Stable regularly tempered classical positional preparation

Fix an integer spatial dimension $d\ge 1$, a reference length ${\mathrm{\ell }}_0>0$ measured in metres, and a bounded Lebesgue-measurable region $\mathrm{\Lambda }\subset {\mathbb{R}}^d$ with dimensionless volume $V=\mathrm{\mid }\mathrm{\Lambda }\mathrm{\mid }<\mathrm{\infty }$. Its physical region is ${\mathrm{\ell }}_0\mathrm{\Lambda }$. The coordinate $x\in \mathrm{\Lambda }$ is dimensionless and represents physical position ${\mathrm{\ell }}_{0}x$. Use the Euclidean norm $\mathrm{\mid }x{\mathrm{\mid }}_2=({\sum }_{j=1}^d{x}_j^2{)}^{1\mathrm{/}2}$ and the maximum norm $\mathrm{\mid }x{\mathrm{\mid }}_{\mathrm{\infty }}={\max }_{1\le j\le d}\mathrm{\mid }{x}_j\mathrm{\mid }$. A finite positional configuration of $N$ classical point particles in $\mathrm{\Lambda }$ is an unordered simple set $\omega =\{x_1,\dots ,x_N\}\subset \mathrm{\Lambda }$; simple means that no point is repeated. Particle number $N$ is a dimensionless nonnegative integer. When $V>0$, the physical number density corresponding to the dimensionless density $N\mathrm{/}V$ is $N\mathrm{/}({\mathrm{\ell }}_0^{d}V)$, in ${\mathrm{m}}^{-d}$; no density $N\mathrm{/}V$ is assigned when $V=0$.

A pair potential is an even measurable function $\phi :{\mathbb{R}}^d\to \mathbb{R}\cup \{+\mathrm{\infty }\}$, where $\phi (x)$ is the interaction energy in joules at physical displacement ${\mathrm{\ell }}_{0}x$ and evenness means $\phi (x)=\phi (-x)$. Set $E_0=1\mathrm{J}$ and $\widehat{\phi }=\phi \mathrm{/}{E}_0$. Whenever $\widehat{\phi }$ is Borel measurable, it is the dimensionless numerical potential used by Superstable regular pair interactions and configurational ensemble notation. For a labelled representative of $\omega$, define the positional pair energy $H_N(\omega )={\sum }_{1\le i<j\le N}\phi (x_i-x_j),H_0=H_1=0.$ If any pair is forbidden by the potential, its energy and hence $H_N$ are $+\mathrm{\infty }$. This model contains no kinetic energy, internal molecular coordinates, quantum exchange terms, or external field.

The interaction is stable if there is a constant $B\ge 0$, measured in joules, such that $H_N(\omega )\ge -BN$ for every finite simple configuration. It has a regular integrable tail if there are $R>0$ and a nonnegative nonincreasing measurable function $\psi :[R,\mathrm{\infty })\to [0,\mathrm{\infty })$, measured in joules, such that $\mathrm{\mid }\phi (x)\mathrm{\mid }\le \psi (\mathrm{\mid }x{\mathrm{\mid }}_2)$ for $\mathrm{\mid }x{\mathrm{\mid }}_2\ge R$ and ${\int }_R^{\mathrm{\infty }}{r}^{d-1}\psi (r)dr<\mathrm{\infty }\mathrm{.}$ A stable potential with this tail condition is the stable regularly tempered class used for the free-cube results on this page. Superstability and a positive divergent core are stronger, separate hypotheses and are not implied by stability.

At temperature $T>0$ measured in kelvins, use the exact SI Boltzmann constant $k_{\mathrm{B}}=1.380649\times {10}^{-23}\mathrm{J}{\mathrm{K}}^{-1}$ of SI quantities and unit conventions and write $\beta =(k_{\mathrm{B}}T{)}^{-1}$ in joules inverse. Thus $\beta H_N$ is dimensionless. Pass the dimensionless numerical inverse temperature $\widehat{\beta }=E_0\beta =E_0\mathrm{/}(k_{\mathrm{B}}T)$ and numerical energy ${\widehat{H}}_N=H_N\mathrm{/}{E}_0$ to the mathematical partition definitions of Superstable regular pair interactions and configurational ensemble notation; for finite energies, $\widehat{\beta }{\widehat{H}}_N=\beta H_N$. For the periodic energy defined below, use the same conversion ${\widehat{H}}_{L,\mathrm{p}\mathrm{e}\mathrm{r}}=H_{L,\mathrm{p}\mathrm{e}\mathrm{r}}\mathrm{/}{E}_0$. Infinite-energy configurations receive zero positive-temperature weight without evaluating $\beta (+\mathrm{\infty })$. The activity $z>0$ is dimensionless relative to the reference intensity ${\mathrm{\ell }}_0^{-d}$. A free-wall finite-volume model contains only positions in $\mathrm{\Lambda }$ and uses $H_N$ above.

For periodic boundary conditions, identify a cubic cell of side $L$ in these dimensionless coordinates with the torus ${\mathbb{R}}^d\mathrm{/}(L{\mathbb{Z}}^d)$. For its half-open representative ${\mathrm{\Lambda }}_L=[0,L{)}^d$ and a finite simple configuration $\omega \subset {\mathrm{\Lambda }}_L$, define the periodic continuation ${\omega }^{\mathrm{p}\mathrm{e}\mathrm{r}}=\{x+Lk:x\in \omega ,k\in {\mathbb{Z}}^d\}$ and $H_{L,\mathrm{p}\mathrm{e}\mathrm{r}}(\omega )=\frac{1}{2}{\sum }_{x\in \omega }{\sum }_{\begin{array}{c}y\in {\omega }^{\mathrm{p}\mathrm{e}\mathrm{r}}\\ y\mathrm{\ne }x\end{array}}\phi (y-x)\mathrm{.}$ Here the inner sum omits only the identical point $y=x$; it includes images of the same particle in other cells. The factor $1\mathrm{/}2$ removes double counting of ordered representatives of each periodic pair. For any fixed $L>0$ and finite $\omega$, the $\mathrm{\mid }k{\mathrm{\mid }}_{\mathrm{\infty }}=m$ image shell has $O_d(m^{d-1})$ terms and, outside finitely many $m$, $\mathrm{\mid }y-x+Lk{\mathrm{\mid }}_2\ge (L\mathrm{/}2)m$. Monotonicity and the finite weighted tail integral imply ${\sum }_m{m}^{d-1}\psi (Lm\mathrm{/}2)<\mathrm{\infty }$, so the far-image tail is absolutely convergent. The remaining near-image set is finite; a term $+\mathrm{\infty }$ makes $H_{L,\mathrm{p}\mathrm{e}\mathrm{r}}=+\mathrm{\infty }$, and otherwise the finite near terms plus the absolutely convergent tail define the sum. Translating a centered half-open cell to $[0,L{)}^d$ preserves every displacement and this energy. Free-wall and periodic preparations are distinct boundary conditions. All limits on this page concern positional configurations and do not by themselves include a momentum integral or a full phase-space energy shell.