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Measurable canonical identities for a dimensionless reference measure

Physical scope

Measurable energy models with a dimensionless reference measure, positive finite partition integral, and compact-parameter local integrable second-moment envelope; no claim is made for an arbitrary phase-space measure or unbounded energy without these hypotheses.

Statement

Let $(X, Σ, μ)$ be a measure space with dimensionless reference measure, let $E : X \to R J$ be a finite physical energy, and let $J_{β} \subset (0, \infty)$ be an open inverse-temperature set in $J^{- 1}$ . Set $E_{*} = 1 J$ , $e = E / E_{*} : X \to R$ , $b = E_{*} β$ , and $J_{b} = E_{*} J_{β}$ ; measurability of $E$ means measurability of the dimensionless numerical coordinate $e$ . Suppose $Z (β) = \int_{X} e^{- β E} d μ$ is positive and finite on $J_{β}$ , and for each compact $K \subset J_{β}$ there is an integrable $G_{K} \geq 0$ with $(1 + ∣ e ∣^{2}) e^{- β E} = (1 + ∣ e ∣^{2}) e^{- b e} \leq G_{K}$ for every $β \in K$ . Under the canonical sampling assumption Canonical sampling assumption for finite and normalized statistical models, define $p_{β} = e^{- β E} / Z$ , $⟨ E ⟩_{β} = \int E p_{β} d μ$ , and ${Var}_{β} (E) = \int (E - ⟨ E ⟩_{β})^{2} p_{β} d μ$ . Then $Z \in C^{2}$ , $- \partial_{β} \log Z = ⟨ E ⟩_{β}$ , $\partial_{β}^{2} \log Z = {Var}_{β} (E)$ , and the relative Gibbs entropy is finite with $S_{β} = - k_{B} \int p_{β} \log p_{β} d μ = k_{B} (\log Z + β ⟨ E ⟩_{β})$ . Hence $F_{s t a t} = - k_{B} T \log Z = ⟨ E ⟩_{β} - T S_{β}$ for $β = (k_{B} T)^{- 1}$ .

Facts & Assumptions

Given: A dimensionless measure space, a measurable energy, positive finite partition integrals, the canonical density, and the stated local integrable envelope.

[F1]

Measure space and integral: use the real measurable energy coordinate $e$ , whose exponential weights are nonnegative; nonnegative integrals use the simple-function supremum, and signed integrals require absolute integrability. No completeness of the measure space is assumed.

[F2]

Canonical sampling assumption for finite and normalized statistical models: with $0 < Z (β) < \infty$ , the density $p_{β} = e^{- β E} / Z$ is the adopted canonical preparation relative to $μ$ .

[L1]

Measurable canonical normalization and identities (current batch 11 draft): for real measurable energy $e$ and real parameter $b \in J_{b}$ , positive finite $\tilde{Z} (b) = \int e^{- b e} d μ$ and the compact-interval integrable envelope $(1 + ∣ e ∣^{2}) e^{- b e}$ give $\tilde{Z} \in C^{2}$ , finite moments, $- \partial_{b} \log \tilde{Z} = ⟨ e ⟩_{b}$ , $\partial_{b}^{2} \log \tilde{Z} = {Var}_{b} (e)$ , and absolutely integrable reference-relative entropy $s = \log \tilde{Z} + b ⟨ e ⟩_{b}$ . The current proof is supplied in that draft; its status is not publication acceptance.

[F3]

SI quantities and unit conventions: $E$ and $⟨ E ⟩$ are in joules, $T$ is in kelvins, and $k_{B}$ is in $J K^{- 1}$ , so $β$ is in $J^{- 1}$ .

[A1]

The reference measure is dimensionless, $Z$ is positive finite on the open parameter set, and $(1 + ∣ e ∣^{2}) e^{- β E}$ has the local integrable envelope stated in the Statement.

Proof

Given: The dimensionless reference measure, the measurable energy, the normalized canonical density, and the local envelope.

1.1F1F2F3L1A1algebra

Apply [L1] with numerical energy $e = E / E_{*}$ and $b = E_{*} β$ . Any compact interval $K_{b} \subset J_{b}$ corresponds to $K_{b} / E_{*} \subset J_{β}$ , so its domination hypothesis is exactly the stated dimensionless envelope. The equality $b e = β E$ identifies $\tilde{Z} (b) = Z (b / E_{*})$ and the normalized density with [F2]. The supplier gives finite first and second moments of $e$ ; hence $⟨ E ⟩_{β} = E_{*} ⟨ e ⟩_{b}$ and ${Var}_{β} (E) = E_{*}^{2} {Var}_{b} (e)$ are finite. All weighted functions and difference quotients are measurable functions of $e$ , so the argument requires no completeness of $μ$ .

2.1F2F3L1step 1.1algebra∎

Since $Z (β) = \tilde{Z} (E_{*} β)$ , substitution of $Δ b = E_{*} Δ β$ in derivative difference quotients multiplies first derivatives by $E_{*}$ and second derivatives by $E_{*}^{2}$ . Thus [L1] yields $Z \in C^{2}$ and the stated log-derivative mean and variance identities in joules and joules squared. The density is unchanged, so its absolutely integrable relative entropy is $s_{β} = \log \tilde{Z} + b ⟨ e ⟩_{b} = \log Z + β ⟨ E ⟩_{β}$ . Multiplying by $k_{B}$ gives $S_{β}$ in $J K^{- 1}$ ; using $β k_{B} = 1 / T$ gives $F_{s t a t} = ⟨ E ⟩_{β} - T S_{β} = - k_{B} T \log Z$ . This is conditional on the declared sampling model and the stated hypotheses of the current draft mathematical supplier.

Measurable canonical identities for a dimensionless reference measure

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources