Moments of the Collision Operator

Boltzmann's famous collision operator for a neutral gas considers only binary collisions, and is, therefore, bilinear in the distribution functions of the two colliding species:

$$C_s(f) =\sum_{s'} C_{ss'}(f_s, f_{s'}),$$ (192)

where $C_{ss'}$ is linear in each of its arguments. Unfortunately, such bilinearity is not strictly valid for the case of Coulomb collisions in a plasma. Because of the long-range nature of the Coulomb interaction, the closest analogue to ordinary two-particle interaction is mediated by Debye shielding, an intrinsically many-body effect. Fortunately, the departure from bilinearity is logarithmic in a weakly coupled plasma, and can, therefore, be neglected to a fairly good approximation (since a logarithm is a comparatively weakly varying function). Thus, from now on, $C_{ss'}$ is presumed to be bilinear.

It is important to realize that there is no simple relationship between the quantity $C_{ss'}$, which describes the effect on species $s$ of collisions with species $s'$, and the quantity $C_{s's}$. The two operators can have quite different mathematical forms (for example, where the masses $m_s$ and $m_{s'}$ are disparate), and they appear in different equations.

Neutral particle collisions are characterized by Boltzmann's collisional conservation laws: the collisional process conserves particles, momentum, and energy at each point. We expect the same local conservation laws to hold for Coulomb collisions in a plasma: the maximum range of the Coulomb force in a plasma is the Debye length, which is assumed to be vanishingly small.

Collisional particle conservation is expressed by

$$\int C_{ss'} \,d^3{\bf v} =0.$$ (193)

Collisional momentum conservation requires that

$$\int m_s\,{\bf v}\,C_{ss'}\,d^3{\bf v} = - \int m_{s'}\,{\bf v}\,C_{s's}\,d^3{\bf v}.$$ (194)

That is, the net momentum exchanged between species $s$ and $s'$ must vanish. It is useful to introduce the rate of collisional momentum exchange, called the collisional friction force, or simply the friction force:

$${\bf F}_{ss'}\equiv \int m_s\,{\bf v}\,C_{ss'}\,d^3{\bf v}.$$ (195)

Clearly, ${\bf F}_{ss'}$ is the momentum-moment of the collision operator. The total friction force experienced by species $s$ is

$${\bf F}_s \equiv \sum_{s'} {\bf F}_{ss'}.$$ (196)

Momentum conservation is expressed in detailed form as

$${\bf F}_{ss'} = -{\bf F}_{s's},$$ (197)

and in non-detailed form as

$$\sum_s {\bf F}_s = {\bf0}.$$ (198)

Collisional energy conservation requires the quantity

$$W_{Lss'} \equiv \int \frac{1}{2}\,m_s\,v^2\,C_{ss'}\,d^3{\bf v}$$ (199)

to be conserved in collisions: i.e.,

$$W_{Lss'} + W_{Ls's} =0.$$ (200)

Here, the $L$-subscript indicates that the kinetic energy of both species is measured in the same ``lab'' frame. Because of Galilean invariance, the choice of this common reference frame does not matter.

An alternative collisional energy-moment is

$$W_{ss'} \equiv \int \frac{1}{2}\,m_s\,w_s^{~2}\,C_{ss'}\,d^3{\bf v}:$$ (201)

i.e., the kinetic energy change experienced by species $s$, due to collisions with species $s'$, measured in the rest frame of species $s$. The total energy change for species $s$ is, of course,

$$W_s \equiv \sum_{s'} W_{ss'}.$$ (202)

It is easily verified that

$$W_{Lss'} = W_{ss'} + {\bf V}_s\cdot{\bf F}_{ss'}.$$ (203)

Thus, the collisional energy conservation law can be written

$$W_{ss'} +W_{s's} +({\bf V}_s-{\bf V}_{s'})\cdot {\bf F}_{ss'} = 0,$$ (204)

or in non-detailed form

$$\sum_s (W_s + {\bf V}_s\cdot{\bf F}_s) = 0.$$ (205)