dot-dash line: Doppler shifted Falkovsky’s expression, blue circles: Svintsov et al expression, orange pluses: numerical integration of Eq. (\[eqpir\]).[]{data-label="figurePi"}](figurePi){width="\columnwidth"}

For a quadratic dispersion relation, $\epsilon_{\bf k} = \hbar^2 k^2/(2m)$, the claim, Eq. (\[noneq-eq\]), is easily verified using the explicit expression for $\Pi^r$, Eq. (\[eqpir\]), by a change of integration variable with a constant shift, ${\bf p} \to {\bf p} + m{\bf v}_1/\hbar$. For graphene with Dirac cone, $\epsilon_{\bf k} = v_F\, \hbar k$, this is no longer true. A variable transform cannot eliminate both $\Phi_{\bf p}$ and $\Phi_{{\bf p}-{\bf q}}$ simultaneously. As a result, Doppler shift of the equilibrium result and nonequilibrium distribution in the Fermi function becomes inequivalent[@svintsov18]. The energy shift needs to be momentum dependent. The linear dependence on $k_x$ times a constant requires a specific assumption on the energy dependence of the relaxation times. According to the usual relaxation time approximation, $\Phi$ is given by (near $K$ or $K'$ points) $$\Phi = - e E \tau_{k} \frac{\partial \epsilon_{\bf k}}{\partial ( \hbar k_x )},$$ where electron carries charge $-e$, $E$ is the applied electric field in $x$ direction, $\tau_k$ is the relaxation time assumed to only depend on the magnitude of ${\bf k}$, and the last factor is 