Dynamical freezing of relaxation to equilibrium

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.