Regular Series


Vol. 23 (1992), No. 3, pp. 177 – 296


A Simple Model System for Irregular Scattering

abstract

In this lecture the basic ideas of classical scattering chaos axe demonstrated with the help of a simple model. The Cantor set of singularities in the deflection function, the invariant set in the phase space and the chaotic structure in the cross section axe presented. Within the semiclassical approximation for the quantum cross section it is shown in which way some signs of the classical chaos can show up in the quantum system.


Random Anisotropy Magnets

abstract

A review of theoretical, experimental and computational work on Random Anisotropy Magnetic systems is given. Monte Carlo simulations of two dimensional spins on a two dimensional lattice in the presence of random anisotropy fields have been carried out. In the absence of randomness this is the familiar \(XY\) model in two dimensions, which has a low temperature phase with algebraic order; this is the well-known Kosterlitz–Thouless phase. In the presence of random anisotropy evidence is presented that there are three phases: a low temperature orientational glass phase, an intermediate temperature Kosterlitz–Thouless phase, and a high temperature disordered paramagnetic phase.


Classical and Quantum Billiards: Integrable, Nonintegrable and Pseudo-Integrable

abstract

Statistical properties of the spectra of quantum two dimensional billiards are shown to be linked to the nature of the dynamics of the corresponding classical systems. Quantised pseudo-integrable billiard exhibits level repulsion, in spite of non chaotic dynamics of its classical counterpart. We conjecture that the level statistics of a quantum pseudo-integrable system depends on the genus of the invariant manifold equivalent to its classical phase space. A model of billiards with finite walls suitable to investigate the problems of chaotic scattering is proposed.


The Coulomb Potential or Scattering Chaos in a Non-Generic Case

abstract

Among the various systems that are analysed in terms of Hamiltonian chaos, the Coulomb potential perturbed by an external electro-magnetic field is of special importance in microscopic physics. We present here a view of the group-theoretical techniques that have been used in order to reduce the non-integrable dynamics to its smallest possible extent. Chaotic diffusion in a driven Coulomb potential is analysed and results presented: fixed points, periodic orbits, deflection functions, inelastic cross-sections. The essentially mixed character of phase space is underlined. As a foreseeable consequence, the deflection function does not display a full self-similar structure after a few generations in the fractal structure.


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