Confined relative Kepler shell and volume entropy preparation

Adopt the relative Hamiltonian, shell $r_0\le \mathrm{\mid }q\mathrm{\mid }\le R$, parameters $\mu ,A,r_0,R,h_{\ast }$, phase-volume function $\mathrm{\Omega }(E)$, and shell measure from Regulated relative Kepler phase volume and shell law. For $E=-e$ in the branch $\frac{A}{R}<e\le \frac{A}{16r_0},\delta :=\frac{e{r}_0}{A}\in (0,1\mathrm{/}16],$ adopt the normalized energy-shell probability $P_E={\mathrm{\Sigma }}_E\mathrm{/}{\mathrm{\Omega }}^{\mathrm{\prime }}(E)$, whose total mass and support on $h=E$ are established by Finite regulated Kepler shell measure is normalized. Adopt the Gibbs-volume entropy convention $S_G(E)=k_B\log \mathrm{\Omega }(E)$, define $1\mathrm{/}{T}_G=d{S}_G\mathrm{/}dE$ where this derivative is positive, and define $C_G=dE\mathrm{/}d{T}_G$ where the derivative exists. These are the selected finite-system preparation and temperature conventions; they are not asserted to be universal operational identifications.

For a physical Newtonian slow-motion regime, additionally require a stated dimensionless tolerance $0<{\delta }_{\mathrm{N}\mathrm{R}}\ll 1$ and $\frac{2A}{\mu r_0{c}^2}\le {\delta }_{\mathrm{N}\mathrm{R}}\mathrm{.}$ On the adopted negative-energy shell, $\mathrm{\mid }p\mathrm{\mid }\mathrm{/}\mu$ then has the corresponding uniform speed bound. This condition restricts the microcanonical branch only; it is not imposed by or transferred to the distinct canonical preparation. No center-of-mass degree of freedom, dynamical wall law, empirical thermalization, or self-consistent relativistic field is included.

The masses, $G$, $r_0$, $R$, $h_{\ast }$, selected energy $E$, and any ${\delta }_{\mathrm{N}\mathrm{R}}$ tolerance are fixed inputs. The reduced mass $\mu$, $A=G{m}_1{m}_2$, relative Hamiltonian, phase-volume function, and entropy/temperature/response quantities are defined constructions of this model; none is asserted to be universal across gravitational frameworks.