Finite-range continuum Gibbs kernels and DLR notion — Examples

Given: An integer $d\ge 1$, constants $ϵ>0$ and $R>0$, and the stability convention in the current continuum-interaction definition.

The current draft Stability, superstability and regular tails defines stability by the existence of a finite $B\ge 0$ such that $U_N(X)\ge -BN$ for every finite position list $X=(x_1,\dots ,x_N)$. For simple configurations this is the same pair-energy bound after enumerating their points. The supplier is a definition; this counterexample uses its stated interface without a proof-status claim.

Integer part: for every real $x$ there is exactly one integer $m$ with $m\le x<m+1$: for every real $x$ there is an integer $m$ with $m\le x<m+1$.

Given: An integer $d\ge 1$ and the stability and cell-superstability conventions in the current continuum-interaction definition.

The current draft Stability, superstability and regular tails fixes the half-open cubic tiling $C_k=\mathrm{\ell }(k+[0,1{)}^d)$ for some $\mathrm{\ell }>0$ and defines cell superstability by the existence of constants $A_{\ast }>0$, $B_{\ast }\ge 0$ such that $U_N(X)\ge A_{\ast }{\sum }_k{n}_k(X{)}^2-B_{\ast }N$ for every finite position list. It defines stability by $U_N(X)\ge -BN$. This counterexample uses these exact interfaces; the supplier is a definition and no proof-status claim is made.

Integer part: for every real $x$ there is exactly one integer $m$ with $m\le x<m+1$: for every real $x$ there is an integer $m$ with $m\le x<m+1$.

Given: Assume AC. Let $d=3$, ${\mathrm{\Lambda }}_L=[0,L{)}^3$ in numerical coordinates, one species with operationally indistinguishable particles, unit-intensity Poisson reference $Q_{{\mathrm{\Lambda }}_L}$, reference scales $E_0>0$ joules and ${\mathrm{\ell }}_0>0$ metres, temperature $T>0$ kelvin, Boltzmann constant $k_B>0$ J/K, reference action $h_{\mathrm{r}\mathrm{e}\mathrm{f}}>0$ J s, particle mass $m_{\mathrm{p}}>0$ kg, chemical potential $\mu \in \mathbb{R}$ joules, and uniform one-body energy $c_{\mathrm{S}\mathrm{I}}>0$ joules per particle. Set ${\beta }_{\mathrm{S}\mathrm{I}}=(k_{B}T{)}^{-1}$, ${\lambda }_T=h_{\mathrm{r}\mathrm{e}\mathrm{f}}\mathrm{/}\sqrt{2\pi m_{\mathrm{p}}{k}_{B}T}$, physical activity $z_c=e^{{\beta }_{\mathrm{S}\mathrm{I}}\mu }{\lambda }_T^{-3}$, $\alpha =\log ({\mathrm{\ell }}_0^3{z}_c)$, and dimensionless $\beta =E_0{\beta }_{\mathrm{S}\mathrm{I}}$, $z=e^{\alpha }={\mathrm{\ell }}_0^3{z}_c$, and $c=c_{\mathrm{S}\mathrm{I}}\mathrm{/}{E}_0$.

The current draft Interacting classical Gibbs equilibrium adoption adopts finite-region Gibbs conditional laws when normalizers are positive and finite; the current draft Classical particle counting and physical observables supplies the $1\mathrm{/}N!$ sector factor for operationally indistinguishable particles. For this model, identify its locally prescribed action reference with $h_{\mathrm{r}\mathrm{e}\mathrm{f}}$ as required by the adopted thermal-wavelength convention of Interacting classical Gibbs equilibrium adoption; this equality is a declared common-reference choice, not a numerical value supplied by the counting postulate. These are an adopted postulate and reference convention, not proven claims.

The current draft Classical Gibbs quantities and SI reference conventions defines ${\beta }_{\mathrm{S}\mathrm{I}}=(k_{B}T{)}^{-1}$ and continuum activity $z_c=e^{{\beta }_{\mathrm{S}\mathrm{I}}\mu }{\lambda }_T^{-3}$ with ${\lambda }_T=h_{\mathrm{r}\mathrm{e}\mathrm{f}}\mathrm{/}\sqrt{2\pi m_{\mathrm{p}}{k}_{B}T}$.

Set the pair potential to $\varphi \equiv 0$. For each uniform energy $c_{\mathrm{S}\mathrm{I}}\ge 0$, additionally adopt the ideal positional grand-sector weight $z^N{e}^{-\beta cN}$ at fixed physical activity $z_c$. Numerical coordinates satisfy $x_{\mathrm{S}\mathrm{I}}={\mathrm{\ell }}_{0}q$, so $z_c^N{d}^{3N}{x}_{\mathrm{S}\mathrm{I}}=z^N{d}^{3N}q$ and ${\beta }_{\mathrm{S}\mathrm{I}}{c}_{\mathrm{S}\mathrm{I}}=\beta c$. Define its standard positional grand partition by ${\mathrm{\Xi }}_L^c={\sum }_{N\ge 0}(z^N\mathrm{/}N!){\int }_{{\mathrm{\Lambda }}_L^N}{e}^{-\beta cN}d{q}_1\cdots d{q}_N$. Define its reduced pressure as $p(c)={\lim }_{L\to \mathrm{\infty }}{L}^{-3}\log {\mathrm{\Xi }}_L^c$ and SI pressure by $P_{\mathrm{S}\mathrm{I}}(c)=(k_{B}T\mathrm{/}{\mathrm{\ell }}_0^3)p(c)$. Here $\alpha ,z,\beta ,c$ are dimensionless. This is an explicit ideal-gas model and pressure convention, conditional on [A1] and [A2], not an empirical claim.

Assume the Axiom of Choice The Axiom of Choice, as required by the unit-Poisson reference and its count-properties theorem.

Qualified interacting continuum thermodynamic and positional ensemble limits restricts its uniform exterior-pressure conclusion to the stated square-count exterior class. A one-body term $cN$ throughout the volume is not an exterior boundary configuration in that class.

Poisson Configuration Reference defines the unit-intensity Poisson reference $Q$ and its bounded-region restrictions $Q_D$.

Stable finite-range continuum specification normalization and consistency states that for bounded Borel $D$ and $N\ge 0$, the $N$-point sector of $Q_D$ is the pushforward of $e^{-\mathrm{\mid }D\mathrm{\mid }}\mathrm{/}N!$ times product Lebesgue measure on ordered simple tuples.

Poisson Reference Counts and Translation Invariance gives $N_{{\mathrm{\Lambda }}_L}\sim \mathrm{P}\mathrm{o}\mathrm{i}\mathrm{s}\mathrm{s}\mathrm{o}\mathrm{n}(L^3)$ and the exact exponential moment $E_Q{e}^{s{N}_{{\mathrm{\Lambda }}_L}}=\exp (L^3(e^s-1))$.

The real exponential function and the number $e$ by a power series defines $\exp$ by its real power series; The exponential addition formula $\exp (x+y)=\exp (x)\exp (y)$ gives $\exp (x+y)=\exp (x)\exp (y)$, and The exponential function is strictly increasing gives strict monotonicity.

The natural logarithm as the inverse of the exponential function defines $\log$ as the inverse of the real exponential, so $\exp (\log x)=x$ for $x>0$.