TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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Nonnegative fixed-Hamiltonian canonical heat capacity

Statement

Under the hypotheses of Finite gravitational Gibbs normalization and differentiable moments, keep the box, particle number, mass, regulator, reference measure, and Hamiltonian fixed while varying $β > 0$ , and set $p_{β} = Z_{N} (β)^{- 1} e^{- β H_{N, a}}$ . For a measurable observable $A : Γ_{N} \to R$ with finite second moment, define $⟨ A ⟩_{β} : = \int_{Γ_{N}} A p_{β} d ν_{N}$ and ${Var}_{β} (A) : = \int_{Γ_{N}} (A - ⟨ A ⟩_{β})^{2} p_{β} d ν_{N}$ . Define $T_{K} = (k_{B} β)^{- 1}$ and $C_{K} = d ⟨ H_{N, a} ⟩_{β} / d T_{K}$ . Then $⟨ H_{N, a} ⟩_{β} = - \partial_{β} \log Z_{N}, \partial_{β} ⟨ H_{N, a} ⟩_{β} = - {Var}_{β} (H_{N, a}), C_{K} = k_{B} β^{2} {Var}_{β} (H_{N, a}) \geq 0.$ This is a response identity for a fixed Hamiltonian and fixed regulator; it does not assert stability when the Hamiltonian or background varies with temperature.

Facts & Assumptions

Given: The finite softened Gibbs model, $β > 0$ , $k_{B} > 0$ , and its temperature convention $T_{K} = (k_{B} β)^{- 1}$ .

[F1]

The finite Gibbs density is normalized, and the partition function is twice differentiable with $Z_{N}^{(j)} = \int (- H_{N, a})^{j} e^{- β H_{N, a}} d ν_{N}$ for $j = 0, 1, 2$ . (Finite gravitational Gibbs normalization and differentiable moments)

Proof

1.1F1algebra

Differentiating $\log Z_{N}$ and using $Z_{N}^{'} = - \int H_{N, a} e^{- β H_{N, a}} d ν_{N}$ gives $- \partial_{β} \log Z_{N} = Z_{N}^{- 1} \int H_{N, a} e^{- β H_{N, a}} d ν_{N} = ⟨ H_{N, a} ⟩_{β}$ .

2.1F1step 1.1algebra

The quotient rule and $Z_{N}^{''} = \int H_{N, a}^{2} e^{- β H_{N, a}} d ν_{N}$ give $\partial_{β} ⟨ H_{N, a} ⟩_{β} = - \frac{Z_{N}^{''}}{Z_{N}} + {(\frac{Z_{N}^{'}}{Z_{N}})}^{2} = - (⟨ H_{N, a}^{2} ⟩_{β} - ⟨ H_{N, a} ⟩_{β}^{2}) = - {Var}_{β} (H_{N, a}) .$ All terms are finite by [F1].

3.1F1step 2.1algebra∎

Since $d T_{K} / d β = - 1 / (k_{B} β^{2})$ , the chain rule yields $C_{K} = (d ⟨ H ⟩ / d β) / (d T_{K} / d β) = k_{B} β^{2} {Var}_{β} (H_{N, a})$ . The variance is the expectation of the nonnegative square $(H_{N, a} - ⟨ H_{N, a} ⟩)^{2}$ , so $C_{K} \geq 0$ .

Depends on

Finite gravitational Gibbs normalization and differentiable moments

Nonnegative fixed-Hamiltonian canonical heat capacity

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources