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General quantum Gibbs normalization and canonical identities

Physical scope

A declared quantum model with a semibounded self-adjoint energy operator, trace-class Gibbs exponential, and local first-and-second-moment envelope; the result concerns the adopted Gibbs preparation and does not derive thermalization or a continuum spectrum.

Statement

Assume the Axiom of Choice and Countable Choice and the canonical Gibbs-preparation assumption The canonical equilibrium-ensemble assumption: the Gibbs preparation of a declared model at assigned temperature. Restrict to its $G = H$ branch: the declared energy operator $H$ is self-adjoint, bounded below, and measured in joules; $T > 0$ is in kelvins, $β = (k_{B} T)^{- 1}$ is in $J^{- 1}$ , and $K_{β} = e^{- β H}$ has $0 < Z_{β} = Tr K_{β} < \infty$ . Suppose also that on an open interval $J_{β} \subset (0, \infty)$ containing $β$ , every compact $C \subset J_{β}$ admits a summable family $a_{j} \geq 0$ with $(1 + ∣ E_{j} ∣^{2}) e^{- γ E_{j}} \leq a_{j}$ for all energy eigenvalues $E_{j}$ and $γ \in C$ . Let $(u_{j})$ be the finite or countable complete orthonormal eigenfamily supplied by Trace-class Gibbs normalization for a semibounded self-adjoint operator, repeated according to multiplicity, with $H u_{j} = E_{j} u_{j}$ , and put $λ_{j} = e^{- β E_{j}} / Z_{β}$ . The supplied trace identity gives $\sum_{j} λ_{j} = 1$ , and these $λ_{j}$ are the eigenvalues of $ρ_{β}$ . Define the spectral mean ${\overset{ˉ}{E}}_{β} : = \sum_{j} λ_{j} E_{j}$ , the variance ${Var}_{β} (H) : = \sum_{j} λ_{j} (E_{j} - {\overset{ˉ}{E}}_{β})^{2}$ , the dimensionless entropy $s_{β} : = - \sum_{j} λ_{j} \log λ_{j}$ , and the physical von Neumann entropy $S_{v N} (ρ_{β}) : = k_{B} s_{β}$ . These are spectral-sum definitions for this commuting Gibbs density; no operator trace of an unbounded product is used. The supplied moment envelope makes these sums finite. Then $- \partial_{β} \log Z_{β} = {\overset{ˉ}{E}}_{β}, \partial_{β}^{2} \log Z_{β} = {Var}_{β} (H), S_{v N} (ρ_{β}) = k_{B} (\log Z_{β} + β {\overset{ˉ}{E}}_{β}) .$ Thus $F_{s t a t} = - k_{B} T \log Z_{β} = {\overset{ˉ}{E}}_{β} - T S_{v N} (ρ_{β})$ . This statement does not extend the Helmholtz interpretation to an effective generator $H - μ N$ .

Facts & Assumptions

Given: A nonzero quantum Hilbert space, a self-adjoint energy operator $H$ bounded below, a positive assigned temperature, the adopted Gibbs preparation, a finite positive trace, and the local summable moment envelope.

[F1]

The canonical equilibrium-ensemble assumption: the Gibbs preparation of a declared model at assigned temperature: for a declared $G = H$ model with self-adjoint semibounded generator and $0 < Tr e^{- β H} < \infty$ , the adopted preparation is $ρ_{β} = e^{- β H} / Z_{β}$ ; the postulate explicitly assumes AC and Countable Choice and does not assert equilibration.

[F2]

Entropy, the Boltzmann constant and the free-energy functional of the declared finite-sector model defines entropy and free energy only on its finite-dimensional sector class; it does not define an entropy functional for general infinite-dimensional sectors. This theorem supplies its own spectral-sum definition above. Dimensions, SI unit conventions and quantity types of the framework and SI quantities and unit conventions give the energy, temperature, and Boltzmann-constant units.

[L1]

Trace-class Gibbs normalization for a semibounded self-adjoint operator (current batch 11 draft): for a semibounded self-adjoint operator with trace-class Gibbs exponential and the stated local moment envelope, the Gibbs weight has a countable energy eigenfamily and the normalized eigenvalue law satisfies the summable canonical mean, variance, and entropy identities. Its full current proof was inspected; its draft status remains conditional.

[F3]

SI quantities and unit conventions: $k_{B}$ is in $J K^{- 1}$ , $T$ in kelvins, and energy in joules; hence $β$ is in $J^{- 1}$ and $β E_{j}$ is dimensionless.

[A1]

The full Axiom of Choice and Countable Choice are assumed at the exact strengths required by [F1] and [L1].

Proof

Given: The $G = H$ reading of the adopted quantum Gibbs preparation, trace finiteness, and the local moment envelope.

1.1F1F3L1A1

By [F1], $K_{β} = e^{- β H}$ is the declared positive Gibbs weight and $ρ_{β} = K_{β} / Z_{β}$ is the model's adopted preparation. The semiboundedness, positive finite trace, AC assumptions and moment envelope in the Statement match the hypotheses of [L1].

2.1F1F2F3L1step 1.1algebra∎

Applying [L1] to the supplied eigenfamily yields the finite spectral mean and variance and the dimensionless entropy identity $s_{β} = \log Z_{β} + β {\overset{ˉ}{E}}_{β}$ . By the definitions in the Statement, $S_{v N} = k_{B} s_{β}$ ; using $β k_{B} = 1 / T$ gives $F_{s t a t} = {\overset{ˉ}{E}}_{β} - T S_{v N} = - k_{B} T \log Z_{β}$ . The deduction is conditional on [L1] and the adopted preparation; it asserts no dynamics.

General quantum Gibbs normalization and canonical identities

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources