Hydrostatic generating functionals and derivative-expansion boundaries

Let $\mathrm{\Sigma }$ be a nonempty compact smooth three-manifold without boundary, let $h$ be a smooth Riemannian metric on $\mathrm{\Sigma }$, and give $\mathbb{R}$ its usual topology and the standard smooth structure generated by the global identity chart ${\mathrm{id}}_{\mathbb{R}}:\mathbb{R}\to \mathbb{R}$. Give $M=\mathbb{R}\times \mathrm{\Sigma }$ the canonical product smooth structure. Adopt a constant $c>0$ in metres per second as the length/time conversion, write $x^0=c{t}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$ for the standard length coordinate on $\mathbb{R}$, where $t_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$ is time in seconds, and use spatial coordinates measured in metres. For a smooth function $N:\mathrm{\Sigma }\to (0,\mathrm{\infty })$, extend $N$ constantly along $\mathbb{R}$ and define the metric on $M$ by $g=-N^{2}d{x}^0\otimes d{x}^0+h$, where $h$ is pulled back from $\mathrm{\Sigma }$. The metric components and lapse $N$ are dimensionless in these length coordinates. This is a static spacetime model: the coefficients are independent of $x^0$, ${\mathrm{\partial }}_{x^0}$ is the selected future timelike Killing field, and its lapse is defined here by $N_{\xi }:=\sqrt{-g(\xi ,\xi )}$ for $\xi ={\mathrm{\partial }}_{x^0}$, so $g({\mathrm{\partial }}_{x^0},{\mathrm{\partial }}_{x^0})=-N^2$ and $N_{\xi }=N$.

The Riemannian volume density of $h$ is denoted by $d{V}_h$; in length coordinates it has units of cubic metres and finite total volume, and no orientation of $\mathrm{\Sigma }$ is required. In a fixed SI gauge, let $\mathrm{\Phi }:\mathrm{\Sigma }\to \mathbb{R}$ be a smooth electrostatic scalar potential in volts, let $q\in \mathbb{R}$ be a fixed charge in coulombs, let $T_0>0$ be a reference temperature in kelvins, and let ${\mu }_0\in \mathbb{R}$ be a reference chemical-potential parameter in joules. The sign convention is that the local electrochemical parameter is ${\mu }_0-q\mathrm{\Phi }$. For the static Killing field $\xi ={\mathrm{\partial }}_{x^0}$, its lapse is $N_{\xi }=N$; fix the Killing-normalized temperature $T_K:=N_{\xi }T$ to be the constant $T_0$, so the local temperature is $T(x)=T_0\mathrm{/}N(x)$. Define the local chemical-potential field by $\mu (x)=({\mu }_0-q\mathrm{\Phi }(x))\mathrm{/}N(x)$ for $x\in \mathrm{\Sigma }$; then $q\mathrm{\Phi }$ and $\mu$ have units of joules.

Let $D\subset (0,\mathrm{\infty })\times \mathbb{R}$ be an open domain for the pair $(T,\mu )$, measured in kelvins and joules, and let $p:D\to \mathbb{R}$ be a supplied $C^2$ pressure function with values in pascals. Require $(T(x),\mu (x))\in D$ at every $x\in \mathrm{\Sigma }$. The function $p$ is model input; this definition does not derive an equation of state or assert that the prescribed data have a material realization.

Smooth manifolds and their smooth charts

The Cartesian product $A\times B:=\{z\in \mathcal{P}(\mathcal{P}(A\cup B)):\mathrm{\exists }a\in A\mathrm{\exists }b\in Bz=(a,b)\}$

Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model

Killing fields of a Lorentzian metric

Products of smooth manifolds have a canonical product smooth structure

Riemannian metric and riemannian manifold

Riemannian volume density

The riemannian volume density is coordinate independent

Riemannian volume is the radon measure of the riemannian density

Measurable integration extends smooth density integration

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)

Use the prescribed static hydrostatic data $\mathrm{\Sigma },h,N,\mathrm{\Phi },q,T_0,{\mu }_0,D,p,d{V}_h$, with local profiles $T=T_0\mathrm{/}N$ and $\mu =({\mu }_0-q\mathrm{\Phi })\mathrm{/}N$. Fix the background fields and parameters, and take an open interval $U\subset \mathbb{R}$ containing ${\mu }_0$ such that $(T_0\mathrm{/}N(x),(r-q\mathrm{\Phi }(x))\mathrm{/}N(x))\in D$ for every $r\in U$ and $x\in \mathrm{\Sigma }$. Adopt the dimensionless effective functional

$W_{\mathrm{h}\mathrm{s}}(T_0,r;h,N,\mathrm{\Phi },q,p):=\log Z_{\mathrm{e}\mathrm{f}\mathrm{f}}=\frac{1}{k_B{T}_0}{\int }_{\mathrm{\Sigma }}N(x)p\text{\hspace{0.17em}⁣}(\frac{T_0}{N(x)},\frac{r-q\mathrm{\Phi }(x)}{N(x)})d{V}_h(x),$

with the integral finite. Here $Z_{\mathrm{e}\mathrm{f}\mathrm{f}}:=\exp (W_{\mathrm{h}\mathrm{s}})$ is notation for the adopted functional; no microscopic trace or Gibbs state is asserted. “Hydrostatic” labels this prescribed static equilibrium response model: the background is time-independent along the chosen Killing field and the Killing-normalized temperature is $T_0$; it does not assert a fluid-equation solution or a thermal state. A “derivative expansion” means a formal ordering of candidate corrections by the number of derivatives of the prescribed background and local thermodynamic variables; this postulate retains its order-zero term only. Each application declares a microscopic length ${\mathrm{\ell }}_{\mathrm{m}\mathrm{i}\mathrm{c}\mathrm{r}\mathrm{o}}>0$ in metres, a microscopic relaxation time ${\tau }_{\mathrm{m}\mathrm{i}\mathrm{c}\mathrm{r}\mathrm{o}}>0$ in seconds, and a positive spatial background scale $L_{\mathrm{b}\mathrm{g}}$ in metres, identifying the physical meaning of each without equating distinct microscopic scales; $L_{\mathrm{b}\mathrm{g}}$ is the shortest declared spatial variation or curvature scale among the prescribed fields. The static model is used only in the declared spatial long-wavelength regime ${\mathrm{\ell }}_{\mathrm{m}\mathrm{i}\mathrm{c}\mathrm{r}\mathrm{o}}\mathrm{/}{L}_{\mathrm{b}\mathrm{g}}\ll 1$. Its integral contains no temporal gradients; any time-dependent extension must separately declare a temporal variation scale $t_{\mathrm{b}\mathrm{g}}>0$ in seconds and satisfy ${\tau }_{\mathrm{m}\mathrm{i}\mathrm{c}\mathrm{r}\mathrm{o}}\mathrm{/}{t}_{\mathrm{b}\mathrm{g}}\ll 1$. No numerical error bound, convergence of a gradient expansion, or microscopic existence claim is asserted. The pressure function $p(T,\mu )$ is separately supplied and is not identified with a pressure derived from a material energy function unless an application states that additional equation-of-state identification.

Static hydrostatic background and equation-of-state data

Thermal vorticity and equilibrium domain

Restricted local equilibrium parcel with adopted material state function

Nabamita Banerjee et al., Constraints on Fluid Dynamics from Equilibrium Partition Functions

Pavel Kovtun, Lectures on hydrodynamic fluctuations in relativistic theories

Fix the background data $\mathrm{\Sigma },h,N,\mathrm{\Phi },q,T_0,p$ of the adopted compact hydrostatic functional and vary only its chemical-potential parameter ${\mu }_0$ inside the open interval $U$ specified by that model. The local temperature profile $T=T_0\mathrm{/}N$ has constant Killing-normalized value $NT=T_0$; this is the model's Tolman profile. Write $p_{\mu }$ and $p_{\mu \mu }$ for the first and second derivatives of the pressure function with respect to its chemical-potential argument, evaluated at $(T_0\mathrm{/}N(x),({\mu }_0-q\mathrm{\Phi }(x))\mathrm{/}N(x))$. Then

${\mathrm{\partial }}_{{\mu }_0}{W}_{\mathrm{h}\mathrm{s}}={\int }_{\mathrm{\Sigma }}\frac{p_{\mu }(T_0\mathrm{/}N,({\mu }_0-q\mathrm{\Phi })\mathrm{/}N)}{k_B{T}_0}d{V}_h,{\mathrm{\partial }}_{{\mu }_0}^2{W}_{\mathrm{h}\mathrm{s}}={\int }_{\mathrm{\Sigma }}\frac{p_{\mu \mu }(T_0\mathrm{/}N,({\mu }_0-q\mathrm{\Phi })\mathrm{/}N)}{N{k}_B{T}_0}d{V}_h\mathrm{.}$

The model chemical-potential susceptibility ${\chi }_{\mathrm{h}\mathrm{s}}:={\mathrm{\partial }}_{{\mu }_0}^2{W}_{\mathrm{h}\mathrm{s}}$ is nonnegative if $p_{\mu \mu }(T_0\mathrm{/}N(x),({\mu }_0-q\mathrm{\Phi }(x))\mathrm{/}N(x))\ge 0$ for every $x\in \mathrm{\Sigma }$. With the stated SI conventions, $W_{\mathrm{h}\mathrm{s}}$ is dimensionless and ${\chi }_{\mathrm{h}\mathrm{s}}$ has units ${\mathrm{J}}^{-2}$. This is a deduction inside the adopted effective model, not an empirical charge-fluctuation claim.

Restricted hydrostatic effective functional

Static hydrostatic background and equation-of-state data

Nabamita Banerjee et al., Constraints on Fluid Dynamics from Equilibrium Partition Functions

Pavel Kovtun, Lectures on hydrodynamic fluctuations in relativistic theories