Relative-shell canonical comparison preparation

Use positive masses $m_1,m_2$ in $\mathrm{k}\mathrm{g}$, reduced mass $\mu =m_1{m}_2\mathrm{/}(m_1+m_2)$, $A=G{m}_1{m}_2>0$ in $\mathrm{J}\mathrm{m}$, relative position $q\in {\mathbb{R}}^3$ in metres, relative momentum $p\in {\mathbb{R}}^3$ in $\mathrm{k}\mathrm{g}\mathrm{m}{\mathrm{s}}^{-1}$, and action reference $h_{\ast }>0$ in $\mathrm{J}\mathrm{s}$. Fix $r_0>0$ and $R>16r_0$. Adopt the relative phase region and Hamiltonian of Regulated relative Kepler phase volume and shell law: ${\mathrm{\Gamma }}_{\mathrm{r}\mathrm{e}\mathrm{l}}=\{(q,p):r_0\le \mathrm{\mid }q\mathrm{\mid }\le R,p\in {\mathbb{R}}^3\},d{\nu }_{\mathrm{r}\mathrm{e}\mathrm{l}}=h_{\ast }^{-3}{d}^{3}q{d}^{3}p,h(q,p)=\frac{\mathrm{\mid }p{\mathrm{\mid }}^2}{2\mu }-\frac{A}{\mathrm{\mid }q\mathrm{\mid }}\mathrm{.}$ There are no center-of-mass variables and no identical-particle factor in this relative phase convention. Adopt a canonical preparation at inverse energy $\beta >0$ with $T_K=(k_B\beta {)}^{-1}$ and density $p_{\beta }(q,p)=Z_{\mathrm{r}\mathrm{e}\mathrm{l}}(\beta {)}^{-1}{e}^{-\beta h(q,p)}$ relative to $d{\nu }_{\mathrm{r}\mathrm{e}\mathrm{l}}$ on all of ${\mathrm{\Gamma }}_{\mathrm{r}\mathrm{e}\mathrm{l}}$. Its finite positive normalizer and normalized density follow from Exact relative-shell canonical normalization and response. Since $h$ is finite throughout this domain, the density is positive for arbitrarily large momentum magnitudes. This is a fixed relative-shell effective Newtonian model, distinct from the microcanonical preparation and Gibbs-volume entropy convention in Confined relative Kepler shell and volume entropy preparation.

Here $m_1,m_2,G,r_0,R,h_{\ast }$ and $\beta$ are fixed inputs; the reduced mass $\mu$, coupling product $A$, and Hamiltonian $h$ are the specified derived constructions. The partition function, density, temperature parameter and response are derived within this selected preparation.

Because canonical momenta are unbounded, this preparation does not have the whole-ensemble slow-motion bound imposed on the separate bounded-energy microcanonical branch. The postulate makes no claim that either distribution is dynamically reached or empirically observed.