Moments of the distribution function

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

$${\bf M}_k({\bf r}, t) = \int {\bf v v\cdots v}\,f_s({\bf r},{\bf v}, t)\,d^3{\bf v},$$ (177)

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

The set $\{{\bf M}_k, k=0,1,2,\cdots\}$ can be viewed as an alternative description of the distribution function, which, indeed, uniquely specifies $f_s$ when the latter is sufficiently smooth. For example, a (displaced) Gaussian distribution 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 names and simple physical interpretations. First, we have the (particle) density,

$$n_s({\bf r},t) = \int f_s({\bf r}, {\bf v},t)\,d^3{\bf v},$$ (178)

and the particle flux density,

$$n_s\,{\bf V}_s({\bf r}, t) = \int {\bf v}\,f_s({\bf r}, {\bf v},t)\,d^3{\bf v}.$$ (179)

The quantity ${\bf V}_s$ is, of course, the flow velocity. Note that the electromagnetic sources, (166)-(167), are determined by these lowest moments:

$$\rho_c = \sum_s e_s n_s,$$ (180)

$${\bf j} = \sum_s e_s n_s\,{\bf V}_s.$$ (181)

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

$${\bf P}_s({\bf r}, t) = \int m_s\,{\bf v}{\bf v}\, f_s({\bf r}, {\bf v},t)\,d^3{\bf v}.$$ (182)

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

$${\bf Q}_s({\bf r}, t) = \int \frac{1}{2}\,m_s\,v^2\,{\bf v}\, f_s({\bf r}, {\bf v},t)\,d^3{\bf v}.$$ (183)

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 assume 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,

$${\bf w}_s\equiv {\bf v} - {\bf V}_s,$$ (184)

in order to write

$${\bf p}_s({\bf r}, t) = \int m_s\,{\bf w}_s{\bf w}_s\, f_s({\bf r}, {\bf v},t)\,d^3{\bf v},$$ (185)

and

$${\bf q}_s({\bf r}, t) = \int \frac{1}{2}\,m_s\,w_s^{~2}\,{\bf w}_s\, f_s({\bf r}, {\bf v},t)\,d^3{\bf v}.$$ (186)

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

$$p_s\equiv \frac{1}{3}\,{\rm Tr}\,({\bf p}_s).$$ (187)

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

$$\frac{3}{2}\,p_s = \int \frac{1}{2}\,m_s\,w_s^{~2} \,f_s\,d^3{\bf v}.$$ (188)

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

$$T_s \equiv \frac{p_s}{n_s}.$$ (189)

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

$${\bf P}_s = {\bf p}_s + m_s n_s\,{\bf V}_s{\bf V}_s,$$ (190)

$${\bf Q}_s = {\bf q}_s + {\bf p}_s\cdot{\bf V}_s + \frac{3}{2}\,p_s\,{\bf V}_s +\frac{1}{2}\,m_s n_s\,V_s^{~2}\, {\bf V}_s.$$ (191)