viscous force is present if the drift velocity $\mathbf{u}$ in the local equilibrium distribution function (\[eq:Local\_equilibrium\]) is a function of the coordinate $\mathbf{x}$. Then the drift term of the Boltzmann equation (\[eq:Boltzmann\]) can be thought of as a force term $$\begin{aligned}
S_{S} & =-vkX_{0,\alpha\beta}\left(\frac{k_{\alpha}k_{\beta}}{k^{2}}-\frac{1}{2}\delta_{\alpha\beta}\right)\lambda\beta\nonumber \\
 & =-\frac{1}{2}kX_{0}\sin\left(2\theta\right)\left(\lambda v\beta\right),\label{eq:Shear_force}\end{aligned}$$ where the stress tensor is given by $$\begin{aligned}
X_{0,\alpha\beta} & =\frac{1}{2}\left(\frac{\partial u_{\alpha}}{\partial x_{\beta}}+\frac{\partial u_{\beta}}{\partial x_{\alpha}}-2\delta_{\alpha\beta}\nabla\cdot\mathbf{u}\right).\end{aligned}$$ In the following, we consider a flow with $\mathbf{u}\left(y\right)=u\left(y\right)\hat{\mathbf{e}}_{x}$ and therefore only include the component $X_{0,xy}$, which is relevant for the calculation of the shear viscosity.

The collision operator is given by $$\begin{aligned}
\left(C\psi\right)_{\mathbf{k}\lambda} & = & \frac{2\pi}{\hbar}\int_{k^{\prime}q}\delta\left(k+k'-\left|\mathbf{k}+\mathbf{q}\right|-\left|\mathbf{k}'-\mathbf{q}\right|\right)\label{eq:Collision_operator}\\
 & \times & \left(1-f_{k}^{\left(0\right)}\right)\left(1-f_{k'}^{\left(0\right)}\right)f_{\left\vert \mathbf{k}+\mathbf{q}\right\vert }^{\left(0\right)}f_{\left\vert \mathbf{k}'-\mathbf{q}\right\vert }^{\left(0\right)}\nonumber \\
 & \times & \left\{ \gamma_{\mathbf{k},\mathbf{k}^{\prime},\mathbf{q}}^{\left(1\right)}\left(\psi_{\mathbf{k}+\mathbf{q},\lambda}+\psi_{\mathbf{k}'-\mathbf{q},\lambda}-\psi_{\mathbf{k}',\lambda}-\psi_{\mathbf{k},\lambda}\right)\right.\nonumber \\
 & + & \left.\gamma_{\mathbf{k},\mathbf{k}^{\prime},\mathbf{q}}^{\left(2\right)}\left(\psi_{\mathbf{k}+\mathbf{q},\lambda}-\psi_{-\mathbf{k}'+\mathbf{q},\bar{\lambda}}+\psi_{-\mathbf{k}',\bar{\lambda}}-\psi_{\mathbf{k},\lambda}\right)\right\} .\nonumber \end{aligned}$$ The matrix elements $\gamma_{\mathbf{k},\mathbf{k}^{\prime},\mathbf{q}}^{\left(1\right)}$, $\gamma_{\mathbf{k},\mathbf{k}^{\prime},\mathbf{q}}^{\left(2\right)}$ can be found in Appendix \[sec:App\_Collision-operator\].

Another important term in the kinetic equation describes the electrostatic forces that arise due to an inhomogeneous distribution of charges. These forces are mediated by a self consistent electric field $\mathbf{E}_{V}$, first introduced by Vlasov [@Vlasov1938]. It reads $$e\mathbf{E}_{V}=-\alpha vN\nabla_{\mathbf{x}}\int d^{2}r'\sum_{\lambda}\int\frac{d^{2}k}{\left(2\pi\right)^{2}}\,\frac{\delta f_{\mathbf{k}\lambda}\left(\mathbf{r}',\omega\right)}{\left|\mathbf{r}-\mathbf{r}'\right|},\label{eq:Vlasov_field}$$ where we have used the abbreviation $\delta f_{\mathbf{k}\lambda}\left(\mathbf{r},\omega\right)=w_{k}\psi_{\mathbf{k},\lambda}\left(\mathbf{r},\omega\right)$. A derivation of the term can be found in Ref. [@KadanoffBaym] (Eqs. (7-3) and (9-16)). Applying a 