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Finite regulated Newtonian gravitational canonical preparation — Examples

1 · Prerequisites

2 · Summary

The companion gives a finite softened two-body canonical instance, a point-particle collapse counterexample in a finite box, a confined Kepler energy with negative Gibbs-volume capacity and nonnegative canonical response under distinct preparations, and a constant-lapse gas normalization. These are exact examples within the stated models; none reports an observation or asserts that a physical system attains the model preparation.

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

None yet.

5 · Examples, experiments and counterexamples

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Prepared finite softened two-body ensemble

Physical scope

One explicit finite box and positive softening instance of the adopted effective Newtonian preparation; establishes a normalized model probability, not observed equilibration or a claim about a real gravitating gas.

Example

Take $B = (0, L)^{3}$ with $L > 0$ , $N = 2$ , equal mass $m > 0$ , Newton constant $G > 0$ , softening length $a > 0$ , action reference $h_{*} > 0$ , and $β > 0$ . Use the finite softened Newtonian preparation and its fixed Hamiltonian and measure.

Facts & Assumptions

Given: The cube has volume $L^{3} \in (0, \infty)$ , $N = 2$ , $a > 0$ , and all remaining physical parameters are positive as required by the adopted preparation.

[F1]

Under those hypotheses the finite regulated preparation has a positive finite partition function and normalized Gibbs density. (Constructed finite canonical gravitational equilibrium)

Verification

1.1givenF1

The box $B = (0, L)^{3}$ is bounded and has positive finite volume, and $N = 2 \geq 2$ with $a > 0$ ; therefore these data satisfy the hypotheses recorded in [F1].

2.1F1step 1.1∎

Apply [F1] to this instance. It gives $0 < Z_{2} (β) < \infty$ and $p_{β} = Z_{2}^{- 1} e^{- β H_{2, a}}$ as a normalized probability density relative to the stated phase reference. This is a constructed model ensemble and does not assert that the system dynamically reaches it.

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Point-particle preparation has no finite-box canonical law

Physical scope

Refutes finite-volume sufficiency for the exact unregulated two-point Newtonian canonical weight, for arbitrary positive masses. The collision limit is outside a controlled weak-field Newtonian regime and is not asserted to describe a real short-distance source.

Statement refuted

The claim refuted is that an outer finite box alone guarantees a normalizable canonical law for the unregulated two-point Newtonian gravity model. For any labelled positive masses $m_{1}, m_{2}$ , bounded box $B \subset R^{3}$ with nonempty interior, and $β > 0$ , define the collision-free phase domain $Γ_{12}^{c f} : = {(q_{1}, q_{2}, p_{1}, p_{2}) \in B^{2} \times R^{6} : q_{1} \neq q_{2}}$ and let $ν_{12}^{c f}$ be the restriction of $ν_{12}$ to this domain. The model in Distinct unregulated point-gravity attempted sampling has $Z_{12} (β) = \int_{Γ_{12}^{c f}} e^{- β H_{12}^{p t}} d ν_{12}^{c f} = + \infty .$ Thus the attempted weight cannot be normalized to a probability density.

Facts & Assumptions

Given: The labelled two-particle phase space, $h_{*}^{- 6}$ reference measure, off-diagonal Hamiltonian, and $β > 0$ from the adopted unregulated model. The integral is taken on the collision-free phase domain, where the Hamiltonian is defined, with the restricted reference measure.

[A1]

The Postulate defines $H_{12}^{p t} = ∣ p_{1} ∣^{2} / (2 m_{1}) + ∣ p_{2} ∣^{2} / (2 m_{2}) - G m_{1} m_{2} / ∣ q_{1} - q_{2} ∣$ on $Γ_{12}^{c f}$ , with $d ν_{12}^{c f}$ the restriction of $h_{*}^{- 6} d^{3} q_{1} d^{3} q_{2} d^{3} p_{1} d^{3} p_{2}$ to that domain. (Distinct unregulated point-gravity attempted sampling)

[F1]

For two equal positive masses in a bounded three-dimensional box with nonempty interior, the point Hamiltonian's canonical partition integral diverges for every $β > 0$ , relative to the identical-particle phase measure. (Point collapse and unscaled gravity nonextensivity)

Counterexample

1.1givenA1F1algebra

If $m_{1} = m_{2} = m$ , the labelled-pair measure restricted to $Γ_{12}^{c f}$ in [A1] is twice the identical-particle reference measure restricted to the same domain, $d ν_{2} = d ν_{12}^{c f} / 2$ . The equal-mass divergence in [F1] therefore also holds for $d ν_{12}^{c f}$ , since multiplication of an infinite nonnegative integral by $2$ leaves it infinite.

1.2givenA1constructalgebra

Suppose $m_{1} \neq m_{2}$ . Choose an interior point $q_{0} \in B$ and a length $ℓ > 0$ metres small enough that, for all sufficiently large integers $n$ , the disjoint cubes $Q_{1, n} = q_{0} + (0, ℓ / n)^{3}$ and $Q_{2, n} = q_{0} + (2 ℓ / n, 0, 0) + (0, ℓ / n)^{3}$ lie in $B$ . For $q_{1} \in Q_{1, n}$ and $q_{2} \in Q_{2, n}$ , their separation satisfies $ℓ / n < ∣ q_{1} - q_{2} ∣ \leq \sqrt{11} ℓ / n$ . Fix a momentum scale $p_{0} > 0$ in $k g m s^{- 1}$ and restrict both momenta to $[- p_{0}, p_{0}]^{3}$ ; there the kinetic energy is at most $K_{0} = 3 p_{0}^{2} / (2 m_{1}) + 3 p_{0}^{2} / (2 m_{2})$ . Hence on this positive-volume phase subset, $H_{12}^{p t} \leq K_{0} - G m_{1} m_{2} n / (\sqrt{11} ℓ)$ and the Boltzmann weight is at least $e^{- β K_{0}} e^{β G m_{1} m_{2} n / (\sqrt{11} ℓ)}$ .

2.1A1step 1.1step 1.2algebra∎

The restricted subset in step 1.2 lies in $Γ_{12}^{c f}$ , so its restricted reference measure is $h_{*}^{- 6} (ℓ / n)^{6} (2 p_{0})^{6}$ . Thus $Z_{12} (β) \geq h_{*}^{- 6} (ℓ / n)^{6} (2 p_{0})^{6} e^{- β K_{0}} e^{β G m_{1} m_{2} n / (\sqrt{11} ℓ)}$ for every sufficiently large $n$ . The right-hand side tends to $+ \infty$ because exponential growth dominates $n^{6}$ ; therefore the full collision-free nonnegative integral is infinite for $m_{1} \neq m_{2}$ . The equal-mass case was established in step 1.1, so no normalized canonical law exists for any positive mass pair in this model.

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Confined negative-capacity Kepler preparation

Physical scope

One energy and shell-radius instance of the finite regulated relative Kepler model under its declared Gibbs-volume convention and weak-field restriction; the canonical comparison remains a distinct preparation with unbounded momenta.

Example

Choose $m_{1}, m_{2} > 0$ , $A = G m_{1} m_{2}$ , reduced mass $μ = m_{1} m_{2} / (m_{1} + m_{2})$ , inner radius $r_{0} > 0$ , and outer radius $R > 32 r_{0}$ . Set $E = - A / (32 r_{0})$ and assume $2 A / (μ r_{0} c^{2}) \leq δ_{N R} ≪ 1$ . Use the two preparations and response conventions of the A-page comparison theorem.

Facts & Assumptions

Given: $R > 32 r_{0}$ , $E = - A / (32 r_{0})$ , $A, μ, r_{0} > 0$ , and the stated weak-field slow-motion condition.

[F1]

Under these hypotheses, the microcanonical Gibbs-volume response is strictly negative, and the separate canonical response for the same relative Hamiltonian is nonnegative. (Regulated gravitational preparations have different responses)

Verification

1.1givenF1algebra

The selected energy has $e = - E = A / (32 r_{0})$ , so $A / R < e < A / (16 r_{0})$ and $δ = e r_{0} / A = 1 / 32$ . The weak-field restriction is the condition required in [F1].

2.1F1step 1.1∎

Apply [F1] to this branch point. It gives $C_{G} < 0$ for the declared Gibbs-volume microcanonical preparation and $C_{K} \geq 0$ for the separate canonical preparation. These are model deductions about distinct finite ensembles, not measurements or evidence that either preparation is realized.

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Fixed lapse two gas normalization

Physical scope

Grand-canonical classical ideal gas on a prescribed flat finite box with constant dimensionless lapse two and a stated Killing-generator normalization; the local temperature parameter is T_K/2 and is not a reported measurement.

Example

Let $B = (0, L)^{3}$ with $L > 0$ , $I_{x^{0}} \subset R$ open, $h_{i j} = δ_{i j}$ , and constant dimensionless lapse $N = 2$ on the prescribed static metric $g = - 4 (d x^{0})^{2} + δ_{i j} d x^{i} d x^{j}$ , with $x^{0} = c t$ . Choose positive particle mass $m > 0$ in kilograms, $c > 0$ in metres per second, action reference $h_{*} > 0$ in joule-seconds, inverse Killing-energy parameter $β_{K} > 0$ in inverse joules, and dimensionless fugacity $z > 0$ . Select the grand-canonical branch, in which particle number is summed rather than fixed. Since the metric coefficients are independent of $x^{0}$ , choose the Killing generator $ξ = \partial_{x^{0}}$ to define $T_{K} = (k_{B} β_{K})^{- 1}$ . This fixes a normalization convention for the generator; $T_{K}$ is not identified with the proper temperature at the wall.

Facts & Assumptions

Given: The spatial box is bounded with positive finite volume, the static metric is prescribed, its lapse is the constant $N = 2$ , the Killing generator has the stated normalization, $m, c, h_{*}, β_{K}, z > 0$ , and the grand-canonical branch is selected.

[F1]

On a prescribed bounded-lapse static background, the selected grand-canonical classical gas has $0 < Z_{1} < \infty$ , $Ξ = e^{z Z_{1}} \in (0, \infty)$ , and $T_{l o c} = T_{K} / N$ . (Constructed static gas redshift and hydrostatics)

Verification

1.1givenF1

The constant lapse satisfies $0 < N_{0} \leq N \leq N_{1} < \infty$ with $N_{0} = N_{1} = 2$ , the flat spatial box has positive finite volume, the classical parameters are positive, and the grand-canonical branch is selected; thus [F1] applies.

2.1F1step 1.1algebra∎

Apply [F1] to the selected branch: $Z_{1}$ is finite and positive and $Ξ = e^{z Z_{1}}$ is finite and positive. Substituting $N = 2$ gives $T_{l o c} = T_{K} / 2$ . These are normalization and parameter identities in the selected model and report no observed temperature.

Finite regulated Newtonian gravitational canonical preparation — Examples

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

5 · Examples, experiments and counterexamples

Prepared finite softened two-body ensemble

Example

Facts & Assumptions

Verification

Point-particle preparation has no finite-box canonical law

Statement refuted

Facts & Assumptions

Counterexample

Confined negative-capacity Kepler preparation

Example

Facts & Assumptions

Verification

Fixed lapse two gas normalization

Example

Facts & Assumptions

Verification

Sources