  We provide evidence of an extremely slow thermalization occurring in the
Discrete NonLinear Schr\"odinger (DNLS) model. At variance with many similar
processes encountered in statistical mechanics - typically ascribed to the
presence of (free) energy barriers - here the slowness has a purely dynamical
origin: it is due to the presence of an adiabatic invariant, which freezes the
dynamics of a tall breather. Consequently, relaxation proceeds via rare events,
where energy is suddenly released towards the background. We conjecture that
this exponentially slow relaxation is a key ingredient contributing to the
non-ergodic behavior recently observed in the negative temperature region of
the DNLS equation.
