• Luttinger liquid
• star graph
• non equilibrium steady state
• effective temperature
• effective potential
• out of equilibrium momentum distribution

The study of non-equilibrium quantum systems is among the most rapidly expanding areas of theoretical physics.
In this work we investigate some general aspects of such systems. The physical model we focus on is a multicomponent system represented by a set of semi-infinite leads (edges) of a star graph. Each lead is attached at infinity to a heat reservoir with fixed temperature and chemical potential. The interaction between the various leads is localized in the vertex of the graph and is defined by a scattering matrix. The bulk properties of the system are determined by a Tomonaga-Luttinger lagrangian. It is well known that the Tomonaga-Luttinger model is exactly solvable on the line and
successfully describes a large class of
one-dimensional quantum fermionic models with gapless excitations and linear spectrum,
it applies to various realistic systems, including nanowire junctions and carbon nanotubes.
By imposing appropriate boundary conditions (the so called current splitting conditions) we exactly solve the model and derive in explicit form the non-equilibrium correlation functions.
By supposing that the only left and right coupling is given by the junction we show that it is possible to define a non-equilibrium
analogue of the Fermi-Dirac distribution. This is a rather interesting fact, since for
systems which are out of equilibrium it is not usually possible to give such a definition.
We then examine the possibility to introduce a kind of effective non-equilibrium
concept of temperature and chemical potential. In particular, we investigate the possibility to
approximate the out of equilibrium distribution with an equilibrium one. Lastly we show that is possible to see the non-equilibrium two point correlation function as an equilibrium one with a distance dependent "temperature".