ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedverified 2026-10-15 (gpt-6-luna, gpt-6.1-sol, and delegated Codex review/repair agents; see completed run evidence)

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Response functions in a formal logarithmic entropy chart

Physical scope

A constructed fixed-composition smooth chart used only to evaluate the preceding definitions; it is not an empirical equation of state or a complete bulk model.

Example

Fix a constitution $c$ and an open dimensionless single-phase subset $O \subseteq D_{c}$ whose physical chart is $Ω = Φ_{c} (O)$ under Entropic temperature, pressure, and molar chemical potentials. Fix positive amounts $n = (n_{1}, \dots, n_{r})$ for which the slice $Ω_{n} : = {(U, V) : (U, V, n) \in Ω}$ is nonempty, and assume $Ω_{n} \subseteq {(U, V) : U, V > 0}$ . Let $a, b > 0$ have units $J K^{- 1}$ , with reference values $U_{0} > 0$ joules and $V_{0} > 0$ cubic metres. Stipulate the physical-coordinate entropy representative on $Ω_{n}$ (denoted by $S_{c} (U, V, n)$ as in Entropic temperature, pressure, and molar chemical potentials) by

$S (U, V) = a \ln (U / U_{0}) + b \ln (V / V_{0}) .$

Applying the definitions on this chart gives

$T = U / a, p = b T / V, C_{V} = a, C_{p} = a + b, κ_{T} = 1 / p, α_{p} = 1 / T .$

This is a formal chart calculation, not measured data and not a claim that this entropy satisfies the track's tagged bulk-scaling postulate for a fixed material type.

Facts & Assumptions

Given: $U, V, U_{0}, V_{0} > 0$ with units $J, m^{3}, J, m^{3}$ respectively; $a, b > 0$ in $J K^{- 1}$ ; fixed composition; the displayed function is stipulated on this chart.

[A1]

Entropic temperature, pressure, and molar chemical potentials: $T = 1 / S_{U}$ and $p = T S_{V}$ , with $V$ held fixed in $S_{U}$ and $U$ held fixed in $S_{V}$ .

[F1]

Thermodynamic response functions on valid local charts: $C_{V} = (\partial U / \partial T)_{V, n}$ , $C_{p} = (\partial H / \partial T)_{p, n}$ with $H = U + p V$ , $κ_{T} = - V^{- 1} (\partial V / \partial p)_{T, n}$ , and $α_{p} = V^{- 1} (\partial V / \partial T)_{p, n}$ where the indicated local charts exist.

[F2]

The natural logarithm as the inverse of the exponential function: $\ln$ denotes the natural logarithm on the positive real numbers.

[L1]

The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t: For positive $x$ , the natural logarithm is differentiable and $\frac{d}{d x} \ln x = 1 / x$ .

Verification

1.1givenA1F2L1algebra

The positive ratios in the stipulated entropy are in the domain of [F2]. By [L1], $S_{U} = a / U$ and $S_{V} = b / V$ . Substitution in [A1] gives $T = U / a > 0$ and $p = b U / (a V) = b T / V$ .

2.1givenA1F1F2L1step 1.1algebra

Solving $T = U / a$ gives $U = a T$ , hence $C_{V} = (\partial U / \partial T)_{V, n} = a$ . The natural-coordinate map $(U, V) \mapsto (T, V)$ has inverse $(T, V) \mapsto (a T, V)$ . The map $(U, V) \mapsto (T, p)$ has inverse $(T, p) \mapsto (a T, b T / p)$ on the positive quadrant. By [F2] and [L1], these maps and inverses have continuous first partial derivatives for positive arguments, so their restrictions to the open slice $Ω_{n}$ give the local charts required by [F1]. Solving $p = b T / V$ gives $(\partial V / \partial p)_{T, n} = - b T / p^{2}$ and $(\partial V / \partial T)_{p, n} = b / p$ . With $V = b T / p$ , [F1] yields $κ_{T} = 1 / p$ and $α_{p} = 1 / T$ .

3.1givenF1step 1.1step 2.1algebra∎

The enthalpy in [F1] is $H = U + p V = a T + b T = (a + b) T$ , so $C_{p} = (\partial H / \partial T)_{p, n} = a + b$ . All displayed units are consistent: $C_{V}, C_{p}$ are $J K^{- 1}$ , $κ_{T}$ is $P a^{- 1}$ , and $α_{p}$ is $K^{- 1}$ .

Response functions in a formal logarithmic entropy chart

Example

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources