Grand canonical particle-exchange preparation

Let $T>0$ be the prescribed temperature control in kelvins, $\beta =(k_{B}T{)}^{-1}$, and let the chemical-potential control $\mu$ have units J per particle under the SI convention SI quantities and unit conventions. In this adopted law, $\mu$ is the energy coefficient conjugate to the dimensionless sector count $N$ in the weight $e^{-\beta (H_N-\mu N)}$; no derivative with respect to integer $N$ is used. For $N\ge 1$, let ${\mathrm{\Gamma }}_N$ and its reference measure be those of Classical particle counting and physical observables, and let $H_N:{\mathrm{\Gamma }}_N\to \mathbb{R}\cup \{+\mathrm{\infty }\}$ be Borel measurable for the sector sigma-algebra ${\mathcal{F}}_N$ generated by the open sets of ${\mathrm{\Gamma }}_N$, with $+\mathrm{\infty }$ marking forbidden states, and use the convention $e^{-\beta (+\mathrm{\infty })}:=0$. In each sector require $H_N(gz)=H_N(z)$ for $m_N^{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$-almost every $z$ and every within-species permutation $g\in G_N$, with $G_N$ and its action as in Classical particle counting and physical observables. Hence each sector Gibbs density admits an invariant representative; the factor $e^{\beta \mu N}$ is constant on that sector and preserves this invariance. Operationally distinguishable labels have trivial permutation factors as in that convention. Define the vacuum sector by ${\mathrm{\Gamma }}_0=\{\ast \}$ with sigma-algebra ${\mathcal{F}}_0=\{\mathrm{\varnothing },\{\ast \}\}$, reference mass $m_0^{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}(\{\ast \})=1$, particle number $N=0$, and vacuum Hamiltonian $H_0(\ast )=0$. On the disjoint union $\mathcal{G}:={⨆}_{N=0}^{\mathrm{\infty }}{\mathrm{\Gamma }}_N$, use the sigma-algebra ${\mathcal{F}}_{\mathcal{G}}:=\{A\subseteq \mathcal{G}:A\cap {\mathrm{\Gamma }}_N\in {\mathcal{F}}_N\text{ for every }N\ge 0\},$ where ${\mathcal{F}}_N$ is the declared measurable structure on sector $N$, and the reference measure $m^{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}(A):={\sum }_{N=0}^{\mathrm{\infty }}{m}_N^{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}(A\cap {\mathrm{\Gamma }}_N)\mathrm{.}$ This disjoint-union reference is sigma-finite because it is a countable sum of the sigma-finite sector references, including the finite vacuum measure. Let $\mathrm{\Xi }(\beta ,\mu ):={\sum }_{N=0}^{\mathrm{\infty }}{\int }_{{\mathrm{\Gamma }}_N}{e}^{-\beta (H_N-\mu N)}d{m}_N^{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$ satisfy $0<\mathrm{\Xi }<\mathrm{\infty }$. Adopt the probability measure on $(\mathcal{G},{\mathcal{F}}_{\mathcal{G}})$ with density $\rho (N,z):={\mathrm{\Xi }}^{-1}{e}^{-\beta (H_N(z)-\mu N)}$ relative to $m^{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$. The finiteness condition is part of the model; it requires control of the sector sum and is not implied by each finite-sector partition function being finite. Particle-exchange preparation is an assumption, not a deduction from fixed-$N$ dynamics. Under the vacuum convention above, the $N=0$ contribution to $\mathrm{\Xi }$ is exactly one.