Peculiar out-of-equilibrium stationary processes may emerge in a Discrete Nonlinear Schroedinger chain with a heat reservoir and a pure mass dissipator acting at its opposite edges. By increasing the temperature of the reservoir, a transition between two distinct regimes occurs. In the low temperature regime the temperature and chemical potential profiles display a smooth shape characteristic of thermodiffusive processes. Remarkably, a regime of negative absolute temperatures emerges spontaneously. In the high temperature regime, localized structures (discrete breathers) naturally arise out of the fluctuations and act as a "thermal wall" that strongly inhibits the flows of energy and mass towards the dissipator. The systems turns into a quasi-insulator.