  We study power dissipation for systems of multiple quantum wires meeting at a
junction, in terms of a current splitting matrix (M) describing the junction.
We present a unified framework for studying dissipation for wires with either
interacting electrons (i.e., Tomonaga-Luttinger liquid wires with Fermi liquid
leads) or non-interacting electrons. We show that for a given matrix M, the
eigenvalues of M^T M characterize the dissipation, and the eigenvectors
identify the combinations of bias voltages which need to be applied to the
different wires in order to maximize the dissipation associated with the
junction. We use our analysis to propose and study some microscopic models of a
dissipative junction which employ the edge states of a quantum Hall liquid.
These models realize some specific forms of the M-matrix whose entries depends
on the tunneling amplitudes between the different edges.
