Superstable regular pair interactions and configurational ensemble notation

Assume AC. Fix an integer dimension $d\ge 1$, use the Euclidean norm $\mathrm{\mid }\cdot {\mathrm{\mid }}_2$ from The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$, define $\mathrm{\mid }x{\mathrm{\mid }}_{\mathrm{\infty }}={\max }_j\mathrm{\mid }{x}_j\mathrm{\mid }$, and use the numerical coordinate conventions in Poisson Configuration Reference. A finite configuration is a finite subset $\omega \subset {\mathbb{R}}^d$ with no repeated points; its number of points is $N(\omega )$. A pair potential is an even Borel function, measurable for the Borel sigma-algebra generated by open sets (The Borel sigma-algebra of a topological space), $\varphi :{\mathbb{R}}^d\to \mathbb{R}\cup \{+\mathrm{\infty }\}$, where even means $\varphi (-x)=\varphi (x)$. Define $H_{\varphi }(\omega )={\sum }_{\{x,y\}\subset \omega }\varphi (x-y),H_{\varphi }(\mathrm{\varnothing })=0.$ The sum is over unordered two-point subsets, so every term is evaluated at a nonzero separation; if any term is $+\mathrm{\infty }$, the sum is $+\mathrm{\infty }$.

The potential is stable if some $B\ge 0$ satisfies $H_{\varphi }(\omega )\ge -BN(\omega )$ for every finite configuration. It is superstable in the Georgii splitting convention if $\varphi ={\varphi }_s+{\varphi }_r$, where ${\varphi }_s$ is stable and ${\varphi }_r\ge 0$ everywhere and, for some $r_0,\delta >0$, ${\varphi }_r(x)\ge \delta$ whenever $\mathrm{\mid }x{\mathrm{\mid }}_{\mathrm{\infty }}\le r_0$. It is lower regular if a nonnegative nonincreasing $\psi :[0,\mathrm{\infty })\to [0,+\mathrm{\infty }]$ satisfies $\varphi (x)\ge -\psi (\mathrm{\mid }x{\mathrm{\mid }}_{\mathrm{\infty }}),{\int }_0^{\mathrm{\infty }}\psi (r)r^{d-1}dr<\mathrm{\infty }\mathrm{.}$ It is regular if it is lower regular and, for some $R<\mathrm{\infty }$, $\varphi (x)\le \psi (\mathrm{\mid }x{\mathrm{\mid }}_{\mathrm{\infty }})$ whenever $\mathrm{\mid }x{\mathrm{\mid }}_{\mathrm{\infty }}\ge R$. These are the conventions used for the interacting results on this page. A genuine divergent positive core is the separate additional condition that some nonincreasing $\chi :(0,r_1]\to (0,\mathrm{\infty })$ satisfies $\varphi (x)\ge \chi (\mathrm{\mid }x{\mathrm{\mid }}_{\mathrm{\infty }})$ for $0<\mathrm{\mid }x{\mathrm{\mid }}_{\mathrm{\infty }}\le r_1$ and ${\int }_0^{r_1}\chi (r)r^{d-1}dr=+\mathrm{\infty }$; it is not implied by superstability.

For $L>0$ put ${\mathrm{\Lambda }}_L=[0,L{)}^d$, $V_L=L^d$, and let $H_{L,\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}}$ be $H_{\varphi }$ for configurations restricted to ${\mathrm{\Lambda }}_L$. For a regular potential and sufficiently large periodic cubes, periodize $\omega \subset {\mathrm{\Lambda }}_L$ by ${\omega }^{\mathrm{p}\mathrm{e}\mathrm{r}}=\{x+Lk:x\in \omega ,k\in {\mathbb{Z}}^d\}$ and set $H_{L,\mathrm{p}\mathrm{e}\mathrm{r}}(\omega )=\frac{1}{2}{\sum }_{x\in \omega }{\sum }_{y\in {\omega }^{\mathrm{p}\mathrm{e}\mathrm{r}},y\mathrm{\ne }x}\varphi (y-x),$ interpreting the value as $+\mathrm{\infty }$ if a positive infinite term occurs. Under this regularity hypothesis, the far image sum is absolutely convergent for all sufficiently large $L$ by Diagonal gap for a decreasing integrable interaction tail ↗; this is the image convention used below. With fixed numerical length and energy units, the canonical positional partition is $Z(L,N;\beta )=\frac{1}{N!}{\int }_{{\mathrm{\Lambda }}_L^N}{e}^{-\beta H_{\varphi }(q_1,\dots ,q_N)}d{q}_1\cdots d{q}_N,Z(L,0;\beta )=1,$ where the tuple notation means $H_{\varphi }(q_1,\dots ,q_N):=H_{\varphi }(\{q_1,\dots ,q_N\})$ when the coordinates are distinct, and $H_{\varphi }(q_1,\dots ,q_N):=+\mathrm{\infty }$ on collision diagonals. The integrand is therefore zero on collisions; these diagonals have product Lebesgue measure zero by the no-collision construction in Poisson Configuration Reference, so this convention leaves all partition integrals unchanged. For $N=0$ the empty tuple has energy zero. Here $\beta >0$, and the grand partition is $\mathrm{\Xi }(L;a,\beta )={\sum }_{N\ge 0}{a}^{N}Z(L,N;\beta )$ for numerical activity $a>0$. The normalized unit-intensity Poisson version is $Z_Q(L;z,\beta )=E_{Q_{{\mathrm{\Lambda }}_L}}[z^{N_{{\mathrm{\Lambda }}_L}}{e}^{-\beta H_{L,\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}}}]$; since the Poisson density contributes $e^{-V_L}$, $\mathrm{\Xi }(L;z,\beta )=e^{V_L}{Z}_Q(L;z,\beta )$. A physical pressure identification is not part of this mathematical definition.

For a probability law $P$ on the configuration space of Poisson Configuration Reference, stationarity means $P\circ T_x^{-1}=P$ for every $x\in {\mathbb{R}}^d$. When $P$ is stationary and $E_P{N}_{[0,1{)}^d}<\mathrm{\infty }$ (finite intensity), write $\rho (P)=E_P{N}_{[0,1{)}^d}$. Use the finite-region relative entropy $D$ defined in Relative Entropy of Probability Measures. If $\varphi$ is stable with constant $B$, then $H_{L,\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}}\ge -B{N}_{{\mathrm{\Lambda }}_L}$, so the negative part of $H_{L,\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}}$ is $P$-integrable for every such $P$. Whenever the limit exists, write $\mathcal{E}(P)={\lim }_{L\to \mathrm{\infty }}{V}_L^{-1}{E}_P{H}_{L,\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}},I(P)={\lim }_{L\to \mathrm{\infty }}{V}_L^{-1}D(P_{{\mathrm{\Lambda }}_L}\mathrm{\parallel }{Q}_{{\mathrm{\Lambda }}_L})\mathrm{.}$ In the variational results below, $\mathrm{\Phi }(P)$ denotes this same positional energy density: $\mathrm{\Phi }(P):=\mathcal{E}(P)$. The cited results establish existence under their stated hypotheses; this definition does not assume either limit exists for every stationary finite-intensity law.