Finite-volume equilibrium ensemble sampling

For a specified finite region and free or periodic boundary preparation, take $T>0$, $\beta =(k_{\mathrm{B}}T{)}^{-1}$ with the SI conventions of Stable regularly tempered classical positional preparation, and the corresponding positional energy. Write $H_{\mathrm{\Lambda },N}^{\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}}=H_N$ for a free region and, for a periodic cube ${\mathrm{\Lambda }}_L$, write $H_{{\mathrm{\Lambda }}_L,N}^{\mathrm{p}\mathrm{e}\mathrm{r}}=H_{L,\mathrm{p}\mathrm{e}\mathrm{r}}$; in each case $N=N(\omega )$. Set ${\mathcal{C}}_{\mathrm{\Lambda },N}^{\mathrm{f}\mathrm{i}\mathrm{n}}=\{\omega \in {\mathcal{C}}_{\mathrm{\Lambda },N}:H_{\mathrm{\Lambda },N}^{\mathrm{b}\mathrm{c}}(\omega )<+\mathrm{\infty }\}$. This set is measurable because the pair potential is measurable and the periodic energy is a countable sum of measurable image terms. On it $H_{\mathrm{\Lambda },N}^{\mathrm{b}\mathrm{c}}$ is real-valued. Specialize the measurable-model canonical preparation of Canonical sampling assumption for finite and normalized statistical models to this set with reference measure ${\mu }_{\mathrm{\Lambda },N}$ restricted to it; adopt $Z_{\mathrm{\Lambda },N}^{\mathrm{b}\mathrm{c}}={\int }_{{\mathcal{C}}_{\mathrm{\Lambda },N}^{\mathrm{f}\mathrm{i}\mathrm{n}}}{e}^{-\beta H_{\mathrm{\Lambda },N}^{\mathrm{b}\mathrm{c}}(\omega )}{\mu }_{\mathrm{\Lambda },N}(d\omega ),P_{\mathrm{\Lambda },N}^{\mathrm{c}\mathrm{a}\mathrm{n}}(d\omega )=\frac{{\mathbf{1}}_{{\mathcal{C}}_{\mathrm{\Lambda },N}^{\mathrm{f}\mathrm{i}\mathrm{n}}}(\omega )e^{-\beta H_{\mathrm{\Lambda },N}^{\mathrm{b}\mathrm{c}}(\omega )}}{Z_{\mathrm{\Lambda },N}^{\mathrm{b}\mathrm{c}}}{\mu }_{\mathrm{\Lambda },N}(d\omega ),$ whenever $0<Z_{\mathrm{\Lambda },N}^{\mathrm{b}\mathrm{c}}<\mathrm{\infty }$. Define the Boltzmann factor to be zero on configurations with energy $+\mathrm{\infty }$; thus no value of an exponential at $+\mathrm{\infty }$ is needed. Separately adopt the grand-canonical law at activity $z>0$ to have weights $z^N{e}^{-\beta H_{\mathrm{\Lambda },N}^{\mathrm{b}\mathrm{c}}}$ on the finite-energy part of each sector and zero elsewhere, relative to the same indistinguishable positional measures. Its normalizer is ${\mathrm{\Xi }}_{\mathrm{\Lambda }}(z,\beta )={\sum }_{N\ge 0}{z}^N{Z}_{\mathrm{\Lambda },N}^{\mathrm{b}\mathrm{c}}$ when finite. Here $z$ is dimensionless relative to the reference intensity in Indistinguishable-particle positional measure, and $\alpha =\log z$.

For the controlled positional shell of Exact-particle-number positional energy shell, adopt the conditional measure $M_n$ only when its shell has positive finite reference measure. These are probability-law preparations used to define the model. They are not derived from Hamiltonian trajectories, and they do not assert that a laboratory apparatus reaches any of them. The inverse-temperature-zero hard-core continuation is governed separately by Retained hard-core continuation at zero inverse temperature.