2.2.2.2 Odd Moments

For the odd moments we obtain

$$\begin{gather*}\begin{split}& \partial_{x_3} \int v_3^2 u^i f(x_3,v_3,v_r) d(v_3... ... & \quad \quad \quad \int Q(f) v_3 u^i d(v_3,v_r) . \end{split}\end{gather*}$$ (2.35)

For the first term we note that by the assumption of isotropy of the symmetric part we have

$$\bigl{\langle} v_3^2 u^i \bigr{\rangle} = \bigg{\langle} \frac{1}... ... + v_3^3) \bigg{\rangle} = \bigg \langle \frac{1}{3} u^{i+1} \bigg \rangle .$$ (2.36)

By using partial integration on the second term we obtain

$$\partial_{v_3} (v_3u^i) = u^i + v_3(iu^{i-1}2v_3) = u^i + 2iu^{i-1}v_3^2 .$$ (2.37)

With symmetric isotropy we get

$$\langle u^i + 2iu^{i-1}v_3^2 \rangle = \bigg \langle u^i \bigg( 1 + \frac{2}{3}i \bigg) \bigg \rangle .$$ (2.38)

In summary we get the following set of equations: For $i=0,1,2$ and $O_{2i+1} = v_3u^i$:

$$\frac{1}{3} \partial_{x_3} M_{2i+2} -\frac{q}{m^{*}} E_{x_3} \big... ...igg( 1 + \frac{2i}{3} \bigg) M_{2i} \bigg ) = \int Q(f) v_3 u^i d(v_3,v_r) .$$ (2.39)

The derivation does not need the full isotropy condition for its validity. Instead, we only need two reduction conditions:

$$\begin{gather*}\begin{split}& \langle v_3^2 \rangle = \bigg \langle \frac{1}{3} ... ...ngle = \bigg \langle \frac{1}{3} v^4 \bigg \rangle . \end{split}\end{gather*}$$ (2.40)

A third reduction condition

$$\langle v_3^2 v^4 \rangle = \bigg \langle \frac{1}{3} v^6 \bigg \rangle$$ (2.41)

is not needed if the sixth moment $M_6$ is redefined as

$$M_6 = 3 \langle v_3^2 r^4 \rangle .$$ (2.42)

So by using the isotropy condition we can eliminate the moment $\langle v_3^2 u^i \rangle$. For the validity of Equation 2.36 the isotropy assumption is a sufficient, but not a necessary condition. A simple counter example is a function of the form $g(\vert k\vert)(\vert k_x\vert + \vert k_y\vert + \vert k_z\vert)$. However, this form does not fulfill cylindrical symmetry. Examples which fulfill cylindrical symmetry can be constructed too, but one has to consider multivariate polynomials up to sixth order for a proof.