Finite-range lattice interaction and conditional energy

Fix an integer dimension $d\ge 1$ and a finite nonempty alphabet $A$. A configuration is a function $\sigma :{\mathbb{Z}}^d\to A$. An interaction list consists of a finite index set $\mathcal{R}$ and, for each $r\in \mathcal{R}$, a finite nonempty shape $S_r\subset {\mathbb{Z}}^d$ containing $0$ and a function ${\mathrm{\Phi }}_r:A^{S_r}\to \mathbb{R}$. The index $r$ is part of the data: two equal shapes with different indices represent two terms and are both counted. Every translate $x+S_r$ contributes once, with value ${\mathrm{\Phi }}_r(({\sigma }_{x+y}{)}_{y\in S_r})$.

The range is $R={\max }_{r\in \mathcal{R}}{\mathrm{diam}}_{\mathrm{\infty }}(S_r)$, with $R=0$ when $\mathcal{R}$ is empty. For finite $\mathrm{\Lambda }\subset {\mathbb{Z}}^d$, define the free-volume energy and the energy conditioned on an exterior configuration $\eta \in A^{{\mathrm{\Lambda }}^c}$ by $H_{\mathrm{\Lambda }}^{\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}}(\sigma )={\sum }_{r\in \mathcal{R}}{\sum }_{x:x+S_r\subseteq \mathrm{\Lambda }}{\mathrm{\Phi }}_r({\sigma }_{x+S_r}),E_{\mathrm{\Lambda }}({\tau }_{\mathrm{\Lambda }}\mid \eta )={\sum }_{r\in \mathcal{R}}{\sum }_{x:(x+S_r)\cap \mathrm{\Lambda }\mathrm{\ne }\mathrm{\varnothing }}{\mathrm{\Phi }}_r(({\tau }_{\mathrm{\Lambda }}\eta {)}_{x+S_r})\mathrm{.}$ Here ${\tau }_{\mathrm{\Lambda }}\in A^{\mathrm{\Lambda }}$, and ${\tau }_{\mathrm{\Lambda }}\eta \in A^{{\mathbb{Z}}^d}$ is the concatenated configuration equal to ${\tau }_{\mathrm{\Lambda }}$ on $\mathrm{\Lambda }$ and to $\eta$ on ${\mathrm{\Lambda }}^c$. These are finite sums: for each finite shape $S_r$, only finitely many anchors $x$ make $x+S_r$ meet $\mathrm{\Lambda }$. Terms meeting $\mathrm{\Lambda }$ but not contained in it appear in the conditional energy and not in the free-volume energy. The definitions are mathematical and assign no physical units; a physical model must state its units separately.

Fix $d\ge 1$. A pair potential is an even Borel function $\varphi :{\mathbb{R}}^d\to \mathbb{R}\cup \{+\mathrm{\infty }\}$. For a finite list of positions $X=(x_1,\dots ,x_N)$, set $U_0=U_1=0$ and $U_N(X)={\sum }_{1\le i<j\le N}\varphi (x_i-x_j)\mathrm{.}$ The value $+\mathrm{\infty }$ is allowed and represents a forbidden configuration; $-\mathrm{\infty }$ is not allowed.

The potential is stable if some finite $B\ge 0$ satisfies $U_N(X)\ge -BN$ for every $N$ and every $X$. Fix a cell side $\mathrm{\ell }>0$ and the half-open cubic tiling $C_k=\mathrm{\ell }(k+[0,1{)}^d)$, $k\in {\mathbb{Z}}^d$, and let $n_k(X)$ count the positions of $X$ in $C_k$. The potential is cell-superstable if there are constants $A_{\ast }>0$ and $B_{\ast }\ge 0$ such that $U_N(X)\ge A_{\ast }{\sum }_{k\in {\mathbb{Z}}^d}{n}_k(X{)}^2-B_{\ast }N$ for every finite configuration; the sum is finite for finite $X$.

For disjoint finite cell-index sets $I,J$, let $W_{I,J}(X)$ be the sum of pair terms with one point in a cell indexed by $I$ and the other in a cell indexed by $J$. The potential is lower regular for this tiling if a nonnegative envelope $\mathrm{\Psi }:{\mathbb{Z}}^d\setminus \{0\}\to [0,\mathrm{\infty })$ exists with ${\sum }_{k\mathrm{\ne }0}\mathrm{\Psi }(k)<\mathrm{\infty }$ and $W_{I,J}(X)\ge -{\sum }_{k\in I}{\sum }_{l\in J}\mathrm{\Psi }(k-l)n_k(X)n_l(X)$ for all such $I,J,X$. The ordered-pair convention in this formula is fixed as written.

A decreasing-integrable tail bound means that for some $R_0\ge 0$ there is a nonnegative nonincreasing function $\psi :[R_0,\mathrm{\infty })\to [0,\mathrm{\infty })$ such that $\mathrm{\mid }\varphi (x)\mathrm{\mid }\le \psi (\mathrm{\mid }x\mathrm{\mid })$ for $\mathrm{\mid }x\mathrm{\mid }\ge R_0$ and ${\int }_{R_0}^{\mathrm{\infty }}\psi (r)r^{d-1}dr<\mathrm{\infty }$. This explicit envelope condition is stronger than bare Lebesgue integrability of an arbitrary signed tail. Stability, cell-superstability, lower regularity, and a tail bound are separate properties; none is inferred from finite range alone.

Use SI units. Let $k_B>0$ be Boltzmann's constant in joules per kelvin, let $T>0$ be absolute temperature in kelvin, and define $\beta =(k_{B}T{)}^{-1}$ in joules${}^{-1}$. For a finite-alphabet lattice model in dimension $d\ge 1$, the lattice spacing $a>0$ is in metres; a spin is a dimensionless label. In the nearest-neighbor Ising convention $H_{\mathrm{\Lambda }}=-J_{\mathrm{I}}{\sum }_{\{i,j\}\subset \mathrm{\Lambda },\mathrm{\mid }i-j{\mathrm{\mid }}_1=1}{s}_i{s}_j-h{\sum }_{i\in \mathrm{\Lambda }}{s}_i,$ the coupling $J_{\mathrm{I}}$ and field parameter $h$ are energies in joules, so $\beta J_{\mathrm{I}}$ and $\beta h$ are dimensionless. Here $h$ is an energy bias per spin, not by itself a magnetic-field strength.

For a finite nonempty region $\mathrm{\Lambda }\subset {\mathbb{Z}}^d$, set $H_{\mathrm{\Lambda }}:=H_{\mathrm{\Lambda }}^{\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}}$ from Finite-range lattice interaction and conditional energy, with each physical interaction value ${\mathrm{\Phi }}_r$ and total energy measured in joules. Thus the partition below uses free boundary conditions: only interaction supports contained in $\mathrm{\Lambda }$ contribute. Exterior-conditioned energies $E_{\mathrm{\Lambda }}(\cdot \mid \eta )$ and their partitions carry a separate exterior label and are not denoted by this $H_{\mathrm{\Lambda }}$ or $Z_{\mathrm{\Lambda }}$. Let $Z_{\mathrm{\Lambda }}={\sum }_{\sigma \in A^{\mathrm{\Lambda }}}{e}^{-\beta H_{\mathrm{\Lambda }}(\sigma )}$ and define dimensionless pressure per site $p_{\mathrm{\Lambda }}=\mathrm{\mid }\mathrm{\Lambda }{\mathrm{\mid }}^{-1}\log Z_{\mathrm{\Lambda }}$. When $p={\lim }_{n\to \mathrm{\infty }}\mathrm{\mid }{B}_n{\mathrm{\mid }}^{-1}\log Z_{B_n}$ exists for cubes $B_n=[-n,n{]}^d\cap {\mathbb{Z}}^d$, define physical pressure $\mathsf{P}=k_{B}Tp\mathrm{/}{a}^d$ in J m${}^{-d}$ (pascals when $d=3$). For a stationary state $\mu$ on a finite alphabet (a probability invariant under every translation $({\theta }_j\sigma {)}_x={\sigma }_{x+j}$), let $H_n=-{\sum }_{\sigma \in A^{B_n}}{\mu }_{B_n}(\sigma )\log {\mu }_{B_n}(\sigma )$ with $0\log 0=0$; when the limit exists, its Shannon entropy density is $h_{\mathrm{S}\mathrm{h}}={\lim }_{n\to \mathrm{\infty }}{H}_n\mathrm{/}\mathrm{\mid }{B}_n\mathrm{\mid }$ in nats per site. The mean energy per site is $u={\sum }_{r\in \mathcal{R}}{\mathbb{E}}_{\mu }[{\mathrm{\Phi }}_r]$ in joules per site. Define reference spin entropy density $s=k_B{h}_{\mathrm{S}\mathrm{h}}\mathrm{/}{a}^d$ and free-energy density $f=u\mathrm{/}{a}^d-Ts$; both are per physical volume. Thus $f$ and $\mathsf{P}$ have units J m${}^{-d}$, while $s$ has units J K${}^{-1}$ m${}^{-d}$; for $d=3$ these are the usual per-cubic-metre units.

For a continuum model in dimension $d$, let $x,p\in {\mathbb{R}}^d$ be position and momentum, let $m_{\mathrm{p}}>0$ be particle mass in kilograms, and use kinetic energy $\mathrm{\mid }p{\mathrm{\mid }}^2\mathrm{/}(2m_{\mathrm{p}})$. Assume Countable Choice for the Lebesgue change-of-variables interface The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$). Tonelli's theorem Tonelli's theorem for nonnegative measurable functions on a sigma-finite product and the one-dimensional Gaussian integral The Gaussian integral ${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{-x^2}dx=\sqrt{\pi }$ give ${\int }_{{\mathbb{R}}^d}{e}^{-\mathrm{\mid }q{\mathrm{\mid }}^2}{d}^{d}q={\pi }^{d\mathrm{/}2}$. Set $c=\sqrt{\beta \mathrm{/}(2m_{\mathrm{p}})}>0$ and apply the nonnegative change-of-variables theorem A C^1 diffeomorphism satisfies the change-of-variables formula for nonnegative Lebesgue measurable functions to the $C^1$ diffeomorphism $q\mapsto cq$ of ${\mathbb{R}}^d$ and $f(q)=e^{-\mathrm{\mid }q{\mathrm{\mid }}^2}$. Since its Jacobian determinant is $c^d$, ${\int }_{{\mathbb{R}}^d}{e}^{-\beta \mathrm{\mid }p{\mathrm{\mid }}^2\mathrm{/}(2m_{\mathrm{p}})}{d}^{d}p=c^{-d}{\pi }^{d\mathrm{/}2}=(2\pi m_{\mathrm{p}}{k}_{B}T{)}^{d\mathrm{/}2}\mathrm{.}$ Under the phase-space reference measure $d^{d}x{d}^{d}p\mathrm{/}{h}_{\mathrm{r}\mathrm{e}\mathrm{f}}^d$, with reference action $h_{\mathrm{r}\mathrm{e}\mathrm{f}}>0$ in joule-seconds, the integrated momentum factor is $h_{\mathrm{r}\mathrm{e}\mathrm{f}}^{-d}{\int }_{{\mathbb{R}}^d}{e}^{-\beta \mathrm{\mid }p{\mathrm{\mid }}^2\mathrm{/}(2m_{\mathrm{p}})}{d}^{d}p={\left(\frac{\sqrt{2\pi m_{\mathrm{p}}{k}_{B}T}}{h_{\mathrm{r}\mathrm{e}\mathrm{f}}}\right)}^d={\lambda }_T^{-d},{\lambda }_T=\frac{h_{\mathrm{r}\mathrm{e}\mathrm{f}}}{\sqrt{2\pi m_{\mathrm{p}}{k}_{B}T}}\mathrm{.}$ For chemical potential $\mu$ in joules, define the classical activity $z_c=e^{\beta \mu }{\lambda }_T^{-d}$, with units ${\mathrm{m}}^{-d}$. Thus ${\lambda }_T$ is a length and $z_c$ is inverse volume. This momentum integration explains the one-particle thermal factor; it does not derive particle indistinguishability or a factorial convention. The choice of $h_{\mathrm{r}\mathrm{e}\mathrm{f}}$ is a reference convention, not a consequence of the classical equations of motion. For continuum interactions, the potential hypotheses and energy meaning are as in Stability, superstability and regular tails.

The parameter status in these formulas is model-relative. $T$, $a$, $J_{\mathrm{I}}$, and $h$ are inputs for the chosen lattice model; $m_{\mathrm{p}}$ and $\mu$ are inputs for the chosen continuum model. Here $h$ is an energy bias per dimensionless spin, not a magnetic-field strength. The reference action $h_{\mathrm{r}\mathrm{e}\mathrm{f}}>0$ is a primitive normalization convention, not a derived quantity or an assertion that it equals Planck's constant. In the current SI, $k_B$ has the exact defining value $1.380649\times {10}^{-23}\mathrm{J}{\mathrm{K}}^{-1}$; it is a fixed SI conversion constant, not a measured fit parameter in this model. The inverse temperature $\beta$, thermal wavelength ${\lambda }_T$, activity $z_c$, pressure and entropy densities, energy density $u$, entropy density $s$, and free-energy density $f$ are defined or derived from those inputs and the state as specified above; their formulas do not calibrate the inputs to a material or assert empirical accuracy.

Fix a classical model with physical energy measured in joules and a positive inverse-temperature parameter $\beta =(k_{B}T{)}^{-1}$ in ${\mathrm{J}}^{-1}$. Its global configuration space is a measurable space $(\mathcal{X},\mathcal{F})$. For each finite region $\mathrm{\Lambda }$, specify a local state measurable space $({\mathcal{X}}_{\mathrm{\Lambda }},{\mathcal{F}}_{\mathrm{\Lambda }})$, an admissible exterior measurable space $({\mathcal{E}}_{\mathrm{\Lambda }},{\mathcal{E}}_{\mathrm{\Lambda }})$, a measurable exterior restriction map $r_{\mathrm{\Lambda }}:(\mathcal{X},\mathcal{F})\to ({\mathcal{E}}_{\mathrm{\Lambda }},{\mathcal{E}}_{\mathrm{\Lambda }})$, and a measurable concatenation map $c_{\mathrm{\Lambda }}:{\mathcal{X}}_{\mathrm{\Lambda }}\times {\mathcal{E}}_{\mathrm{\Lambda }}\to \mathcal{X}$. Require $r_{\mathrm{\Lambda }}(c_{\mathrm{\Lambda }}(x,\eta ))=\eta$ for every $x\in {\mathcal{X}}_{\mathrm{\Lambda }}$ and $\eta \in {\mathcal{E}}_{\mathrm{\Lambda }}$, and write $x\eta :=c_{\mathrm{\Lambda }}(x,\eta )$. Specify a jointly measurable conditional energy $H_{\mathrm{\Lambda }}:{\mathcal{X}}_{\mathrm{\Lambda }}\times {\mathcal{E}}_{\mathrm{\Lambda }}\to \mathbb{R}\cup \{+\mathrm{\infty }\}$, where $+\mathrm{\infty }$ marks a forbidden local state, and a measurable kernel $\eta \mapsto m_{\mathrm{\Lambda }}^{\eta }$ of positive $\sigma$-finite reference measures on $({\mathcal{X}}_{\mathrm{\Lambda }},{\mathcal{F}}_{\mathrm{\Lambda }})$; $\sigma$-finite means that ${\mathcal{X}}_{\mathrm{\Lambda }}$ is a countable union of measurable sets of finite $m_{\mathrm{\Lambda }}^{\eta }$-measure. The reference normalization and any units carried by its coordinates are part of the model data. For a thermodynamic partition function, choose the reference normalization so $Z_{\mathrm{\Lambda }}^{\eta }$ is dimensionless. The product $\beta H_{\mathrm{\Lambda }}$ is dimensionless.

For $\eta \in {\mathcal{E}}_{\mathrm{\Lambda }}$, define $Z_{\mathrm{\Lambda }}^{\eta }:={\int }_{{\mathcal{X}}_{\mathrm{\Lambda }}}{e}^{-\beta H_{\mathrm{\Lambda }}(x\mid \eta )}d{m}_{\mathrm{\Lambda }}^{\eta }(x),{\gamma }_{\mathrm{\Lambda }}(C\mid \eta ):=(Z_{\mathrm{\Lambda }}^{\eta }{)}^{-1}{\int }_{{\mathcal{X}}_{\mathrm{\Lambda }}}{\mathbf{1}}_C(x\eta )e^{-\beta H_{\mathrm{\Lambda }}(x\mid \eta )}d{m}_{\mathrm{\Lambda }}^{\eta }(x),C\in \mathcal{F},$ with $e^{-\beta (+\mathrm{\infty })}:=0$. Adopt these conditional probability kernels only when $0<Z_{\mathrm{\Lambda }}^{\eta }<\mathrm{\infty }$ for every finite $\mathrm{\Lambda }$ and admissible $\eta$, and when their dependence on $\eta$ is measurable. A probability measure $\mu$ on $(\mathcal{X},\mathcal{F})$ is a DLR state for this adopted model if, for every finite $\mathrm{\Lambda }$ and every $C\in \mathcal{F}$, $\mu (C)={\int }_{\mathcal{X}}{\gamma }_{\mathrm{\Lambda }}(C\mid r_{\mathrm{\Lambda }}(y))d\mu (y)\mathrm{.}$ A specification is a family of these probability kernels that is proper when ${\gamma }_{\mathrm{\Lambda }}(r_{\mathrm{\Lambda }}^{-1}(D)\mid \eta )={\mathbf{1}}_D(\eta )$ for every $D\in {\mathcal{E}}_{\mathrm{\Lambda }}$ and $\eta \in {\mathcal{E}}_{\mathrm{\Lambda }}$, and consistent when ${\gamma }_{\mathrm{\Lambda }}{\gamma }_{\mathrm{\Delta }}f={\gamma }_{\mathrm{\Lambda }}f$ for every bounded measurable $f:\mathcal{X}\to \mathbb{R}$ and finite $\mathrm{\Delta }\subseteq \mathrm{\Lambda }$. Here ${\gamma }_{\mathrm{\Lambda }}(f\mid \eta ):={\int }_{\mathcal{X}}f(y){\gamma }_{\mathrm{\Lambda }}(dy\mid \eta )$, and composition is explicitly $({\gamma }_{\mathrm{\Lambda }}{\gamma }_{\mathrm{\Delta }}f)(\eta ):={\int }_{\mathcal{X}}{\gamma }_{\mathrm{\Delta }}(f\mid r_{\mathrm{\Delta }}(y)){\gamma }_{\mathrm{\Lambda }}(dy\mid \eta )$. These identities are required when a DLR specification is asserted; normalization alone does not imply them.

In the finite-alphabet lattice instance, let $A$ be finite and nonempty, $\mathcal{X}=A^{{\mathbb{Z}}^d}$ with the cylinder-generated Borel $\sigma$-algebra, ${\mathcal{X}}_{\mathrm{\Lambda }}=A^{\mathrm{\Lambda }}$ with its power-set $\sigma$-algebra, and ${\mathcal{E}}_{\mathrm{\Lambda }}=A^{{\mathrm{\Lambda }}^c}$. The restriction map is $r_{\mathrm{\Lambda }}(\sigma )=\sigma {\mathrm{\mid }}_{{\mathrm{\Lambda }}^c}$ and concatenation is coordinatewise patching. For the finite-range interaction of Finite-range lattice interaction and conditional energy, use counting measure on $A^{\mathrm{\Lambda }}$ and the energy $E_{\mathrm{\Lambda }}(\cdot \mid \eta )$ in joules; the resulting kernels are exactly those in Lattice Gibbs kernels and DLR states. The consistency result Conditional Gibbs kernel consistency supplies their properness and consistency. The cylinder-measure extension in Finite-alphabet product and probability compactness uses the Countable Choice axiom The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$) to construct an infinite-volume probability; no finite conditional sum spends that choice assumption.

When particle-number sectors are present, use the exact interface of Classical particle counting and physical observables. For species counts $N_a$ with total $N={\sum }_a{N}_a\ge 1$ in $d$ spatial dimensions, the phase-space definition Classical configuration, cotangent phase and reference measures sets the labelled reference to $m_N=c_N{\mathrm{\Lambda }}_N$ with arbitrary $c_N>0$. This postulate chooses $c_N=a_0^{-dN}$, where $a_0>0$ is a prescribed reference action in joule-seconds, and uses the species factor $1\mathrm{/}{\prod }_a{N}_a!$ in the normalized quotient reference. Equivalently, the symmetric labelled integral uses $({\prod }_a{N}_a!a_0^{dN}{)}^{-1}dqdp$; the factorial is included once, and the phase-volume dimension $(\mathrm{J}\mathrm{s}{)}^{dN}$ makes this reference dimensionless. For the vacuum sector $N=0$, define the zero-fold phase space to be a singleton with unit Dirac reference measure (consistent with $0!=1$ and $a_0^0=1$); in a finite region with nonempty exterior, the local vacuum still has the energy $H_{\mathrm{\Lambda }}(\mathrm{\varnothing }\mid \eta )$ in its Gibbs weight. If the thermal-wavelength convention of Classical Gibbs quantities and SI reference conventions is used, set $h_{\mathrm{r}\mathrm{e}\mathrm{f}}=a_0$; omit the species factorial for operationally distinguishable labels. These are adopted reference and counting conventions, not consequences of Hamiltonian mechanics or evidence of equilibration. No empirical parameter, measured transition temperature, or uncertainty is asserted here.

This is a model postulate. Normalization, stationarity of a Hamiltonian flow, and finite-volume calculations do not prove equilibration or establish that the model describes nature.

Let $S$ be a finite nonempty set, $H:S\to \mathbb{R}$, and $\beta >0$. Define $Z={\sum }_{s\in S}{e}^{-\beta H(s)}$ and $\gamma (s)=Z^{-1}{e}^{-\beta H(s)}$. For every probability vector $p$ on $S$, with $0\log 0=0$, its Shannon entropy $H_{\mathrm{S}\mathrm{h}}(p)=-{\sum }_{s}p(s)\log p(s)$ and relative entropy $D(p\mathrm{\parallel }\gamma )={\sum }_{p(s)>0}p(s)\log (p(s)\mathrm{/}\gamma (s))$ satisfy $D(p\mathrm{\parallel }\gamma )=-H_{\mathrm{S}\mathrm{h}}(p)+\beta {\sum }_{s}p(s)H(s)+\log Z\ge 0.$ Equality holds exactly when $p=\gamma$. Equivalently, $\gamma$ is the unique minimizer of ${\sum }_{s}p(s)H(s)-{\beta }^{-1}{H}_{\mathrm{S}\mathrm{h}}(p)$, and the minimum is $-{\beta }^{-1}\log Z$.

Let $S$ be finite nonempty, let $H_0,M:S\to \mathbb{R}$, fix $\beta >0$, and set $H_t=H_0-tM$ and $Z(t)={\sum }_{s\in S}{e}^{-\beta H_t(s)}$ for $t\in \mathbb{R}$. For the normalized finite Gibbs law ${\gamma }_t(s)=Z(t{)}^{-1}{e}^{-\beta H_t(s)}$, $\frac{d}{dt}\log Z(t)=\beta {\gamma }_t(M),\frac{d^2}{d{t}^2}\log Z(t)={\beta }^2{\mathrm{Var}}_{{\gamma }_t}(M)\ge 0,$ where ${\mathrm{Var}}_{\gamma }(M):=\gamma (M^2)-\gamma (M{)}^2$. For fixed $H:S\to \mathbb{R}$, the corresponding inverse-temperature identities are $\frac{d}{d\beta }\log Z(\beta )=-{\gamma }_{\beta }(H),\frac{d^2}{d{\beta }^2}\log Z(\beta )={\mathrm{Var}}_{{\gamma }_{\beta }}(H)\ge 0.$ These are finite-state identities; they do not justify differentiating an infinite-volume limit.

Assume the Axiom of Countable Choice. Let $d\ge 1$, let $A$ be a finite nonempty set, and put $X=A^{{\mathbb{Z}}^d}$. Fix a bijection $(i_n{)}_{n\ge 1}:{\mathbb{N}}_{>0}\to {\mathbb{Z}}^d$ and give $X$ the metric $d_X(\sigma ,\eta )={\sum }_{n\ge 1}{2}^{-n}{\mathbf{1}}_{\{{\sigma }_{i_n}\mathrm{\ne }{\eta }_{i_n}\}}\mathrm{.}$ For a finite $F\subseteq {\mathbb{Z}}^d$ and $B\subseteq A^F$, write $[F,B]=\{\sigma \in X:\sigma {\mathrm{\mid }}_F\in B\}$ and let $\mathcal{C}=\{[F,B]:F\subseteq {\mathbb{Z}}^d\text{ finite},B\subseteq A^F\}$ be the cylinder algebra. A family $({\nu }_F{)}_{F\subseteq {\mathbb{Z}}^d\text{ finite}}$ of probability measures on $A^F$ is consistent if $({\pi }_E^F{)}_{\ast }{\nu }_F={\nu }_E$ whenever $E\subseteq F$, where ${\pi }_E^F:A^F\to A^E$ is restriction. Such a family defines ${\mu }_0([F,B])={\nu }_F(B)$; consistency makes this a well-defined normalized finitely additive probability on $\mathcal{C}$. Then $X$ is compact, the Borel sigma-algebra is $\sigma (\mathcal{C})$, and ${\mu }_0$ extends uniquely to a probability on $\sigma (\mathcal{C})$. Moreover, every sequence of probabilities on this sigma-algebra has a subsequence whose cylinder probabilities converge to those of a probability measure; along that subsequence integrals of every continuous $f:X\to \mathbb{R}$ converge.

Assume Countable Choice for the product-measure extension framework of The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$) and Finite-alphabet product and probability compactness. Work in the finite-alphabet configuration space $X=A^{{\mathbb{Z}}^d}$ with its cylinder-generated Borel sigma-algebra. Let the finite-range interaction and its conditional energy be as in Finite-range lattice interaction and conditional energy, and fix $\beta >0$. For a finite $\mathrm{\Lambda }\subset {\mathbb{Z}}^d$ and exterior configuration $\eta \in A^{{\mathrm{\Lambda }}^c}$, put $Z_{\mathrm{\Lambda }}^{\eta }={\sum }_{\tau \in A^{\mathrm{\Lambda }}}{e}^{-\beta E_{\mathrm{\Lambda }}(\tau \mid \eta )},{\gamma }_{\mathrm{\Lambda }}(f\mid \eta )=\frac{{\sum }_{\tau \in A^{\mathrm{\Lambda }}}f(\tau \eta )e^{-\beta E_{\mathrm{\Lambda }}(\tau \mid \eta )}}{Z_{\mathrm{\Lambda }}^{\eta }}$ for bounded measurable $f:X\to \mathbb{R}$. Here ${\gamma }_{\mathrm{\Lambda }}f$ denotes the function $\eta \mapsto {\gamma }_{\mathrm{\Lambda }}(f\mid \eta )$, composition is operator composition, and $\mu {\gamma }_{\mathrm{\Lambda }}(f)=\int {\gamma }_{\mathrm{\Lambda }}(f\mid \eta )d\mu (\eta )$. A proper kernel leaves every exterior-measurable event fixed. A specification is a family $({\gamma }_{\mathrm{\Lambda }}{)}_{\mathrm{\Lambda }\text{ finite}}$ of proper probability kernels satisfying ${\gamma }_{\mathrm{\Lambda }}{\gamma }_{\mathrm{\Delta }}={\gamma }_{\mathrm{\Lambda }}$ whenever $\mathrm{\Delta }\subseteq \mathrm{\Lambda }$. A DLR state is a probability measure $\mu$ on $X$ satisfying $\mu {\gamma }_{\mathrm{\Lambda }}=\mu$ for every finite $\mathrm{\Lambda }$.

For the existence and uniqueness results below, a specification has finite range when for every finite $\mathrm{\Delta }$ and every cylinder event $C$ the function $\eta \mapsto {\gamma }_{\mathrm{\Delta }}({\mathbf{1}}_C\mid \eta )$ depends on only finitely many coordinates of $\eta$. The displayed finite-range Gibbs formula has this property because only finitely many interaction terms meet $\mathrm{\Delta }$, and a cylinder indicator depends on only finitely many additional coordinates.

The formula is the candidate Gibbs specification; Conditional Gibbs kernel consistency ↗ establishes its positive finite normalization, measurability, properness and consistency under the stated hypotheses. The Countable Choice assumption is used for the product-measure construction, not for any finite conditional sum.

Assume the finite-alphabet product-measure convention and conditional Gibbs formula of Lattice Gibbs kernels and DLR states, a finite-range interaction, and $\beta >0$. For every finite $\mathrm{\Lambda }$ and exterior $\eta$, the displayed partition $Z_{\mathrm{\Lambda }}^{\eta }$ is finite and strictly positive, the formula ${\gamma }_{\mathrm{\Lambda }}(\cdot \mid \eta )$ defines a measurable proper probability kernel, and the family satisfies ${\gamma }_{\mathrm{\Lambda }}{\gamma }_{\mathrm{\Delta }}={\gamma }_{\mathrm{\Lambda }}$ whenever finite $\mathrm{\Delta }\subseteq \mathrm{\Lambda }$.

Assume Countable Choice, a finite nonempty alphabet $A$, and a proper consistent finite-range specification $\gamma$ on $X=A^{{\mathbb{Z}}^d}$, with finite range as defined in Lattice Gibbs kernels and DLR states. No translation invariance is required. For every exhaustion ${\mathrm{\Lambda }}_n\uparrow {\mathbb{Z}}^d$ and every sequence of exterior configurations ${\eta }_n$, the measures ${\gamma }_{{\mathrm{\Lambda }}_n}(\cdot \mid {\eta }_n)$ extended with the fixed exterior have a cylinder-convergent subsequence. Every such subsequential limit is a DLR state. Consequently the DLR state set is nonempty, convex, and closed under cylinder convergence. No uniqueness is asserted.

The cited Princeton notes make different broad claims about projective consistency in distinct model sections: the O$(N)$ existence discussion at PDF pp. 16–17 says its finite-volume Gibbs laws do not satisfy the Kolmogorov-extension consistency hypothesis, while the Ising/DLR discussion at pp. 31–32 says the finite-volume family is consistent in that sense. Because those passages concern different model sections, they do not by themselves establish a contradiction for the finite-alphabet specification in this theorem. Neither passage is used here. This proof obtains a DLR state from a cylinder-convergent subsequence, local kernel consistency, and measure extension; it does not treat the finite-volume laws as a projectively consistent family.

Assume Countable Choice, a translation-invariant finite-range interaction on ${\mathbb{Z}}^d$ with finite alphabet, and a nonempty DLR state set. Then there exists at least one translation-stationary DLR state. This assertion does not imply uniqueness or ergodicity.

For probability vectors $p,q$ on a finite set $A$, there is a coupling $(X,Y)$ with $\mathrm{Pr}(X\mathrm{\ne }Y)=\frac{1}{2}{\sum }_{a\in A}\mathrm{\mid }p(a)-q(a)\mathrm{\mid }$. For a finite region $\mathrm{\Lambda }$ with strictly positive single-site Gibbs conditionals, the random-scan heat-bath chain has the finite-volume Gibbs law as its unique stationary distribution. This chain is an auxiliary finite-state proof device, not a physical time evolution.