Boundary dissipation applied to the Discrete Nonlinear Schrodinger Equation is a practical procedure for the distillation of Breather states, that can be interpreted as small atomic condensates or localized light beams in BEC and quantum optics, respectively. The appearance of this kind of solutions, far from being a peculiar nonlinear effect, is shown to be related to the thermodynamics of the DNLSE, where isolated breathers are special cases of a much wider class of equilibrium states at negative temperature.