Finite regulated Newtonian gravitational canonical preparation

Work in SI units. Adopt positive Newton constant $G$ in ${\mathrm{m}}^3\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{s}}^{-2}$, common particle mass $m$ in $\mathrm{k}\mathrm{g}$, positions $q_i$ in $\mathrm{m}$, momenta $p_i$ in $\mathrm{k}\mathrm{g}\mathrm{m}{\mathrm{s}}^{-1}$, energy $H$ in joules, Boltzmann constant $k_B$ in $\mathrm{J}{\mathrm{K}}^{-1}$, and inverse-energy parameter $\beta >0$ in ${\mathrm{J}}^{-1}$. Fix an action reference $h_{\ast }>0$ in $\mathrm{J}\mathrm{s}$; it is a phase-counting convention, not an assertion that $h_{\ast }$ equals Planck's constant. These conventions follow Thermodynamics-GR fields, currents, reference normalization and units and Finite gravitational phase and entropy reference.

For an integer $N\ge 2$, take a bounded measurable confining region $B\subset {\mathbb{R}}^3$ with $0<\mathrm{\mid }B\mathrm{\mid }<\mathrm{\infty }$ and adopt the ordered phase space ${\mathrm{\Gamma }}_N=B^N\times {\mathbb{R}}^{3N}$ and identical-particle reference measure $d{\nu }_N=d^{3N}q{d}^{3N}p\mathrm{/}(N!h_{\ast }^{3N})$ of Finite gravitational phase and entropy reference. Fix a softening length $a>0$ in metres and adopt the Hamiltonian $H_{N,a}(q,p)={\sum }_{i=1}^N\frac{\mathrm{\mid }{p}_i{\mathrm{\mid }}^2}{2m}-{\sum }_{1\le i<j\le N}\frac{G{m}^2}{\sqrt{\mathrm{\mid }{q}_i-q_j{\mathrm{\mid }}^2+a^2}},$ which is the model defined in Finite softened gravitational Hamiltonian and domain. The preparation assigns canonical weight $e^{-\beta H_{N,a}}$ relative to ${\nu }_N$. Its normalizer is $Z_B(\beta ):={\int }_{B^N\times {\mathbb{R}}^{3N}}{e}^{-\beta H_{N,a}}d{\nu }_N$. To check finiteness for this general region, enclose $B$ in a bounded cube $Q$ and apply Finite gravitational Gibbs normalization and differentiable moments on $Q$ with the same parameters and Hamiltonian formula. Restriction of its nonnegative integral gives $Z_B\le Z_Q<\mathrm{\infty }$. Moreover, the integrand is strictly positive everywhere, and $B^N$ times any positive-volume bounded momentum cube has positive measure, so $Z_B>0$. Thus the density $p_{\beta }=Z_B^{-1}{e}^{-\beta H_{N,a}}$ is normalized. These are mathematical consequences of the specified weight, not proofs of the physical preparation. For box instances, the further variational and response conclusions are stated in Constructed finite canonical gravitational equilibrium. Confinement specifies the finite position region; no wall regularity or wall dynamics is used in this calculation.

Within this formulation, $N,m,G,a,B,h_{\ast }$ and $\beta$ are fixed model/preparation inputs. The displayed $H_{N,a}$ is the defined effective Hamiltonian; $Z_N$, the normalized density, mean energy, entropy and response are derived mathematical quantities. Their definition does not make them universal properties of gravitating matter.

This is an adopted effective Newtonian statistical model in a stated weak-field, slow-motion regime. The unbounded momentum domain is part of the canonical mathematical preparation and does not give a uniform speed bound on every point in its support. The postulate asserts neither dynamical equilibration, an observed gravitating gas, nor stability of a self-consistently gravitating relativistic solution.