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.

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.

In SI units, take two labelled masses $m_1,m_2>0$ in $\mathrm{k}\mathrm{g}$, positions $q_1,q_2$ in a bounded box $B\subset {\mathbb{R}}^3$ with nonempty interior, momenta $p_1,p_2$ in $\mathrm{k}\mathrm{g}\mathrm{m}{\mathrm{s}}^{-1}$, and $G>0$ in ${\mathrm{m}}^3\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{s}}^{-2}$. Fix $h_{\ast }>0$ in $\mathrm{J}\mathrm{s}$ as the phase-action reference and $\beta >0$ in ${\mathrm{J}}^{-1}$, 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 ${\mathrm{\Gamma }}_{12}=B^2\times {\mathbb{R}}^6$ with reference measure $d{\nu }_{12}=h_{\ast }^{-6}{d}^3{q}_1{d}^3{q}_2{d}^3{p}_1{d}^3{p}_2\mathrm{.}$ On the off-diagonal domain $q_1\mathrm{\ne }{q}_2$, adopt the unregulated point Hamiltonian $H_{12}^{\mathrm{p}\mathrm{t}}(q,p)=\frac{\mathrm{\mid }{p}_1{\mathrm{\mid }}^2}{2m_1}+\frac{\mathrm{\mid }{p}_2{\mathrm{\mid }}^2}{2m_2}-\frac{G{m}_1{m}_2}{\mathrm{\mid }{q}_1-q_2\mathrm{\mid }}\mathrm{.}$ Attempt canonical sampling with weight $e^{-\beta H_{12}^{\mathrm{p}\mathrm{t}}}$, 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 $d{\nu }_{12}\mathrm{/}2$; this constant convention does not alter whether the integral converges.

The masses, $G$, box, action reference and $\beta$ are fixed inputs of this attempted model; $H_{12}^{\mathrm{p}\mathrm{t}}$ 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.

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.

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.

Use the same positive relative parameters $\mu ,A,r_0,R,h_{\ast }$ and fixed Hamiltonian $h(q,p)=\mathrm{\mid }p{\mathrm{\mid }}^2\mathrm{/}(2\mu )-A\mathrm{/}\mathrm{\mid }q\mathrm{\mid }$ for both preparations in Confined relative Kepler shell and volume entropy preparation and Relative-shell canonical comparison preparation. Let $R>32r_0$, choose $E=-e=-A\mathrm{/}(32r_0)$, and assume the microcanonical physical applicability condition $\frac{2A}{\mu r_0{c}^2}\le {\delta }_{\mathrm{N}\mathrm{R}}\ll 1.$ Then the Gibbs-volume microcanonical response satisfies $C_G(E)<0$, while the separate canonical preparation has $C_K(\beta )\ge 0$ for every $\beta >0$. For its density $p_{\beta }$, define $⟨g{⟩}_{\beta }:=\int g{p}_{\beta }d{\nu }_{\mathrm{r}\mathrm{e}\mathrm{l}}$ and ${\mathrm{Var}}_{\beta }(h):=\int (h-⟨h{⟩}_{\beta }{)}^2{p}_{\beta }d{\nu }_{\mathrm{r}\mathrm{e}\mathrm{l}}$. The microcanonical energy lies on the strictly negative-capacity branch and obeys the stated uniform slow-speed bound. The canonical model has unbounded momentum support, so this theorem makes no whole-ensemble slow-speed claim about it. The signs concern different finite-system preparations and do not contradict one another; neither sign establishes that nature realizes the model or that either preparation is dynamically attained.

Let $M\ge 0$ be an integer, let $E_1,\dots ,E_M\in \mathbb{R}$, $\beta >0$, and $\mu \in \mathbb{R}$. Define the finite index set ${\mathcal{N}}_M=\{0,1{\}}^M$ and the finite-dimensional Hilbert space ${\mathcal{H}}_M={\mathbb{C}}^{{\mathcal{N}}_M}$ with orthonormal basis $\mathrm{\mid }{n}_1,\dots ,n_M⟩$. Define diagonal operators $H$ and $N$ on this basis by $H\mathrm{\mid }\mathbf{n}⟩=\left({\sum }_{j=1}^M{E}_j{n}_j\right)\mathrm{\mid }\mathbf{n}⟩,N\mathrm{\mid }\mathbf{n}⟩=\left({\sum }_{j=1}^M{n}_j\right)\mathrm{\mid }\mathbf{n}⟩\mathrm{.}$ For a matrix on ${\mathcal{H}}_M$, trace means the finite sum of its diagonal entries in this basis. Then $\mathrm{\Xi }=\mathrm{Tr}{e}^{-\beta (H-\mu N)}={\prod }_{j=1}^M(1+e^{-\beta (E_j-\mu )})\in (0,\mathrm{\infty })\mathrm{.}$ The normalized diagonal state with weights ${\mathrm{\Xi }}^{-1}{e}^{-\beta ({\sum }_j{E}_j{n}_j-\mu {\sum }_j{n}_j)}$ has, for each $1\le j\le M$, occupation mean $⟨n_j⟩=\frac{1}{e^{\beta (E_j-\mu )}+1}\mathrm{.}$ For $M=0$, the empty product and trace are $1$, and there are no occupation marginals. This is a finite-dimensional algebraic identity and uses no choice principle, curved-space spectrum, or gravitational field equation.

For the classical branch, prescribe a smooth static metric on a neighborhood of $I_{x^0}\times \bar{B}$, where $I_{x^0}\subset \mathbb{R}$ is an open interval and $B\subset {\mathbb{R}}^3$ is an open bounded coordinate box with compact closure and positive volume: $g=-\mathcal{N}(x{)}^2(d{x}^0{)}^2+h_{ij}(x)d{x}^{i}d{x}^j,x^0=ct\mathrm{.}$ Here $h_{ij}$ is a smooth positive-definite spatial metric whose eigenvalues are bounded above and away from zero, and the dimensionless lapse satisfies $0<{\mathcal{N}}_0\le \mathcal{N}(x)\le {\mathcal{N}}_1<\mathrm{\infty }$. The static chart and observer congruence are distinct objects: with $\xi ={\mathrm{\partial }}_{x^0}$, the static unit observer is $u={\mathcal{N}}^{-1}\xi$, satisfying $g(u,u)=-1$, under the conventions of Thermodynamics-GR fields, currents, reference normalization and units. This geometry is prescribed and is not assumed to solve an Einstein equation with gas stress as source.

Take particle mass $m>0$ in kilograms, $c>0$ in metres per second, orthonormal momentum $\widehat{p}\in {\mathbb{R}}^3$ in $\mathrm{k}\mathrm{g}\mathrm{m}{\mathrm{s}}^{-1}$, and adopt the massive-particle dispersion function $E(\widehat{p})=\sqrt{m^2{c}^4+c^2\mathrm{\mid }\widehat{p}{\mathrm{\mid }}^2}$ in joules as a model input. The Killing energy is the derived function $H_K(x,\widehat{p})=\mathcal{N}(x)E(\widehat{p})$, and the action reference $h_{\ast }>0$ is in joule-seconds. For a classical ideal gas with no interparticle potential and inverse Killing-energy parameter ${\beta }_K>0$ in inverse joules, choose exactly one of these alternatives: (i) a fixed-particle canonical branch with an integer $M\ge 1$ and the $1\mathrm{/}M!$ phase-counting convention; or (ii) a grand-canonical branch with dimensionless fugacity $z>0$, in which particle number is summed using the same $1\mathrm{/}M!$ convention. The finite proper-phase one-particle integral and the corresponding fixed-particle and grand-canonical normalizers are supplied by Static relativistic finite gas normalization and hydrostatics. If a Killing-normalized chemical parameter ${\mu }_K$ in joules is used for the classical gas, it belongs to the grand-canonical branch and is related by $z=e^{{\beta }_K{\mu }_K}$; the fixed-particle branch has no fugacity parameter. There is no interaction term or gas-induced change of the metric in either branch.

As a distinct finite quantum regulator, one may instead specify a finite mode set $J=\{1,\dots ,M\}$, $M\ge 0$, positive dimensionless lapse values ${\mathcal{N}}_j$, and momenta ${\widehat{p}}_j\in {\mathbb{R}}^3$. Adopt mode energies $E_j={\mathcal{N}}_j\sqrt{m^2{c}^4+c^2\mathrm{\mid }{\widehat{p}}_j{\mathrm{\mid }}^2}$, canonical parameter ${\beta }_K>0$ in inverse joules, and chemical-potential parameter ${\mu }_K\in \mathbb{R}$ in joules, together with the finite occupation basis $n_j\in \{0,1\}$. The associated finite trace and occupation identity are given by Finite static fermion-mode trace and occupation identity. This is an explicitly supplied spectral cutoff, not a claim that arbitrary curved spacetime has these modes. Neither branch reports an observation or asserts actual thermalization.

The metric $h_{ij}$, lapse $\mathcal{N}$, region, $m,c,h_{\ast }$, canonical parameters and (in the finite-mode branch) the mode list are supplied inputs. The local energy, Killing-energy weight, partition functions, normalized state, local densities and response identities are derived constructions under the stated mathematical results; no universal equation of state or metric backreaction is assumed.

Assume the prescribed static background in Fixed static ideal gas or finite fermion regulator preparation and consider its two alternative classical branches. Let $Z_1({\beta }_K)={\int }_B{\int }_{{\mathbb{R}}^3}{e}^{-{\beta }_K\mathcal{N}(x)E(\widehat{p})}\frac{d{V}_h(x)d^3\widehat{p}}{h_{\ast }^3},d{V}_h=\sqrt{\det (h_{ij}(x))}{d}^{3}x\mathrm{.}$ Then $0<Z_1<\mathrm{\infty }$. In the fixed-particle canonical branch, for the selected integer $M\ge 1$, the normalizer is $Z_M=Z_1^M\mathrm{/}M!$. In the grand-canonical branch, for dimensionless fugacity $z>0$, the normalizer is $\mathrm{\Xi }=e^{z{Z}_1}$.

For either classical branch, define its coefficient $\alpha$ by $\alpha =M\mathrm{/}{Z}_1$ in the fixed-particle branch and $\alpha =z$ in the grand-canonical branch. Its proper local expected particle phase-space density is $f_{\alpha }(x,\widehat{p})=\alpha h_{\ast }^{-3}{e}^{-{\beta }_K\mathcal{N}(x)E(\widehat{p})}\mathrm{.}$ Define proper local number density $n=\int f_{\alpha }{d}^3\widehat{p}$, energy density $\epsilon =\int E{f}_{\alpha }{d}^3\widehat{p}$, and isotropic pressure $P=\frac{1}{3}\int \widehat{p}\cdot {\mathrm{\nabla }}_{\widehat{p}}E{f}_{\alpha }{d}^3\widehat{p}$. With $T_K=(k_B{\beta }_K{)}^{-1}$, both branches satisfy $T_{\mathrm{l}\mathrm{o}\mathrm{c}}(x)=\frac{T_K}{\mathcal{N}(x)},P=n{k}_B{T}_{\mathrm{l}\mathrm{o}\mathrm{c}},{\mathrm{\partial }}_{i}P=-(\epsilon +P){\mathrm{\partial }}_i\log \mathcal{N}\mathrm{.}$ Only in the grand-canonical branch, if $z=e^{{\beta }_K{\mu }_K}$ for a Killing-normalized chemical parameter ${\mu }_K\in \mathbb{R}$ measured in joules, rewriting the local weight as $e^{-{\beta }_K\mathcal{N}(E-{\mu }_{\mathrm{l}\mathrm{o}\mathrm{c}})}$ gives ${\mu }_{\mathrm{l}\mathrm{o}\mathrm{c}}={\mu }_K\mathrm{/}\mathcal{N}$.

In the finite fermion-mode branch, let $E_j={\mathcal{N}}_j{ϵ}_j$ with ${ϵ}_j=\sqrt{m^2{c}^4+c^2\mathrm{\mid }{\widehat{p}}_j{\mathrm{\mid }}^2}$, ${\mathcal{N}}_j>0$, ${\beta }_K>0$, and ${\mu }_K\in \mathbb{R}$. The normalized diagonal state has $⟨n_j⟩=\frac{1}{e^{{\beta }_K(E_j-{\mu }_K)}+1}=\frac{1}{e^{{\beta }_j({ϵ}_j-{\mu }_j)}+1},{\beta }_j={\beta }_K{\mathcal{N}}_j,{\mu }_j=\frac{{\mu }_K}{{\mathcal{N}}_j}\mathrm{.}$ Both conclusions are conditional model deductions on the fixed background or finite supplied mode set. They assert no Einstein equation, gas backreaction, arbitrary curved-space spectrum, or observed equilibrium.