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Suitable Langevin thermostats are introduced which are able to control boththe temperature and the chemical potential of a one-dimensional lattice ofnonlinear Schr\"odinger oscillators. The resulting non-equilibrium stationarystates are then investigated in two limit cases (low temperatures and largeparticle densities), where the dynamics can be mapped onto that of acoupled-rotor chain with an external torque. As a result, an effective kineticdefinition of temperature can be introduced and compared with the generalmicrocanonical (global) definition.

Off-equilibrium Langevin dynamics of the discrete nonlinear Schroedinger chain