Weakly nonergodic dynamics in the Gross--Pitaevskii lattice

The microcanonical Gross--Pitaevskii (aka semiclassical Bose-Hubbard) lattice model dynamics is characterized by a pair of energy and norm densities. The grand canonical Gibbs distribution fails to describe a part of the density space, due to the boundedness of its kinetic energy spectrum. We define Poincare equilibrium manifolds and compute the statistics of microcanonical excursion times off them. The tails of the distribution functions quantify the proximity of the many-body dynamics to a weakly-nonergodic phase, which occurs when the average excursion time is infinite. We find that a crossover to weakly-nonergodic dynamics takes place inside the nonGibbs phase, being unnoticed by the largest Lyapunov exponent. In the ergodic part of the non-Gibbs phase, the Gibbs distribution should be replaced by an unknown modified one. We relate our findings to the corresponding integrable limit, close to which the actions are interacting through a short range coupling network.