Finite gravitational phase and entropy reference — Examples

Fix $E=-e<0$ and take the sequence ${\epsilon }_k=A\mathrm{/}(e(k+1))$, $k\ge 1$. For each cutoff domain $r\ge {\epsilon }_k$, the momentum sublevel at radius $r$ is the ball $\mathrm{\mid }p\mathrm{\mid }\le \sqrt{2\mu (A\mathrm{/}r-e)}$ for ${\epsilon }_k\le r\le A\mathrm{/}e$. Applying [F2] to the momentum ball and the position shell and using [F4] gives ${\mathrm{\Omega }}_{\epsilon }(-e)=\frac{(4\pi )(4\pi \mathrm{/}3)(2\mu {)}^{3\mathrm{/}2}}{h_{\ast }^3}{\int }_{\epsilon }^{A\mathrm{/}e}{r}^2(A\mathrm{/}r-e{)}^{3\mathrm{/}2}dr\mathrm{.}$ As $k\to \mathrm{\infty }$, these sublevel sets increase to the full relative phase sublevel because every allowed position has $r>0$; [F3] therefore gives $\mathrm{\Omega }(-e)={\lim }_{k\to \mathrm{\infty }}{\mathrm{\Omega }}_{{\epsilon }_k}(-e)$. With $s=er\mathrm{/}A$ this is $\mathrm{\Omega }(-e)=C{e}^{-3\mathrm{/}2}{J}_0,C=\frac{(4\pi )(4\pi \mathrm{/}3)(2\mu {)}^{3\mathrm{/}2}{A}^3}{h_{\ast }^3},J_0={\int }_0^1{s}^{1\mathrm{/}2}(1-s{)}^{3\mathrm{/}2}ds\mathrm{.}$ The integral $J_0$ is finite and positive because its integrand is bounded by $s^{1\mathrm{/}2}$ near zero and by $(1-s{)}^{3\mathrm{/}2}$ near one.

Since $E=-e$, differentiation gives $\omega (E)={\mathrm{\Omega }}^{\mathrm{\prime }}(E)=\frac{3}{2}C{J}_0{e}^{-5\mathrm{/}2}>0$. The Gibbs-volume entropy therefore has $T_G=2e\mathrm{/}(3k_B)$ and $C_G=dE\mathrm{/}d{T}_G=-3k_B\mathrm{/}2$.

For any fixed $\mathrm{\Delta }E>0$, the surface-density entropy has $\omega (E)\mathrm{\Delta }E=(\frac{3}{2}C{J}_0\mathrm{\Delta }E)e^{-5\mathrm{/}2}$, so $T_B=2e\mathrm{/}(5k_B)$ and $C_B=dE\mathrm{/}d{T}_B=-5k_B\mathrm{/}2$. Both temperatures are positive under their declared conventions, but $C_G\mathrm{\ne }{C}_B$.

This explicit six-dimensional relative model contradicts convention-independent finite-system heat capacity. Its bound-energy phase volume is finite at each $E<0$; no canonical normalizability, finite outer box, or empirical preparation is claimed.

For every $(q_1,q_2)\in B^2$, the pair attraction is between $-G{m}^2\mathrm{/}a$ and $0$, so the Hamiltonian has the exact form covered by [F1]. The box has volume $L^3>0$ and finite, so all hypotheses apply.

By [F1], $0<Z_2(\beta )<\mathrm{\infty }$ and $p_{\beta }=Z_2^{-1}{e}^{-\beta H}$ is a probability density relative to the stated dimensionless phase measure. This verifies the finite softened two-body preparation for every $\beta >0$; it makes no claim about the distinct point model or dynamical equilibration.

The box $B\subset {\mathbb{R}}^3$ is bounded and has nonempty interior since each $b_j-a_j>0$. The phase variables, two equal positive masses, point Hamiltonian and reference measure are exactly those of [F1]; no short-distance cutoff is imposed.

By [F1], $Z_2(\beta )=+\mathrm{\infty }$ for this finite box and every $\beta >0$. This contradicts the refuted universal finiteness claim, so a finite outer box alone does not cure point collapse.

Fix $L>0$, take $C=[0,L{]}^3$, and set $B_N=[0,L{N}^{1\mathrm{/}3}{]}^3$ for every integer $N\ge 1$. Then $C\subseteq B_N$, $\mathrm{\mid }C\mathrm{\mid }=L^3>0$, and $V_N=L^{3}N$, so this is a bounded-box sequence satisfying the softened concentration hypotheses in [F1].

Clause 2 of [F1] gives $\log Z_N\mathrm{/}N\ge \beta c(N-1)-\log N+K_{\beta }\to +\mathrm{\infty }$, hence $f_N=-{\beta }^{-1}\log Z_N\mathrm{/}N\to -\mathrm{\infty }$. Since $V_N=L^{3}N$, also $g_N=f_N\mathrm{/}{L}^3\to -\mathrm{\infty }$. Neither normalization has a finite real thermodynamic-limit value along this sequence, contradicting the claimed guarantee.

Fix a grid spacing larger than the hard-core radius and take $N=n^3$ particles, one in each small cube around the cubic grid sites. The enclosing box has side proportional to $n=N^{1\mathrm{/}3}$ and the product of the small cubes is an admissible positive-volume position set, matching [F1].

By [F1], $f_N=F_N\mathrm{/}N=-{\beta }^{-1}\log Z_N\mathrm{/}N\to -\mathrm{\infty }$. Here $V_N=\mathrm{\mid }{B}_n\mathrm{\mid }=d^{3}N$, so $g_N=F_N\mathrm{/}{V}_N=d^{-3}{f}_N\to -\mathrm{\infty }$. Neither normalization has a finite real thermodynamic-limit value, contradicting the claimed guarantee.

Since $R>32r_0$, $A\mathrm{/}R<A\mathrm{/}(32r_0)=e$; also $e=A\mathrm{/}(32r_0)\le A\mathrm{/}(16r_0)$. Thus the branch assumptions hold and $\delta =1\mathrm{/}32\le 1\mathrm{/}16$.

Applying [F1] gives $C_G<0$ for the explicit Gibbs-volume entropy convention. Applying [F2] to the same relative Hamiltonian but its canonical preparation gives $C_K\ge 0$. These are different preparations and response definitions, so the two signs do not contradict one another.

Differentiating the displayed bath entropy gives $S_b^{\mathrm{\prime }}(E_b)=C_b\mathrm{/}(E_b-E_0)$, so $T_b=1\mathrm{/}{S}_b^{\mathrm{\prime }}(E_b)=(E_b-E_0)\mathrm{/}{C}_b$ and $d{E}_b\mathrm{/}d{T}_b=C_b$. The chosen $E_b$ makes $T_b=T_{\ast }$.

Since $C_b=\mathrm{\mid }{C}_G\mathrm{\mid }\mathrm{/}2$, the reciprocal-capacity sum is $1\mathrm{/}{C}_G+1\mathrm{/}{C}_b=-1\mathrm{/}\mathrm{\mid }{C}_G\mathrm{\mid }+2\mathrm{/}\mathrm{\mid }{C}_G\mathrm{\mid }=1\mathrm{/}\mathrm{\mid }{C}_G\mathrm{\mid }>0$. At the matched-temperature critical point, [F1] gives $S_{\mathrm{t}\mathrm{o}\mathrm{t}}^{\mathrm{\prime }\mathrm{\prime }}=-\frac{1}{T_{\ast }^2}(\frac{1}{C_G}+\frac{1}{C_b})=-\frac{1}{T_{\ast }^2\mathrm{\mid }{C}_G\mathrm{\mid }}<0.$ Thus the combined entropy has a strict local maximum in this exchange direction.

The selected effective law $\dot{x}=L{S}_{\mathrm{t}\mathrm{o}\mathrm{t}}^{\mathrm{\prime }}(x)$ has nonnegative entropy production and exponentially attracts this local maximum by [F1]. This is a constructed mathematical reservoir model, not empirical data or a generic gravitational stability conclusion.