Hydrostatic susceptibility and Tolman variables

Given: A compact nonempty smooth spatial manifold $\mathrm{\Sigma }$ without boundary, smooth $h,N,\mathrm{\Phi }$, $N>0$, fixed $q\in \mathbb{R}$ and $T_0>0$, $p\in C^2(D)$, and an open interval $U$ around ${\mu }_0$ on which all local $(T,\mu )$ profiles remain in $D$; hold these background inputs fixed while varying ${\mu }_0$.

Restricted hydrostatic effective functional: $W_{\mathrm{h}\mathrm{s}}=(k_B{T}_0{)}^{-1}{\int }_{\mathrm{\Sigma }}Np(T_0\mathrm{/}N,({\mu }_0-q\mathrm{\Phi })\mathrm{/}N)d{V}_h$ on the specified parameter interval, with the background held fixed.

Static hydrostatic background and equation-of-state data: $N$ is smooth and strictly positive, $d{V}_h$ is the Riemannian volume density on compact $\mathrm{\Sigma }$ with ${\mathrm{vol}}_h(\mathrm{\Sigma })<\mathrm{\infty }$, and $p$ is $C^2$ on an open domain containing the local profiles.

Riemannian volume density: in local coordinates $d{V}_h=\sqrt{\det (h_{ij})}\mathrm{\mid }d{x}^{1}d{x}^{2}d{x}^3\mathrm{\mid }$ and its coordinate coefficient is positive.

Riemannian volume is the radon measure of the riemannian density associates the Borel measure ${\mathrm{vol}}_h$ to $d{V}_h$. Under ${\mathrm{A}\mathrm{C}}_{\omega }$ [A2], Measurable integration extends smooth density integration defines integration of Borel functions against this measure and gives linearity and order preservation for integrable functions. Since ${\mathrm{vol}}_h(\mathrm{\Sigma })<\mathrm{\infty }$, every bounded Borel $g:\mathrm{\Sigma }\to \mathbb{R}$ is integrable and $\mathrm{\mid }{\int }_{\mathrm{\Sigma }}gd{V}_h\mathrm{\mid }\le \mathrm{\parallel }g{\mathrm{\parallel }}_{\mathrm{\infty }}{\mathrm{vol}}_h(\mathrm{\Sigma })$.

A closed bounded interval in $\mathbb{R}$ is compact (Heine-Borel in ${\mathbb{R}}^n$: with the Euclidean metric a subset of ${\mathbb{R}}^n$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line, claim 3), and a finite product of compact spaces with the product topology is compact (A product of finitely many compact spaces is compact in the product topology).

A continuous image of a compact space is compact (A continuous image of a compact space is compact; a continuous real-valued map on a nonempty compact space attains a maximum and a minimum; and a continuous bijection from a compact space to a Hausdorff space is a homeomorphism, claim 1).

A continuous function from a compact metric space to a metric space is uniformly continuous (Heine-Cantor: a continuous map from a compact metric space to any metric space is uniformly continuous).

For a real function continuous on a closed interval and differentiable in its interior, the mean value theorem gives a point where its derivative equals the interval difference quotient (The mean value theorem, as the case $g(x)=x$ of Cauchy's: for $f$ continuous on $[a,b]$ with $a<b$ and differentiable on $(a,b)$ there is $c\in (a,b)$ with $f(b)-f(a)=f^{\mathrm{\prime }}(c)(b-a)$).

The chain rule for total derivatives applies to the profile map $r\mapsto G(r,x)$ and the $C^2$ pressure function $p$ (The chain rule for total derivatives: $D(g\circ f)(a)=Dg(f(a))\circ Df(a)$).

Continuous real-valued functions on a nonempty compact space attain their extrema (A continuous image of a compact space is compact; a continuous real-valued map on a nonempty compact space attains a maximum and a minimum; and a continuous bijection from a compact space to a Hausdorff space is a homeomorphism, claim 2).

The pointwise convexity condition $p_{\mu \mu }(T_0\mathrm{/}N(x),({\mu }_0-q\mathrm{\Phi }(x))\mathrm{/}N(x))\ge 0$ holds for every $x\in \mathrm{\Sigma }$ when asserting the nonnegative conclusion.

Adopt ${\mathrm{A}\mathrm{C}}_{\omega }$ as defined in The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), the explicit assumption of the intrinsic density-integration supplier [F4].

The adopted SI Boltzmann constant satisfies $k_B>0$.