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.