Fig. \[fig:The-matrix-elements\]. To find closed expressions for the non-local transport coefficients, the scattering rates are approximated by Eq. (\[eq:Linear\_approximation\]) for $m>2$. In principle, the numerically exact scattering rates up to an arbitrary $m$ can be included. Here, the rates for $m>2$ will be assumed to follow Eq.\[eq:Linear\_approximation\] in order to keep the algebraic efforts at a minimum. The projections of the force terms (\[eq:Electric\_force\])-(\[eq:Shear\_force\]) onto collinear zero modes read $$\Braket{S_{E}|\chi_{\mathbf{k},\lambda}}=-\frac{eE_{0}}{2\hbar^{2}\beta v}\delta_{\left|m\right|,1}\left[\begin{array}{c}
\frac{\log(2)}{\pi}\\
0\\
0
\end{array}\right],\label{eq:electric_force_projected}$$ $$\Braket{S_{T}|\chi_{\mathbf{k},\lambda}}=\frac{\left|\nabla T\right|k_{B}\pi^{4}}{v\beta\hbar^{2}}\delta_{\left|m\right|,1}\left[\begin{array}{c}
0\\
\frac{\pi}{6}\\
\frac{9\zeta(3)}{2\pi}
\end{array}\right],\label{eq:Thermal_force_projected}$$ $$\begin{aligned}
\Braket{S_{S}|\chi_{\mathbf{k},\lambda}} & =-\frac{iX_{0}}{4\left(v\beta\hbar\right)^{2}}\textrm{sign}\left(m\right)\delta_{\left|m\right|,2}\left[\begin{array}{c}
0\\
\frac{\pi}{6}\\
\frac{9\zeta\left(3\right)}{2\pi}
\end{array}\right].\label{eq:viscous_projected}\end{aligned}$$ For the Vlassov term (\[eq:Vlasov\_term\]) on finds $$\begin{aligned}
\left\langle \psi_{\mathbf{k},\lambda}^{\left(m,s\right)}|\mathcal{V}|\psi_{\mathbf{k},\lambda}^{\left(m',s'\right)}\right\rangle  & = & i\alpha N\left(e^{-i\vartheta_{\bm{q}}}\delta_{m,1}+e^{i\vartheta_{\bm{q}}}\delta_{m,-1}\right)\nonumber \\
 &  & \times\frac{\delta_{1,s}\delta_{1,s'}\delta_{m',0}}{2v^{2}\beta^{3}\hbar^{3}}\left[\begin{array}{c}
\frac{\log(2)^{2}}{\pi^{2}}\\
0\\
0
\end{array}\right]\label{eq:Vlasov_projected}\end{aligned}$$ The non-equilibrium part of the distribution function expanded in the subset of colinear zero modes becomes $$\psi_{\mathbf{k},\lambda}=\sum_{m=-\infty}^{\infty}\sum_{s=1}^{3}a_{m,s}\left(\omega,\mathbf{q}\right)\chi_{\mathbf{k},\lambda}^{\left(m,s\right)}.\label{eq:Distr_dunction_expanded}$$ Together, the expressions (\[eq:Kin\_Eq\]), (\[eq:Liouville\_projected\]), (\[eq:Linear\_approximation\]), (\[eq:electric\_force\_projected\])-(\[eq:viscous\_projected\]), (\[eq:Vlasov\_projected\]) and (\[eq:Distr\_dunction\_expanded\]) provide a linearized kinetic equation restricted to the basis of collinear zero modes that becomes exact for small values of the fine structure constant $\alpha$. Since no assumptions on the spatial dependencies were made, except that they are be within the limits of the applicability of the kinetic equation, this expansion can be used to derive the non-local transport coefficients in 