be neglected. This translates to neglecting the off-diagonal components of the distribution function in the band representation, which is then given by its diagonal elements: $$f_{\mathbf{k}\lambda}=\left\langle \psi_{\lambda,\mathbf{k}}^{\dagger}\psi_{\lambda,\mathbf{k}}\right\rangle .$$ The quantum Boltzmann equation then reads $$\left(\partial_{t}+\mathbf{v}_{\mathbf{k}\lambda}\cdot\nabla_{\mathbf{r}}+\mathbf{F}\left(\mathbf{r},t\right)\cdot\nabla_{\mathbf{k}}+C\right)f_{\mathbf{k}\lambda}\left(\mathbf{r},t\right)=0.\label{eq:Boltzmann}$$ Here, $\mathbf{v}_{\mathbf{k}\lambda}=\partial\varepsilon_{\mathbf{k}\lambda}/\partial\mathbf{k}$ is the group velocity and $$\mathbf{F}\left(\mathbf{r},t\right)=\mathbf{F}_{{\rm ext}}\left(\mathbf{r},t\right)+e\mathbf{E}_{V}\left(\mathbf{r},t\right)$$ is the sum of external forces, in our case due to an external electric field, and the self consistent electric field $\mathbf{E}_{V}$ that is produced by an inhomogeneous distribution of charges - the so called Vlasov field[@Vlasov1938]. It will be dealt with at the end of this section. $C$ represents the central part of the kinetic theory - the Boltzmann collision operator describingelectron-electron Coulomb scattering. Details on the derivation of $C$ are summarized in Appendix \[sec:App\_Collision-operator\], based on Refs. [@Fritz2008; @Kiselev2019b].

Studying the linear response to $\mathbf{F}_{{\rm ext}}$, we expand the distribution function around the local equilibrium distribution $f_{k\lambda}^{\left(0\right)}$ $$f_{\mathbf{k}\lambda}\left(\mathbf{r},t\right)=f_{k\lambda}^{\left(0\right)}+w_{k}\psi_{\mathbf{k}\lambda}\left(\mathbf{r},t\right).\label{eq:Distribution_expanded}$$ where $f_{k,\lambda}^{0}$ is given by $$f_{k\lambda}^{\left(0\right)}=\frac{1}{e^{\beta\left(\varepsilon_{\mathbf{k}\lambda}-\mathbf{u}\cdot\mathbf{k}\right)}+1}.\label{eq:Local_equilibrium}$$ The product $w_{k}\equiv f_{k}^{\left(0\right)}\left(1-f_{k}^{\left(0\right)}\right)$ , that will soon play the role of a weight function in the scalar product, does not depend on $\lambda$, and the corresponding index is dropped in Eq. (\[eq:Distribution\_expanded\]) and in the following.

Performing a Fourier transformation $\psi{}_{\mathbf{k}\lambda}\left(\mathbf{r},t\right)\rightarrow\psi_{\mathbf{k}\lambda}\left(\mathbf{q},\omega\right)$ to frequency and momentum space, we obtain the 