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The conductive state is the no-flow equilibrium in which heat conduction alone maintains a linear temperature profile across a layer with fixed boundary temperatures. Its hydrostatic pressure follows by integrating $P_b^{\prime }=-g{\rho }_b$, with mass density obtained from the equation of state. Under the Boussinesq approximation, deviations from this state are the variables used in the linear Rayleigh-Bénard convection problem.

For constant positive $\lambda =p_r/p_g$ and molecular gas constant $\mathcal{R}=k/({\mu }_m{m}_p)$, the gas–radiation equation of state gives $\rho =4\sigma T^3/(3c\lambda \mathcal{R})$. Vertical hydrostatic equilibrium in harmonic central gravity therefore gives $T=T_0(1-z^2/H^2)$ with $T_0={\Omega }^2{H}^2/[8(1+\lambda )\mathcal{R}]$. The mass density and pressure powers are three and four. Integrating both faces yields $\Sigma =(32/35){\rho }_{0}H$ and $\int pdz=\Sigma {\Omega }^2{H}^2/9$. With constant opacity, radiative diffusion produces a linear flux and height-independent dynamic viscosity $\mu =4c\lambda /[9\kappa (1+\lambda )]$. This is an interior formal profile; its low-density surface requires a photospheric treatment beyond diffusion.

Fresh water reaches maximum mass density near $4{}^{\circ }\mathrm{C}$ at atmospheric pressure. A useful local equation of state isso water becomes less dense when its temperature moves either above or below $T_m$.

Neglecting interactions and thermal pairs, the Fermi-Dirac distribution $f(p)=[\exp ((\epsilon (p)-{\mu }_{\mathrm{kin}})/(k_{B}T))+1{]}^{-1}$ with kinetic energy $\epsilon (p)=\sqrt{m_e^2{c}^4+p^2{c}^2}-m_e{c}^2$ determines the Electron contribution to a stellar equation of state. The kinetic chemical potential is fixed by the electron number density, not an independently chosen pressure fraction:The electron kinetic internal energy per volume is $({\pi }^2{\hslash }^3{)}^{-1}{\int }_0^{\infty }{p}^2\epsilon (p)f(p)dp$. In the nondegenerate limit $P_e=n_e{k}_{B}T$; at zero temperature the integrals become the equation of state of a cold electron gas. These are limits of one component and must not be added together as independent electron pressures. When electron-positron thermal pair abundance matters, both charge species must be included, with charge neutrality fixing their net density.

For a perfect fluid in general relativity of constant equation of state $P=w\rho$, conservation gives $\rho ={\rho }_0{a}^{-3(1+w)}$. In units $8\pi G=1$, the Friedmann equation becomes ${\dot{a}}^2/2+V(a)=0$. Allowed scales have $V\le 0$, and the Friedmann acceleration equation is $\ddot{a}=-V^{\prime }(a)$. The curvature term shifts the potential vertically; positive cosmological constant bends it downward and negative cosmological constant bends it upward.

A dilute particle suspension can supply both excess mass density and heat. To first order its equation of state is $\rho /{\rho }_0=1+\gamma \varphi -\theta$, where $\varphi$ is the particle volume fraction and $\theta$ is fractional thermal expansion. Particle deposition flux reduces $\varphi$, while particle heating increases $\theta$. The competition can reverse the reduced gravity and create unstable density stratification.

Let ${\rho }_0>0$ be a density threshold and let $a,\mu$ be the radius and enclosed mass where a monotonically decreasing stellar density reaches that threshold. Then $\mu \ge 4\pi {\rho }_0{a}^3/3$. A local compactness bound $\mu /a<4/9$ implies $a<(3\pi {\rho }_0{)}^{-1/2}$ and $\mu <4/(9\sqrt{3\pi {\rho }_0})$, independently of the equation of state above the threshold. A total-mass bound also requires suitable control of the known low-density envelope.

The endpoint of a gas-liquid phase coexistence curve, where the distinct liquid and gas states merge. For a smooth fluid equation of state, the critical isotherm has $p_v=p_{vv}=0$ at this point; this thermodynamic meaning differs from a stationary critical point of an arbitrary function.