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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.

Point-particle preparation has no finite-box canonical law

Statement refuted

Facts & Assumptions

Counterexample

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