give rise to a rapid equilibration to the subset of modes given in Eq. (\[eq:Collinear\_modes\]) which then dominate the long-time dynamics.

In order to proceed, the matrix elements of Eq. (\[eq:Kin\_Eq\]) in this basis must be calculated. The matrix elements of the Liouville operator $\mathcal{L}$ are given by

$$\left\langle \chi_{\mathbf{k},\lambda}^{\left(m,s\right)}\left|\mathcal{L}\right|\chi_{\mathbf{k},\lambda}^{\left(m',s'\right)}\right\rangle =\left(-i\omega\delta_{m,m'}+\frac{1}{2}ivq\left(e^{-i\vartheta_{\mathbf{q}}}\delta_{m,m'+1}+e^{i\vartheta_{\mathbf{q}}}\delta_{m,m'-1}\right)\right)\left(v\beta\hbar\right)^{-2}L_{s,s'},\label{eq:Liouville_projected}$$

where $\vartheta_{\mathbf{q}}$ is the polar angle of the wave-vector $\mathbf{q}$ and $$L=\left[\begin{array}{ccc}
\frac{\log(2)}{\pi} & 0 & 0\\
0 & \frac{\log(2)}{\pi} & \frac{\pi}{6}\\
0 & \frac{\pi}{6} & \frac{9\zeta(3)}{2\pi}
\end{array}\right].$$ The rows and columns of the matrix notation refer to the mode index $s$ of Eq. (\[eq:Collinear\_modes\]).

We calculate the matrix elements of the collision operator $\mathcal{C}$ numerically (some values are given in Appendix \[sec:App\_Matrix-elements\]). Due to the rotational invariance of the low-energy Dirac Hamiltonian (\[eq:nl\_Hamiltonian\]), they are diagonal in the angular harmonic representation. Most importantly, the matrix elements rapidly approach a linear behavior for large $\left|m\right|$: $$\left\langle \chi_{\mathbf{k},\lambda}^{\left(m,s\right)}\left|\mathcal{C}\right|\chi_{\mathbf{k},\lambda}^{\left(m',s'\right)}\right\rangle =\frac{\delta_{m,m'}}{v^{2}\beta^{3}\hbar^{3}}\left(\left|m\right|\gamma_{s,s'}-\eta_{s,s'}\right).\label{eq:Linear_approximation}$$ $\gamma_{s,s'}$ and $\eta_{s,s'}$ are numerical coefficients that are listed below Eqs. (\[eq:linear\_scattering\_rates\_charge\]) and (\[eq:linear\_scattering\_rates\_energy\]). This surprising result is due to the linear Dirac spectrum of the system. It allows to solve the Boltzmann equation exactly, as will be seen later. The linear behavior of the scattering rates is also shown in 