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Moments of Distribution Function

The $ k$ th velocity space moment of the (ensemble-averaged) distribution function $ f_s({\bf r}, {\bf v}, t)$ is written

$\displaystyle {\bf M}_k({\bf r}, t) = \int {\bf v v\cdots v}\,f_s({\bf r},{\bf v}, t)\,d^3{\bf v},$ (4.4)

with $ k$ factors of $ {\bf v}$ . Clearly, $ {\bf M}_k$ is a tensor of rank $ k$ (Riley 1974).

The set $ {\bf M}_k$ , for $ k=0,1,2,\cdots$ , can be viewed as an alternative description of the distribution function that uniquely specifies $ f_s$ when the latter is sufficiently smooth. For example, a (displaced) Gaussian distribution function is uniquely specified by three moments: $ M_0$ , the vector $ {\bf M}_1$ , and the scalar formed by contracting $ {\bf M}_2$ .

The low-order moments all have simple physical interpretations. First, we have the particle number density,

$\displaystyle n_s({\bf r},t) = \int f_s({\bf r}, {\bf v},t)\,d^3{\bf v},$ (4.5)

and the particle flux density,

$\displaystyle n_s\,{\bf V}_s({\bf r}, t) = \int {\bf v}\,f_s({\bf r}, {\bf v},t)\,d^3{\bf v}.$ (4.6)

The quantity $ {\bf V}_s$ is, of course, the flow velocity. The constitutive relations, (3.1) and (3.2), are determined by these lowest moments. In fact,

$\displaystyle \rho$ $\displaystyle = \sum_s e_s\, n_s,$ (4.7)
$\displaystyle {\bf j}$ $\displaystyle = \sum_s e_s \,n_s\,{\bf V}_s.$ (4.8)

The second-order moment, describing the flow of momentum in the laboratory frame, is called the stress tensor, and takes the form

$\displaystyle {\bf P}_s({\bf r}, t) = \int m_s\,{\bf v}{\bf v}\, f_s({\bf r}, {\bf v},t)\,d^3{\bf v}.$ (4.9)

Finally, there is an important third-order moment measuring the energy flux density,

$\displaystyle {\bf Q}_s({\bf r}, t) = \int \frac{1}{2}\,m_s\,v^2\,{\bf v}\, f_s({\bf r}, {\bf v},t)\,d^3{\bf v}.$ (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, $ {\bf p}_s$ , whereas the energy flux density becomes the heat flux density, $ {\bf q}_s$ . We introduce the relative velocity,

$\displaystyle {\bf w}_s\equiv {\bf v} - {\bf V}_s,$ (4.11)

in order to write

$\displaystyle {\bf p}_s({\bf r}, t) = \int m_s\,{\bf w}_s{\bf w}_s\, f_s({\bf r}, {\bf v},t)\,d^3{\bf v},$ (4.12)


$\displaystyle {\bf q}_s({\bf r}, t) = \int \frac{1}{2}\,m_s\,w_s^{\,2}\,{\bf w}_s\, f_s({\bf r}, {\bf v},t)\,d^3{\bf v}.$ (4.13)

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

$\displaystyle p_s\equiv \frac{1}{3}\,{\rm Tr}\,({\bf p}_s).$ (4.14)

In fact, $ (3/2)\,p_s$ is the kinetic energy density of species $ s$ : that is,

$\displaystyle \frac{3}{2}\,p_s = \int \frac{1}{2}\,m_s\,w_s^{\,2} \,f_s\,d^3{\bf v}.$ (4.15)

In thermodynamic equilibrium, the distribution function becomes a Maxwellian characterized by some temperature $ T$ , and Equation (4.15) yields $ p=n\,T$ . It is, therefore, natural to define the (kinetic) temperature as

$\displaystyle T_s \equiv \frac{p_s}{n_s}.$ (4.16)

Of course, the moments measured in the two different frames are related. By direct substitution, it is easily verified that

$\displaystyle {\bf P}_s$ $\displaystyle ={\bf p}_s + m_s\, n_s\,{\bf V}_s{\bf V}_s,$ (4.17)
$\displaystyle {\bf Q}_s$ $\displaystyle = {\bf q}_s + {\bf p}_s\cdot{\bf V}_s + \frac{3}{2}\,p_s\,{\bf V}_s +\frac{1}{2}\,m_s\, n_s\,V_s^{\,2}\, {\bf V}_s.$ (4.18)

next up previous
Next: Moments of Collision Operator Up: Plasma Fluid Theory Previous: Introduction
Richard Fitzpatrick 2016-01-23