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Finite entropy integral does not force a heat-capacity power bound of order one or higher

Physical scope

A formal fixed-composition positive-temperature response chart used to refute an inference from integral convergence alone; it is not a measured material or a complete global bulk-family model.

Statement refuted

For a positive-temperature fixed- $V, n$ chart, write $θ = T / T_{*}$ . For a dimensionless real exponent $q \geq 1$ , the notation $C_{V} (T) = O (θ^{q})$ as $θ ↓ 0$ means that there exist a constant $M > 0$ measured in $J K^{- 1}$ and a dimensionless $δ \in (0, 1]$ such that $∣ C_{V} (T) ∣ \leq M θ^{q}$ whenever $0 < θ < δ$ . Finite convergence of the entropy integral alone does not imply such a bound for any real $q \geq 1$ , even when the entropy has a finite limit and the local response-chart identities apply. The displayed branch is a formal chart calculation, not a claim that a particular substance realizes the chart.

The displayed entropy is $S^{S I}$ in physical SI coordinates. Take one named species and the positive-amount open neighborhood specified here, and let $O_{c}$ be the numerical preimage of that product patch under $Φ_{c} (\hat{U}, \hat{V}, \hat{n}) = (\hat{U} J, \hat{V} m^{3}, \hat{n} m o l)$ . For this formal state-domain model, let $E_{c} : = O_{c}$ be the set of preparation labels, designate each label as an equilibrium preparation, and let $π_{c} : E_{c} \to R^{3}$ be the identity, so $D_{c} = O_{c}$ . At each label $e = (\hat{U}, \hat{V}, \hat{n})$ assign a one-point mechanical model of mass $m_{*} \hat{n}$ kilograms at the origin with zero center-of-mass velocity, where $m_{*} > 0$ kilograms is the fixed mass assigned to one mole; this supplies the positive-mass and rest-frame data required by Entropic temperature, pressure, and molar chemical potentials. Set $S_{c} = S^{S I} \circ Φ_{c}$ . The derivatives below are those of $S^{S I} = S_{c} \circ Φ_{c}^{- 1}$ on $Φ_{c} (O_{c})$ , exactly as in Entropic temperature, pressure, and molar chemical potentials. This supplies a numerical-to-SI chart and equilibrium-label realization for the formal witness, with no assertion of a measured material or a global constitutive extension.

Facts & Assumptions

Given: A fixed- $V, n$ positive-temperature chart on $0 < T < T_{*}$ , with $T_{*} > 0$ kelvins and finite limiting entropy as $T ↓ 0$ .

[F1]

Thermodynamic response functions on valid local charts: On a valid response chart $C_{V} = (\partial U / \partial T)_{V, n}$ , in $J K^{- 1}$ .

[F2]

Entropic temperature, pressure, and molar chemical potentials: On a smooth positive-temperature entropy chart $T = 1 / S_{U}$ , with the other physical coordinates held fixed.

[F3]

SI quantities and unit conventions: $U$ is in joules, $S$ in $J K^{- 1}$ , $T$ in kelvins, and $C_{V}$ in $J K^{- 1}$ .

[F4]

Concave macroscopic fundamental entropy: On a designated open single-phase chart, $S$ is $C^{2}$ , concave, and has finite $S_{U} > 0$ ; these are adopted local model premises.

[L1]

Real powers for positive bases, with the zero-base positive-exponent convention and Continuity and derivatives of positive-base real powers: For $a > 0$ and $x \in R$ , $a^{x} = \exp (x \log a)$ ; for $α \in R$ , $(x^{α})^{'} = α x^{α - 1}$ on $x > 0$ .

[L2]

Chain sum product and composition rules for Banach derivatives: The sum, scalar-multiple, and composition rules apply to the displayed differentiable real-valued functions of $T$ on their open positive-temperature interval.

[L3]

The natural logarithm as the inverse of the exponential function, Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm, and The exponential function is strictly increasing: $\log$ is strictly increasing with $\log 1 = 0$ ; $\exp (\log x) = x$ for $x > 0$ ; and $\exp$ is strictly increasing on $R$ .

[L4]

Ordered field, The reals form a totally ordered field, and Order is preserved by adding a constant and by adding inequalities: The real numbers form an ordered field; positive elements are closed under multiplication, the product of a positive and negative real is negative, and adding the same real preserves the order.

[L5]

Convex, strictly convex, concave, strictly concave, and midpoint-convex real functions on an interval: A real function is concave on an interval exactly when its negative is convex, using the convex-combination inequality.

[L6]

A twice-differentiable function on an open interval is convex if and only if its second derivative is nonnegative: For a twice-differentiable scalar function on an open interval, a nonnegative second derivative is equivalent to convexity.

[L7]

A continuous function on $[a, b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion: Every continuous function on a nondegenerate closed interval is Riemann integrable there.

[L8]

Newton–Leibniz needs only continuity on $[a, b]$ , differentiability on $(a, b)$ , and a Riemann-integrable extension of the interior derivative: A Riemann-integrable function on a closed interval that agrees with the derivative of a continuous primitive on the interior integrates to the primitive's endpoint difference.

[A1]

Choose $T_{*} > 0$ kelvins, $C_{*} > 0$ measured in $J K^{- 1}$ , a finite $S_{0}$ in $J K^{- 1}$ , $U_{0} > 0$ joules, $V_{0} > 0$ cubic metres, and $n_{0} > 0$ moles. On the open chart $U_{0} < U < U_{0} + \frac{2}{3} C_{*} T_{*}, V_{0} / 2 < V < 3 V_{0} / 2, n_{0} / 2 < n < 3 n_{0} / 2,$ define $θ (U) : = [3 (U - U_{0}) / (2 C_{*} T_{*})]^{2 / 3} \in (0, 1)$ and $S_{c} (U, V, n) : = S_{0} + 2 C_{*} θ (U)^{1 / 2} .$ This finite function is $C^{2}$ , independent of $V, n$ , and has $S_{U} = 1 / (T_{*} θ) > 0$ and $S_{U U} = - 1 / (C_{*} T_{*}^{2} θ^{5 / 2}) = - 1 / (T^{2} C_{V}) < 0$ throughout the chart. Thus it satisfies the local smoothness, positive-temperature, and concavity conditions used here. Put $T : = T_{*} θ (U)$ and $θ : = T / T_{*}$ ; solving for $U$ gives the fixed- $V, n$ response branch $U (T) = U_{0} + \frac{2}{3} C_{*} T_{*} θ^{3 / 2}, S (T) = S_{0} + 2 C_{*} θ^{1 / 2}, 0 < T < T_{*} .$ Equivalently, the corresponding entropy-coordinate energy chart is $U_{c} (s, V, n) = U_{0} + \frac{2}{3} C_{*} T_{*} {(\frac{s - S_{0}}{2 C_{*}})}^{3}, S_{0} < s < S_{0} + 2 C_{*} .$ It has $T = \partial_{s} U_{c} = T_{*} [(s - S_{0}) / (2 C_{*})]^{2}$ , whose explicit inverse is $s = S_{0} + 2 C_{*} \sqrt{T / T_{*}}$ for $0 < T < T_{*}$ . Hence the $(T, V, n)$ natural-coordinate chart is open and valid on the stated patch; its inverse is given explicitly by the branch formulas. The constructed chart is a formal local response model only; it reports no material data and asserts no global extension.

Counterexample

On the branch in [A1], the displayed maps $U (T)$ and $T (U)$ are mutually inverse $C^{1}$ maps on the stated open intervals, so they give a valid local fixed- $V, n$ response chart. The entropy chart uses the absolute-temperature definition $T = 1 / S_{U}$ ; the response identity needed below follows directly from the explicit branch formulas.

1.1givenA1F1F2F3F4L1L2L5L6algebra

Repeated use of the power derivative [L1] and scalar chain, sum, and product rules [L2] shows that $S_{c}$ is $C^{2}$ on the positive open chart. Differentiating $θ (U)$ gives $θ_{U} = (C_{*} T_{*})^{- 1} θ^{- 1 / 2}$ , so $S_{U} = C_{*} θ^{- 1 / 2} θ_{U} = 1 / (T_{*} θ)$ and $S_{U U} = - (C_{*} T_{*}^{2})^{- 1} θ^{- 5 / 2} < 0$ . At fixed $V, n$ , [L6] applied to $- S_{c}$ shows that $S_{c}$ is concave as a function of $U$ ; because $S_{c}$ is independent of $V, n$ , the same convex-combination inequality holds on the full product chart. Thus the entropy is concave there and $S_{U}$ is finite and positive, as required by [F4]. By [F2], the absolute temperature is $T = 1 / S_{U} = T_{*} θ$ . Differentiating the explicit branch functions $U (T), S (T)$ in [A1] by [L1,L2] gives $U_{T} = C_{*} θ^{1 / 2}, S_{T} = \frac{C_{*}}{T_{*}} θ^{- 1 / 2} .$ The response definition [F1] now gives $C_{V} = U_{T} = C_{*} θ^{1 / 2} = T S_{T}$ . The displayed inverse branch and these continuous derivatives give a valid local response chart.

2.1givenA1step 1.1L8algebra

For $0 < ϵ < T$ , the integrand $C_{V} (t) / t$ is continuous on $[ϵ, T]$ and is Riemann integrable there by [L7]; it agrees on its interior with the derivative of $2 C_{*} \sqrt{t / T_{*}}$ . By [L8], $\int_{ϵ}^{T} \frac{C_{V} (t)}{t} d t = 2 C_{*} (\sqrt{T / T_{*}} - \sqrt{ϵ / T_{*}}) .$ As $ϵ ↓ 0$ , this tends to $2 C_{*} \sqrt{θ} = S (T) - S_{0}$ , so the improper entropy integral converges and the entropy has the finite limit $S_{0}$ .

3.1givenA1L1L3L4step 1.1algebrachoose∎

Fix any dimensionless real exponent $q \geq 1$ , any proposed $M > 0$ in $J K^{- 1}$ , and any dimensionless $δ \in (0, 1]$ . Since $0 < θ < 1$ implies $\log θ < 0$ by [L3], ordered-field arithmetic [L4] gives $(q - 1) \log θ \leq 0$ and hence $q \log θ \leq \log θ$ . Thus $θ^{q} = \exp (q \log θ) \leq \exp (\log θ) = θ$ by [L1,L3]. Choose $0 < θ < \min {δ, (C_{*} / M)^{2}, 1}$ . Then $M θ^{q} \leq M θ < C_{*} \sqrt{θ} = C_{V} (T)$ , so the proposed bound fails at this temperature. As $M$ and $δ$ were arbitrary, no constants in the stated big- $O$ definition work for this $q$ ; since $q \geq 1$ was arbitrary, the assertion fails for every real exponent $q \geq 1$ .

Remarks

Tong's §4.5 says that finite entropy-integral convergence entails a low-temperature heat-capacity power $T^{n}$ for some $n \geq 1$ or faster, without specifying an integer exponent. This branch refutes that general power-bound inference from integrability alone, while its own heat capacity still tends to zero. It does not refute a separate physical model theorem with additional low-temperature spectral or constitutive hypotheses, and it is not an empirical statement about any material.

Finite entropy integral does not force a heat-capacity power bound of order one or higher

Statement refuted

Facts & Assumptions

Counterexample

Remarks

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