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Representation And Equilibrium — Examples

1 · Prerequisites

2 · Summary

The formal charts calculate response derivatives and entropy-maximizing exchange conditions under explicitly stipulated functions. They are not measurements or empirical equations of state. The retained-volume example shows that equal temperature does not supply pressure equality when volume exchange is constrained. The finite-dimensional tag model checks that tagged scaling and additivity alone do not establish homogeneity for one fixed tag.

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

None yet.

5 · Examples, experiments and counterexamples

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

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 $K^{- 1}$ .

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

Stable equilibrium with retained unequal volumes

Physical scope

A constructed two-subsystem entropy-maximization example with fixed volumes; it illustrates the effect of retained constraints and is not an empirical model of a named substance.

Example

For each of two separated fixed-composition subsystems $i \in {A, B}$ , let $O_{i} \subseteq D_{c_{i}}$ be an open dimensionless single-phase subset with physical chart $Ω_{i} = Φ_{c_{i}} (O_{i})$ as in Entropic temperature, pressure, and molar chemical potentials. Fix positive amounts $n_{i}$ with nonempty physical slices $Ω_{i, n_{i}} : = {(U_{i}, V_{i}) : (U_{i}, V_{i}, n_{i}) \in Ω_{i}}$ . Fix $U_{0} > 0$ joules and $V_{0} > 0$ cubic metres, and set $V_{A} = V_{0}$ and $V_{B} = 2 V_{0}$ . Assume $Ω_{i, n_{i}} \subseteq (0, 2 U_{0}) \times (0, \infty)$ and $(U, V_{i}) \in Ω_{i, n_{i}}$ for every $0 < U < 2 U_{0}$ and each $i$ . Let each physical-coordinate entropy representative be $S_{i} (U_{i}, V_{i}) = a \ln (U_{i} / U_{0}) + b \ln (V_{i} / V_{0})$ on its slice, where $a, b > 0$ are in $J K^{- 1}$ . The isolated compound permits energy exchange but retains the volume and mole partitions, and its adopted total entropy is $S_{A} + S_{B}$ . For total energy $2 U_{0}$ , stipulate that every split $0 < U_{A} < 2 U_{0}$ , $U_{B} = 2 U_{0} - U_{A}$ with these retained volumes and amounts is admitted, and that these are exactly the feasible endpoints. As additional model input, designate the split $U_{A} = U_{B} = U_{0}$ as a stable equilibrium. This designation must satisfy Stable equilibrium requires an attained global entropy maximum, which supplies a necessary maximum criterion rather than a converse. The unique entropy-maximizing energy split is $U_{A} = U_{B} = U_{0}$ , so $T_{A} = T_{B} = U_{0} / a$ , while $p_{A} = b T_{A} / V_{0}$ and $p_{B} = b T_{B} / (2 V_{0}) = p_{A} / 2$ . Thus this model's stable equilibrium under energy exchange does not imply pressure equality when volume exchange is retained.

Facts & Assumptions

Given: $a, b > 0$ in $J K^{- 1}$ , $U_{0} > 0$ in joules, $V_{0} > 0$ in cubic metres, two independent separated subsystems, $U_{A} + U_{B} = 2 U_{0}$ , retained $V_{A} = V_{0}$ , $V_{B} = 2 V_{0}$ and mole amounts, the stated physical-slice restrictions and exact feasible endpoints, and the additional model designation of the equal-energy split as stable equilibrium.

[A1]

Stable equilibrium requires an attained global entropy maximum: The adopted postulate requires every designated stable equilibrium to be an attained global maximum of the summed entropy over the exact feasible set; attainment is assumed, and no uniqueness is supplied.

[F1]

Isolated compound and feasible equilibrium set: The feasible set fixes the declared totals and retained partitions; an exchange is allowed only when the protocol permits it.

[F2]

Additive separated systems and tagged bulk scaling and Concave macroscopic fundamental entropy: The separated compound uses additive total energy and the sum of the constituent entropies. These postulates do not assert an empirical equation of state.

[F3]

Entropic temperature, pressure, and molar chemical potentials: $T = 1 / S_{U}$ and $p = T S_{V}$ on the smooth positive-temperature chart.

[F4]

The natural logarithm as the inverse of the exponential function: $\ln$ is the natural logarithm on the positive reals.

[F5]

Interior exchange conditions from entropy maximization: Pressure equality follows from entropy maximization only when the corresponding volume exchange is locally permitted (and thermal exchange supplies equal temperatures).

[L1]

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

Verification

1.1givenA1F1F2F4L1algebra

The stipulated feasible-set model, slice restrictions and total-energy constraint make the feasible energy splits exactly $u = U_{A} \in (0, 2 U_{0})$ and $U_{B} = 2 U_{0} - u$ . The ratios in $f (u) = a \ln (u / U_{0}) + a \ln ((2 U_{0} - u) / U_{0})$ are positive, so [F4] applies. By [L1], $f^{'} (u) = a (1 / u - 1 / (2 U_{0} - u))$ and $f^{''} (u) = - a / u^{2} - a / (2 U_{0} - u)^{2} < 0$ . The derivative is positive for $u < U_{0}$ , zero at $U_{0}$ , and negative for $u > U_{0}$ ; hence $u = U_{0}$ is the unique global maximizer on the exact feasible set, so the additional stable-equilibrium designation satisfies the necessary criterion in [A1]. The entropy tends to $- \infty$ at both excluded endpoints.

2.1givenF1F3F5step 1.1algebra∎

At the maximizing split, [F3] gives $T_{A} = U_{A} / a = U_{0} / a = T_{B}$ . It gives $p_{A} = b T_{A} / V_{0}$ and $p_{B} = b T_{B} / (2 V_{0})$ , hence $p_{B} = p_{A} / 2 \neq p_{A}$ since $b, T_{A}, V_{0} > 0$ . The volume partition is retained by [F1], so no volume-exchange equality is implied by [F5].

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

Tagged bulk scaling need not be homogeneity within one fixed tag

Physical scope

A formal constitutive-data countermodel to a logical implication between the framework's tagged scaling and entropy-additivity postulates; it is not an empirical model of a material.

Statement refuted

The tagged bulk-scaling relation of Concave macroscopic fundamental entropy together with the additive coordinate composition of Additive separated systems and tagged bulk scaling forces the fixed-tag identity $S_{c} (λ x) = λ S_{c} (x)$ for each material constitution $c$ .

Facts & Assumptions

Given: A dimensionless positive tag $τ$ , positive dimensionless state coordinates $\hat{U}, \hat{V}, \hat{n}$ , and the following formal family; their physical coordinates $U, V, n$ use joules, cubic metres, and moles.

[A1]

Additive separated systems and tagged bulk scaling: Independent separated systems may have a composition map that adds the admitted extensive coordinates, and the bulk scaling map may send a state of tag $c$ to a distinct tag $c^{(λ)}$ .

[A2]

Concave macroscopic fundamental entropy: Entropy is finite and concave on its specified convex domains, additive for admitted independent separated systems, and satisfies the tagged relation $S_{c^{(λ)}} (δ_{λ} x) = λ S_{c} (x)$ . The tag $c^{(λ)}$ is not silently identified with $c$ .

[F1]

Equilibrium state domains: The equilibrium domain uses dimensionless coordinates, with physical values $(U, V, n)$ supplied by its diagonal SI coordinate map. A formal positive chart may be specified as an open subset of those dimensionless coordinates.

[F2]

SI quantities and unit conventions: $U, V, n$ have units joules, cubic metres, and moles; entropy has units $J K^{- 1}$ .

Counterexample

1.1givenF1F2algebra

Let tags be $τ > 0$ , compose them by $τ_{A} \oplus τ_{B} = τ_{A} + τ_{B}$ , and scale by $τ^{(λ)} = λ τ$ . For each tag use the open dimensionless state domain $D_{τ} = (0, \infty)^{3}$ with coordinates $x = (\hat{U}, \hat{V}, \hat{n})$ and physical map $Φ_{τ} (x) = (\hat{U} J, \hat{V} m^{3}, \hat{n} m o l)$ . Choose $a, b, q > 0$ in $J K^{- 1}$ and define $S_{τ} (x) = a \hat{U} + b \hat{V} + q τ$ . The corresponding physical-coordinate representative is $S_{τ}^{S I} (U, V, n) = a U / (1 J) + b V / (1 m^{3}) + q τ$ . By [F1] the dimensionless domain is open and positive, and [F2] makes every entropy term have units $J K^{- 1}$ . The function is finite, $C^{2}$ , affine and hence concave; its physical energy derivative is $S_{τ, U}^{S I} = a / (1 J) > 0$ .

2.1A1A2step 1.1algebra

For two separated states, define the compound tag as $τ_{A} + τ_{B}$ and the composition map to add their dimensionless state coordinates. Its physical coordinates are therefore the sums of the constituent physical coordinates. Then $S_{τ_{A} + τ_{B}} (x_{A} + x_{B}) = S_{τ_{A}} (x_{A}) + S_{τ_{B}} (x_{B})$ , so the model satisfies the stated coordinate and entropy additivity.

2.2A1A2step 1.1algebra

Under scaling, $S_{λ τ} (λ x) = a λ \hat{U} + b λ \hat{V} + q λ τ = λ S_{τ} (x)$ , so the tagged scaling relation in [A2] holds for every $λ > 0$ .

3.1givenA1A2step 1.1step 2.2algebra∎

For the same fixed tag $τ$ , however, $S_{τ} (2 x) = 2 a \hat{U} + 2 b \hat{V} + q τ$ whereas $2 S_{τ} (x) = 2 a \hat{U} + 2 b \hat{V} + 2 q τ$ . Since $q τ > 0$ , these values differ. Thus tagged scaling plus the stated additivity does not imply fixed-tag homogeneity.

Representation And Equilibrium — Examples

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

5 · Examples, experiments and counterexamples

Response functions in a formal logarithmic entropy chart

Example

Facts & Assumptions

Verification

Stable equilibrium with retained unequal volumes

Example

Facts & Assumptions

Verification

Tagged bulk scaling need not be homogeneity within one fixed tag

Statement refuted

Facts & Assumptions

Counterexample

Sources