Moments of Collision Operator

Boltzmann's collision operator for a neutral gas considers only binary collisions, and is, therefore, bilinear in the distribution functions of the two colliding species. (See Section 3.4.) In other words,

$$C_s(f) =\sum_{s'} C_{ss'}(f_s, f_{s'}),$$ (4.19)

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 modified by Debye shielding, which is 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 (because a logarithm is a comparatively weakly varying function). (See Section 3.10.) 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 distinct mathematical forms (for example, where the masses $m_s$ and $m_{s'}$ are significantly different), and they do not appear in the same equations.

Neutral particle collisions are characterized by Boltzmann's collisional conservation laws. (See Section 3.5.) In fact, the collisional process conserves particles, momentum, and energy at each point in space. We expect the same local conservation laws to hold for Coulomb collisions in a plasma, because 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 as

$$\int C_{ss'} \,d^3{\bf v} =0.$$ (4.20)

Collisional momentum conservation requires that

$$\int m_s\,{\bf v}\,C_{ss'}\,d^3{\bf v} = - \int m_{s'}\,{\bf v}\,C_{s's}\,d^3{\bf v}.$$ (4.21)

In other words, there is zero net momentum exchanged between species $s$ and $s'$. It is useful to introduce the rate of collisional momentum exchange, which is called the collisional friction force (per unit volume), or simply the friction force:

$${\bf F}_{ss'}\equiv \int m_s\,{\bf v}\,C_{ss'}\,d^3{\bf v}.$$ (4.22)

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

$${\bf F}_s \equiv \sum_{s'} {\bf F}_{ss'}.$$ (4.23)

Momentum conservation is expressed in detailed form as

$${\bf F}_{ss'} = -{\bf F}_{s's},$$ (4.24)

and in non-detailed form as

$$\sum_s {\bf F}_s = {\bf0}.$$ (4.25)

Collisional energy conservation requires the quantity

$$W_{ss'} \equiv \int \frac{1}{2}\,m_s\,v^2\,C_{ss'}\,d^3{\bf v}$$ (4.26)

to be conserved in collisions. In other words,

$$W_{ss'} + W_{s's} =0.$$ (4.27)

An alternative collisional energy-moment is

$$w_{ss'} \equiv \int \frac{1}{2}\,m_s\,u_s^{2}\,C_{ss'}\,d^3{\bf v}.$$ (4.28)

This is the rate of kinetic energy change (per unit volume) experienced by species $s$, due to collisions with species $s'$, measured in the rest frame of species $s$. The total rate of energy change for species $s$ is

$$w_s \equiv \sum_{s'} w_{ss'}.$$ (4.29)

It is easily verified that

$$W_{ss'} = w_{ss'} + {\bf V}_s\cdot{\bf F}_{ss'}.$$ (4.30)

Thus, the collisional energy conservation law can be written in detailed form as

$$w_{ss'} +w_{s's} +({\bf V}_s-{\bf V}_{s'})\cdot {\bf F}_{ss'} = 0,$$ (4.31)

or in non-detailed form as

$$\sum_s (w_s + {\bf V}_s\cdot{\bf F}_s) = 0.$$ (4.32)