Next: Moments of the Collision
Up: Plasma Fluid Theory
Previous: Introduction
The th moment of the (ensemble averaged) distribution function
is written

(177) 
with factors of . Clearly, is a tensor of rank .
The set
can be viewed as an alternative description of the distribution function, which,
indeed, uniquely specifies when the latter is sufficiently smooth. For example,
a (displaced) Gaussian distribution is uniquely specified by three
moments: , the vector , and the scalar formed by contracting
.
The loworder moments all have names and simple physical interpretations.
First, we have the (particle) density,

(178) 
and the particle flux density,

(179) 
The quantity is, of course, the flow velocity. Note that
the electromagnetic sources, (166)(167), are determined by these lowest
moments:
The secondorder moment, describing the flow of momentum in the
laboratory frame, is called the stress tensor, and denoted by

(182) 
Finally, there is an important thirdorder moment
measuring the energy flux density,

(183) 
It is often convenient to measure the second and thirdorder moments in the restframe of the species under consideration. In this case, the
moments assume different names: the stress tensor measured in the restframe
is called the pressure tensor, , whereas the energy flux
density becomes the heat flux density, . We introduce the
relative velocity,

(184) 
in order to write

(185) 
and

(186) 
The trace of the pressure tensor measures the ordinary (or ``scalar'') pressure,

(187) 
Note that is the kinetic energy density of species :

(188) 
In thermodynamic equilibrium, the distribution function becomes a Maxwellian
characterized by some temperature , and Eq. (188) yields . It
is, therefore, natural to define the (kinetic) temperature as

(189) 
Of course, the moments measured in the two different frames are related.
By direct substitution, it is easily verified that
Next: Moments of the Collision
Up: Plasma Fluid Theory
Previous: Introduction
Richard Fitzpatrick
20110331