A numerical exploration of the statistical behavior of the discretized nonlinear Schroedinger equation
In this dissertation, we consider the equilibrium as well as near-equilibrium statistical behavior of the discretized nonlinear Schrödinger equation (NLS). We create a modified version of the Metropolis algorithm for generating empirical distributions that approximate the mixed ensemble Gibbs distribution for the NLS. The mixed ensemble is canonical in energy and microcanonical in particle number invariant. After generating and analyzing many such empirical distributions spanning a full range of equilibrium behaviors, we study their near-equilibrium responses to perturbations via linear response theory. This leads us to the discovery of a regime in which near-equilibrium ensembles resist relaxation toward equilibrium when evolved under the NLS dynamics. Within this regime, perturbed mean observables relax in two stages; they undergo a rapid disruption followed by an extremely slow equilibration. In some cases of the latter stage, there is no observable rate of decay towards equilibrium. We propose that quasiperiodicity of individual solutions may be the dynamical mechanism that underlies this two stage behavior. We exhibit a direct correspondence between the two stage regime and the regime within which quasiperiodicity prevails.