Scope and empirical status of the many-body and equilibrium models

The page declares and works with the finite-mode and spatial realizations of a declared fixed-$N$ many-body model (The declared fixed-N many-body model: species sector, one-particle data, interaction and the sector energy observable): a species sector of P6, a one-particle energy operator, a pair interaction and the self-adjoint restriction of the labelled Hamiltonian to the adopted symmetric or antisymmetric sector. The page also treats separately declared number-variable free-mode spaces, including the one-mode and countable-mode occupation models of Thermal occupation means of the declared free-mode models: Bose and Fermi factors, product convergence and dilute remainders, under reading (a)(ii) of The canonical equilibrium-ensemble assumption: the Gibbs preparation of a declared model at assigned temperature; these are additional model declarations, not realizations supplied by the fixed-$N$ definition. Every result is conditional on its own stated model and parameters:

• the non-interacting finite-mode model is diagonal in the occupation basis with the stated occupation eigenvalues, partition sum and ground energy (Diagonalisation of the non-interacting finite-mode sector: occupation eigenvalues, partition sum and ground energy);
• the one-body readout of a bounded self-adjoint one-particle quantity has mean equal to $N$ times the pairing of the one-body reduced preparation with that quantity (The one-body readout law: sector means of summed bounded one-particle quantities and their reduced forms);
• for two orthonormal orbitals satisfying the stated regularity and a repulsive Coulomb pair interaction, the spatially antisymmetric (spin-triplet) sector preparation has mean pair energy $D-X$, no greater than the spatially symmetric (spin-singlet) preparation's $D+X$, where $D$ and $X\ge 0$ are the direct and exchange integrals of the cited theorem; the splitting is $2X$, and the triplet's mean pair energy is strictly lower exactly when $X>0$ (Direct and exchange contributions to a two-fermion mean energy and the repulsive-Coulomb ordering);
• in the finite-sector class the canonical preparation is the unique minimiser of the free-energy functional at the assigned $\beta =1\mathrm{/}(k_{B}T)$ (The canonical preparation as the unique free-energy minimiser of the declared finite-sector model);
• in the separately declared number-variable mode models, the free-mode thermal occupation means are the Bose and Fermi factors with the stated product convergence criterion and dilute remainders (Thermal occupation means of the declared free-mode models: Bose and Fermi factors, product convergence and dilute remainders);
• in the finite-sector class, the model's unitary flow preserves the entropy of every declared sector preparation and never maps a preparation different from the canonical one to it (Unitary evolution preserves entropy and cannot produce the canonical preparation from a different one).

The canonical ensemble is the adopted assumption of The canonical equilibrium-ensemble assumption: the Gibbs preparation of a declared model at assigned temperature; the maximum-entropy and free-energy arguments are motivations inside the model, and the free-energy minimum theorem is a statement about the functional, not a derivation of the ensemble from P1-P4. The Boltzmann constant is the exact SI defining constant, while the temperature, chemical potential, masses, couplings and any potential parameters are assigned model parameters with uncertainty (Entropy, the Boltzmann constant and the free-energy functional of the declared finite-sector model, Dimensions, SI unit conventions and quantity types of the framework). The assignment of a concrete sample, mode family or detector to these model objects is a tested identification under P7 (Preparation and apparatus postulate P7: declared preparations, isolation and sampling assumptions).

The page does not assert: thermalization or equilibration of any preparation (the opposite conservation statement is proved for the closed model); a thermodynamic limit, pressure, volume, phase transitions or condensation; entropy or free energy for infinite-dimensional sectors, for which $\rho \log \rho$ need not be trace class; any spin-statistics theorem, which remains the adopted sector assignment of P6; any experimental value, count or measured level splitting. The mode models are declared idealizations: a finitely many-mode or product-of-modes reading of a free system, not a derivation from a spatial continuum model. The exchange computation of the page compares two declared preparations of one model; it does not claim that any atom's spectrum equals the difference computed.

The physical reading of the Gibbs distribution and of the relations among $U$, $S$ and $F$ follows the treatments of Binney-Skinner and Burmistrov-Lunkin at the indicated locators; both are textbook and lecture-note presentations whose thermodynamic identifications are heuristic model assumptions rather than mathematical derivations, and the present page therefore records them as adopted assumptions. Burmistrov-Lunkin work with $\beta$ as a formal inverse-energy parameter and set $k_B=1$ implicitly; the SI reading $\beta =1\mathrm{/}(k_{B}T)$ with the exact BIPM value of $k_B$ used here is our own convention, not a claim made at their locators. No source supplies measured temperature-dependent data for the declared mode models on this page, and no experiment item is created from these sources.