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Euler and Gibbs-Duhem identities under same-tag homogeneity

Physical scope

Conditional local identities on a positive-temperature C2 energy-representation chart when the same fixed material tag satisfies degree-one homogeneity on that chart; the separately designated common-bulk constitutive-law branch supplies this condition through its entropy transport on local inverse charts with common species, intensive parameters, units and energy/entropy references. The general tagged-scaling postulate alone does not provide it.

Statement

Let $C$ be the open positive-temperature domain of a local C2 energy representation $U_{c} (s, V, n)$ for a fixed material constitution $c$ , with $V > 0$ , $n_{j} > 0$ , and conjugates $T = U_{c, s}$ , $p = - U_{c, V}$ , and $μ_{j} = U_{c, n_{j}}$ . Let $W \subseteq C$ be open. In addition to the framework suppliers, assume the same-tag homogeneity hypothesis: for every $q = (s, V, n_{1}, \dots, n_{r}) \in W$ there is an $ε_{q} \in (0, 1)$ such that

$U_{c} (λ s, λ V, λ n_{1}, \dots, λ n_{r}) = λ U_{c} (s, V, n_{1}, \dots, n_{r})$

for every $λ > 0$ with $∣ λ - 1 ∣ < ε_{q}$ and $λ q \in C$ . Then, throughout $W$ ,

$U_{c} = T s - p V + \sum_{j = 1}^{r} μ_{j} n_{j},$

and on the same chart

$s d T - V d p + \sum_{j = 1}^{r} n_{j} d μ_{j} = 0.$

The same-tag hypothesis is supplied, in particular, when the inverse states $Ψ (q) : = (U_{c} (q), V, n)$ for $q \in W$ lie in the designated common-law cone of Common constitutive entropy law for a designated bulk family, expressed in physical SI coordinate values using the fixed unit conversion. On this branch the adopted entropy transport implies the energy homogeneity locally by uniqueness of the local inverse; no global inverse or global energy-domain cone is assumed. These are conditional model deductions. The general tagged relation in Concave macroscopic fundamental entropy alone does not identify the scaled system with one fixed constitutive law.

Facts & Assumptions

Given: The local C2 energy representation on $C$ , the open subdomain $W$ , and the same-tag homogeneity hypothesis in the Statement, either assumed independently or supplied by the designated common-law branch as specified there. Coordinates are entropy $s$ in $J K^{- 1}$ , volume $V$ in $m^{3}$ , and whole-system amounts $n_{j}$ in $m o l$ .

[A1]

Additional same-tag premise: for each $q \in W$ there is an open interval $∣ λ - 1 ∣ < ε_{q}$ on which $U_{c} (λ q) = λ U_{c} (q)$ whenever $λ q \in C$ . The common-law branch in [A2] supplies this premise on the corresponding local inverse subdomain.

[A2]

Common constitutive entropy law for a designated bulk family: For the supplied bulk branch, the inverse states $Ψ (W)$ lie in its designated open single-phase cone, after the fixed SI unit conversion. Coordinate-preserving entropy transport and tagged scaling give $S_{c} (λ x) = λ S_{c} (x)$ there. The material law has fixed species and intensive parameters and common units and energy/entropy reference conventions; distinct physical system tags are retained. This premise is required only for the assertion that the framework supplies [A1], not for the general implication from independently assumed [A1].

[F1]

Additive separated systems and tagged bulk scaling: The scaling map may send $c$ to a different material tag $c^{(λ)}$ ; the statement explicitly does not identify those tags.

[F2]

Concave macroscopic fundamental entropy: The entropy scaling statement is $S_{c^{(λ)}} (δ_{λ} x) = λ S_{c} (x)$ for the tagged family; it does not state fixed-tag homogeneity of the energy representation.

[F3]

Local energy representation and thermodynamic conjugates: On its local inverse chart, $U_{c, s} = T_{c}$ , $U_{c, V} = - p_{c}$ , and $U_{c, n_{j}} = μ_{c, j}$ . It supplies a unique local C1 inverse from the draft local-rank graph interface and derives C2 regularity from the C2 entropy and its positive energy derivative; draft readiness does not constitute independent proof acceptance.

[F4]

Differentials of the thermodynamic potentials: $d U_{c} = T d s - p d V + \sum_{j} μ_{j} d n_{j}$ on the same chart, conditional on [F3].

[L1]

Euler identity for a differentiable degree-one homogeneous function: A C1 function with degree-one homogeneity along each local ray satisfies $f (x) = D f (x) [x]$ .

[L2]

Chain sum product and composition rules for Banach derivatives: Differentiating finite scalar products obeys the product rule.

Proof

1.1givenA2F3construct

On the supplied branch [A2], fix $q = (s, V, n) \in W$ and $x = Ψ (q)$ in the original-coordinate uniqueness neighborhood of the local inverse [F3]. By openness, for $λ$ in some open interval about $1$ , both $λ q \in W$ and $λ x$ remain in that uniqueness neighborhood and in the common-law entropy cone. Its fixed unit conversion commutes with multiplication by $λ$ . Thus [A2] gives $S_{c} (λ U_{c} (q), λ V, λ n) = λ S_{c} (x) = λ s$ . By uniqueness of the inverse [F3], $U_{c} (λ q) = λ U_{c} (q)$ throughout this interval. Consequently [A2] supplies [A1] locally, without a global inverse claim.

2.1givenA1L1step 1.1

For the general conclusion assume [A1], which step 1.1 supplies when [A2] applies. Since $W$ is open, at each $q \in W$ shrink the homogeneity interval so that $λ q \in W$ throughout it. The local-ray hypotheses of [L1] therefore apply to $U_{c}$ restricted to $W$ , giving $U_{c} = s U_{c, s} + V U_{c, V} + \sum_{j} n_{j} U_{c, n_{j}}$ .

3.1F3step 2.1algebra

Substitute the conjugate derivatives from [F3] into step 2.1 to obtain $U_{c} = T s - p V + \sum_{j} μ_{j} n_{j}$ .

4.1givenL2step 3.1algebra

Differentiate the identity in step 3.1 on the C2 chart using [L2]. The right side has differential $T d s + s d T - p d V - V d p + \sum_{j} (μ_{j} d n_{j} + n_{j} d μ_{j})$ .

5.1F4step 4.1algebra

By [F4], the differential of the left side is $T d s - p d V + \sum_{j} μ_{j} d n_{j}$ . Equating these differentials and cancelling the common terms yields $s d T - V d p + \sum_{j} n_{j} d μ_{j} = 0$ .

6.1givenA1A2F1F2step 1.1step 3.1step 5.1∎

The general tagged premises [F1]–[F2] alone do not discharge [A1]. On the separately designated common-law branch, [A2] and the local inverse argument in step 1.1 do discharge it, so steps 3.1–5.1 give the full Euler and Gibbs–Duhem identities there. These are identities of local state functions with the stated rest-frame SI and reference conventions; no empirical validation or exact finite-size scaling is inferred.

Remarks

The extra common-law bulk postulate supplies the constitutive identification absent from the general tagged family. The theorem also retains its general conditional implication for any independently stipulated same-tag local energy law. The batch-8 countermodel concerns the broader tagged premises and lies outside this added branch; it remains nonlogical context. Neither local entropy inversion nor the differential identities assert global invertibility, admissibility across phase boundaries, or universal extensivity for finite, surface-sensitive, long-range or gravitating systems.

Euler and Gibbs-Duhem identities under same-tag homogeneity

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources