PostulateStatement: AI-adaptedProof: Not applicablePipeline-generatedprecheck pending‡ sources checked 2026-10-23

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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.
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Local simple-fluid equilibrium and Gibbs closure

Physical scope

An adopted local-equilibrium state relation for a smooth single-component simple-fluid branch with positive density and temperature. No global equilibrium, global thermodynamic chart, universal material validity or empirical measurement is asserted.

Postulate

Use a smooth single-component, nonpolar fluid branch in a Cartesian inertial frame. Let $U \subset (0, \infty) \times R$ be an open state domain with coordinates $v$ and $s$ , where $v$ is specific volume and $s$ is specific entropy. Adopt a specific internal-energy function $e : U \to R$ of class $C^{2}$ , meaning that its coordinate partial derivatives through total order two exist and are continuous. At a state $(v, s)$ , define the temperature and thermodynamic pressure by $T (v, s) = \frac{\partial e}{\partial s} (v, s), p (v, s) = - \frac{\partial e}{\partial v} (v, s), ρ = \frac{1}{v},$ where each partial derivative holds the other state coordinate fixed. Assume the local material states lie in $U$ and $T > 0$ . Local equilibrium here means that the fields $e, T, p$ at each event are assigned by these state functions of the local $v, s$ .

In the physical SI interpretation, $v$ is in $m^{3} k g^{- 1}$ , $s$ in $J k g^{- 1} K^{- 1}$ , $e$ in $J k g^{- 1}$ , $T$ in $K$ , $p$ in $P a$ , and $ρ$ in $k g m^{- 3}$ . Positivity of $e_{s s}$ , global invertibility of $s \mapsto T$ , convexity, and the dynamical entropy inequality are separate assumptions or postulates; none follows from this local equilibrium adoption. This closure is an additional physical model choice, not a consequence of global equilibrium thermodynamics.

Postulate references

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, §§6.1.1 and 6.1.4, printed pp. 221–226

Facts & Assumptions

Given: A local state $(v, s) \in U$ , positive density $ρ = 1 / v$ , and the $C^{2}$ state function in the Postulate.

[L1]

Specific fundamental relation and valid response charts: on an open $U \subset (0, \infty) \times R$ , $e \in C^{2} (U)$ , $T = e_{s}$ , $p = - e_{v}$ , and $ρ = 1 / v$ ; each partial derivative holds the other state coordinate fixed. That mathematical definition imposes no constitutive claim about an actual fluid. Its current batch-10 carrier and complete definition were inspected, and step1Decision currently reports ready; this records authoring readiness only, not independent proof acceptance.

[L2]

Classical frames, primitive quantities and SI units: the current density $ρ$ , temperature $T$ , specific energy $e$ , specific entropy $s$ , and pressure $p$ have the SI types and units stated above. Its current batch-3 carrier was inspected, and step1Decision currently reports ready.

[A1]

The pointwise use of the mathematical state relation as local material response is an adopted physical premise. No global inversion or empirical validity is included.

Local simple-fluid equilibrium and Gibbs closure

Postulate

Postulate references

Facts & Assumptions

Depends on

Dependency tree · 2 levels

Sources