the collision integral’s matrix elements with respect to the modes $\chi_{\mathbf{k},\lambda}^{\left(m,s\right)}$ (see Fig. \[fig:The-matrix-elements\]) shows, that the relaxation rates of these modes grow linearly with increasing $m$: $$\tau_{\varepsilon/c,m}^{-1}\sim\left|m\right|,\label{eq:summary_scattering_rates}$$ for large $m$ (see sections \[subsec:Collinear-zero-modes\] and \[subsec:Scattering-times\]). This unusual behavior allows us to solve the (linearized) Boltzmann equation exactly in the limit of a small $\alpha$. The details of this solution are given in Sec. \[subsec:Non-local-transport-coefficients\].

![The matrix elements of the collision operator $\mathcal{C}$ of Eq. (\[eq:Collision\_operator\]) with respect to the collinear zero modes $\chi_{\mathbf{k},\lambda}^{\left(m,s\right)}=\lambda^{m}e^{im\theta}\left\{ 1,\lambda,\lambda\beta v\hbar k\right\} $ of Eq. (\[eq:Collinear\_modes\]) grow linearly with increasing angular harmonic numbers $m$. The linear fits of Eqs. (\[eq:linear\_scattering\_rates\_charge\]), (\[eq:linear\_scattering\_rates\_energy\]) are plotted as solid red and green lines. The linear behavior of the matrix elements and scattering rates allows to solve the quantum Boltzmann equation exactly.\[fig:The-matrix-elements\]](Pics/Non_local_transport/lin_matrixelements)

It is an important feature of graphene at the neutrality point, that the hydrodynamic modes excited by electric and thermal fields decouple in linear response, and in the absence of magnetic fields [@Mueller2008; @Mueller2009]. The modes are characterized by the distinct scattering, with all of them following Eq. (\[eq:summary\_scattering\_rates\]). Using our full solution of the Boltzmann equation, the non-local, i.e. wave-vector-dependent, charge and thermal conductivities as well as 