Based on \cite{geier2015cumulant}, collision operator  $\vec{C}$:
\begin{equation*}
\vec{C}\left( \vec{f} \right) = \matr{M}^{-1} \vec{G}^{-1}\left( \matr{S}\left(\vec{\kum}^{(eq)} - \vec{G}\left(\matr{M}\vec{f}\right) \right)\right),
\end{equation*}
where
\input{\path/d3q27_cum_matS1}
$\oA,\oB,\cdots, \oJ \in \left(0,2\right).$

The nonlinear operator $\vec{G}$ (with its inverse $\vec{G}^{-1}$) transforms the raw moment vector $\vec{\mu}$ defined by matrix $\matr{M}$ to the cumulant vector
$$
	\vec{\kum} = \vec{G}(\vec{\mu}) = \left( \kum_{(0,0,0)}, \kum_{(1,0,0)}, \kum_{(0,1,0)}, \cdots, \kum_{(2,2,2)}\right)^T
$$ 
as:
\input{\path/d3q27_cum_kappa}

The equilibrium cumulant vector $\vec{\kum}^{(eq)}$ is defined by
\input{\path/d3q27_cum_eq}

