  Non-equilibrium steady states of quantum fields on star graphs are explicitly
constructed. These states are parametrized by the temperature and the chemical
potential, associated with each edge of the graph. Time reversal invariance is
spontaneously broken. We study in this general framework the transport
properties of the Schroedinger and the Dirac systems on a star graph, modeling
a quantum wire junction. The interaction, which drives the system away from
equilibrium, is localized in the vertex of the graph. All point-like vertex
interactions, giving rise to self-adjoint Hamiltonians possibly involving the
minimal coupling to a static electromagnetic field in the ambient space, are
considered. In this context we compute the exact electric steady current and
the non-equilibrium charge density. We investigate also the heat transport and
derive the Casimir energy density away from equilibrium. The appearance of
Friedel type oscillations of the charge and energy densities along the edges of
the graph is established. We focus finally on the noise power and discuss the
non-trivial impact of the point-like interactions on the noise.
