathrm{e}\mathrm{r}}(z)={\sum }_{N\ge 0}\int w_{L,0}^{\mathrm{p}\mathrm{e}\mathrm{r}}d{\mu }_{{\mathrm{\Lambda }}_L,N}$. This is the periodic retained-admissibility convention; it is not an unrestricted zero-beta limit of a shell law.

For a specified finite region and free or periodic boundary preparation, take $T>0$, $\beta =(k_{\mathrm{B}}T{)}^{-1}$ with the SI conventions of Stable regularly tempered classical positional preparation, and the corresponding positional energy. Write $H_{\mathrm{\Lambda },N}^{\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}}=H_N$ for a free region and, for a periodic cube ${\mathrm{\Lambda }}_L$, write $H_{{\mathrm{\Lambda }}_L,N}^{\mathrm{p}\mathrm{e}\mathrm{r}}=H_{L,\mathrm{p}\mathrm{e}\mathrm{r}}$; in each case $N=N(\omega )$. Set ${\mathcal{C}}_{\mathrm{\Lambda },N}^{\mathrm{f}\mathrm{i}\mathrm{n}}=\{\omega \in {\mathcal{C}}_{\mathrm{\Lambda },N}:H_{\mathrm{\Lambda },N}^{\mathrm{b}\mathrm{c}}(\omega )<+\mathrm{\infty }\}$. This set is measurable because the pair potential is measurable and the periodic energy is a countable sum of measurable image terms. On it $H_{\mathrm{\Lambda },N}^{\mathrm{b}\mathrm{c}}$ is real-valued. Specialize the measurable-model canonical preparation of Canonical sampling assumption for finite and normalized statistical models to this set with reference measure ${\mu }_{\mathrm{\Lambda },N}$ restricted to it; adopt $Z_{\mathrm{\Lambda },N}^{\mathrm{b}\mathrm{c}}={\int }_{{\mathcal{C}}_{\mathrm{\Lambda },N}^{\mathrm{f}\mathrm{i}\mathrm{n}}}{e}^{-\beta H_{\mathrm{\Lambda },N}^{\mathrm{b}\mathrm{c}}(\omega )}{\mu }_{\mathrm{\Lambda },N}(d\omega ),P_{\mathrm{\Lambda },N}^{\mathrm{c}\mathrm{a}\mathrm{n}}(d\omega )=\frac{{\mathbf{1}}_{{\mathcal{C}}_{\mathrm{\Lambda },N}^{\mathrm{f}\mathrm{i}\mathrm{n}}}(\omega )e^{-\beta H_{\mathrm{\Lambda },N}^{\mathrm{b}\mathrm{c}}(\omega )}}{Z_{\mathrm{\Lambda },N}^{\mathrm{b}\mathrm{c}}}{\mu }_{\mathrm{\Lambda },N}(d\omega ),$ whenever $0<Z_{\mathrm{\Lambda },N}^{\mathrm{b}\mathrm{c}}<\mathrm{\infty }$. Define the Boltzmann factor to be zero on configurations with energy $+\mathrm{\infty }$; thus no value of an exponential at $+\mathrm{\infty }$ is needed. Separately adopt the grand-canonical law at activity $z>0$ to have weights $z^N{e}^{-\beta H_{\mathrm{\Lambda },N}^{\mathrm{b}\mathrm{c}}}$ on the finite-energy part of each sector and zero elsewhere, relative to the same indistinguishable positional measures. Its normalizer is ${\mathrm{\Xi }}_{\mathrm{\Lambda }}(z,\beta )={\sum }_{N\ge 0}{z}^N{Z}_{\mathrm{\Lambda },N}^{\mathrm{b}\mathrm{c}}$ when finite. Here $z$ is dimensionless relative to the reference intensity in Indistinguishable-particle positional measure, and $\alpha =\log z$.

For the controlled positional shell of Exact-particle-number positional energy shell, adopt the conditional measure $M_n$ only when its shell has positive finite reference measure. These are probability-law preparations used to define the model. They are not derived from Hamiltonian trajectories, and they do not assert that a laboratory apparatus reaches any of them. The inverse-temperature-zero hard-core continuation is governed separately by Retained hard-core continuation at zero inverse temperature.

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 $\mathrm{\Phi }(P)$:

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.

Assume AC. Let $d\ge 2$, $\phi$ be stable with the regular integrable tail of Stable regularly tempered classical positional preparation, and assume its dimensionless representative $\widehat{\phi }=\phi \mathrm{/}{E}_0$ is Borel measurable, as required by the current mathematical supplier. Take $T>0$ and $z>0$ as in Finite-volume equilibrium ensemble sampling, set $E_0=1\mathrm{J}$, $\beta =(k_{B}T{)}^{-1}$, and $\widehat{\beta }=E_0\beta >0$. For cubes ${\mathrm{\Lambda }}_L=[-L\mathrm{/}2,L\mathrm{/}2{)}^d$, let ${\mathrm{\Xi }}_L(z,\beta )$ be the free-wall grand partition function specified by Finite-volume equilibrium ensemble sampling. Then the positional grand pressure per physical $d$-volume $p_L(z,T)=\frac{k_{B}T}{{\mathrm{\ell }}_0^d\mathrm{\mid }{\mathrm{\Lambda }}_L\mathrm{\mid }}\log {\mathrm{\Xi }}_L(z,\beta )$ has a finite limit as $L\to \mathrm{\infty }$. Its units are $\mathrm{J}{\mathrm{m}}^{-\mathrm{d}}$; for $d=3$ this is SI pressure in pascals. In the dimensionless reference-volume convention of Canonical–grand-canonical thermodynamic duality, let $P(\alpha ,\widehat{\beta })={\lim }_{L\to \mathrm{\infty }}\mathrm{\mid }{\mathrm{\Lambda }}_L{\mathrm{\mid }}^{-1}\log {\mathrm{\Xi }}_L(e^{\alpha },\widehat{\beta })$, where the supplier uses numerical energies in units of $E_0$. Then $p=(k_{B}T\mathrm{/}{\mathrm{\ell }}_0^d)P(\log z,\widehat{\beta })$. This is a conditional model deduction, not a measurement or an assertion that every real gas realizes the preparation.

Assume AC, the ultrafilter extension principle, DC, and HB, as required by the current variational-pressure supplier. Let $E_0=1\mathrm{J}$, $\widehat{\phi }=\phi \mathrm{/}{E}_0$, and assume $\widehat{\phi }$ is Borel measurable. Require the exact superstable regular hypotheses of Superstable regular pair interactions and configurational ensemble notation for this dimensionless potential: a stable part plus a nonnegative core part bounded below by a positive constant on an ${\mathrm{\ell }}^{\mathrm{\infty }}$-ball, with the supplier's lower and upper regular tail conditions. Use $\widehat{H}=H\mathrm{/}{E}_0$; then $\widehat{\beta }\widehat{H}=\beta H$ when $\widehat{\beta }=E_0\beta$.

Use the stationary finite-intensity law space ${\mathcal{P}}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}}^1$ and the supplier's $\rho (P)$ (expected count in a dimensionless unit cube), $I(P)$ (specific relative entropy against the unit-intensity Poisson law), and $\widehat{\mathrm{\Phi }}(P)$ (limiting numerical positional energy density) from Superstable regular continuum grand pressure and variational formula. Define $\nu =\sup \{\rho (P):P\in {\mathcal{P}}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}}^1,I(P)<\mathrm{\infty },\widehat{\mathrm{\Phi }}(P)<\mathrm{\infty }\},e_0(\rho )=\inf \{\widehat{\mathrm{\Phi }}(P):P\in {\mathcal{P}}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}}^1,\rho (P)=\rho ,I(P)<\mathrm{\infty }\},$ with the supremum of an empty set equal to $0$ and the infimum of an empty set equal to $+\mathrm{\infty }$. Here $\rho$ is particle count per dimensionless reference volume, corresponding to physical number density $\rho \mathrm{/}{\mathrm{\ell }}_0^d$ in ${\mathrm{m}}^{-d}$. Let $D\subset [0,\mathrm{\infty })$ be an interval of positive length and let $\widehat{e}\in \mathbb{R}$ be a numerical energy per dimensionless reference volume. Require $\inf D<\nu$ and $\widehat{e}>{\inf }_{\rho \in D}{e}_0(\rho )$, and set $e=E_0\widehat{e}$ in joules per dimensionless reference volume. The interval-event supplier Interacting density-interval and upper-energy event limit supplies a multiplier pair $(\widehat{\beta },z)$ with $\widehat{\beta }\ge 0$ and $z>0$ for this same $(D,\widehat{e})$. If $\widehat{\beta }>0$, take $\beta =\widehat{\beta }\mathrm{/}{E}_0$ in ${\mathrm{J}}^{-1}$ and $T=E_0\mathrm{/}(k_{\mathrm{B}}\widehat{\beta })>0$ in kelvins. If $\widehat{\beta }=0$, use only the formal retained-hard-admissibility branch below; it is not a finite-temperature preparation.

For integers $n\ge 0$, set $L_n=2n+1$, ${\mathrm{\Lambda }}_n=[-L_n\mathrm{/}2,L_n\mathrm{/}2{)}^d$, and $V_n=\mathrm{\mid }{\mathrm{\Lambda }}_n\mathrm{\mid }=L_n^d$. Define $A_n=\{\omega :N_{{\mathrm{\Lambda }}_n}(\omega )\mathrm{/}{V}_n\in D,H_{L_n,\mathrm{p}\mathrm{e}\mathrm{r}}(\omega )\mathrm{/}(E_0{V}_n)\le \widehat{e}\},$ where $H_{L_n,\mathrm{p}\mathrm{e}\mathrm{r}}$ is the physical positional energy in joules with the image convention of Stable regularly tempered classical positional preparation. Equivalently, the upper-energy event is $H_{L_n,\mathrm{p}\mathrm{e}\mathrm{r}}\mathrm{/}{V}_n\le e$. For each $n$ with positive event probability, set $R_n(B)=Q_{{\mathrm{\Lambda }}_n}(B\cap A_n)\mathrm{/}{Q}_{{\mathrm{\Lambda }}_n}(A_n)$ on the finite count sigma-algebra. Equivalently, with $M_n={\sum }_{N\ge 0}{\mu }_{{\mathrm{\Lambda }}_n,N}$, $R_n(B)=M_n(B\cap A_n)\mathrm{/}{M}_n(A_n)$, since $Q_{{\mathrm{\Lambda }}_n}=e^{-V_n}{M}_n$; use the count-sigma restriction of these positional measures. This defines the density-interval/upper-energy conditional preparation.

For the positive-$\widehat{\beta }$ branch, the finite-volume grand and conditional laws use the adopted positional preparation of Finite-volume equilibrium ensemble sampling at the matching $T$ and $z$. For the $\widehat{\beta }=0$ branch, require a nonempty hard-core displacement set $B=\{x:\phi (x)=+\mathrm{\infty }\}$ and use the formal retained-admissibility continuation of Retained hard-core continuation at zero inverse temperature; do not interpret $0\cdot (+\mathrm{\infty })$. Assume the singleton-minimizer postulate Selected grand-variational phase uniqueness holds on its stated finite-intensity stationary-law domain at this exact supplier-selected multiplier pair and branch.

For either branch, Interacting density-interval and upper-energy event limit gives eventual event positivity, the constrained entropy-rate limit, and common-minimizer accumulation points; at $\widehat{\beta }=0$, Zero-inverse-temperature-parameter limit with forbidden pairs retained identifies the matching retained grand law and minimizer set. Under the stated singleton assumption, both the conditioned laws and the corresponding periodic grand laws converge locally to $P^{\ast }$ for every measurable local observable $f$ supported in a bounded region $A$ and satisfying $\mathrm{\mid }f\mathrm{\mid }\le c(1+N_A)$. The energy-density center $e$ and widths for this model are in joules per dimensionless reference volume; the corresponding physical $d$-volume density has units $\mathrm{J}{\mathrm{m}}^{-\mathrm{d}}$. This is an interval-density and upper-energy result, not a fixed-particle-number or shrinking-shell theorem.

Here local observables are evaluated on the periodic continuation of the prepared torus configuration. This is the same periodic boundary preparation, with no new physical sampling assumption; a configuration is not extended by the empty configuration outside its fundamental cube.

Assume AC and $T>0$, with $\beta =(k_{B}T{)}^{-1}$, and take $z>0$ as in Finite-volume equilibrium ensemble sampling. Under the stable regularly tempered positional model of Stable regularly tempered classical positional preparation and the adopted ensemble weights of Finite-volume equilibrium ensemble sampling, assume the dimensionless potential $\widehat{\phi }=\phi \mathrm{/}{E}_0$ is Borel measurable as required by the current mathematical carriers. Set $E_0=1\mathrm{J}$ and $\widehat{\beta }=E_0\beta$; write $F(\rho ,\beta )$ and $P(\alpha ,\beta )$ for the mathematical rates evaluated at $\widehat{\beta }$. Use free-wall cubes ${\mathrm{\Lambda }}_L=[0,L{)}^d$ with dimensionless volume $V_L=L^d$. Write $Z_L(N;\beta )=\int e^{-\beta H_N}d{\mu }_{{\mathrm{\Lambda }}_L,N}$, extending the integrand by zero at infinite energy, for their canonical positional partitions. Let $F(\rho ,\beta )$ be the canonical log-partition rate per dimensionless reference volume, defined on positive interior attainable densities by $V_L^{-1}\log Z_L(N_L;\beta )\to F(\rho ,\beta )$ for every sequence $N_L\mathrm{/}{V}_L\to \rho$ and extended to the closed effective density domain by the upper-semicontinuous concave extension specified in Canonical–grand-canonical thermodynamic duality, with $F(\rho ,\beta )=-\mathrm{\infty }$ outside that closed domain. Let $P(\alpha ,\beta )={\lim }_{L\to \mathrm{\infty }}{V}_L^{-1}\log {\sum }_{N\ge 0}{e}^{\alpha N}{Z}_L(N;\beta ),\alpha =\log z\mathrm{.}$ Then $P(\alpha ,\beta )={\sup }_{\rho \ge 0}\{F(\rho ,\beta )+\alpha \rho \},$ and, for every $\rho$ in the closed effective density domain of $F$, $F(\rho ,\beta )={\inf }_{\alpha \in \mathbb{R}}\{P(\alpha ,\beta )-\alpha \rho \}\mathrm{.}$ The density $\rho$ is particle count per dimensionless reference volume, corresponding to physical number density $\rho \mathrm{/}{\mathrm{\ell }}_0^d$ in ${\mathrm{m}}^{-d}$. In physical units, $p=(k_{B}T\mathrm{/}{\mathrm{\ell }}_0^d)P$ and the positional Helmholtz free-energy density is $f_{\mathrm{p}\mathrm{o}\mathrm{s}}=-(k_{B}T\mathrm{/}{\mathrm{\ell }}_0^d)F$; both are per physical $d$-volume, in $\mathrm{J}{\mathrm{m}}^{-\mathrm{d}}$ (pascals when $d=3$). The fixed $1\mathrm{/}N!$ reference normalization is that of Indistinguishable-particle positional measure. This duality concerns limiting thermodynamic functions only. It does not imply local ensemble equivalence, uniqueness of a phase, or convergence at excluded density endpoints.