  The functional renormalization group (FRG) provides a flexible tool to study
correlations in low-dimensional electronic systems. In this paper, we present a
novel FRG approach to the steady-state of quantum wires out of thermal
equilibrium. Our method is correct up to second order in the two-particle
interaction and accounts for inelastic scattering. We combine semi-analytic
solutions of the flow equations with MPI parallelization techniques, which
allows us to treat systems of up to 60 lattice sites. The equilibrium limit is
well-understood and serves as a benchmark. We compute effective distribution
functions, the local density of states, and the steady-state current and
demonstrate that all of these quantities depend strongly on the choice of the
cutoff employed within the FRG. Non-equilibrium is plagued by the lack of
physical arguments in favor of a certain cutoff as well as by the appearance of
secular higher-order terms which are only partly included in our approach. This
demonstrates the inadequacy of a straightforward second-order FRG scheme to
study interacting quantum wires out of equilibrium in the absence of a natural
cutoff choice.
