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Chaos and Thermalization in the one-dimensional Bose-Hubbard model in the classical-field approximation

In this thesis, we present a comprehensive study of chaos and thermalization of the one-dimensional Bose-Hubbard Model (BHM) within the classical field approximation. Two quantitative measures are compared: the ensemble-averaged Finite-time Maximal Lyapunov exponent, a measures of chaos and the normalized spectral entropy, a measure of the distance between the numerical time-averaged momentum distribution and the one predicted by thermodynamics. A threshold for chaos is found, which depends on two parameters, the nonlinearity and the total energy-per-particle. Below the threshold, the dynamics are regular, while far above the threshold, complete thermalization is observed, as measured by the normalized spectral entropy. We study individual resonances in the Bose-Hubbard model to determine the criterion for chaos. The criterion based on Chirikov's method of overlapping resonances diverges in the thermodynamic limit, in contrast to the criterion parameters inferred from numerical calculations, signifying the failure of the standard Chirikov's approach. The Ablowitz-Ladik lattice is one of several integrable models that are close to the BHM. We outline the method of Inverse Scattering Transform and generate the integrals of motion of the Ablowitz-Ladik lattice. Furthermore, we discuss the possible role of these quantities in the relaxation dynamics of the BHM.