The canonical equilibrium-ensemble assumption: the Gibbs preparation of a declared model at assigned temperature

(a) Adopted readings and framework data. Adopt the state and composition postulate State and composition postulate P1: Hilbert spaces, preparations and composite systems (P1), the observable and probability postulate Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes (P2), the autonomous-dynamics postulate Autonomous dynamics postulate P4: self-adjoint generators and unitary evolution (P4) and the preparation and apparatus postulate Preparation and apparatus postulate P7: declared preparations, isolation and sampling assumptions (P7), together with the Boltzmann-constant and temperature conventions of Entropy, the Boltzmann constant and the free-energy functional of the declared finite-sector model. Declare one of the following model readings of the same adopted Gibbs assignment:

(i) a fixed-$N$ model of The declared fixed-N many-body model: species sector, one-particle data, interaction and the sector energy observable, on its sector space, with Gibbs generator $G:=H_N^{\sigma }$; or

(ii) a separately declared number-variable mode model, such as a one-mode bosonic Fock space, a fermionic occupation space, or a declared countable independent-mode space, with its declared assigned observables and energy operator $H$. The model declares a self-adjoint Gibbs generator $G$ on that space: either $G=H$, or a separately supplied self-adjoint realization of the effective generator $G=H-\mu \mathcal{N}$, where $\mathcal{N}$ is the declared particle-number operator and $\mu \in \mathbb{R}$ is an assigned chemical potential in joules. The subtraction is an operator with its supplied domain, not an assertion that a formal unbounded difference is self-adjoint.

In either reading assume $G$ is bounded below and give its spectral PVM $E_G$ (Spectral theorem for unbounded self-adjoint operators (PVM form), Projection valued measure). Let $T>0$ be an assigned temperature in kelvin and let $\beta :=1\mathrm{/}(k_{B}T)$ in ${\mathrm{J}}^{-1}$ with the assigned Boltzmann constant $k_B$. The Gibbs assignment to either model, and the chemical-potential identification in (ii), are additional physical preparation assumptions made here; they are not derived from P1-P4 or from the fixed-$N$ construction. In (ii) no restriction on the assigned real $\mu$ is imposed beyond the declared generator and trace existence conditions; the mode-specific convergence conditions are proved separately. For reading (ii), $G$ specifies the Gibbs weights. If the assigned autonomous dynamics is generated by $H$, stationarity of the Gibbs density requires its separate dynamical or commutation hypotheses; this assignment does not identify $H-\mu \mathcal{N}$ with the physical evolution generator. The physical Helmholtz free-energy vocabulary of the entropy definition continues to apply only to its stated nonzero finite-sector class and is not extended to every effective generator by this postulate. Assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The Axiom of Choice), the choice hypotheses of the spectral calculus.

(b) Existence hypothesis and the Gibbs preparation. The model assumes that the partition trace is strictly positive and finite: $0<Z:=\mathrm{Tr}{e}^{-\beta G}<\mathrm{\infty },$ where $\mathrm{Tr}B:={\sum }_{f\in \mathcal{B}}⟨Bf,f⟩$ for a bounded positive $B$ and an orthonormal basis $\mathcal{B}$, with value allowed to be $+\mathrm{\infty }$. Such a basis exists under the declared AC by the maximal orthonormal-family argument using Zorn's lemma and the zero-complement criterion of Parseval equivalences for an orthonormal family. This trace is basis independent: for $C:=B^{1\mathrm{/}2}$ of Positive square root, the defining sum is ${\sum }_{f\in \mathcal{B}}\mathrm{\parallel }Cf{\mathrm{\parallel }}^2$, whose invariance is The Hilbert–Schmidt norm is basis independent. When the sum is finite, $C$ is Hilbert-Schmidt (Hilbert–Schmidt operator and Hilbert–Schmidt norm), so $B=C^2$ is trace class by Trace class iff product of two Hilbert Schmidt operators, and its ordinary trace is this same diagonal sum by Trace is absolutely convergent and basis independent. Apply this to the bounded positive operator $e^{-\beta G}$, since the declared Gibbs generator $G$ is bounded below. The model adopts as the canonical or Gibbs preparation of the declared model at temperature $T$ the density operator ${\rho }_{\beta }:=Z^{-1}{e}^{-\beta G}\mathrm{.}$ If the sector is zero-dimensional, its partition trace is $Z=0$, no trace-one density exists, and no canonical density or equilibrium free energy is assigned. On a nonzero finite-dimensional model space, $Z$ is a positive finite sum ${\sum }_j{e}^{-\beta E_j}$ over the eigenvalues of $G$; no energy sign restriction is needed. There ${\rho }_{\beta }$ is the canonical operator of Finite Gibbs free-energy minimum, and the strict positivity and finiteness hypothesis is automatic. In reading (i), $Z=Z_N$ and this is precisely the finite-sector Gibbs assignment.

Outside the finite-dimensional class the existence hypothesis is a genuine restriction. For a declared non-interacting fixed-$N$ model with a supplied countably infinite orthonormal eigenmode basis and real mode energies ${\epsilon }_j$ bounded below, the complete construction and trace criterion of Fixed-particle-number occupation bases and the countable-mode partition criterion give $Z_N=h_N(q)$ for bosonic statistics or $Z_N=e_N(q)$ for fermionic statistics, where $q_j=e^{-\beta {\epsilon }_j}$ and $h_N,e_N$ are the corresponding nonnegative occupation-series sums. For every fixed $N\ge 1$, either trace is finite exactly when ${\sum }_j{q}_j<\mathrm{\infty }$. This is a trace on the fixed-$N$ sector and permits negative mode energies.

For the separately declared full bosonic Fock models of reading (ii), the single-mode calculation of Single-mode thermal trace and the product criterion of Countable-mode product criterion provide the separate mathematical full-Fock traces, with their original positive mode-energy hypotheses: they sum over every particle number, including the vacuum, giving $(1-e^{-\beta \epsilon }{)}^{-1}$ for one mode and ${\prod }_j(1-e^{-\beta {\epsilon }_j}{)}^{-1}$ for countably many modes. Their physical Gibbs reading is the additional assignment of (a)(ii), not a consequence of a mathematical trace formula. Those full-Fock traces are not substituted for the fixed-$N$ partition trace of reading (i).

(c) The equilibrium assumption and thermal means. The model assumes that, for a declared sample of the species at the assigned temperature $T$ and in the absence of further preparation information, the preparation description of the sample is ${\rho }_{\beta }$ in the sense of P1. For an assigned sharp observable $(A,u_A)$ with spectral PVM $E_A$, define its outcome law ${\mu }_{{\rho }_{\beta }}^A(B):=\mathrm{tr}({\rho }_{\beta }{E}_A(B))$ as in Assigned sharp observables and the outcome law of the framework clause (b). If and only if the absolute first moment is finite, ${\int }_{\mathbb{R}}\mathrm{\mid }a\mathrm{\mid }d{\mu }_{{\rho }_{\beta }}^A(a)<\mathrm{\infty },$ the model defines the thermal mean at temperature $T$ by $⟨A{⟩}_{\beta }:={\int }_{\mathbb{R}}ad{\mu }_{{\rho }_{\beta }}^A(a),$ read in the assigned unit $u_A$. For bounded $A$ this equals $\mathrm{tr}({\rho }_{\beta }A)=\mathrm{tr}(A{\rho }_{\beta })$. For unbounded $A$, if the mean exists and the literal product $A{\rho }_{\beta }$ is defined on the whole Hilbert space and is trace class, the mean also equals $\mathrm{tr}(A{\rho }_{\beta })$ and $\mathrm{tr}(\overline{{\rho }_{\beta }A})$; these conditional product identities are proved by the density eigenexpansion in Spectral first moments and trace products with a density operator. First-moment existence alone imposes no such product or operator-square-domain condition. When the absolute first moment is infinite the model asserts no finite thermal mean. The identification of the parameter $T$ with the thermodynamic temperature of a sample, and of $\beta$ with its inverse temperature, is part of this assumption; the stated conservation and equilibrium properties of ${\rho }_{\beta }$ under the flow of P4, and the extremal property of ${\rho }_{\beta }$ among preparations, are separate conditional results of this pair.

(d) Motivation and status. The assumption is motivated, but not established, by the maximum-entropy reasoning of the cited sources: among preparations with a prescribed mean energy, the Gibbs weights maximize the entropy functional, and the finite-dimensional comparison theorem makes ${\rho }_{\beta }$ the minimizer of the corresponding functional built from $G$. Its physical Helmholtz free-energy reading applies in the stated finite fixed-sector class; an effective-generator interpretation in a different mode model is a separate declared reading. That reasoning is an argument inside mathematical statistics and thermodynamics; it does not derive the physical assumption that a sample is described by ${\rho }_{\beta }$. The postulate is an adopted statistical-mechanical assumption of the model, in the sense of the content model's distinction between adopted assumptions and derived consequences.

(e) Scope. The postulate asserts nothing about how a sample reaches the Gibbs preparation; in particular it asserts no thermalization, no equilibration time, no ergodicity, no master-equation or Markov approximation, no thermodynamic limit, no pressure, volume or phase-transition structure, and no measured temperature or expectation. The requirement $0<Z<\mathrm{\infty }$ is part of the assumption; the model does not define a canonical preparation when $Z=0$ or the partition trace diverges. Any comparison with a laboratory sample requires the declared preparation, isolation, detection and sampling models of P7, and agreement of a thermal prediction with data is evidence about the combined model and apparatus assumptions, not a proof of the framework or of this postulate.

• The finite-sector case. For a nonzero finite-dimensional sector the existence hypothesis is a positive finite sum and hence automatic (and likewise for any separately declared nonzero finite-dimensional mode space); the only content of clause (b) there is the assignment of the Gibbs weights to the declared energies.
• Relation to P4. The postulate adds a preparation assumption to the dynamics; by itself P4 describes the unitary evolution of whatever preparation is declared, and does not single out ${\rho }_{\beta }$. The precise sense in which unitary evolution cannot produce ${\rho }_{\beta }$ from a different preparation is a separate theorem of this pair.