Interacting marked phase-space and reference measures

Assume AC. Fix $d\ge 1$ and let $\varphi :{\mathbb{R}}^d\to \mathbb{R}\cup \{+\mathrm{\infty }\}$ be the numerical even Borel pair potential. Assume the stability, cell-superstability, lower-regularity, and decreasing-integrable-tail conditions of Stability, superstability and regular tails, with its fixed cell side $\mathrm{\ell }$, and the superstable regular interaction, unit-coordinate, and periodic-image conventions of Superstable regular pair interactions and configurational ensemble notation. These are model hypotheses; the definitions do not prove that a particular material satisfies them. Use the finite-region phase-space and statistical-state conventions of Statistical states, densities, observables and expectations.

Let $E_0>0$ joules be the energy unit, ${\mathrm{\ell }}_0>0$ metres the length unit, $m>0$ kilograms the particle mass, and $p_{\ast }>0$ kg m s${}^{-1}$ the momentum reference. A physical position vector ${\mathbf{q}}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}\in {\mathbb{R}}^d$ in metres has dimensionless numerical coordinate $x={\mathbf{q}}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}\mathrm{/}{\mathrm{\ell }}_0$. From here on, $q\subseteq {\mathrm{\Lambda }}_L=[0,L{)}^d$ denotes a finite configuration of these numerical coordinates, so the physical cube has side ${\mathrm{\ell }}_{0}L$ and its points are ${\mathrm{\ell }}_{0}x$. Each momentum mark $p_x\in {\mathbb{R}}^d$ is an SI momentum vector in kg m s${}^{-1}$. Write $V_L=L^d$ for numerical volume. Fix also a primitive positive scalar conversion parameter $k_B$ with units J K${}^{-1}$, called the Boltzmann constant: the thermal-energy scale assigned to temperature $T$ is $k_{B}T$ joules. Its value is fixed as part of the model data, not derived by this mathematical definition. If $T>0$ kelvin, put ${\beta }_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}=1\mathrm{/}(k_{B}T)$ in J${}^{-1}$ and $b={\beta }_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}{E}_0=E_0\mathrm{/}(k_{B}T)>0$.

The potential $\varphi$ in the two interaction suppliers is numerical in units of $E_0$ and takes dimensionless displacements as input. For a physical displacement $\mathbf{r}\in {\mathbb{R}}^d$ measured in metres, its pair-potential map is $V_{\mathrm{p}\mathrm{a}\mathrm{i}\mathrm{r}}(\mathbf{r}):=E_0\varphi (\mathbf{r}\mathrm{/}{\mathrm{\ell }}_0)$ joules. For a finite positional configuration $q\subseteq {\mathrm{\Lambda }}_L$, set $U_{L,\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}}(q):=H_{\varphi }(q)$ using Superstable regular pair interactions and configurational ensemble notation, and set $U_{L,\mathrm{p}\mathrm{e}\mathrm{r}}(q):=H_{L,\mathrm{p}\mathrm{e}\mathrm{r}}(q)$ using that item's periodic-image sum. The periodic definition is used only for periods for which that sum is defined; the large-$L$ results below use sufficiently large periods as supplied there. Thus $E_0{U}_{L,bc}$ is the dimensional pair energy in either boundary convention.

Let $Q_L$ be the unit-intensity Poisson probability on positional configurations in ${\mathrm{\Lambda }}_L=[0,L{)}^d$, defined by Poisson Configuration Reference. Its $N$-particle conditional law $Q_{L,N}$ is uniform on the unordered simple $N$-point configurations; in labelled coordinates it is $L^{-dN}d{x}_1\cdots d{x}_N$ on the no-collision set. A finite marked configuration in this sector is a simple set $\xi =\{(x,p_x):x\in q\}$ with $q\subseteq {\mathrm{\Lambda }}_L$, $N(q)=N$, and $p_x\in {\mathbb{R}}^d$. Equip the finite marked sector with the sigma-algebra generated by the counts $N_{A\times C}(\xi )=\mathrm{\#}\{(x,p_x)\in \xi :x\in A,p_x\in C\}$ for bounded Borel $A\subseteq {\mathbb{R}}^d$ and Borel $C\subseteq {\mathbb{R}}^d$. Let ${\lambda }_d$ be Lebesgue measure on the SI numerical momentum coordinates and set $r:=p_{\ast }^{-d}{\lambda }_d$. For each fixed $q=\{x_1,\dots ,x_N\}$, define $r^{\otimes q}$ as the image of $r^{\otimes N}$ under $(p_1,\dots ,p_N)\mapsto \{(x_j,p_j):1\le j\le N\}$; this is independent of the enumeration of $q$ because $r^{\otimes N}$ is permutation-invariant. The cell-code representation in Poisson Configuration Reference supplies a measurable enumeration on each finite sector, so $q\mapsto r^{\otimes q}$ is a measurable kernel for the marked-count sigma-algebra. The finite phase reference is $R_{L,N}(dq,dp):=Q_L(dq)r^{\otimes q}(dp)1_{\{N(q)=N\}}\mathrm{.}$ It is sigma-finite, meaning the marked sector is covered by countably many measurable sets of finite reference measure: for each integer $k\ge 1$, restrict to marks satisfying $\mathrm{\mid }{p}_x\mathrm{\mid }\le k{p}_{\ast }$ for every one of the $N$ points. These sets exhaust the sector, and their reference measures are at most $Q_L(N(q)=N)r(B_k{)}^N<\mathrm{\infty }$, where $B_k=\{p:\mathrm{\mid }p\mathrm{\mid }\le k{p}_{\ast }\}$ satisfies $r(B_k)\le (2k{)}^d$ because it lies in the box $[-k{p}_{\ast },k{p}_{\ast }{]}^d$ of Lebesgue volume $(2k{p}_{\ast }{)}^d$ by the box-volume supplier in the dependency closure of Poisson Configuration Reference. The factor $e^{-V_L}\mathrm{/}N!$ from the Poisson sector is retained. In physical coordinates the corresponding one-particle product scale is $h_{\mathrm{r}\mathrm{e}\mathrm{f}}^d={\mathrm{\ell }}_0^d{p}_{\ast }^d$, an action to the power $d$; it is a reference convention, not a derived quantum constant.

For a marked configuration define the total dimensionless energy ${\mathcal{H}}_{L,bc}(q,p)=U_{L,bc}(q)+{\sum }_{x\in q}K(p_x),K(p)=\frac{\mathrm{\mid }p{\mathrm{\mid }}^2}{2m{E}_0}\mathrm{.}$ The corresponding physical Hamiltonian is $H_{L,bc}^{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}:=E_0{\mathcal{H}}_{L,bc}$ joules, so ${\beta }_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}{H}_{L,bc}^{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}=b{\mathcal{H}}_{L,bc}$. For $b>0$, Gaussian mark normalization, moments, and exponential controls gives ${\kappa }_b=\int e^{-bK}r(dp)=(2\pi m{E}_0\mathrm{/}(b{p}_{\ast }^2){)}^{d\mathrm{/}2}$ and the probability kernel ${\nu }_b(dp)={\kappa }_b^{-1}{e}^{-bK(p)}r(dp)$. Whenever the positional canonical partition is finite and positive, nonnegative product integration Tonelli's theorem for nonnegative measurable functions on a sigma-finite product gives exact identities $Z_{Q,L,N,bc}^{\mathrm{p}\mathrm{h}}(b)=\int e^{-b{\mathcal{H}}_{L,bc}}d{R}_{L,N}={\kappa }_b^N{Z}_{Q,L,N,bc}^{\mathrm{p}\mathrm{o}\mathrm{s}}(b),C_{L,N,bc}^{\mathrm{p}\mathrm{h}}=C_{L,N,bc}^{\mathrm{p}\mathrm{o}\mathrm{s}}{\nu }_b^{\otimes q}\mathrm{.}$ Here $Z_{Q,L,N,bc}^{\mathrm{p}\mathrm{o}\mathrm{s}}=E_{Q_L}[1_{\{N(q)=N\}}{e}^{-b{U}_{L,bc}}]$ and the canonical probabilities are the normalized respective integrands. Under a full grand preparation of activity $z_{\mathrm{p}\mathrm{h}}$, integrating each momentum changes the positional activity to $z=z_{\mathrm{p}\mathrm{h}}{\kappa }_b$; thus a positional activity $z=e^{\alpha }$ corresponds to $z_{\mathrm{p}\mathrm{h}}=e^{\alpha }\mathrm{/}{\kappa }_b$. These are exact definitions and normalizations; they do not assert dynamical equilibration or finite-volume grand convergence. Write ${\mathbb{T}}_L^d:={\mathbb{R}}^d\mathrm{/}L{\mathbb{Z}}^d$ for the periodic position space, represented by ${\mathrm{\Lambda }}_L$; translations act modulo $L$, and the translation-average is Lebesgue measure on ${\mathrm{\Lambda }}_L$ divided by $V_L$. For periodic boundary conditions, simultaneous translation of all positions modulo the period preserves $R_{L,N}$ and ${\mathcal{H}}_{L,\mathrm{p}\mathrm{e}\mathrm{r}}$: the conditional position law is uniform on ${\mathbb{T}}_L^d$, the mark reference is invariant under relabelling points, and the image energy depends only on pair displacements. Hence normalized canonical and finite band restrictions are invariant under every such translation.