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Bound relative Kepler entropies give different heat capacities

Statement refuted

For the six-dimensional bound relative Kepler model defined below, the Gibbs-volume and fixed-width surface-density entropy conventions give the same heat capacity at every bound energy $E < 0$ .

Facts & Assumptions

Given: The relative phase space $Γ_{r e l} = (R^{3} ∖ {0}) \times R^{3}$ , with $q$ in metres and $p$ in $k g m s^{- 1}$ , and relative point-Kepler Hamiltonian $h (q, p) = \frac{∣ p ∣^{2}}{2 μ} - \frac{A}{∣ q ∣}, μ, A, h_{*} > 0,$ where $μ$ is in kilograms, $A$ in $J m$ , and $h_{*}$ in $J s$ . Its dimensionless phase measure is $d ν_{r e l} = h_{*}^{- 3} d^{3} q d^{3} p$ , with no center-of-mass factor or $1 / 2!$ . For $E < 0$ , define $Ω (E) : = ν_{r e l} {h \leq E}$ and, where $ω (E) : = d Ω / d E > 0$ , set $S_{G} (E) : = k_{B} \log Ω (E)$ and $S_{B} (E) : = k_{B} \log (ω (E) Δ E)$ for fixed $Δ E > 0$ in joules. Where $d S_{G} / d E$ and $d S_{B} / d E$ are positive, define $T_{G}^{- 1} = d S_{G} / d E$ and $T_{B}^{- 1} = d S_{B} / d E$ ; where the temperature derivatives are nonzero, define $C_{G} = d E / d T_{G}$ and $C_{B} = d E / d T_{B}$ . Thus the two conventions are explicitly applied to this relative measure, rather than inferred from the $N$ -particle measure in Finite gravitational phase and entropy reference.

[F1]

The phase reference defines $S_{G} = k_{B} \log Ω (E)$ and, when $ω (E) = Ω^{'} (E) > 0$ , the distinct convention $S_{B} = k_{B} \log (ω (E) Δ E)$ for fixed $Δ E > 0$ . (Finite gravitational phase and entropy reference)

[F2]

Radial integrals on bounded shells satisfy $\int_{a \leq ∣ q ∣ \leq b} F (∣ q ∣) d q = 4 π \int_{a}^{b} F (r) r^{2} d r$ for continuous $F$ . (Radial integration over a bounded Euclidean shell)

[F3]

Increasing nonnegative measurable functions pass to their pointwise limit under the integral. (Monotone convergence for the integral)

[F4]

Tonelli identifies the nonnegative phase integral with its iterated momentum and position integrals. (Tonelli's theorem for nonnegative measurable functions on a sigma-finite product)

Counterexample

1.1F1F2F3F4givenconstruct

Fix $E = - e < 0$ and take the sequence $ε_{k} = A / (e (k + 1))$ , $k \geq 1$ . For each cutoff domain $r \geq ε_{k}$ , the momentum sublevel at radius $r$ is the ball $∣ p ∣ \leq \sqrt{2 μ (A / r - e)}$ for $ε_{k} \leq r \leq A / e$ . Applying [F2] to the momentum ball and the position shell and using [F4] gives $Ω_{ε} (- e) = \frac{(4 π) (4 π / 3) (2 μ)^{3 / 2}}{h_{*}^{3}} \int_{ε}^{A / e} r^{2} (A / r - e)^{3 / 2} d r .$ As $k \to \infty$ , these sublevel sets increase to the full relative phase sublevel because every allowed position has $r > 0$ ; [F3] therefore gives $Ω (- e) = \lim_{k \to \infty} Ω_{ε_{k}} (- e)$ . With $s = e r / A$ this is $Ω (- e) = C e^{- 3 / 2} J_{0}, C = \frac{(4 π) (4 π / 3) (2 μ)^{3 / 2} A^{3}}{h_{*}^{3}}, J_{0} = \int_{0}^{1} s^{1 / 2} (1 - s)^{3 / 2} d s .$ The integral $J_{0}$ is finite and positive because its integrand is bounded by $s^{1 / 2}$ near zero and by $(1 - s)^{3 / 2}$ near one.

2.1F1step 1.1algebra

Since $E = - e$ , differentiation gives $ω (E) = Ω^{'} (E) = \frac{3}{2} C J_{0} e^{- 5 / 2} > 0$ . The Gibbs-volume entropy therefore has $T_{G} = 2 e / (3 k_{B})$ and $C_{G} = d E / d T_{G} = - 3 k_{B} / 2$ .

3.1F1step 2.1algebra

For any fixed $Δ E > 0$ , the surface-density entropy has $ω (E) Δ E = (\frac{3}{2} C J_{0} Δ E) e^{- 5 / 2}$ , so $T_{B} = 2 e / (5 k_{B})$ and $C_{B} = d E / d T_{B} = - 5 k_{B} / 2$ . Both temperatures are positive under their declared conventions, but $C_{G} \neq C_{B}$ .

4.1step 1.1step 2.1step 3.1∎

This explicit six-dimensional relative model contradicts convention-independent finite-system heat capacity. Its bound-energy phase volume is finite at each $E < 0$ ; no canonical normalizability, finite outer box, or empirical preparation is claimed.

Bound relative Kepler entropies give different heat capacities

Statement refuted

Facts & Assumptions

Counterexample

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