Canonical thermal preparation

For a fixed $N$, controls $\lambda$, measurable classical Hamiltonian $H_{N,\lambda }:{\mathrm{\Gamma }}_N\to \mathbb{R}\cup \{+\mathrm{\infty }\}$, measurable for the phase Borel sigma-algebra and taking energy values in joules, and the declared physical reference measure $m_N^{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$ of Classical particle counting and physical observables, choose a prescribed thermodynamic-temperature control $T>0$ in kelvins and $\beta =(k_{B}T{)}^{-1}$ in J${}^{-1}$ using the SI temperature convention of SI quantities and unit conventions. For every within-species permutation $g\in G$ of Classical particle counting and physical observables, require $H_{N,\lambda }(gz)=H_{N,\lambda }(z)$ for $m_N^{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$-almost every $z$. Thus its Gibbs density has an invariant representative and represents a quotient state under the declared indistinguishable-particle convention. For operationally distinguishable labels the corresponding group is trivial and its factorial is omitted, as specified by that convention. If $0<Z_N(\beta ,\lambda ):={\int }_{{\mathrm{\Gamma }}_N}{e}^{-\beta H_{N,\lambda }}d{m}_N^{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}<\mathrm{\infty },$ adopt the canonical state with density ${\rho }_{\beta ,\lambda }=Z_N(\beta ,\lambda {)}^{-1}{e}^{-\beta H_{N,\lambda }}$ relative to $m_N^{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$. Set $e^{-\beta (+\mathrm{\infty })}=0$. Normalizability is a required condition on the chosen model; bounded-below energy alone does not imply it. This probability law is an adopted preparation, not a consequence of deterministic mechanics or evidence of laboratory equilibration.