The Euler equations

When the gas is elastic and very near to macroscopic equilibrium, it is reasonable to assume that at every spatial location the distribution function can be approximated with a Maxwell-Boltzmann distribution function, with parameters varying in time and spatial location. This is the so called assumption of local thermodynamic equilibrium.

If the particles of the gas have no internal degrees of freedom, then the first velocity moments $m$, $m\mathbf{c}$ and $mc^2/2$ ($5$ scalar quantities) are the only summation invariants that depend on the particle velocities. The Maxwell equations for these quantities (precisely Equations (2.122)) extremely simplify with this local equilibrium assumption:

where it must be noted that for gravity or electromagnetic fields (as well for most external forces) the term $\langle a_i c_i \rangle$ vanishes.

The above three equations constitute the Euler equations for a perfect gas. They are the simplest form of fluid equations and are valid only when the assumption of local thermodynamic equilibrium can be assumed.