The th moment of the ensemble-average kinetic equation is obtained by multiplying the above equation by powers of and integrating over velocity space. The flow term is simplified by pulling the divergence outside the velocity integral. The acceleration term is treated by partial integration. Note that these two terms couple the th moment to the th and th moments, respectively.
Making use of the collisional conservation laws, the zeroth moment of Eq. (208)
yields the continuity equation for species :
The interpretation of Eqs. (209)-(211) as conservation laws is
straightforward. Suppose that is some physical quantity (e.g.,
total number of particles, total energy, ...), and
is its density:
Applying this reasoning to Eq. (209), we see that is indeed the species- particle flux density, and that there are no local sources or sinks of species- particles. From Eq. (210), we see that the stress tensor is the species- momentum flux density, and that the species- momentum is changed locally by the Lorentz force and by collisional friction with other species. Finally, from Eq. (211), we see that is indeed the species- energy flux density, and that the species- energy is changed locally by electrical work, energy exchange with other species, and frictional heating.