Constructed finite canonical gravitational equilibrium

Fix $N\ge 2$, a bounded measurable box $B\subset {\mathbb{R}}^3$ with $0<\mathrm{\mid }B\mathrm{\mid }<\mathrm{\infty }$, $m,G,a,k_B,h_{\ast }>0$, $a>0$, and $\beta >0$. Adopt the physical preparation in Finite regulated Newtonian gravitational canonical preparation with its fixed Hamiltonian $H_{N,a}$ and reference measure ${\nu }_N$. Conditional on the exact current mathematical interfaces in Finite gravitational Gibbs normalization and differentiable moments, Unique finite gravitational Gibbs variational optimum, and Nonnegative fixed-Hamiltonian canonical heat capacity, the partition function $Z_N(\beta )=\int e^{-\beta H_{N,a}}d{\nu }_N$ is finite and positive, and $p_{\beta }=Z_N^{-1}{e}^{-\beta H_{N,a}}$ is a normalized density.

For a measurable function $g$ with finite expectation, write $⟨g{⟩}_{\rho }:=\int g\rho d{\nu }_N$ and $⟨g{⟩}_{\beta }:=\int g{p}_{\beta }d{\nu }_N$; define ${\mathrm{Var}}_{\beta }(H_{N,a}):=\int (H_{N,a}-⟨H_{N,a}{⟩}_{\beta }{)}^2{p}_{\beta }d{\nu }_N$. For every measurable probability density $\rho$ relative to ${\nu }_N$ with $0\log 0:=0$, $\int \mathrm{\mid }\rho \log \rho \mathrm{\mid }d{\nu }_N<\mathrm{\infty }$, and $\int \mathrm{\mid }{H}_{N,a}\mathrm{\mid }\rho d{\nu }_N<\mathrm{\infty }$, its reference-relative entropy $S(\rho )=-k_B\int \rho \log \rho d{\nu }_N$ satisfies $S(\rho )-k_B\beta ⟨H_{N,a}{⟩}_{\rho }\le k_B\log Z_N\mathrm{.}$ Equality holds exactly at $\rho =p_{\beta }$ almost everywhere; among densities with the same mean energy as $p_{\beta }$, this is the unique entropy maximizer. With $T_K=(k_B\beta {)}^{-1}$ and the Hamiltonian, box, and regulator held fixed, $C_K:=\frac{d⟨H_{N,a}{⟩}_{\beta }}{d{T}_K}=k_B{\beta }^2{\mathrm{Var}}_{\beta }(H_{N,a})\ge 0.$ These are deductions about the specified model and density class. They do not assert dynamical equilibration, a self-consistent relativistic solution, or a thermodynamic limit.