Interacting classical Gibbs equilibrium adoption

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.