Fourier transform to Eq. (\[eq:Vlasov\_field\]) one finds $$\begin{aligned}
e\mathbf{E}_{V} & = & -i\mathbf{q}\alpha vN\sum_{\lambda}\int\frac{d^{2}k}{\left(2\pi\right)^{2}}\,\frac{2\pi\delta f_{\mathbf{k}\lambda}\left(\mathbf{q},\omega\right)}{q}.\end{aligned}$$ The corresponding term in the Boltzmann equation, linearized in $\psi_{\mathbf{k},\lambda}$, reads $$\mathcal{V}=e\mathbf{E}_{V}\cdot\mathbf{v}_{\mathbf{k}\lambda}.\label{eq:Vlasov_term}$$

Collinear zero modes\[subsec:Collinear-zero-modes\]
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In this section, we summarize how Eq. (\[eq:Kin\_Eq\]) is solved in the limit of a small fine structure constant. A standard way to deal with an integral equation like (\[eq:Kin\_Eq\]) is to expand the function $\psi_{\mathbf{k},\lambda}$ into a set of suitable basis functions. The choice of this basis is facilitated by the fact that for small values of the graphene fine structure constant $\alpha$, the collision operator (\[eq:Collision\_operator\]) logarithmically diverges if the velocities of involved particles are parallel to each other. This is a consequence of the linear single particle spectrum, and the resulting momentum independent velocity of massless Dirac particles. Intuitively speaking, the scattering is enhanced, because particles traveling in the same direction interact with each other over a particularly long period of time. A more mathematical picture of this so-called collinear scattering anomaly is presented in Appendix (\[sec:App\_Collinear-scattering\]). It is convenient to write the collision operator as a sum of the collinear part $\mathcal{C}_{c}$ and the non-collinear part $\mathcal{C}_{nc}$: $$\mathcal{C}=\log{\left(1/\alpha\right)}\mathcal{C}_{c}+\mathcal{C}_{nc}.\label{eq:Collinear_non_collinear_separation}$$ The factor $\log{\left(1/\alpha\right)}$ is large at small 