Isothermal-isobaric joint preparation

Fix integers $N,d\ge 1$, a prescribed temperature control $T>0$ in kelvins, $\beta =(k_{B}T{)}^{-1}$ as in SI quantities and unit conventions, and external pressure $P_{\mathrm{e}\mathrm{x}\mathrm{t}}>0$ in J m${}^{-d}$, a fixed dimensionless shape $Q_1\subseteq {\mathbb{R}}^d$ that is Lebesgue measurable, meaning $Q_1\in \mathcal{L}({\mathbb{R}}^d)$, and has Lebesgue measure $\mathrm{\mid }{Q}_1{\mathrm{\mid }}_d:={\lambda }_d(Q_1)=1$ (Lebesgue measurable sets, the family $\mathcal{L}({\mathbb{R}}^n)$, and the restricted set function ${\lambda }_n$), and a positive volume reference $V_0$ in m${}^d$. Write $y=(y_1,\dots ,y_N)\in Q_1^N$, take $dy$ to be product Lebesgue measure on $Q_1^N$, take $dp$ to be Lebesgue measure on ${\mathbb{R}}^{dN}$, and take $dV$ to be Lebesgue measure on $(0,\mathrm{\infty })$. Identify the position domain at volume $V>0$ by $q_i=V^{1\mathrm{/}d}{y}_i$. The Jacobian is $V^N$, which is the full position-volume factor because $\mathrm{\mid }{Q}_1{\mathrm{\mid }}_d=1$. Adopt the joint reference $d{\nu }_N=\frac{dV}{V_0}\frac{V^{N}dydp}{N!a_0^{dN}},$ where $a_0>0$ is the action reference in J s. The factors $dV\mathrm{/}{V}_0$ and $V^{N}dydp\mathrm{/}{a}_0^{dN}$ are dimensionless. Let $H_N:(0,\mathrm{\infty })\times Q_1^N\times {\mathbb{R}}^{dN}\to \mathbb{R}\cup \{+\mathrm{\infty }\}$ be a product-measurable energy-valued Hamiltonian (in joules) for this reference, with $+\mathrm{\infty }$ marking forbidden states. Require $H_N(V,g\cdot (y,p))=H_N(V,y,p)$ for ${\nu }_N$-almost every $(V,y,p)$ and every within-species permutation $g$, acting by simultaneous permutation of the position and momentum particle coordinates. For the displayed single-species factor this group is $S_N$; for several species it is ${\prod }_a{S}_{N_a}$ as in Classical particle counting and physical observables. The volume coordinate is unchanged by this action, so the Gibbs density admits an invariant representative on the labelled lift. Set $e^{-\beta (+\mathrm{\infty }+P_{\mathrm{e}\mathrm{x}\mathrm{t}}V)}:=0$. The product reference ${\nu }_N$ is sigma-finite: $Q_1^N$ has finite measure, while $(0,\mathrm{\infty })$ and ${\mathbb{R}}^{dN}$ have countable finite-measure exhaustions; the factor $V^N$ is finite on each set where $V$ is bounded. Assume $0<{\mathrm{\Delta }}_N:={\int }_{(0,\mathrm{\infty })\times Q_1^N\times {\mathbb{R}}^{dN}}{e}^{-\beta (H_N(V,y,p)+P_{\mathrm{e}\mathrm{x}\mathrm{t}}V)}d{\nu }_N<\mathrm{\infty }\mathrm{.}$ Adopt the probability density ${\rho }_N(V,y,p):={\mathrm{\Delta }}_N^{-1}{e}^{-\beta (H_N(V,y,p)+P_{\mathrm{e}\mathrm{x}\mathrm{t}}V)}$ relative to ${\nu }_N$. The displayed $N!$ is the single-species convention; for several indistinguishable species with fixed counts $N_a$, replace it by ${\prod }_a{N}_a!$ as in Classical particle counting and physical observables. The factor $dV\mathrm{/}{V}_0$ is an explicit volume-reference convention; a different piston state measure gives a different preparation. This postulate includes no piston kinetic energy unless it is explicitly included in the Hamiltonian, and does not claim a smooth wall dynamics.