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Countable canonical identities under a local moment envelope

Physical scope

Discrete countable energy lists with a specified enumeration and a uniform local summable second-moment envelope on an open positive inverse-temperature interval; no lower-energy-bound-only or arbitrary trace claim is made.

Statement

Under the canonical sampling assumption Canonical sampling assumption for finite and normalized statistical models, let $I$ be a specified nonempty countable microstate list with finite energies $E_{i}$ in joules, counted once ( $g_{i} = 1$ ), and let $J_{β} \subset (0, \infty)$ be an open inverse-temperature interval in $J^{- 1}$ . Set $E_{*} = 1 J$ , $e_{i} = E_{i} / E_{*} \in R$ , $b = E_{*} β$ , and $J_{b} = E_{*} J_{β} \subset (0, \infty)$ ; $e_{i}$ and $b$ are dimensionless numerical coordinates. For $β \in J_{β}$ , define $Z (β) = \sum_{i} e^{- β E_{i}}$ and, when $0 < Z (β) < \infty$ , $p_{i} (β) = e^{- β E_{i}} / Z (β)$ . Suppose every compact interval $K \subset J_{β}$ admits a summable sequence $a_{i} \geq 0$ with $(1 + ∣ e_{i} ∣^{2}) e^{- β E_{i}} = (1 + ∣ e_{i} ∣^{2}) e^{- b e_{i}} \leq a_{i}$ for all $i$ and $β \in K$ . Define ${\overset{ˉ}{E}}_{β} = \sum_{i} p_{i} E_{i}$ and ${Var}_{β} (E) = \sum_{i} p_{i} (E_{i} - {\overset{ˉ}{E}}_{β})^{2}$ . Then these moments are finite, $Z$ is $C^{2}$ , $- \partial_{β} \log Z = {\overset{ˉ}{E}}_{β}$ , $\partial_{β}^{2} \log Z = {Var}_{β} (E)$ , and the dimensionless Gibbs entropy $s_{β} : = - \sum_{i} p_{i} \log p_{i} = \log Z + β {\overset{ˉ}{E}}_{β}$ is finite. Define the physical Gibbs entropy by $S_{β} : = k_{B} s_{β}$ in $J K^{- 1}$ . Then $F_{s t a t} : = - k_{B} T \log Z = {\overset{ˉ}{E}}_{β} - T S_{β}$ for $β = (k_{B} T)^{- 1}$ . A lower bound on the energies alone does not supply the summable envelope or finite partition function.

Facts & Assumptions

Given: A specified countable energy list, a positive finite canonical partition sum on $J_{β}$ , and the local summable second-moment envelope.

[F1]

Countable canonical model: use its real energy coordinates $e_{i}$ , real parameter $b \in J_{b}$ , and $g_{i} = 1$ . Its nonnegative series is the supremum of finite subsums; normalized probabilities and absolute-moment expectations are defined only at positive finite $\tilde{Z} (b) = \sum_{i} e^{- b e_{i}} = Z (b / E_{*})$ .

[F2]

Canonical sampling assumption for finite and normalized statistical models: for this normalized countable model, $p_{i} = e^{- β E_{i}} / Z$ is the adopted preparation law.

[L1]

Summable canonical differentiation (current batch 11 draft): for real $e_{i}, b$ and $g_{i} = 1$ , the local envelope $(1 + ∣ e_{i} ∣^{2}) e^{- b e_{i}} \leq a_{i}$ on each compact interval implies positive finite $\tilde{Z} \in C^{2}$ , $- \partial_{b} \log \tilde{Z} = ⟨ e ⟩_{b}$ , $\partial_{b}^{2} \log \tilde{Z} = {Var}_{b} (e)$ , and absolutely convergent 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: $E_{i}$ and ${\overset{ˉ}{E}}_{β}$ are in joules, $T$ is in kelvins, and $k_{B}$ is in $J K^{- 1}$ , so $β$ is in $J^{- 1}$ .

[A1]

The energy list and degeneracy convention are fixed while $β$ varies; every compact interval in $J_{β}$ has the stated summable envelope.

Proof

Given: The countable model and its local envelope on the open interval $J_{β}$ .

1.1F1F2F3L1A1algebra

Apply [L1] to the real coordinates $e_{i}, b$ of [F1] with $g_{i} = 1$ . For any compact interval $K_{b} \subset J_{b}$ , its inverse image $K_{b} / E_{*} \subset J_{β}$ is compact, and the stated dimensionless envelope becomes exactly $(1 + ∣ e_{i} ∣^{2}) e^{- b e_{i}} \leq a_{i}$ there. Thus [L1] gives $0 < \tilde{Z} (b) < \infty$ and finite first and second numerical energy moments on $J_{b}$ . Because $b e_{i} = β E_{i}$ , the normalized probabilities are precisely the adopted counted-once law [F2]. Multiplication by $E_{*}$ gives ${\overset{ˉ}{E}}_{β} = E_{*} ⟨ e ⟩_{b}$ and ${Var}_{β} (E) = E_{*}^{2} {Var}_{b} (e)$ .

2.1F2F3L1step 1.1algebra∎

Since $Z (β) = \tilde{Z} (E_{*} β)$ , substituting $Δ b = E_{*} Δ β$ in the derivative difference quotients gives $\partial_{β} \log Z = E_{*} \partial_{b} \log \tilde{Z}$ and $\partial_{β}^{2} \log Z = E_{*}^{2} \partial_{b}^{2} \log \tilde{Z}$ . The derivatives are continuous by [L1], hence $Z \in C^{2}$ and the stated joule and joule-squared mean and variance identities follow. The same probabilities have absolutely convergent entropy $s_{β} = \log \tilde{Z} + b ⟨ e ⟩_{b} = \log Z + β {\overset{ˉ}{E}}_{β}$ . Defining $S_{β} = k_{B} s_{β}$ and using $β k_{B} = 1 / T$ gives $F_{s t a t} = {\overset{ˉ}{E}}_{β} - T S_{β} = - k_{B} T \log Z$ . These are conditional model identities using the stated envelope and current draft mathematical supplier.

Countable canonical identities under a local moment envelope

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources