(4.33) |

The two forms are equivalent because flow in velocity space under the Lorentz force is incompressible: that is,

(4.34) |

Thus, Equation (4.1) becomes

The rearrangement of the flow term is, of course, trivial, because is independent of .

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
:

Likewise, the first moment gives the

Finally, the contracted second moment yields the

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:

(4.39) |

If is conserved then must evolve according to

(4.40) |

where is the flux density of , and is the local rate per unit volume at which is created, or exchanged with other entities in the fluid. According to the previous equation, the density of at some point changes because there is net flow of towards or away from that point (characterized by the divergence term), or because of local sources or sinks of (characterized by the right-hand side).

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.