Moments of the kinetic equation

We obtain fluid equations by taking appropriate moments of the ensemble-average kinetic equation, (175). In the following, we suppress all ensemble-average over-bars for ease of notation. It is convenient to rearrange the acceleration term,

$${\bf a}_s\cdot\nabla_v f_s = \nabla_v\cdot({\bf a}_s\,f_s).$$ (206)

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

$$\nabla_v\cdot {\bf a}_s = 0.$$ (207)

Thus, Eq. (175) becomes

$$\frac{\partial f_s}{\partial t} + \nabla\cdot({\bf v}\,f_s) +\nabla_v\cdot({\bf a}_s\,f_s) = C_s(f).$$ (208)

The rearrangement of the flow term is, of course, trivial, since ${\bf v}$ is independent of ${\bf r}$.

The $k$th moment of the ensemble-average kinetic equation is obtained by multiplying the above equation by $k$ powers of ${\bf v}$ 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 $k$th moment to the $(k+1)$th and $(k-1)$th moments, respectively.

Making use of the collisional conservation laws, the zeroth moment of Eq. (208) yields the continuity equation for species $s$:

$$\frac{\partial n_s}{\partial t} + \nabla\cdot(n_s\,{\bf V}_s) = 0.$$ (209)

Likewise, the first moment gives the momentum conservation equation for species $s$:

$$\frac{\partial(m_s n_s\,{\bf V}_s)}{\partial t} + \nabla\cdo... ...f P}_s -e_s n_s({\bf E} + {\bf V}_s\times {\bf B}) ={\bf F}_s.$$ (210)

Finally, the contracted second moment yields the energy conservation equation for species $s$:

$$\frac{\partial}{\partial t}\left( \frac{3}{2}\,p_s + \frac{1... ...s n_s\,{\bf E}\cdot{\bf V}_s = W_s + {\bf V}_s\cdot {\bf F}_s.$$ (211)

The interpretation of Eqs. (209)-(211) as conservation laws is straightforward. Suppose that $G$ is some physical quantity (e.g., total number of particles, total energy, ...), and $g({\bf r}, t)$ is its density:

$$G = \int g\,d^3{\bf r}.$$ (212)

If $G$ is conserved then $g$ must evolve according to

$$\frac{\partial g}{\partial t} + \nabla\cdot {\bf g} = {\mit\Delta} g,$$ (213)

where ${\bf g}$ is the flux density of $G$, and ${\mit\Delta}g$ is the local rate per unit volume at which $G$ is created or exchanged with other entities in the fluid. Thus, the density of $G$ at some point changes because there is net flow of $G$ towards or away from that point (measured by the divergence term), or because of local sources or sinks of $G$ (measured by the right-hand side).

Applying this reasoning to Eq. (209), we see that $n_s\,{\bf V}_s$ is indeed the species-$s$ particle flux density, and that there are no local sources or sinks of species-$s$ particles.⁹ From Eq. (210), we see that the stress tensor ${\bf P}_s$ is the species-$s$ momentum flux density, and that the species-$s$ momentum is changed locally by the Lorentz force and by collisional friction with other species. Finally, from Eq. (211), we see that ${\bf Q}_s$ is indeed the species-$s$ energy flux density, and that the species-$s$ energy is changed locally by electrical work, energy exchange with other species, and frictional heating.