Physics theoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

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Gibbs energy, entropy and potential identities

Physical scope

Canonical preparation identities for finite Hamiltonians or trace-class self-adjoint Gibbs operators with finite form energy; adopted thermodynamic identification only, with no empirical claim.

Statement

Under the adopted canonical preparation Canonical quantum thermal preparation, let $k_{B} > 0$ and $β = 1 / (k_{B} T) > 0$ , with $β$ in J $^{- 1}$ , $H$ in J, $T$ in K, $k_{B}$ in J/K, entropy in J/K, and free energy in J. In a finite-dimensional model on a Hilbert space of dimension $d \geq 1$ , for $σ = e^{- β H} / Z_{β}$ , $U = Tr (σ H)$ , and $F_{β} = - β^{- 1} \log Z_{β}$ , $S (σ) = k_{B} (\log Z_{β} + β U), F_{β} = U - T S (σ) .$ The same identities hold for a self-adjoint Hamiltonian bounded below when its Gibbs exponential is trace class on an open positive- $β$ interval and the state has the finite form energy required by Self-adjoint trace-Gibbs normalization and form-energy optimization. These are mathematical model identities under an adopted preparation rule; they do not establish that a physical system is equilibrated.

Facts & Assumptions

Given: The Axiom of Choice; $k_{B} > 0$ ; the canonical preparation postulate; and either a finite Hermitian energy matrix on a space of dimension $d \geq 1$ or a self-adjoint bounded-below $H$ with the actual trace-class/open-interval and finite-form-energy hypotheses.

[A1]

The Axiom of Choice: full Choice is retained for the unbounded spectral-theorem branch.

[F1]

Canonical quantum thermal preparation: the adopted canonical state is $e^{- β H} / Z_{β}$ , with $β = 1 / (k_{B} T)$ and $H$ an energy observable.

[F2]

Finite Gibbs free-energy and entropy optimization: for finite $H$ , the Gibbs state has entropy $k_{B} (\log Z + β Tr (σ H))$ and free energy $F_{β} = Tr (σ H) - T S (σ)$ .

[F3]

Self-adjoint trace-Gibbs normalization and form-energy optimization: under the stated self-adjoint, lower-bound, trace-class/open-interval, Choice, and finite form-energy conditions, the Gibbs state has finite entropy and obeys $S (ρ_{β}) = k_{B} (\log Z_{β} + β E_{H} (ρ_{β}))$ .

Proof

1.1givenF1F2algebra

In finite dimension, [F2] gives $S (σ) = k_{B} (\log Z_{β} + β U)$ . Multiplying by $T$ and using $k_{B} T = β^{- 1}$ gives $T S = β^{- 1} \log Z_{β} + U$ ; since $F_{β} = - β^{- 1} \log Z_{β}$ , rearrangement yields $F_{β} = U - T S$ . The units match: both $U$ and $T S$ are joules.

2.1givenA1F1F3algebra∎

In the self-adjoint trace-class case, [F3] supplies the entropy identity with $U = E_{H} (ρ_{β})$ , the scalar spectral first moment. The same multiplication by $T$ and rearrangement prove the displayed free-energy identity on this stated form-energy domain. No trace of an undefined unbounded product is used.

Gibbs energy, entropy and potential identities

Statement

Facts & Assumptions

Proof

Depends on

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