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

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Confined coupled quadratic canonical systems

Physical scope

Finite classical continuous canonical model with positive diagonal mass matrix, strictly positive stiffness matrix, fixed positive action reference and beta>0; the Lebesgue product calculations assume ACω; no unpinned zero mode or empirical equilibration claim.

Statement

Assume ${A C}_{ω}$ as defined in The Axiom of Countable Choice ( ${A C}_{ω}$ ) for the Euclidean Lebesgue product measure used below. Let $n \geq 1$ , let $M = diag (m_{1}, \dots, m_{n})$ with $m_{i} > 0$ in kg, and let $K$ be a real symmetric positive-definite stiffness matrix with entries in J m $^{- 2}$ . For displacement $q \in R^{n}$ in metres and momentum $p \in R^{n}$ in kg m s $^{- 1}$ , define $H (q, p) = \frac{1}{2} p^{T} M^{- 1} p + \frac{1}{2} q^{T} K q .$ Adopt the canonical probability density proportional to $e^{- β H}$ relative to $d q d p / h_{0}^{n}$ , where $β > 0$ has units J $^{- 1}$ and the fixed action reference $h_{0} > 0$ has units J s. Under the preparation of Declared example preparations and thermodynamic interpretation, the dimensionless partition function and moments are $Z = {(\frac{2 π}{β h_{0}})}^{n} \sqrt{\frac{\det M}{\det K}}, E [q] = E [p] = 0, E [q q^{T}] = β^{- 1} K^{- 1}, E [p p^{T}] = β^{- 1} M, E [q p^{T}] = 0.$ The model mean energy is $U = E [H] = n / β$ . With $k_{B} > 0$ in J K $^{- 1}$ and $T > 0$ in K from Declared example preparations and thermodynamic interpretation, so that $β = (k_{B} T)^{- 1}$ , define the fixed-parameter heat-capacity response by $C_{h a r m} : = {(\frac{\partial U}{\partial T})}_{M, K, h_{0}} = n k_{B} .$ It has units J K $^{- 1}$ . This model has no volume variable, so $C_{h a r m}$ is not a fixed-volume derivative.

Facts & Assumptions

Given: ${A C}_{ω}$ , $n \geq 1$ , $M$ positive diagonal, $K$ symmetric positive definite, $β > 0$ , fixed $h_{0} > 0$ , and the displayed continuous-coordinate canonical model.

[F1]

Declared example preparations and thermodynamic interpretation: in each named model with finite positive $Z$ , the canonical probability is $Z^{- 1} e^{- β H}$ relative to the specified reference; $U = E H$ and the fixed-reference thermodynamic interpretation is adopted separately from the probability preparation.

[F2]

Gaussian covariance for positive quadratic forms and a boundary counterexample: assuming ${A C}_{ω}$ , for a symmetric positive-definite quadratic form on $R^{n}$ , the normalized Gaussian integral is $(2 π / β)^{n / 2} (\det K)^{- 1 / 2}$ , with mean zero and covariance $β^{- 1} K^{- 1}$ .

[F3]

Under ${A C}_{ω}$ , Euclidean phase-space Lebesgue measure is the product of the $q$ and $p$ Lebesgue measures; Tonelli factors nonnegative densities and Fubini factors their integrable moments (On Borel subsets of R^{m+n}, the product lambda_m times lambda_n agrees with lambda_{m+n}, Tonelli's theorem for nonnegative measurable functions on a sigma-finite product, Fubini's theorem for L^1 functions on a sigma-finite product).

[A1]

The canonical density is a model preparation, not a claim that an isolated trajectory equilibrates or that a laboratory realizes this law.

Proof

1.1givenF1F2F3algebra

By [F1], the phase-space weight factors as $e^{- β p^{T} M^{- 1} p / 2} e^{- β q^{T} K q / 2}$ . The product-measure identity and Tonelli [F3] justify multiplying the two nonnegative Lebesgue integrals. Applying [F2] to the $q$ factor gives normalization $(2 π / β)^{n / 2} (\det K)^{- 1 / 2}$ and covariance $β^{- 1} K^{- 1}$ .

1.2givenF2algebra

The momentum quadratic form has symmetric positive-definite matrix $M^{- 1}$ . Applying [F2] with $K = M^{- 1}$ gives its integral $(2 π / β)^{n / 2} \sqrt{\det M}$ and covariance of $p$ equal to $β^{- 1} (M^{- 1})^{- 1} = β^{- 1} M$ .

2.1A1F3step 1.1step 1.2algebra

Multiplying the two integrals from steps 1.1–1.2 and dividing by the reference factor $h_{0}^{n}$ gives $Z = (2 π / (β h_{0}))^{n} \sqrt{\det M / \det K}$ . The phase-space Lebesgue measure is the product of its $q$ and $p$ factors by [F3]; the normalized canonical density therefore factors into independent $q$ and $p$ laws. Each factor is even, so its mean is zero, and its covariance is the one computed in steps 1.1–1.2.

3.1A1F1step 1.1step 1.2algebra∎

By [F1], taking the expectation of $H$ and summing the diagonal traces of the two covariance identities gives $U = \frac{1}{2} E [p^{T} M^{- 1} p] + \frac{1}{2} E [q^{T} K q] = \frac{n}{2 β} + \frac{n}{2 β} = \frac{n}{β} .$ The adopted relation $β^{- 1} = k_{B} T$ makes $U = n k_{B} T$ . Holding the model parameters $M, K, h_{0}$ fixed therefore gives $C_{h a r m} = (\partial U / \partial T)_{M, K, h_{0}} = n k_{B}$ , with units J K $^{- 1}$ .

Remarks

Positive definiteness is a confinement condition. If $K$ has a translation zero mode, the full-space configurational integral diverges and this normalized ensemble does not exist without an additional constraint or confinement.

Confined coupled quadratic canonical systems

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources