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Stable-tempered free-cube grand pressure

Physical scope

Positive-temperature grand-canonical positional preparations in cubic free-wall boxes for stable even Borel pair potentials with the explicitly stated decreasing integrable tail.

Statement

Assume AC. Let $d \geq 2$ , $φ$ be stable with the regular integrable tail of Stable regularly tempered classical positional preparation, and assume its dimensionless representative $\hat{φ} = φ / E_{0}$ is Borel measurable, as required by the current mathematical supplier. Take $T > 0$ and $z > 0$ as in Finite-volume equilibrium ensemble sampling, set $E_{0} = 1 J$ , $β = (k_{B} T)^{- 1}$ , and $\hat{β} = E_{0} β > 0$ . For cubes $Λ_{L} = [- L / 2, L / 2)^{d}$ , let $Ξ_{L} (z, β)$ be the free-wall grand partition function specified by Finite-volume equilibrium ensemble sampling. Then the positional grand pressure per physical $d$ -volume $p_{L} (z, T) = \frac{k_{B} T}{ℓ_{0}^{d} ∣ Λ_{L} ∣} \log Ξ_{L} (z, β)$ has a finite limit as $L \to \infty$ . Its units are $J m^{- d}$ ; for $d = 3$ this is SI pressure in pascals. In the dimensionless reference-volume convention of Canonical–grand-canonical thermodynamic duality, let $P (α, \hat{β}) = \lim_{L \to \infty} ∣ Λ_{L} ∣^{- 1} \log Ξ_{L} (e^{α}, \hat{β})$ , where the supplier uses numerical energies in units of $E_{0}$ . Then $p = (k_{B} T / ℓ_{0}^{d}) P (\log z, \hat{β})$ . This is a conditional model deduction, not a measurement or an assertion that every real gas realizes the preparation.

Facts & Assumptions

[F1]

Stable regularly tempered classical positional preparation: if $B$ joules is a stability constant for $H_{N}$ , then $\hat{B} = B / E_{0}$ gives ${\hat{H}}_{N} = H_{N} / E_{0} \geq - \hat{B} N$ for the dimensionless potential. The tail has a nonnegative decreasing envelope $ψ$ in joules with $\int_{R}^{\infty} r^{d - 1} ψ (r) d r < \infty$ . Set $\hat{ψ} = ψ / E_{0}$ ; then $∣ \hat{φ} (x) ∣ \leq \hat{ψ} (∣ x ∣_{2})$ and $\hat{ψ}$ has the same finite weighted tail integral. Since $∣ x ∣_{2} \geq ∣ x ∣_{\infty}$ and $\hat{ψ}$ is nonincreasing, $\hat{ψ} (∣ x ∣_{2}) \leq \hat{ψ} (∣ x ∣_{\infty})$ , so the mathematical supplier's maximum-norm tail bound holds. That supplier uses numerical energy; $\hat{H} = H / E_{0}$ and $\hat{β} = E_{0} β$ preserve $β H = \hat{β} \hat{H}$ .

[A1]

Finite-volume equilibrium ensemble sampling: the finite-volume grand law uses $Ξ_{Λ} (z, β) = \sum_{N \geq 0} z^{N} Z_{Λ, N}$ ; each $Z_{Λ, N}$ integrates $e^{- β H_{N}}$ only on finite-energy configurations and assigns zero weight elsewhere. In identifying this with the mathematical partition, define its extended-real factor as zero at $\hat{H} = + \infty$ ; on finite energies $e^{- \hat{β} \hat{H}} = e^{- β H}$ .

[L1]

Canonical–grand-canonical thermodynamic duality: the current carrier's statement defines the dimensionless grand rate from the free-cube activity sum, gives its limit under the stable regular tail hypotheses at positive numerical $\hat{β}$ , and states both Legendre--Fenchel identities for the specified closed effective-domain extension. The physical-to-numerical energy conversion is $\hat{β} = E_{0} β$ . The item and draft proof text are present, and its scoped batch-18 proof-contract entry passes; current owner Step 1 proof-readiness is recorded and independent review remains pending, so the conclusion remains conditional.

Proof

Given: AC, the stable regularly tempered Borel pair model, the adopted finite-volume positional grand preparation, and the current draft free-cube supplier statement in [L1]. No kinetic partition factor or empirical premise is used.

1.1givenF1A1L1algebra

By [A1], the physical positional partition sum is $Ξ_{L} (z, β) = \sum_{N \geq 0} z^{N} Z_{Λ_{L}, N}$ . Set $α = \log z$ ; then $z^{N} = e^{α N}$ . Translate the centered cube by $L / 2$ in each coordinate to $[0, L)^{d}$ ; this preserves Lebesgue measure and every pair displacement, so the positional partition is exactly the free-cube sum of [L1]. The dimensionless energy conversion in [F1] matches its numerical $\hat{β}$ . To verify the all-radius lower-regularity hypothesis of [L1], apply stability to the two-point configuration ${0, x}$ : $\hat{φ} (x) \geq - 2 \hat{B}$ for $x \neq 0$ . Choose a finite $C \geq \max {2 \hat{B}, \hat{ψ} (R), 0}$ also satisfying $\hat{φ} (0) \geq - C$ ; this is possible since that value is real or $+ \infty$ . Define $Ψ (r) = C$ for $0 \leq r < R$ and $Ψ (r) = \hat{ψ} (r)$ for $r \geq R$ . Then $Ψ$ is nonnegative and nonincreasing, $\hat{φ} (x) \geq - Ψ (∣ x ∣_{\infty})$ everywhere, and $\int_{0}^{\infty} Ψ (r) r^{d - 1} d r = C R^{d} / d + \int_{R}^{\infty} \hat{ψ} (r) r^{d - 1} d r < \infty$ . Together with the maximum-norm upper-tail bound in [F1], this supplies the exact regularity required by [L1].

2.1givenF1L1step 1.1algebra∎

The conditional conclusion of [L1] gives a finite limit $P (\log z, \hat{β})$ for $∣ Λ_{L} ∣^{- 1} \log Ξ_{L}$ . Multiplying by $k_{B} T / ℓ_{0}^{d}$ , which is independent of $L$ , converts the dimensionless reference-volume rate to pressure per physical volume.

Stable-tempered free-cube grand pressure

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources