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No cyclic device can have as its sole effect the extraction of heat from one reservoir and its complete conversion into work.

Kinetic theory describes many-particle systems through a distribution on phase space, retaining both position and velocity. In the number-density convention, $f(t,\mathbf{x},\mathbf{v})d^{3}x{d}^{3}v$ is the expected number of particles in a small position–velocity region. For identical particles of mass $m$, its integrals give the number density $n$ and bulk velocity $\mathbf{u}$:This $f$ is not normalized to one: its integral over the whole phase space gives the total expected particle number. Transport moves particles along trajectories, while collision or diffusion terms change the distribution. The Boltzmann equation, Hamiltonian Liouville equation and Fokker-Planck equation describe different physical assumptions, rather than interchangeable collision laws.

An ideal gas is a motivating example: the pressure on a wall comes from microscopic momentum transfers, even when the bulk velocity is zero. The Maxwell-Boltzmann velocity distribution separates this random molecular motion from bulk motion; see ideal gas pressure from wall collisions. The attractive feature of kinetic theory is that a distribution can describe two opposing populations where one average velocity hides almost everything. It also tells us when averaging into a continuum approximation loses essential information: compare the intermolecular mean free path with the apparatus scale using the Knudsen number. ^[ref]

The Lagrangian perturbation compares a variable on the same moving fluid element before and after a displacement. Taylor expansion of the background value at the displaced position gives the displayed relation to an Eulerian perturbation. For an adiabatic ideal gas, $\Delta (p/{\rho }^{\gamma })=0$ even when the equilibrium entropy varies spatially.

An imposed spherical profile $\rho ={\rho }_c(1-r/R)$ gives $M=\pi {\rho }_c{R}^3/3$ in hydrostatic equilibrium. For an ideal gas with zero surface pressure, $P=P_c(1-x{)}^2(1+2x-9x^2/5)$ and $T=T_c(1-x)(1+2x-9x^2/5)$, where $x=r/R$, $P_c=5G{M}^2/(4\pi R^4)$ and $T_c=5GM/(12\mathcal{R}R)$. The density has a central cusp and temperature initially rises outward, so positive outward radiative diffusion with regular positive nuclear heating cannot satisfy this imposed profile near the centre.

Continuity of temperature, pressure and mass density matches the ideal gas envelope pressure $\mathcal{R}{\rho }_b{T}_c$ to the nonrelativistic electron degeneracy pressure $K_{\mathrm{NR}}{\rho }_b^{5/3}$. Substituting this base mass density into the radiative-zero white-dwarf envelope law gives the luminosity-temperature scaling behind Mestel's cooling law. The matched point is a transition approximation; the cold degeneracy law does not represent finite-temperature corrections there exactly.

When $E-\mu \gg k_{B}T$, either quantum occupation law reduces to the classical factor

The mole fraction of species $i$ is $x_i=N_i/{\sum }_j{N}_j$, where $N_i$ is its amount in moles. It is dimensionless and equals the volume mixing ratio for an ideal gas mixture.

The negative hydrogen ion supplies important continuum opacity in cool hydrogen-rich stellar atmospheres. A useful local approximate law is $\kappa ={\kappa }_0{\rho }^{1/2}{T}^9$; it is not valid over arbitrary temperatures and compositions. In an ideal gas power-law opacity radiative envelope matched to ${\nabla }_{\mathrm{phot}}=1/8$, it gives ${\nabla }_{\mathrm{rad}}=-1/3+(11/24)(T/T_{\mathrm{eff}}{)}^{9/2}$, rising rapidly toward convective instability.

A gas is nondegenerate when its occupation numbers are low enough that quantum statistics reduce to the Maxwell-Boltzmann distribution. A common criterion is $n{\lambda }_{\mathrm{th}}^3\ll 1$, where ${\lambda }_{\mathrm{th}}$ is the thermal de Broglie wavelength.