Melting of trapped few-particle systems

In small confined systems predictions for the melting point strongly depend on the choice of quantity and on the way it is computed, even yielding divergent and ambiguous results. We present a very simple quantity that allows us to control these problems-the variance of the block averaged interparticle distance fluctuations.     

One-particle states of a fissionable nucleus

Perturbation theory is used to derive semiclassical expressions for the one-particle energy levels in the transition state for all geometric shapes of the fissionable nucleus which are of practical interest.Translated from Izvestiya VUZ. Fizika, No. 4, pp. 52–55, April, 1970.     

One-particle theory of Ion-Channel Laser

Following the work of Chen and Dawson (1991) on Ion-Channel Laser (ICL), we study the amplification mechanism of the ICL by using the one-particle model proposed by them, and methods of secular perturbation theory. We calculate the conditions for resonance in the wave-particle interaction, and estimate the gain expected. The resonant Electro-Magnetic wave frequency appears to depend strongly on the amplitudes of oscillations of the beam particles. This may significantly reduce the amplification, and limit the use of ICL to relatively low frequencies, up to Infra-Red, where high output power (~GW) is achievable. Analytical results are confirmed by numerical simulations     

Rashba billiards

We study the energy levels of non-interacting electrons confined to move in two-dimensional billiard regions and having a spin-dependent dynamics due to a finite Rashba spin splitting. The free space Green's function for such Rashba billiards is constructed analytically and used to find the area and perimeter contributions to the density of states, as well as the corresponding smooth counting function. We show that, in contrast to systems with spin-rotational invariance, Rashba billiards always possess a negative energy spectrum. A semi-classical analysis is presented to interpret the singular behavior of the density of states at certain negative energies for circular Rashba billiards. Our detailed analysis of the spin structure of circular Rashba billiards reveals a finite out-of-plane spin projection for electron eigenstates.     

One-particle subspaces in the stochasticXY model

We study the spectrum of the generatorH of the Glauber dynamics for a model of planar rotators on a lattice in the case of a high temperature 1/. We construct two so-called one-particle subspacesH _± forH and describe the spectrum of the generator in these subspaces. As a consequence we find time asymptotics of the correlations for the Glauber dynamics.     

Inelastic gravitational billiards

We present an experimental investigation of gravitational billiards where the particle undergoes inelastic collisions with its boundary. The motion is mapped for an inelastic particle contained within parabolic, wedge, and hyperbolic boundaries. For the parabola, stable orbits are found and the wedge demonstrates a characteristic instability for its vertex angle. In the instance of the hyperbola, there are several features of the dynamics similar to the parabola at low driving and the wedge for higher driving. However, the low driving case for a hyperbola can only be completely understood by considering inelasticity effects predicted by a numerical simulation and the observation that the velocity dependent inelasticity allows the particle to sample several nearby trajectories for fixed driving.     

Quantum mushroom billiards

We report the first large-scale statistical study of very high-lying eigenmodes (quantum states) of the mushroom billiard proposed by L. A. Bunimovich [Chaos 11, 802 (2001)]. The phase space of this mixed system is unusual in that it has a single regular region and a single chaotic region, and no KAM hierarchy. We verify Percival's conjecture to high accuracy (1.7%). We propose a model for dynamical tunneling and show that it predicts well the chaotic components of predominantly regular modes. Our model explains our observed density of such superpositions dying as E(-1/3) (E is the eigenvalue). We compare eigenvalue spacing distributions against Random Matrix Theory expectations, using 16,000 odd modes (an order of magnitude more than any existing study). We outline new variants of mesh-free boundary collocation methods which enable us to achieve high accuracy and high mode numbers (approximately 10(5)) orders of magnitude faster than with competing methods.     

Rational billiards and algebraic curves

We associate an algebraic curve to a rational triangular billiard and study some of its properties.     

Stickiness in mushroom billiards

We investigate the dynamical properties of chaotic trajectories in mushroom billiards. These billiards present a well-defined simple border between a single regular region and a single chaotic component. We find that the stickiness of chaotic trajectories near the border of the regular region occurs through an infinite number of marginally unstable periodic orbits. These orbits have zero measure, thus not affecting the ergodicity of the chaotic region. Notwithstanding, they govern the main dynamical properties of the system. In particular, we show that the marginally unstable periodic orbits explain the periodicity and the power-law behavior with exponent gamma=2 observed in the distribution of recurrence times.     

Caustics for inner and outer billiards

With a plane closed convex curve,T, we associate two area preserving twist maps: the (classical) inner billiard inT and the outer billiard in the exterior ofT. The invariant circles of these twist maps correspond to certain plane curves: the inner and the outer caustics ofT. We investigate how the shape ofT determines the possible location of caustics, establish the existence of open regions which are free of caustics, and estimate fro below the size of these regions in terms of the geometry ofT.Partially supported by NSF.Partially supported by NSF Grant DMS 9017995.     

Dual polygonal billiards and necklace dynamics

We study the orbits of the dual billiard map about a polygonal table using the technique of necklace dynamics. Our main result is that for a certain class of tables, called the quasi-rational polygons, the dual billiard orbits are bounded. This implies that for the subset of rational tables (i.e. polygons with rational vertices) the dual billiard orbits are periodic.Partially supported by NSF Grant DMS 88-02643     

Chaos in composite billiards

The mechanisms and features of the chaotic behavior in billiards with ray splitting (refraction) are considered. In contrast to ordinary billiards, the law of motion in composite billiards that is coded with a sequence of ray visits to different media is shown to be deterministically chaotic. The analysis is performed in terms of a geometrical-dynamical approach in which a symmetric phase space is used instead of the ordinary Hamiltonian phase space. The chaotization elements in composite billiards of a general position are studied. The dynamics of rays in ring billiards consisting of two concentric media with different refractive indices is considered.     

Optical billiards for atoms

One of the central paradigms for classical and quantum chaos in conservative systems is the two-dimensional billiard in which particles are confined to a closed region in the plane, undergoing elastic collisions with the walls and free motion in between. We report the first realization of billiards using ultracold atoms bouncing off beams of light. These beams create the desired spatial pattern, forming an "optical billiard." We find excellent agreement between theory and our experimental demonstration of chaotic and stable motion in optical billiards, establishing a new testing ground for classical and quantum chaos.     

Quantization of fractal systems: One-particle excitation states

We consider a self-similar chain of quantum harmonic oscillators as a model of quantum field theory on a fractal supporter. In terms of this model a mass generation mechanism for one-particle excitations is proposed.     

Phase space scars and quantum billiards

A semiclassical expression is derived for the spectral Wigner function of ergodic billiards in terms of a sum over contributions from classical periodic orbits. It represents a generalization of a similar formula by Berry, which does not immediately apply to billiard systems. These results are a natural generalization of Gutzwiller's trace formula for the density of states. Our theory clarifies the origin of scars in the eigenfunctions of billiard systems. However, in its present form, it is unable to predict what states will be dominated by individual periodic orbits. Finally, we compare some of the predictions of our theory with numerical results from the stadium. Within the limitations of numerical resolution, we find agreement between the two.     

Numerical experiments on billiards

We investigate decay properties of correlation functions in a class of chaotic billiards. First we consider the statistics of Poincaré recurrences (induced by a partition of the billiard): the results are in agreement with theoretical bounds by Bunimovich, Sinai, and Bleher, and are consistent with a purely exponential decay of correlations out of marginality. We then turn to the analysis of the velocity-velocity correlation function: except for intermittent situations, the decay is purely exponential, and the decay rates scale in a simple way with the (uniform) curvature of the dispersing arcs. A power-law decay is instead observed when the system is equivalent to an infinite-horizon Lorentz gas. Comments are given on the behaviour of other types of correlation functions, whose decay, during the observed time scale, appears slower than exponential.     

Transport in polygonal billiards

We present in this work a numerical study of the dynamics of ensembles of point particles within a polygonal billiard chain. This billiard is a system with no exponential instability. Our numerical results suggest that some members of the family exhibit normal diffusive behavior while others present anomalous diffusion. Our conclusions are drawn from the numerical evaluation of the mean square displacement, the velocity autocorrelation function and its spectral analysis. Furthermore we analyze the properties of the incoherent scattering function. The super Burnett coefficient seems to be ill defined in all systems. The multifractal analysis of the spectrum of the velocity autocorrelation functions is also reported. Finally, we study the heat conduction in our polygonal chain.     

Scars in nonintegrable and rational billiards

We numerically study quantum mechanical features of the Bunimovich stadium billiard and the rational billiards which approach the former as the number of their sides increases. The statistics of energy levels and eigenfunctions of the rational billiards becomes indistinguishable from that of the Bunimovich stadium billiard below a certain energy. This fact contradicts the classical picture in which the Bunimovich stadium billiard is chaotic, but the rational billiard is pseudointegrable. It is numerically confirmed that the wave functions do not detect the fine structure, which is much smaller than the wavelength.     

Proximity-induced subgaps in andreev billiards

We examine the density of states of an Andreev billiard and show that any billiard with a finite upper cutoff in the path length distribution P(s) will possess an energy gap on the scale of the Thouless energy. An exact quantum mechanical calculation for different Andreev billiards gives good agreement with the semiclassical predictions when the energy dependent phase shift for Andreev reflections is properly taken into account. Based on this new semiclassical Bohr-Sommerfeld approximation of the density of states, we derive a simple formula for the energy gap. We show that the energy gap, in units of Thouless energy, may exceed the value predicted earlier from random matrix theory for chaotic billiards.