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Distinct unregulated point-gravity attempted sampling

Physical scope

Two labelled positive Newtonian point masses in a bounded box with interior and no short-distance cutoff; canonical weight is attempted without assuming a finite normalization. Collision configurations lie outside any asserted controlled weak-field model.

Postulate

In SI units, take two labelled masses $m_{1}, m_{2} > 0$ in $k g$ , positions $q_{1}, q_{2}$ in a bounded box $B \subset R^{3}$ with nonempty interior, momenta $p_{1}, p_{2}$ in $k g m s^{- 1}$ , and $G > 0$ in $m^{3} k g^{- 1} s^{- 2}$ . Fix $h_{*} > 0$ in $J s$ as the phase-action reference and $β > 0$ in $J^{- 1}$ , with the unit conventions of Thermodynamics-GR fields, currents, reference normalization and units and Finite gravitational phase and entropy reference. The labelled-pair phase space is $Γ_{12} = B^{2} \times R^{6}$ with reference measure $d ν_{12} = h_{*}^{- 6} d^{3} q_{1} d^{3} q_{2} d^{3} p_{1} d^{3} p_{2} .$ On the off-diagonal domain $q_{1} \neq q_{2}$ , adopt the unregulated point Hamiltonian $H_{12}^{p t} (q, p) = \frac{∣ p_{1} ∣^{2}}{2 m_{1}} + \frac{∣ p_{2} ∣^{2}}{2 m_{2}} - \frac{G m_{1} m_{2}}{∣ q_{1} - q_{2} ∣} .$ Attempt canonical sampling with weight $e^{- β H_{12}^{p t}}$ , but do not assume its integral is finite or that the normalized probability law exists. If $m_{1} = m_{2}$ , the identical-particle reference measure of Finite gravitational phase and entropy reference is $d ν_{12} / 2$ ; this constant convention does not alter whether the integral converges.

The masses, $G$ , box, action reference and $β$ are fixed inputs of this attempted model; $H_{12}^{p t}$ is defined by the formula. A finite partition function or probability law is not a primitive input and is not postulated.

This is a distinct unregulated model, not the softened preparation with $a$ set to zero after normalization. Arbitrarily small separations are not asserted to remain in a controlled Newtonian weak-field regime. The attempted weight is a model specification, not a reported physical preparation or an assertion of successful thermalization.

Distinct unregulated point-gravity attempted sampling

Postulate

Depends on

Dependency tree · 2 levels

Sources