  Non-equilibrium steady states (NESS), in the sense of D. Ruelle, of Boson
systems with Bose--Einstein condensation (BEC) are investigated with the aid of
the C^*-algebraic method. The model consists of a quantum particle and several
bosonic reservoirs. We show that the mean entropy production rate is strictly
positive, independent of phase differences provided that the temperatures or
the chemical potentials of reservoirs are different. Moreover, Josephson
currents occur without entropy production, if the temperatures and the chemical
potentials of reservoirs are identical.
