Classical chaos onset

Analysis of this 7feff using the techniques of nonlinear classical dynamics reveals the structure of phase space (mapped as a continuous function of the conserved quantities E, Ka, and Kb) and the qualitative nature of the classical trajectory that corresponds to every eigenstate in every polyad. This analysis reveals qualitative changes, or bifurcations, in the dynamics, the onset of classical chaos, and the fraction of phase space associated with each qualitatively distinct class of regular (quasiperiodic) and chaotic trajectories. 

Ray Kapral came to Toronto from the United States in 1969. His research interests center on theories of rate processes both in systems close to equilibrium, where the goal is the development of a microscopic theory of condensed phase reaction rates,89 and in systems far from chemical equilibrium, where descriptions of the complex spatial and temporal reactive dynamics that these systems exhibit have been developed.90 He and his collaborators have carried out research on the dynamics of phase transitions and critical phenomena, the dynamics of colloidal suspensions, the kinetic theory of chemical reactions in liquids, nonequilibrium statistical mechanics of liquids and mode coupling theory, mechanisms for the onset of chaos in nonlinear dynamical systems, the stochastic theory of chemical rate processes, studies of pattern formation in chemically reacting systems, and the development of molecular dynamics simulation methods for activated chemical rate processes. His recent research activities center on the theory of quantum and classical rate processes in the condensed phase91 and in clusters, and studies of chemical waves and patterns in reacting systems at both the macroscopic and mesoscopic levels. 

The plan of Chapter 5 is the following. In order to get a feeUng for the dynamics of the kicked molecule, we approximate it by a one-dimensional schematic model by restricting its motion to rotation in the x, z) plane and ignoring motion of the centre of mass. In this approximation the kicked molecule becomes the kicked rotor, probably the most widely studied model in quantum chaology. This model was introduced by Casati et al. in 1979. The classical mechanics of the kicked rotor is discussed in Section 5.1. Section 5.2 presents Chirikov s overlap criterion, which can be applied generally to estimate analytically the critical control parameter necessary for the onset of chaos. We use it here to estimate the onset of chaos in the kicked rotor model. The quantum mechanics of the kicked rotor is discussed in Section 5.3. In Section 5.4 we show that the results obtained for the quantum kicked rotor model are of immediate... 

Before we conclude this section we have to ask an important question Is it possible to observe experimentally the onset of global chaos predicted to occur at K = Kc On the classical level the answer to this... 

Figure 1. Computed13 discrete stick spectrum (a) and the fit by (2.17) (solid line) to the distribution of fluctuations in that spectrum, (b). The initial state is at an energy (indicated by an arrow in (a)) above the onset of large-scale chaos in the classical dynamics of the Hamiltonian. The accessed final states span a wide range of energies about , and there are no very strong or many very weak transitions.

---