Phase-space distribution function ($f(\mathbf{x},\mathbf{p})$)

A phase-space distribution function gives the occupation density of particles as a function of position and momentum. For a spatially homogeneous gas it depends only on momentum.

A phase-space distribution function constant inside a prescribed region and zero outside. The one-dimensional example in the title has a velocity plateau and distributional derivatives at its edges. Its simple kinetic response can support undamped collective modes, illustrating why absence of growing modes does not imply Landau damping. For electrons with fixed ions, $ϵ(\omega ,k)=1-{\omega }_p^2/({\omega }^2-k^2{V}^2)$, so collective roots lie outside the populated velocity interval.

An interval of particle velocities on which an equilibrium phase-space distribution function is constant. Its zero derivative removes the resonant imaginary part of the linear dielectric function for waves with phase velocity inside the interval. This permits a completely undamped electron acoustic wave in suitable single-population, nonthermal electron distributions. Zero slope alone is necessary but does not guarantee that the real dispersion relation has a root.

For $N$ identical particles with symmetric, unit-normalized phase-space distribution function $D_N$, the number-normalized two-particle marginal is $f_2(z_1,z_2)=N(N-1)\int D_{N}d{z}_3\cdots d{z}_N$. Its integral over $z_2$ is $(N-1)f(z_1)$ and its full integral is $N(N-1)$. The factorial convention counts ordered distinct pairs and absorbs the combinatorial coefficient in the first BBGKY hierarchy equation. It does not imply statistical independence of the pair.

The spatial marginal $n_2=\int f_2{d}^{3}v{d}^3{v}_{\star }$ of a number-normalized two-particle distribution function. It counts ordered distinct pairs, so its integral over both positions is $N(N-1)$. Exchange symmetry gives $n_2(\mathbf{x},{\mathbf{x}}_{\star })=n_2({\mathbf{x}}_{\star },\mathbf{x})$. In a homogeneous isotropic fluid it is $n^{2}g(|{\mathbf{x}}_{\star }-\mathbf{x}|)$, where $g$ is the radial distribution function.

The number density is the number of particles per unit volume. For a homogeneous species with $g$ internal states and phase-space distribution $f(\mathbf{p})$ in natural units,

Bacterial density is the number of bacteria per unit length, area or volume in a spatial model. It is a field variable, distinct from the organisms themselves.

Column density is the number density integrated along a line of sight. Multiplication by an absorption cross-section gives an optical depth. A small column can result from a small local density, a small path length, or a small fraction of atoms in the absorbing state, so density inference requires these quantities to be constrained.

Electron number density is the number of free electrons per unit volume. In fully ionized stellar matter, $n_e=\rho /({\mu }_e{m}_u)$, with mass density $\rho$ and mean molecular weight per electron ${\mu }_e$.

The energy density is energy per unit volume. A homogeneous particle species with one-particle energy $E(\mathbf{p})$ has

Energy density per unit spectral coordinate, such as $U_{\nu }$ per frequency or $U_{\lambda }$ per wavelength. Conservation of energy in corresponding bins requires $U_{\lambda }=U_{\nu }|d\nu /d\lambda |$; therefore their peak positions need not map by $\lambda =c/\nu$.

For an isotropic gas, averaging the momentum flux over directions gives