  One of the most fascinating experimental achievements of the last decade was
the realization of Bose-Einstein Condensation (BEC) of ultra-cold atoms in
optical lattices (OL's). The extraordinary level of control over these
structures allows us to investigate complex solid state phenomena and the
emerging field of ``atomtronics'' promises a new generation of nanoscale
devices. It is therefore of fundamental and technological importance to
understand their dynamical properties. Here we study the outgoing atomic flux
of BECs loaded in an one-dimensional OL with leaking edges, using a mean field
description provided by the Discrete Non-Linear Schrodinger Equation (DNLSE).
We demonstrate that the atom population inside the OL decays in avalanches of
size $J$. For intermediate values of the interatomic interaction strength their
distribution ${\cal P}(J)$ follows a power law i.e. ${\cal
P}(J)\sim1/J^{\alpha}$ characterizing systems at phase transition. This scale
free behaviour of ${\cal P}(J)$ reflects the complexity and the hierarchical
structure of the underlying classical mixed phase space. Our results are
relevant in a variety of contexts (whenever DNLSE is adequate), most
prominently the light emmitance from coupled non-linear optics waveguides.
