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We investigate out-of-equilibrium stationary processes emerging in a DiscreteNonlinear Schroedinger chain in contact with a heat reservoir (a bath) attemperature $T_L$ and a pure dissipator (a sink) acting on opposite edges. Weobserve two different regimes. For small heat-bath temperatures $T_L$ andchemical-potentials, temperature profiles across the chain display anon-monotonous shape, remain remarkably smooth and even enter the region ofnegative absolute temperatures. For larger temperatures $T_L$, the transport ofenergy is strongly inhibited by the spontaneous emergence of discretebreathers, which act as a thermal wall. A strongly intermittent energy flux isalso observed, due to the irregular birth and death events of the breathers.The corresponding statistics exhibits the typical signature of rare events ofprocesses with large deviations. In particular, the breather lifetime is foundto be ruled by a stretched-exponential law.

A Chain, a Bath, a Sink and a Wall