Transport properties of a two-lead Luttinger liquid junction out of   equilibrium: fermionic representation
  The electrical current through an arbitrary junction connecting quantum wires of spinless interacting fermions is calculated in fermionic representation. The wires are adiabatically attached to two reservoirs at chemical potentials differing by the applied voltage bias. The relevant scale-dependent contributions in perturbation theory in the interaction up to infinite order are evaluated and summed up. The result allows one to construct renormalization group equations for the conductance as a function of voltage (or temperature, wire length). There are two fixed points at which the conductance follows a power law in terms of a scaling variable $\Lambda$, which equals the bias voltage $V$, if $V$ is the largest energy scale compared to temperature $T$ and inverse wire length $L^{-1}$, and interpolates between these quantities in the crossover regimes. 
D.N. Aristov and P. W\"olfle
cond-mat.str-el