Cumulants and large deviations of the current through non-equilibrium steady states

Using a generalisation of the detailed balance for systems maintained out of equilibrium by contact with 2 reservoirs at unequal temperatures or at unequal densities, we recover the fluctuation theorem for the large deviation funtion of the current. For large diffusive systems, we show how the large deviation funtion of the current can be computed using a simple additivity principle. The validity of this additivity principle and the occurence of phase transitions are discussed in the framework of the macroscopic fluctuation theory.