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Controlled canonical–grand thermodynamic duality

Physical scope

Stable regularly tempered classical positional Borel pair models at positive temperature in free cubes, for the closed concave extension of the attainable canonical rate; no local-observable equivalence is asserted.

Statement

Assume AC and $T > 0$ , with $β = (k_{B} T)^{- 1}$ , and take $z > 0$ as in Finite-volume equilibrium ensemble sampling. Under the stable regularly tempered positional model of Stable regularly tempered classical positional preparation and the adopted ensemble weights of Finite-volume equilibrium ensemble sampling, assume the dimensionless potential $\hat{φ} = φ / E_{0}$ is Borel measurable as required by the current mathematical carriers. Set $E_{0} = 1 J$ and $\hat{β} = E_{0} β$ ; write $F (ρ, β)$ and $P (α, β)$ for the mathematical rates evaluated at $\hat{β}$ . Use free-wall cubes $Λ_{L} = [0, L)^{d}$ with dimensionless volume $V_{L} = L^{d}$ . Write $Z_{L} (N; β) = \int e^{- β H_{N}} d μ_{Λ_{L}, N}$ , extending the integrand by zero at infinite energy, for their canonical positional partitions. Let $F (ρ, β)$ be the canonical log-partition rate per dimensionless reference volume, defined on positive interior attainable densities by $V_{L}^{- 1} \log Z_{L} (N_{L}; β) \to F (ρ, β)$ for every sequence $N_{L} / V_{L} \to ρ$ and extended to the closed effective density domain by the upper-semicontinuous concave extension specified in Canonical–grand-canonical thermodynamic duality, with $F (ρ, β) = - \infty$ outside that closed domain. Let $P (α, β) = \lim_{L \to \infty} V_{L}^{- 1} \log \sum_{N \geq 0} e^{α N} Z_{L} (N; β), α = \log z .$ Then $P (α, β) = \sup_{ρ \geq 0} {F (ρ, β) + α ρ},$ and, for every $ρ$ in the closed effective density domain of $F$ , $F (ρ, β) = \inf_{α \in R} {P (α, β) - α ρ} .$ The density $ρ$ is particle count per dimensionless reference volume, corresponding to physical number density $ρ / ℓ_{0}^{d}$ in $m^{- d}$ . In physical units, $p = (k_{B} T / ℓ_{0}^{d}) P$ and the positional Helmholtz free-energy density is $f_{p o s} = - (k_{B} T / ℓ_{0}^{d}) F$ ; both are per physical $d$ -volume, in $J m^{- d}$ (pascals when $d = 3$ ). The fixed $1 / N!$ reference normalization is that of Indistinguishable-particle positional measure. This duality concerns limiting thermodynamic functions only. It does not imply local ensemble equivalence, uniqueness of a phase, or convergence at excluded density endpoints.

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; its Euclidean tail is bounded by a decreasing integrable envelope $ψ$ in joules. Set $\hat{ψ} = ψ / E_{0}$ ; then $∣ \hat{φ} (x) ∣ \leq \hat{ψ} (∣ x ∣_{2}) \leq \hat{ψ} (∣ x ∣_{\infty})$ , so the mathematical suppliers' maximum-norm tail condition holds, with the weighted tail integral still finite. For the lower-regularity condition at all radii, two-particle stability gives $\hat{φ} (x) \geq - 2 \hat{B}$ for $x \neq 0$ . Choose a finite $M \geq \max (2 \hat{B}, \hat{ψ} (R))$ also satisfying $\hat{φ} (0) \geq - M$ , possible because the potential never takes $- \infty$ . Extend the maximum-norm envelope by $\tilde{ψ} (r) = M$ for $0 \leq r < R$ and $\tilde{ψ} (r) = \hat{ψ} (r)$ for $r \geq R$ . It is nonnegative and nonincreasing with finite weighted integral, bounds $- \hat{φ}$ everywhere, and bounds $∣ \hat{φ} ∣$ for $∣ x ∣_{\infty} \geq R$ . Thus both lower and upper regularity hold. Densities are measured per dimensionless reference volume.

[A1]

Finite-volume equilibrium ensemble sampling: at exact $N$ the positional partition integrates the Boltzmann factor on finite-energy configurations and gives infinite-energy configurations zero weight; on the specified free cube the grand partition is $\sum_{N} z^{N} Z_{L} (N; β)$ with $α = \log z$ . In the mathematical comparison, extend $e^{- \hat{β} \hat{H}}$ by zero at $\hat{H} = + \infty$ ; for finite energies $\hat{β} \hat{H} = β H$ .

[F2]

Indistinguishable-particle positional measure: each canonical sector uses $μ_{Λ, N} = d x_{1} \dots d x_{N} / N!$ , fixing the positional reference normalization.

[L1]

Canonical free-cube limit at every interior attainable density: the current statement gives $V_{L}^{- 1} \log Z (L, N_{L}; \hat{β}) \to F (ρ, \hat{β})$ for free cubes and every sequence $N_{L} / V_{L} \to ρ$ at positive interior attainable density, under its stable regular Borel convention; excluded endpoints are not asserted. The carrier and draft proof text are present and the scoped batch-18 proof-contract check passes, but current owner Step 1 proof-readiness is recorded and independent review remains pending, so this premise remains conditional.

[L2]

Canonical–grand-canonical thermodynamic duality: the current statement defines the free-cube grand pressure rate and states both Legendre--Fenchel identities for the closed effective-domain extension, with pressure existence delegated to its decreasing-integrable-tail supplier. Its item and draft proof text are present and the scoped batch-18 proof-contract check passes, but current owner Step 1 proof-readiness is recorded and independent review remains pending; this result is used conditionally.

Proof

Given: The stable tail, Borel and numerical-energy conventions, finite-volume sector definitions, exact canonical limits [L1], and the actual draft closed-concave duality statement [L2]. The chosen reference length, $E_{0} = 1 J$ , and $1 / N!$ convention are fixed across all volumes.

1.1givenF1A1F2L1L2algebra

By [A1] and [F2], the free-wall partition on $Λ_{L} = [0, L)^{d}$ is precisely the activity sum over the canonical positional sectors used in [L2], with $α = \log z$ ; [F1] and [L1] supply the stable-tail hypotheses and the canonical rate $F$ on its interior. In the mathematical partition, $\hat{β} = E_{0} β$ and $\hat{H} = H / E_{0}$ , so the Boltzmann factors agree.

2.1givenL1L2step 1.1

The conditional mathematical conclusion [L2] gives both displayed Legendre–Fenchel identities, including only the stipulated closed concave endpoint extension.

3.1givenF2L2step 2.1∎

Multiplication by $k_{B} T / ℓ_{0}^{d}$ converts the dimensionless log-partition rate to physical pressure, while multiplication of the canonical rate by $- k_{B} T / ℓ_{0}^{d}$ gives the stated positional free-energy density; the common measure normalization is fixed by [F2]. Since [L2] is a statement about thermodynamic functions, it supplies no local convergence or phase-uniqueness claim.

Controlled canonical–grand thermodynamic duality

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources