behaviour of those injected electrons (via electron-electron interactions). The transmission function can be determined self-consistently with the charge distribution, if the latter is treated in a time-independent mean-field manner (neglecting single electron effects). We note that the same nonlinear scattering theory was also derived for resonant level models [@Humphrey-Linke2005; @Nakpathomkun-Xu-Linke2010] using functional RG to treat single-electron charging effects [@Meden2013].

The scattering theory for the heat current is based on the observation that an electron leaving reservoir $i$ at energy ${\epsilon}$ is carrying heat ${\epsilon}- \mu_i$ out of that reservoir [@Butcher1990], where $\mu_i$ is the reservoir’s chemical potential. Thus, a reservoir is cooled by removing an electron above the Fermi surface, but heated by removing a electron below the Fermi surface. It is convenient to treat empty states below a reference chemical potential (which we define as ${\epsilon}=0$), as “holes”. Then we do not need to keep track of a full Fermi sea of electrons, but only the holes in that Fermi sea. Then the heat-currents out of reservoirs L and R and into the quantum system are $$\begin{aligned}
J_L  \! &=& \!
{1 \over h} \sum_\mu \int_0^\infty {\rm d}{\epsilon}\, ({\epsilon}-  \mu{ e^{\operatorname{-}} }V_L) \,  
{\cal T}^{\mu\mu}_{RL}({\epsilon})  \, \big[f_L^\mu ({\epsilon}) - 