Physics theoremStatement: AI-adaptedProof: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)

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Local Gibbs and parcel first law in the adopted material

Physical scope

Conditional local-equilibrium continuum material with the supplied C2 fixed-composition state function; local parcel calculus only, not a microscopic finite-particle process or global gravitating extensivity.

Statement

In the adopted fixed-composition local-equilibrium material model of Restricted local equilibrium parcel with adopted material state function, let $X \subset (0, \infty) \times R$ be its open state domain, let $e \in C^{2} (X, R)$ include the rest-energy contribution, and set $T = e_{σ_{p}}$ , $p = - e_{v}$ , $ε = n e$ , and $μ = e + p v - T σ_{p}$ , with $n = 1 / v$ and $s = n σ_{p}$ as in Rest fluid state, frame and geometric current definitions. The model's local state identities are $d ε = T d s + μ d n, ε + p = T s + μ n, d p = s d T + n d μ .$ For a local continuum parcel with real extensive marker variable $N > 0$ , proper volume $V > 0$ , entropy $S \in R$ , and $(V / N, S / N) \in X$ , its adopted energy is $E = N e (V / N, S / N)$ . On this open parcel domain it obeys $d E = T d S - p d V + μ d N, E = T S - p V + μ N .$ The count variable is treated continuously as part of the local-equilibrium continuum model; these differentials are not claims about discrete microscopic transitions. The result is conditional model calculus and does not establish that a chosen state function describes a particular material.

Facts & Assumptions

Given: A nonempty open state domain $X \subset (0, \infty) \times R$ , a supplied $e \in C^{2} (X, R)$ , a state $(v, σ_{p}) \in X$ , and a local-equilibrium continuum parcel with $N, V > 0$ and $(V / N, S / N) \in X$ .

[F1]

Restricted local equilibrium parcel with adopted material state function (draft physical postulate): for a selected fixed-composition branch, adopt a supplied $C^{2}$ state function $e (v, σ_{p})$ , $T = e_{σ_{p}} > 0$ , energy per marker including rest energy, and the parcel assignment $E = N e (V / N, S / N)$ . Its applicability is a model assumption; it asserts no universal or measured equation of state.

[F2]

Local Gibbs, Euler, parcel first law and response identities (draft mathematical supplier): for such an open $X$ and $C^{2}$ function $e$ , with $v = 1 / n$ , $σ_{p} = s / n$ , $T = e_{σ_{p}}$ , $p = - e_{v}$ , $ε = n e$ , and $μ = e + p v - T σ_{p}$ , it states $d ε = T d s + μ d n$ , $ε + p = T s + μ n$ , $d p = s d T + n d μ$ , and on the positive open parcel domain it proves $d E = T d S - p d V + μ d N$ and $E = T S - p V + μ N$ . These identities apply at exactly the stated open-domain and $C^{2}$ hypotheses.

[F3]

Rest fluid state, frame and geometric current definitions: on a nonempty open spacetime set, a $C^{2}$ state map $(v, σ_{p})$ takes values in a nonempty open $X \subset (0, \infty) \times R$ and defines $n = 1 / v$ , $s = n σ_{p}$ , $ε = n e$ , $T = e_{σ_{p}}$ , $p = - e_{v}$ , and $μ = e + p v - T σ_{p}$ . The local differential calculation below uses this state definition and the adopted material model.

Proof

1.1givenF1F2F3algebra

By the state definitions, $d e = e_{v} d v + e_{σ_{p}} d σ_{p} = - p d v + T d σ_{p}$ , while $v = 1 / n$ and $σ_{p} = s / n$ give $d v = - d n / n^{2}$ and $d σ_{p} = d s / n - s d n / n^{2}$ . Therefore $d ε = d (n e) = e d n + n d e = T d s + (e + p v - T σ_{p}) d n = T d s + μ d n$ . Also $ε + p = n (e + p v) = n (T σ_{p} + μ) = T s + μ n$ .

1.2givenF1F2F3constructalgebra

On the open set $D = {(S, V, N) : V > 0, N > 0, (V / N, S / N) \in X}$ , the parcel energy is a $C^{2}$ composition. Differentiating at fixed other parcel coordinates gives $\partial_{S} E = e_{σ_{p}} = T$ , $\partial_{V} E = e_{v} = - p$ , and $\partial_{N} E = e - (V / N) e_{v} - (S / N) e_{σ_{p}} = e + p v - T σ_{p} = μ$ . Thus $d E = T d S - p d V + μ d N$ .

2.1F2F3step 1.1algebra

Differentiate $ε + p = T s + μ n$ and substitute the first identity in step 1.1. The terms $T d s$ and $μ d n$ cancel, leaving $d p = s d T + n d μ$ . This is a state-function identity on the open domain.

3.1givenF1F3algebra∎

Using $μ = e + p v - T σ_{p}$ and $V = N v$ , $S = N σ_{p}$ , we have $T S - p V + μ N = N (T σ_{p} - p v + e + p v - T σ_{p}) = N e = E$ . This is a deduction within the adopted continuum material model, not an empirical statement.

Local Gibbs and parcel first law in the adopted material

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources