The th moment of the kinetic equation is obtained by multiplying the previous 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. 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 Equation (4.35) yields the continuity equation for species :
The interpretation of Equations (4.36)-(4.38) as conservation laws is straightforward. Suppose that is some physical quantity (for instance, the total number of particles, the total energy, and so on), and is its density:
Applying this reasoning to Equation (4.36), we see that is indeed the species- particle flux density, and that there are no local sources or sinks of species- particles.4.1 From Equation (4.37), it is apparent 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 Equation (4.38), 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.