Finite gravitational phase and entropy reference

Use SI dimensions for the numerical model parameters: Newton constant $G$ has units ${\mathrm{m}}^3\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{s}}^{-2}$, each mass $m_i$ has units $\mathrm{k}\mathrm{g}$, each position $q_i$ has units $\mathrm{m}$, each momentum $p_i$ has units $\mathrm{k}\mathrm{g}\mathrm{m}{\mathrm{s}}^{-1}$, energy $H$ has units $\mathrm{J}$, Boltzmann constant $k_B$ has units $\mathrm{J}{\mathrm{K}}^{-1}$, and inverse energy $\beta$ has units ${\mathrm{J}}^{-1}$. Fix an action reference $h_{\ast }>0$ in $\mathrm{J}\mathrm{s}$; it is a normalization convention and is not identified with Planck's constant.

For an integer $N\ge 1$, any Lebesgue measurable position region $B\subseteq {\mathbb{R}}^3$, and equal particle mass $m>0$, define the ordered phase space ${\mathrm{\Gamma }}_N=B^N\times {\mathbb{R}}^{3N}$ with coordinates $(q_1,\dots ,q_N,p_1,\dots ,p_N)$. This phase space and measure are allowed even when $B$ is unbounded or has infinite volume; finite canonical preparations separately require a positive finite partition integral. The dimensionless reference measure for identical classical particles is $d{\nu }_N=\frac{d^{3N}q{d}^{3N}p}{N!h_{\ast }^{3N}}\mathrm{.}$ For a measurable real Hamiltonian $H:{\mathrm{\Gamma }}_N\to \mathbb{R}$ and $\beta >0$ with $0<Z_N(\beta ):={\int }_{{\mathrm{\Gamma }}_N}{e}^{-\beta H}d{\nu }_N<\mathrm{\infty }$, define the canonical density $p_{\beta }=Z_N(\beta {)}^{-1}{e}^{-\beta H}$ relative to ${\nu }_N$. A probability density relative to ${\nu }_N$ is a measurable function $\rho :{\mathrm{\Gamma }}_N\to [0,\mathrm{\infty })$ with ${\int }_{{\mathrm{\Gamma }}_N}\rho d{\nu }_N=1$. Set $0\log 0:=0$. For such a density with ${\int }_{{\mathrm{\Gamma }}_N}\mathrm{\mid }\rho \log \rho \mathrm{\mid }d{\nu }_N<\mathrm{\infty }$, define its reference-relative entropy by $S(\rho )=-k_B{\int }_{{\mathrm{\Gamma }}_N}\rho \log \rho d{\nu }_N\mathrm{.}$

When $H$ is fixed and the Gibbs-volume function $\mathrm{\Omega }(E):={\nu }_N\{H\le E\}$ satisfies $0<\mathrm{\Omega }(E)<\mathrm{\infty }$, define the Gibbs-volume entropy by $S_G(E)=k_B\log \mathrm{\Omega }(E)$. Where $\omega (E):={\mathrm{\Omega }}^{\mathrm{\prime }}(E)>0$, define the distinct surface-entropy convention, for a fixed energy width $\mathrm{\Delta }E>0$ measured in the same units as $E$, by $S_B(E)=k_B\log (\omega (E)\mathrm{\Delta }E)$. Since ${\nu }_N$ and $\mathrm{\Omega }$ are dimensionless, $\omega$ has units inverse to energy and $\omega \mathrm{\Delta }E$ is dimensionless. For either $\alpha \in \{G,B\}$, where $S_{\alpha }$ is differentiable and $S_{\alpha }^{\mathrm{\prime }}(E)>0$, define $T_{\alpha }(E):=[S_{\alpha }^{\mathrm{\prime }}(E){]}^{-1}$; where also $T_{\alpha }^{\mathrm{\prime }}(E)\mathrm{\ne }0$, define $C_{\alpha }(E):=dE\mathrm{/}d{T}_{\alpha }=1\mathrm{/}{T}_{\alpha }^{\mathrm{\prime }}(E)$. These are finite-system conventions, and their temperatures and capacities need not agree.

The measure is dimensionless because $d^{3N}q{d}^{3N}p$ and $h_{\ast }^{3N}$ have the same units. The definition supplies no physical assertion that the chosen Hamiltonian describes nature and no mechanism of dynamical equilibration.

Let $d\ge 1$ be an integer, $m>0$, and $0<{\beta }_0\le {\beta }_1<\mathrm{\infty }$. For every $\beta \in [{\beta }_0,{\beta }_1]$, ${\int }_{{\mathbb{R}}^d}{e}^{-\beta \mathrm{\mid }p{\mathrm{\mid }}^2\mathrm{/}(2m)}dp={\left(\frac{2\pi m}{\beta }\right)}^{d\mathrm{/}2}\mathrm{.}$ For every integer $s\ge 0$, the polynomially weighted Gaussian has a uniform finite bound ${\sup }_{\beta \in [{\beta }_0,{\beta }_1]}{\int }_{{\mathbb{R}}^d}(1+\mathrm{\mid }p{\mathrm{\mid }}^{2s})e^{-\beta \mathrm{\mid }p{\mathrm{\mid }}^2\mathrm{/}(2m)}dp<\mathrm{\infty }\mathrm{.}$

Let $0\le a<b<\mathrm{\infty }$ and let $F:[a,b]\to \mathbb{R}$ be continuous. The closed radial shell $A_{a,b}=\{x\in {\mathbb{R}}^3:a\le \mathrm{\mid }x\mathrm{\mid }\le b\}$ is Jordan measurable, and ${\int }_{A_{a,b}}F(\mathrm{\mid }x\mathrm{\mid })dx=4\pi {\int }_a^{b}F(r)r^{2}dr\mathrm{.}$ The angular area measure used below is the pushforward of $\sin \theta d\theta d\varphi$ on $[0,\pi ]\times [0,2\pi ]$ under $(\theta ,\varphi )\mapsto (\sin \theta \cos \varphi ,\sin \theta \sin \varphi ,\cos \theta )$; its total mass is $4\pi$.

Use the phase space and dimensionless reference measure of Finite gravitational phase and entropy reference. Fix an integer $N\ge 1$, a bounded Lebesgue measurable $B\subseteq {\mathbb{R}}^3$ with positive finite volume, a common mass $m>0$, a gravitational coupling $G>0$, and a softening length $a>0$. On ${\mathrm{\Gamma }}_N=B^N\times {\mathbb{R}}^{3N}$ define $H_{N,a}(q,p)={\sum }_{i=1}^N\frac{\mathrm{\mid }{p}_i{\mathrm{\mid }}^2}{2m}+U_a(q),U_a(q)=-{\sum }_{1\le i<j\le N}\frac{G{m}^2}{\sqrt{\mathrm{\mid }{q}_i-q_j{\mathrm{\mid }}^2+a^2}}\mathrm{.}$ The positions and momenta are mathematical phase variables; the softening is a fixed regulator parameter. The Hamiltonian is measurable and its potential is bounded because each pair denominator is at least $a$. A point-particle model is a different Hamiltonian and is not included at $a=0$.

Use SI units, positive masses $m_1,m_2$ in $\mathrm{k}\mathrm{g}$, reduced mass $\mu =m_1{m}_2\mathrm{/}(m_1+m_2)$, gravitational parameter $A=G{m}_1{m}_2$ in $\mathrm{J}\mathrm{m}$, relative position $q\in {\mathbb{R}}^3$ in $\mathrm{m}$, relative momentum $p\in {\mathbb{R}}^3$ in $\mathrm{k}\mathrm{g}\mathrm{m}{\mathrm{s}}^{-1}$, and a fixed action reference $h_{\ast }>0$ in $\mathrm{J}\mathrm{s}$. Assume $r_0>0$ and $R>16r_0$. The relative phase domain is ${\mathrm{\Gamma }}_{\mathrm{r}\mathrm{e}\mathrm{l}}=\{(q,p):r_0\le \mathrm{\mid }q\mathrm{\mid }\le R,p\in {\mathbb{R}}^3\},d{\nu }_{\mathrm{r}\mathrm{e}\mathrm{l}}=\frac{d^{3}q{d}^{3}p}{h_{\ast }^3},$ with no center-of-mass variables or factor $1\mathrm{/}2!$. The relative Hamiltonian is $h(q,p)=\frac{\mathrm{\mid }p{\mathrm{\mid }}^2}{2\mu }-\frac{A}{\mathrm{\mid }q\mathrm{\mid }}\mathrm{.}$

For $E=-e<0$ with $A\mathrm{/}R<e\le A\mathrm{/}(16r_0)$, put $\delta =e{r}_0\mathrm{/}A\in (0,1\mathrm{/}16]$ and $J(\delta )={\int }_{\delta }^1{s}^{1\mathrm{/}2}(1-s{)}^{3\mathrm{/}2}ds,\mathrm{\Omega }(E)={\nu }_{\mathrm{r}\mathrm{e}\mathrm{l}}\{(q,p)\in {\mathrm{\Gamma }}_{\mathrm{r}\mathrm{e}\mathrm{l}}:h(q,p)\le E\}\mathrm{.}$ The radial integration formula of Radial integration over a bounded Euclidean shell gives the explicit phase volume $\mathrm{\Omega }(-e)=\frac{(4\pi )(4\pi \mathrm{/}3)(2\mu {)}^{3\mathrm{/}2}{A}^3}{h_{\ast }^3}{e}^{-3\mathrm{/}2}J(\delta )\mathrm{.}$ This is finite and positive on the stated branch.

Define $S^2=\{u\in {\mathbb{R}}^3:\mathrm{\mid }u\mathrm{\mid }=1\}$ and $p_{\ast }(r,E)=\sqrt{2\mu (E+A\mathrm{/}r)}$ for $r_0\le r\le A\mathrm{/}e$. Let $d\sigma$ be the angular area measure of total mass $4\pi$ from Radial integration over a bounded Euclidean shell. Push the finite measure $(\mu p_{\ast }\mathrm{/}{h}_{\ast }^3)d^{3}qd\sigma (u)$ on $\{r_0\le \mathrm{\mid }q\mathrm{\mid }\le A\mathrm{/}e\}\times S^2$ forward by $(q,u)\mapsto (q,p_{\ast }(\mathrm{\mid }q\mathrm{\mid },E)u)$; call the resulting measure ${\mathrm{\Sigma }}_E$. Its candidate normalized shell law is $P_E={\mathrm{\Sigma }}_E\mathrm{/}{\mathrm{\Omega }}^{\mathrm{\prime }}(E)$. The separate result Finite regulated Kepler shell measure is normalized ↗ proves ${\mathrm{\Omega }}^{\mathrm{\prime }}(E)>0$, finite total mass, and $P_E(\{h=E\})=1$. Thus the energy-shell preparation is explicitly defined; it is not inferred from phase-volume counting alone.

The entropy convention attached to this model is Gibbs volume, $S_G(E)=k_B\log \mathrm{\Omega }(E)$, with $1\mathrm{/}{T}_G=d{S}_G\mathrm{/}dE$ and $C_G=dE\mathrm{/}d{T}_G$ where defined. Other finite-system entropy conventions are distinct.

Let $I_{x^0}\subseteq \mathbb{R}$ be an open interval and $B\subset {\mathbb{R}}^3$ an open bounded coordinate box with compact closure and positive volume. On a neighborhood of $I_{x^0}\times \bar{B}$, with $x^0=ct$ assigned metres and $t$ in seconds, prescribe the smooth static metric $g=-\mathcal{N}(x{)}^2(d{x}^0{)}^2+h_{ij}(x)d{x}^{i}d{x}^j,$ where $h_{ij}$ is positive definite with eigenvalues bounded above and away from zero, and the lapse satisfies $0<{\mathcal{N}}_0\le \mathcal{N}(x)\le {\mathcal{N}}_1<\mathrm{\infty }$. The static unit observer is $u={\mathcal{N}}^{-1}{\mathrm{\partial }}_{x^0}$, so $g(u,u)=-1$; this is a prescribed background, not a solution sourced by the gas.

Use SI units: particle mass $m>0$ is in $\mathrm{k}\mathrm{g}$, $c>0$ is in $\mathrm{m}{\mathrm{s}}^{-1}$, orthonormal momentum $\widehat{p}$ is in $\mathrm{k}\mathrm{g}\mathrm{m}{\mathrm{s}}^{-1}$, and local energy $E$ is in $\mathrm{J}$. In any local orthonormal coframe, $p_i=e_i{}^a{\widehat{p}}_a$ and $h_{ij}={\sum }_a{e}_i{}^a{e}_j{}^a$, so $d{V}_h{d}^3\widehat{p}=d^{3}x{d}^3{p}_i$; orthonormal-frame changes are rotations and preserve $d^3\widehat{p}$. This defines a fibre measure without assuming a global frame. Use the fixed action reference $h_{\ast }>0$ in $\mathrm{J}\mathrm{s}$ and ${\beta }_K>0$ in ${\mathrm{J}}^{-1}$: $E(\widehat{p})=\sqrt{m^2{c}^4+c^2\mathrm{\mid }\widehat{p}{\mathrm{\mid }}^2},H_K(x,\widehat{p})=\mathcal{N}(x)E(\widehat{p}),Z_1({\beta }_K)={\int }_B{\int }_{{\mathbb{R}}^3}{e}^{-{\beta }_K{H}_K}\frac{d{V}_h{d}^3\widehat{p}}{h_{\ast }^3}\mathrm{.}$ Then $0<Z_1<\mathrm{\infty }$, with all polynomial momentum moments and compact-positive-${\beta }_K$ derivatives finite. For each fixed particle count $M\ge 1$, the classical ideal-gas preparation with Gibbs factor $1\mathrm{/}M!$ has $Z_M=Z_1^M\mathrm{/}M!$; a grand fugacity preparation with dimensionless $z>0$ has $\mathrm{\Xi }=\exp (z{Z}_1)$.

Define $f(x,\widehat{p})=z{h}_{\ast }^{-3}{e}^{-{\beta }_K{H}_K}$ and the proper local densities $n(x)=\int f{d}^3\widehat{p},\epsilon (x)=\int Ef{d}^3\widehat{p},P(x)=\frac{1}{3}\int \widehat{p}\cdot {\mathrm{\nabla }}_{\widehat{p}}Ef{d}^3\widehat{p}\mathrm{.}$ Their units are ${\mathrm{m}}^{-3}$, $\mathrm{J}{\mathrm{m}}^{-3}$, and $\mathrm{J}{\mathrm{m}}^{-3}$, respectively. With ${\beta }_{\mathrm{l}\mathrm{o}\mathrm{c}}={\beta }_K\mathcal{N}(x)$ and $T_K=(k_B{\beta }_K{)}^{-1}$, the exact model identities are $T_{\mathrm{l}\mathrm{o}\mathrm{c}}(x)=\frac{1}{k_B{\beta }_{\mathrm{l}\mathrm{o}\mathrm{c}}}=\frac{T_K}{\mathcal{N}(x)},P=n{k}_B{T}_{\mathrm{l}\mathrm{o}\mathrm{c}},{\mathrm{\partial }}_{i}P=-(\epsilon +P){\mathrm{\partial }}_i\log \mathcal{N}\mathrm{.}$ If $z=e^{{\beta }_K{\mu }_K}$ with ${\mu }_K$ in joules, the local chemical parameter is ${\mu }_{\mathrm{l}\mathrm{o}\mathrm{c}}={\mu }_K\mathrm{/}\mathcal{N}$. No Einstein equation, self-consistent gas backreaction, interacting quantum trace, or arbitrary curved-space mode spectrum is asserted.

Use the box, mass $m$, integer $N\ge 1$, regulator $a>0$, and Hamiltonian $H_{N,a}=K_N+U_a$ of Finite softened gravitational Hamiltonian and domain, with $K_N={\sum }_i\mathrm{\mid }{p}_i{\mathrm{\mid }}^2\mathrm{/}(2m)$. Put $C_N=G{m}^{2}N(N-1)\mathrm{/}(2a)$ and use the reference measure of Finite gravitational phase and entropy reference. For every $\beta >0$, $0<\frac{\mathrm{\mid }B{\mathrm{\mid }}^N}{N!h_{\ast }^{3N}}{\left(\frac{2\pi m}{\beta }\right)}^{3N\mathrm{/}2}\le Z_N(\beta )\le e^{\beta C_N}\frac{\mathrm{\mid }B{\mathrm{\mid }}^N}{N!h_{\ast }^{3N}}{\left(\frac{2\pi m}{\beta }\right)}^{3N\mathrm{/}2}<\mathrm{\infty }\mathrm{.}$ The density $p_{\beta }=Z_N^{-1}{e}^{-\beta H_{N,a}}$ is a positive probability density. For each integer $j\ge 0$, ${\int }_{{\mathrm{\Gamma }}_N}\mathrm{\mid }{H}_{N,a}{\mathrm{\mid }}^j{e}^{-\beta H_{N,a}}d{\nu }_N$ is finite, and $Z_N$ is $C^{\mathrm{\infty }}$ on $(0,\mathrm{\infty })$ with $Z_N^{(j)}(\beta )={\int }_{{\mathrm{\Gamma }}_N}(-H_{N,a}{)}^j{e}^{-\beta H_{N,a}}d{\nu }_N\mathrm{.}$ On every compact positive $\beta$ interval, one integrable majorant works for each fixed derivative order.

Use the dimensionless phase reference of Finite gravitational phase and entropy reference and the softened Hamiltonian of Finite softened gravitational Hamiltonian and domain.

1. For two equal positive masses in a bounded three-dimensional box with nonempty interior, the distinct point Hamiltonian $H_2^{\mathrm{p}\mathrm{t}}(q,p)=\frac{\mathrm{\mid }{p}_1{\mathrm{\mid }}^2+\mathrm{\mid }{p}_2{\mathrm{\mid }}^2}{2m}-\frac{G{m}^2}{\mathrm{\mid }{q}_1-q_2\mathrm{\mid }}$ makes the canonical partition integral infinite for every $\beta >0$.

2. For fixed $a>0$, let $B_N$ be a sequence of bounded boxes containing one fixed cube $C$ of volume $v>0$ and diameter $D<\mathrm{\infty }$, with constants $b_1,b_2>0$ such that $b_{1}N\le \mathrm{\mid }{B}_N\mathrm{\mid }\le b_{2}N$. The softened canonical partition functions satisfy $\frac{\log Z_N(\beta )}{N}\ge \beta c(N-1)-\log N+K_{\beta },c=\frac{G{m}^2}{2\sqrt{D^2+a^2}}>0,$ where $K_{\beta }$ is independent of $N$. Hence $\log Z_N\mathrm{/}N\to +\mathrm{\infty }$ and the free energy per particle $-{\beta }^{-1}\log Z_N\mathrm{/}N$ tends to $-\mathrm{\infty }$.

3. For the unscaled point attraction $-G{m}^2\mathrm{/}r$ restricted to the hard-core position domain $D_N=\{q\in B_n^N:\mathrm{\mid }{q}_i-q_j\mathrm{\mid }>a\text{ for }i\mathrm{\ne }j\}$ with fixed $a>0$, there is a fixed-density sequence of integers $n\ge 2$, $N=n^3$, boxes $B_n=(0,nd{)}^3$ of side $O(n)$ for fixed $d>a$, and positive-volume admissible position sets on which the interaction energy is at most $-c_0{N}^{5\mathrm{/}3}$ for some $c_0>0$. Its canonical partition functions therefore also satisfy $\log Z_N\mathrm{/}N\to +\mathrm{\infty }$.

For the sequences in clauses 2 and 3, write $F_N(\beta )=-{\beta }^{-1}\log Z_N(\beta )$. In clause 2, the free energy per particle $F_N\mathrm{/}N$ tends to $-\mathrm{\infty }$. Since $b_1\le \mathrm{\mid }{B}_N\mathrm{\mid }\mathrm{/}N\le b_2$, the free energy per volume $F_N\mathrm{/}\mathrm{\mid }{B}_N\mathrm{\mid }$ also tends to $-\mathrm{\infty }$. Clause 3 gives the same per-particle limit along its fixed-density sequence, and there $\mathrm{\mid }{B}_n\mathrm{\mid }\mathrm{/}N=d^3$, so the per-volume free energy also tends to $-\mathrm{\infty }$. Thus finite boxes at each particle count, fixed softening in clause 2, and hard-core exclusion with unscaled attraction in clause 3 do not ensure a finite real thermodynamic-limit value for either normalized free energy in these sequences. These results do not address differently scaled interactions or every possible gravitational thermodynamic limit.

Under Regulated relative Kepler phase volume and shell law, for every $E=-e$ with $A\mathrm{/}R<e\le A\mathrm{/}(16r_0)$, the phase volume is finite and positive and has an ordinary derivative on the interior of this energy interval. At the endpoint $e=A\mathrm{/}(16r_0)$, write ${\mathrm{\Omega }}^{\mathrm{\prime }}(E)$ for the right derivative in $E$ from within the interval. In both cases the pushed-forward shell measure ${\mathrm{\Sigma }}_E$ has total mass ${\mathrm{\Omega }}^{\mathrm{\prime }}(E)>0$, so $P_E={\mathrm{\Sigma }}_E\mathrm{/}{\mathrm{\Omega }}^{\mathrm{\prime }}(E)$ is a probability measure supported on the exact energy surface $\{(q,p):h(q,p)=E\}$.