Van Hove regions and uniformly tempered exterior preparations

Use the dimensionless coordinates, simple configurations, and pair potential of Stable regularly tempered classical positional preparation, with $d\ge 2$ as stated in this physical scope. The boundary $\mathrm{\partial }\mathrm{\Lambda }$ is the topological boundary in the Euclidean topology of ${\mathbb{R}}^d$, and $\mathrm{dist}(x,\mathrm{\partial }\mathrm{\Lambda })={\inf }_{y\in \mathrm{\partial }\mathrm{\Lambda }}\mathrm{\mid }x-y{\mathrm{\mid }}_2$. For a bounded measurable region $\mathrm{\Lambda }$, define its two-sided layer of width $s>0$ by ${\mathrm{\partial }}_s\mathrm{\Lambda }=\{x\in {\mathbb{R}}^d:\mathrm{dist}(x,\mathrm{\partial }\mathrm{\Lambda })\le s\}\mathrm{.}$ A sequence $({\mathrm{\Lambda }}_k{)}_{k\ge 1}$ of bounded measurable regions is van Hove if $0<\mathrm{\mid }{\mathrm{\Lambda }}_k\mathrm{\mid }<\mathrm{\infty }$, $\mathrm{\mid }{\mathrm{\Lambda }}_k\mathrm{\mid }\to \mathrm{\infty }$, and, for every fixed $s>0$, $\frac{\mathrm{\mid }{\mathrm{\partial }}_s{\mathrm{\Lambda }}_k\mathrm{\mid }}{\mathrm{\mid }{\mathrm{\Lambda }}_k\mathrm{\mid }}⟶0.$ This is the two-sided Euclidean boundary-layer condition used by the mathematical shape suppliers; it controls points on both sides of $\mathrm{\partial }{\mathrm{\Lambda }}_k$. Increasing volume alone does not imply it. A free wall permits particles only in ${\mathrm{\Lambda }}_k$ and includes no exterior particles. A periodic wall uses the specified periodic cell and its image convention.

For the deterministic exterior class, let $C=[-1\mathrm{/}2,1\mathrm{/}2{)}^d$ and ${\mathrm{\Lambda }}_n=[-n-1\mathrm{/}2,n+1\mathrm{/}2{)}^d$, with dimensionless volume $V_n=\mathrm{\mid }{\mathrm{\Lambda }}_n\mathrm{\mid }$. A configuration $\zeta \subset {\mathbb{R}}^d$ is locally finite if $\mathrm{\mid }\zeta \cap K\mathrm{\mid }<\mathrm{\infty }$ for every bounded set $K\subset {\mathbb{R}}^d$. For such $\zeta$, write $N_{C+i}(\zeta )=\mathrm{\mid }\zeta \cap (C+i)\mathrm{\mid }$. The parameter $t$ is dimensionless. Given $0\le t<\mathrm{\infty }$, define $\mathrm{\Omega }(t)=\{\zeta :{\sum }_{i\in {\mathrm{\Lambda }}_n\cap {\mathbb{Z}}^d}{N}_{C+i}(\zeta {)}^2\le t{V}_n\text{ for every integer }n\ge 0\}\mathrm{.}$ For an interior finite configuration $\omega \subset {\mathrm{\Lambda }}_n$ and $\zeta \in \mathrm{\Omega }(t)$, its exterior interaction is $W_n(\omega ,\zeta )={\sum }_{x\in \omega }{\sum }_{y\in \zeta \setminus {\mathrm{\Lambda }}_n}\phi (x-y)\mathrm{.}$ This sum is well-defined in $\mathbb{R}\cup \{+\mathrm{\infty }\}$. Indeed, Cauchy--Schwarz and the defining bound for $\mathrm{\Omega }(t)$ give $\mathrm{\mid }\zeta \cap {\mathrm{\Lambda }}_m\mathrm{\mid }\le \sqrt{t}{V}_m=O_t(m^d)$. Since $\omega$ is finite and bounded, for each fixed $x\in \omega$ and all sufficiently large dyadic shells $2^k\le \mathrm{\mid }y{\mathrm{\mid }}_2<2^{k+1}$, one has $\mathrm{\mid }x-y{\mathrm{\mid }}_2\ge c{2}^k$ for some $c>0$. The absolute interaction on that shell is at most a constant times $2^{kd}\psi (c{2}^k)$. These bounds are summable: monotonicity gives ${\int }_{c{2}^{k-1}}^{c{2}^k}{r}^{d-1}\psi (r)dr\ge C_d{2}^{kd}\psi (c{2}^k)$ for a constant $C_d>0$, and the tail integral is finite. Only finitely many near exterior points remain; if one contributes $+\mathrm{\infty }$, set $W_n=+\mathrm{\infty }$, and otherwise the near sum and absolutely convergent tail define $W_n$. A positive infinite value has zero Boltzmann weight. The class $\mathrm{\Omega }(t)$ is a fixed uniform bound on exterior cell counts, not permission for an arbitrary volume-filling external potential. The van Hove and uniformly tempered exterior conditions are separate boundary regimes.