On chaos and anomalous diffusion in classical and quantum mechanical systems

The phenomenon of dynamically induced anomalous diffusion in both the classical and quantum kicked rotor is investigated in this Dissertation. We discuss the capability of the quantum mechanical version of the system to reproduce for extended periods the corresponding classical chaotic behavior.; Chapter 1 introduces the discussion. The objectives of the work are stated, the encountered problems are described and a sketch of the procedure adopted to solve these problems is given.; In Chapter 2 we review the procedure to obtain a diffusion equation from either a deterministically driven or noise driven system of equations. We discuss separately the cases in which normal and anomalous diffusion are obtained and show how the properties of the correlation function for the irrelevant variables ultimately determines the nature of the diffusion process.; In Chapter 3 we apply the results obtained in Chapter 2 to the study of the classical kicked rotor for both the case of normal and anomalous diffusion. The statistical properties of the diffusing variable are discussed and, in the anomalous case, the presence of strong peaks delimiting its distribution is interpreted to be a consequence of the dynamical nature of the process. We show how a study of the peaks' dynamics allows for a complete characterization of the diffusion process.; Chapter 4 deals with the quantum kicked rotor. Normal and anomalous diffusion are discussed from a statistical point of view. A procedure to rigorously define the correspondence between the classical and quantum kicked rotor is defined. The statistical features of the diffusing variable are found to be very close to the classical case, including the presence of the peaks. The numerical advantages afforded by restricting the numerical analysis to a study of the peaks are discussed. However, quantum-induced features in the peaks' decay are found in the form of a logarithmic modulation of the expected inverse-power law time decay of their population. A model interpreting those features as a consequence of quantum mechanical coherence is proposed.; In Chapter 6 we draw some conclusions and discuss the implications of the obtained results.