In its most general form the Boltzmann equation is a seven-dimensional nonlinear integro-differential equation. The solutions of the Boltzmann equation provide a full description of the phase-space distribution function at all times. In most cases, however, it is next to impossible to solve the full Boltzmann equation and one has to resort to various approximate methods to describe the spatial and temporal evolution of macroscopic quantities characterizing the gas.

Transport equations for macroscopic molecular averages are obtained by taking velocity moments of the Boltzmann equation. This seemingly straightforward technique runs into considerable difficulties because the governing equations for the components of the n-th velocity moment also depend on components of the (n + 1)-th moment. In order to get a closed transport equation system, one has to use closing relations (expressing a higher-order velocity moment of the distribution function in terms of the components of lower moments) and thus make implicit assumptions about the distribution function.

Moment Equations

Velocity Moments

We start by examining the physical interpretation of the various velocity moments of the phase-space distribution function.

Macroscopic variables, such as number density, average flow velocity, kinetic pressure, and so on, can be considered as average values of molecular properties.