Non-equilibrium steady states (NESS) of Boson systems with different phases are investigated with the aid of the C^*-algebraic method. The system consists of sample and several free or pure-hopping bosonic reservoirs coupled with each other. Initially, the sample and reservoirs are uncoupled and in equilibrium with temperatures, local densities, and phases. At t=0, the reservoirs couple with each other through the sample. NESS are constructed as time limit of the composition of initial state and time evolution of coupled system. In NESS, Josephson currents and the mean entropy production rates are calculated. The mean entropy production rate does not depend on phase differences. As consequence, the currents provided to phase differences produces no entropy. 查看全文>>