Restricted hydrostatic effective functional

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