Conductance scaling of junctions of Luttinger-liquid wires out of equilibrium

We develop the renormalization group theory of the conductances of N-lead junctions of spinless Luttinger-liquid wires as functions of bias voltages applied to N independent Fermi-liquid reservoirs. Based on the perturbative results up to second order in the interaction we demonstrate that the conductances obey scaling. The corresponding renormalization group $\beta$ functions are derived up to second order.