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.

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,

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

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