Threshold for Chaos and Thermalization in One-Dimensional Mean-Field   Bose-Hubbard Model
  We study the threshold for chaos and its relation to thermalization in the 1D mean-field Bose-Hubbard model, which in particular describes atoms in optical lattices. We identify the threshold for chaos, which is finite in the thermodynamic limit, and show that it is indeed a precursor of thermalization. Far above the threshold, the state of the system after relaxation is governed by the usual laws of statistical mechanics. 
Amy C. Cassidy, Douglas Mason, Vanja Dunjko, Maxim Olshanii