Physics theoremStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck pendingverified 2026-10-23 (gpt-6.1-sol)

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.

Continuum entropy residual and Newtonian admissibility

Physical scope

Conditional smooth local-equilibrium continuum calculation with positive temperature, the stated local mass and energy-balance hypotheses, and Newtonian/Fourier closure. Entropy production is a pointwise coefficient test, not an empirical validation or a proof of the adopted entropy postulate.

Statement

Assume the complete smooth material, state, contact and mass/momentum/angular-momentum/total-energy balance hypotheses of the local Gibbs-entropy identity, including ${A C}_{ω}$ and positive temperature, as stated in Facts & Assumptions. Use a $C^{2}$ local state relation $e (v, s)$ with $v = 1 / ρ$ , $T = e_{s} (v, s) > 0$ , and $p = - e_{v} (v, s)$ . Let $σ = - p I_{3} + τ$ , define $L_{i j} = \partial_{x_{j}} u_{i}$ , $D = (L + L^{T}) / 2$ , $θ = tr D$ , and set $G = ρ D_{t} s + div (q / T) - ρ r / T, D_{t} = \partial_{t} + u \cdot \nabla .$ For the adopted Newtonian/Fourier form $τ = 2 μ D_{0} + ζ θ I_{3}$ , define $D_{0} = D - θ I_{3} / 3$ , $A : B : = \sum_{i, j = 1}^{3} A_{i j} B_{i j}$ for $A, B \in R^{3 \times 3}$ , and $∣ D_{0} ∣^{2} : = D_{0} : D_{0} = \sum_{i, j = 1}^{3} (D_{0})_{i j}^{2}$ . Set $sym (K) : = (K + K^{T}) / 2$ and $q = - K \nabla T$ . The residual is $G = \frac{2 μ ∣ D_{0} ∣^{2} + ζ θ^{2}}{T} + \frac{(\nabla T)^{T} sym (K) \nabla T}{T^{2}} .$ At a fixed state, if the algebraic coefficient family is tested for every symmetric $D$ , every $g = \nabla T \in R^{3}$ , and every $T > 0$ , this residual is nonnegative exactly when $μ \geq 0$ , $ζ \geq 0$ , and $x^{T} sym (K) x \geq 0$ for every $x \in R^{3}$ . These conditions make the adopted closure compatible with the separate continuum entropy inequality. In SI units, $G$ has units $W m^{- 3} K^{- 1}$ . The theorem does not derive the balance or entropy postulates, infer material coefficients, or assert empirical validity.

Statement references

David Tong, Fluid Mechanics, Part II lecture notes, §3.1.2, printed pp. 61–63

David Tong, Fluid Mechanics, Part III lecture notes, §§4.4.2–4.4.3, printed pp. 146–149

An Introduction to Continuum Mechanics, Second Edition, Chapter 6, §§6.1.4, 6.5.1–6.5.3 and 6.7.2, printed pp. 224–245 and 251–252

Facts & Assumptions

Given: The smooth material motion and positive-density fields, the $C^{2}$ state relation, local-equilibrium closure, ${A C}_{ω}$ , and the contact and material-balance assumptions specified below.

[A1]

Single-component nonpolar continuum and material balances: the physical postulate adopts no-source mass conservation, local contact traction $σ n$ , linear and angular momentum balances, total-energy balance, and a separate entropy inequality on its smooth positive-density material regions. It does not derive these assumptions. Its current batch-3 carrier and source-backed formulation were inspected, and step1Decision currently reports ready.

[A2]

Local simple-fluid equilibrium and Gibbs closure: at each admissible state, $v = 1 / ρ$ , $e \in C^{2}$ , $T = e_{s} > 0$ , and $p = - e_{v}$ ; this is an adopted local state relation. The current batch-11 carrier states this relation explicitly.

[A3]

Newtonian stress and Fourier conductive closure: the adopted stress and heat-flux closures are $τ = 2 μ D_{0} + ζ θ I_{3}$ and $q = - K \nabla T$ , with specified state laws and nonnegative $μ, ζ, sym K$ . The current batch-11 carrier states this closure explicitly.

[L1]

Gibbs cancellation and entropy-production identity: under ${A C}_{ω}$ , an open time interval, a bounded $C^{1}$ material region, a $C^{2}$ orientation-preserving motion, positive $C^{1}$ density and state fields with $v = 1 / ρ$ , $T = e_{s} > 0$ , $p = - e_{v}$ , a $C^{1}$ stress split $σ = - p I_{3} + τ$ , the local contact hypothesis, continuity equation, and separate linear/angular-momentum and total-energy integral balances, it concludes $σ = σ^{T}$ , $ρ D_{t} e = σ : D - div q + ρ r$ , and the displayed Gibbs residual identity. The complete displayed Gibbs hypotheses remain binding, and its proof step 3.1 supplies time recentering.

[L2]

Newtonian/Fourier dissipation and exact coefficient conditions: for ${A C}_{ω}$ , $T > 0$ , symmetric $D$ , arbitrary $g \in R^{3}$ , $τ = 2 μ D + λ (tr D) I_{3}$ , $q = - K g$ , and $ζ = λ + 2 μ / 3$ , the residual decomposes as $(2 μ ∣ D_{0} ∣^{2} + ζ θ^{2}) / T + g^{T} sym (K) g / T^{2}$ , and it is nonnegative for all $D, g, T$ iff $μ, ζ \geq 0$ and $sym (K)$ is positive semidefinite. The algebraic test and its conditional Gibbs interpretation retain exactly these hypotheses.

[A4]

At a point, the algebraic test varies $D$ and $g$ independently; it is not a claim that every such pair occurs in a particular global solution.

Proof

1.1A1A2A3L1algebra

Under the exact motion, contact, continuity and energy-balance hypotheses in [L1], the Gibbs identity gives $G = τ : D / T - q \cdot \nabla T / T^{2}$ . Substitute the postulated closures from [A3]. Since $D = D_{0} + θ I_{3} / 3$ , $tr D_{0} = 0$ , $D_{0} : I_{3} = 0$ , and $I_{3} : I_{3} = 3$ , contraction gives $τ : D = 2 μ ∣ D_{0} ∣^{2} + ζ θ^{2}$ . Also $- q \cdot \nabla T = (\nabla T)^{T} K \nabla T = (\nabla T)^{T} sym (K) \nabla T$ , because a real scalar quadratic form annihilates the skew-symmetric part. This yields the stated formula.

2.1A1A3L2step 1.1algebra∎

Apply the exact coefficient equivalence [L2] with $g = \nabla T$ , $λ = ζ - 2 μ / 3$ , and $T > 0$ . It gives nonnegativity for every independent local $D, g$ exactly under the three stated conditions. The adopted coefficient range in [A3] satisfies those conditions; thus this closure is consistent with the separately adopted entropy inequality in [A1]. This is a conditional model deduction, not a proof of either physical postulate.

Continuum entropy residual and Newtonian admissibility

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources