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# 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,

 (4.19)

where 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, is presumed to be bilinear.

It is important to realize that there is no simple relationship between the quantity , which describes the effect on species of collisions with species , and the quantity . The two operators can have quite distinct mathematical forms (for example, where the masses and 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

 (4.20)

Collisional momentum conservation requires that

 (4.21)

In other words, there is zero net momentum exchanged between species and . It is useful to introduce the rate of collisional momentum exchange, which is called the collisional friction force, or simply the friction force:

 (4.22)

Clearly, is the momentum-moment of the collision operator. The total friction force experienced by species is

 (4.23)

Momentum conservation is expressed in detailed form as

 (4.24)

and in non-detailed form as

 (4.25)

Collisional energy conservation requires the quantity

 (4.26)

to be conserved in collisions. In other words,

 (4.27)

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

An alternative collisional energy-moment is

 (4.28)

This is the kinetic energy change experienced by species , due to collisions with species , measured in the rest frame of species . The total energy change for species is

 (4.29)

It is easily verified that

 (4.30)

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

 (4.31)

or in non-detailed form as

 (4.32)

Next: Moments of Kinetic Equation Up: Plasma Fluid Theory Previous: Moments of Distribution Function
Richard Fitzpatrick 2016-01-23