(3.125) | ||

(3.126) |

Now, from Equation (3.106),

(3.127) |

Moreover,

Hence, it is easily demonstrated that

According to Equations (3.115) and (3.116),

(3.132) | ||

(3.133) |

where we have integrated the first equation by parts, making use of Equation (3.110). Thus, we deduce from Equations (3.130) and (3.131) that

The quantities
and
are known as *Rosenbluth potentials* (Rosenbluth, MacDonald, and Judd 1957), and can easily be seen to satisfy

(3.136) | ||

(3.137) |

where denotes a velocity-space Laplacian operator. The former result follows because , and the latter because . In particular, if is isotropic in velocity space then we obtain

Suppose that is a Maxwellian distribution of characteristic number density , mean flow velocity zero, and temperature . In other words,

In this case, Equation (3.138) reduces to

(3.141) |

where , and . Hence, requiring to be finite at , we can integrate the previous expression to give

where

(3.143) |

is a so-called

(3.144) |

when , and

(3.145) |

when . Equation (3.139) yields

(3.146) |

which can be integrated, subject to the constraint that be finite at , to give

According to Equations (3.128), (3.129), (3.142), and (3.147),

where

Finally, it follows from Equations (3.114), (3.134), (3.135), (3.148), and (3.149) that

Suppose that is a Maxwellian distribution of characteristic number density , mean flow velocity zero, and temperature . In other words,

(3.154) |

It follows that

(3.155) |

Hence, Equations (3.113) and (3.153) yield

where

and use has been made of the fact that . The ensemble-averaged kinetic equation, Equation (3.9), can thus be written in the form

where

(3.159) |

is the ensemble-averaged Lorentz force. In deriving Equation (3.158), we have made use of the easily proved result

(3.160) |

According to Equation (3.158), collisions with particles of type give rise to a velocity dependent effective force, , acting on individual particles of type . As expected, this force is zero if the temperatures of the two species are equal. On the other hand, if particles of type have a higher kinetic temperature than particles of type (i.e., if ) then the collisional force acts to speed up the latter particles--in other words, the force always acts in the same direction as the particle's instantaneous velocity. [This follows because .] Conversely, if particles of type have a lower kinetic temperature than particles of type , then the collisional force acts to slow down the latter particles--in other words, the force always acts in the opposite direction to the particle's instantaneous velocity. In both cases, the collisional force is clearly acting to equalize the kinetic temperatures.

Suppose that is a Maxwellian distribution of characteristic number density , mean flow velocity , and temperature . In other words,

It follows that

(3.162) |

Hence, Equation (3.153) yields

(3.163) |

which implies that

(3.164) |

where

As before, collisions with particles of type give rise to a velocity dependent effective force, , acting on individual particles of type . In particular, if is parallel to , then

We conclude that particles of type moving parallel to the mean drift velocity (of particles of type relative to particles of type ) experience a velocity dependent force due to collisions with particles of type , which acts to reduce their speed. Of course, this has the effect of reducing the drift velocity.

It is easily demonstrated that
as
. Hence, Equations (3.157) and (3.166) yield
and
, respectively, in the limit
, implying that collisions only have a relatively weak effect on high speed
particles. In fact, collisions are unable to prevent an imposed electric field from accelerating super-thermal particles (whose number is generally only a very small
fraction of the total number of particles) to relativistic speeds (Rose and Clark 1961). Such particles
are known as *runaway particles*.