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Regulated gravitational preparations have different responses

Physical scope

Same fixed relative Kepler Hamiltonian and radial shell, with Gibbs-volume microcanonical versus canonical preparation. At E=−A/(32r0), R>32r0, and 2A/(μr0c²)≤δNR≪1, the microcanonical branch is slow-motion and has CG<0; the canonical model has CK≥0 but retains unbounded momentum tails. No equivalence or thermalization is asserted.

Statement

Use the same positive relative parameters $μ, A, r_{0}, R, h_{*}$ and fixed Hamiltonian $h (q, p) = ∣ p ∣^{2} / (2 μ) - A / ∣ q ∣$ for both preparations in Confined relative Kepler shell and volume entropy preparation and Relative-shell canonical comparison preparation. Let $R > 32 r_{0}$ , choose $E = - e = - A / (32 r_{0})$ , and assume the microcanonical physical applicability condition $\frac{2 A}{μ r_{0} c^{2}} \leq δ_{N R} ≪ 1.$ Then the Gibbs-volume microcanonical response satisfies $C_{G} (E) < 0$ , while the separate canonical preparation has $C_{K} (β) \geq 0$ for every $β > 0$ . For its density $p_{β}$ , define $⟨ g ⟩_{β} : = \int g p_{β} d ν_{r e l}$ and ${Var}_{β} (h) : = \int (h - ⟨ h ⟩_{β})^{2} p_{β} d ν_{r e l}$ . The microcanonical energy lies on the strictly negative-capacity branch and obeys the stated uniform slow-speed bound. The canonical model has unbounded momentum support, so this theorem makes no whole-ensemble slow-speed claim about it. The signs concern different finite-system preparations and do not contradict one another; neither sign establishes that nature realizes the model or that either preparation is dynamically attained.

Facts & Assumptions

Given: The two adopted preparations use the same $μ, A, r_{0}, R, h_{*}$ and fixed relative Hamiltonian; $R > 32 r_{0}$ ; $E = - A / (32 r_{0})$ ; and the stated $δ_{N R}$ condition.

[A1]

The microcanonical preparation adopts the finite energy shell, Gibbs-volume entropy, and $C_{G} = d E / d T_{G}$ convention. (Confined relative Kepler shell and volume entropy preparation)

[A2]

The canonical preparation uses the same fixed relative Hamiltonian and shell, canonical $β > 0$ , and $T_{K} = (k_{B} β)^{- 1}$ . (Relative-shell canonical comparison preparation)

[A3]

Its relative density is $p_{β} = Z_{r e l}^{- 1} e^{- β h}$ on $r_{0} \leq ∣ q ∣ \leq R$ and all $p \in R^{3}$ , with $Z_{r e l} > 0$ . (Relative-shell canonical comparison preparation)

[L1]

The canonical expectation and variance in the Statement are integrals relative to the normalized density $p_{β}$ and measure $d ν_{r e l}$ .

[F1]

On the mathematical branch $E = - e$ , $A / R < e \leq A / (16 r_{0})$ , the Gibbs-volume heat capacity is strictly negative. (Negative Gibbs-volume capacity in a finite regulated shell)

[F2]

For the same relative Hamiltonian with the canonical preparation, $Z_{r e l}$ is positive and finite and $C_{K} = k_{B} β^{2} {Var}_{β} (h) \geq 0$ for every $β > 0$ . (Exact relative-shell canonical normalization and response)

Proof

1.1givenA1F1algebra

Set $e = A / (32 r_{0})$ . Since $R > 32 r_{0}$ , $A / R < A / (32 r_{0}) = e$ , while $e < A / (16 r_{0})$ , so [F1] applies with $δ = e r_{0} / A = 1 / 32$ . It gives $C_{G} (E) < 0$ for the adopted Gibbs-volume convention.

2.1givenA1step 1.1algebra

On the microcanonical energy surface $h = - e$ , $∣ p ∣^{2} / (2 μ) = A / ∣ q ∣ - e \leq A / r_{0}$ . Hence $(∣ p ∣ / μ)^{2} \leq 2 A / (μ r_{0}) \leq δ_{N R} c^{2}$ . Thus the selected microcanonical branch has the uniform slow-speed qualification stated in the theorem.

2.2A2F2L1step 1.1algebra

The canonical preparation keeps the same $h$ and shell fixed while varying $β$ . By [F2], its response is $C_{K} = k_{B} β^{2} {Var}_{β} (h) \geq 0$ . This sign uses the canonical variance [L1] and the $T_{K}$ convention, not the Gibbs-volume derivative.

3.1A3F2constructstep 2.2algebra∎

To check the support qualification, fix any momentum threshold $P_{*} > 0$ and width $Δ p > 0$ . The shell $Q = {q : r_{0} \leq ∣ q ∣ \leq R}$ and momentum annulus $A = {p : P_{*} < ∣ p ∣ < P_{*} + Δ p}$ both have positive finite Lebesgue measure. On $Q \times A$ , $h (q, p) \leq (P_{*} + Δ p)^{2} / (2 μ)$ , so [A3] gives $p_{β} (q, p) \geq Z_{r e l}^{- 1} e^{- β (P_{*} + Δ p)^{2} / (2 μ)} > 0$ . Therefore this annulus has positive canonical probability for every $P_{*}$ ; the momentum support is unbounded. The two response signs use distinct preparations and do not entail equality of response functions.

Regulated gravitational preparations have different responses

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources