Mode-count balance in a ballistic junction

Let $T_{ij}$ count transmitted modes from injector $j$ to collector $i$, excluding reflection $R_{ii}$. Scattering conservation gives ${\sum }_{i\ne j}{T}_{ij}=N_j-R_{jj}$; unitarity also gives the corresponding row balance. The conductance matrix has zero row sum, making equal lead potentials carry no current, and zero column sum, conserving total current.