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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:

(192) 
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 longrange nature of the Coulomb interaction, the closest analogue
to ordinary twoparticle interaction is mediated by Debye shielding, an intrinsically
manybody 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, 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 different mathematical forms (for example, where the masses
and 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

(193) 
Collisional momentum conservation requires that

(194) 
That is, the net momentum exchanged between species and must vanish.
It is useful to introduce the rate of collisional momentum exchange, called the
collisional friction force, or simply the friction force:

(195) 
Clearly, is the momentummoment of the collision operator.
The total friction force experienced by species is

(196) 
Momentum conservation is expressed in detailed form as

(197) 
and in nondetailed form as

(198) 
Collisional energy conservation requires the quantity

(199) 
to be conserved in collisions: i.e.,

(200) 
Here, the 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 energymoment is

(201) 
i.e., 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, of course,

(202) 
It is easily verified that

(203) 
Thus, the collisional energy conservation law can be written

(204) 
or in nondetailed form

(205) 
Next: Moments of the Kinetic
Up: Plasma Fluid Theory
Previous: Moments of the Distribution
Richard Fitzpatrick
20110331