Ergodicity-breaking in Discrete Nonlinear Schrödinger Equation

Statistical mechanics of the discrete nonlinear Schrödinger equation shows a phase transition in the energy and norm density parameter space and the phase transition is characterized by the formation of long-lived excitations in the non-Gibbsian regime. It is believed that this phase transition line, seperates Gibbsian and non-Gibbsian regimes, is connected with the non-ergodicity. We address this problem by following a trajectory in the phase space and recording the instant the trajectory pierces an ergodic Poincare section. The probability distribution of the time intervals between two following piercings (excursion times) is analaysed to calculate the exponent. Our analysis shows that the phase transition line is not the actual one which separates the ergodic to non-ergodic transition in the dynamics.