The
th velocity space moment of the (ensemble-averaged) distribution function
is written
![$\displaystyle {\bf M}_k({\bf r}, t) = \int {\bf v v\cdots v}\,f_s({\bf r},{\bf v}, t)\,d^3{\bf v},$](img1012.png) |
(4.4) |
with
factors of
. Clearly,
is a tensor of rank
(Riley 1974).
The set
, for
,
can be viewed as an alternative description of the distribution function that uniquely specifies
when the latter is sufficiently smooth. For example,
a (displaced) Gaussian distribution function is uniquely specified by three
moments:
, the vector
, and the scalar formed by contracting
.
The low-order moments all have simple physical interpretations.
First, we have the particle number density,
![$\displaystyle n_s({\bf r},t) = \int f_s({\bf r}, {\bf v},t)\,d^3{\bf v},$](img1018.png) |
(4.5) |
and the particle flux density,
![$\displaystyle n_s\,{\bf V}_s({\bf r}, t) = \int
{\bf v}\,f_s({\bf r}, {\bf v},t)\,d^3{\bf v}.$](img1019.png) |
(4.6) |
The quantity
is, of course, the flow velocity. The constitutive relations, (3.1) and (3.2), are determined by these lowest
moments. In fact,
The second-order moment, describing the flow of momentum in the
laboratory frame, is called the stress tensor, and takes the form
![$\displaystyle {\bf P}_s({\bf r}, t) = \int
m_s\,{\bf v}{\bf v}\, f_s({\bf r}, {\bf v},t)\,d^3{\bf v}.$](img1022.png) |
(4.9) |
Finally, there is an important third-order moment
measuring the energy flux density,
![$\displaystyle {\bf Q}_s({\bf r}, t) = \int
\frac{1}{2}\,m_s\,v^2\,{\bf v}\, f_s({\bf r}, {\bf v},t)\,d^3{\bf v}.$](img1023.png) |
(4.10) |
It is often convenient to measure the second- and third-order moments in the rest-frame of the species under consideration. In this case, the
moments have different names. The stress tensor measured in the rest-frame
is called the pressure tensor,
, whereas the energy flux
density becomes the heat flux density,
. We introduce the
relative velocity,
![$\displaystyle {\bf u}_s\equiv {\bf v} - {\bf V}_s,$](img1026.png) |
(4.11) |
in order to write
![$\displaystyle {\bf p}_s({\bf r}, t) = \int
m_s\,{\bf u}_s{\bf u}_s\, f_s({\bf r}, {\bf v},t)\,d^3{\bf v},$](img1027.png) |
(4.12) |
and
![$\displaystyle {\bf q}_s({\bf r}, t) = \int
\frac{1}{2}\,m_s\,u_s^{2}\,{\bf u}_s\, f_s({\bf r}, {\bf v},t)\,d^3{\bf v}.$](img1028.png) |
(4.13) |
The trace of the pressure tensor measures the ordinary (or scalar) pressure,
![$\displaystyle p_s\equiv \frac{1}{3}\,{\rm Tr}\,({\bf p}_s).$](img1029.png) |
(4.14) |
In fact,
is the kinetic energy density of species
: that is,
![$\displaystyle \frac{3}{2}\,p_s = \int \frac{1}{2}\,m_s\,u_s^{2} \,f_s\,d^3{\bf v}.$](img1031.png) |
(4.15) |
In thermodynamic equilibrium, the distribution function becomes a Maxwellian
characterized by some temperature
, and Equation (4.15) yields
. It
is, therefore, natural to define the (kinetic) temperature as
![$\displaystyle T_s \equiv \frac{p_s}{n_s}.$](img1033.png) |
(4.16) |
Of course, the moments measured in the two different frames are related.
By direct substitution, it is easily verified that