A peculiar Maxwell’s Demon observed in a time-dependent stadium-like billiard

The dynamics of a driven stadium-like billiard is considered using the formalism of discrete mappings. The model presents a resonant velocity that depends on the rotation number around fixed points and external boundary perturbation which plays an important separation rule in the model. We show that particles exhibiting Fermi acceleration (initial velocity is above the resonant one) are scaling invariant with respect to the initial velocity and external perturbation. However, initial velocities below the resonant one lead the particles to decelerate therefore unlimited energy growth is not observed. This phenomenon may be interpreted as a specific Maxwell’s Demon which may separate fast and slow billiard particles.     

Properties of Some Chaotic Billiards with Time-Dependent Boundaries

A dispersing billiard (Lorentz gas) and focusing billiards (in the form of a stadium) with time-dependent boundaries are considered. The problem of a particle acceleration in such billiards is studied. For the Lorentz gas two cases of the time-dependence are investigated: stochastic perturbations of the boundary and its periodic oscillations. Two types of focusing billiards with periodically forced boundaries are explored: stadium with strong chaotic properties and a near-rectangle stadium. It is shown that in all cases billiard particles can reach unbounded velocities. Average velocities of the particle ensemble as functions of time and the number of collisions are obtained.     

Mechanism of Fermi acceleration in dispersing billiards with time-dependent boundaries

The paper is devoted to the problem of Fermi acceleration in Lorentz-type dispersing billiards whose boundaries depend on time in a certain way. Two cases of boundary oscillations are considered: the stochastic case, when a boundary changes following a random function, and a regular case with a boundary varied according to a harmonic law. Analytic calculations show that the Fermi acceleration takes place in such systems. The first and second moments of the velocity increment of a billiard particle, alongside the mean velocity in a particle ensemble as a function of time and number of collisions, have been investigated. Velocity distributions of particles have been obtained. Analytic and numerical calculations have been compared. Zh. éksp. Teor. Fiz. 116, 1781–1797 (November 1999)     

Fermi Acceleration in driven relativistic billiards

We show numerical experiments of driven billiards using special relativity. We have the remarkable fact that for the relativistic driven circular and annular concentric billiards, depending on initial conditions and parameters, we observe Fermi Acceleration, absent in the Newtonian case. The velocity for these cases tends to the speed of light very quickly. We find that for the annular eccentric billiard the initial velocity grows for a much longer time than the concentric annular billiard until it asymptotically reach c.     

Dynamic modeling and simulation of a real world billiard

Gravitational billiards provide an experimentally accessible arena for testing formulations of nonlinear dynamics. We present a mathematical model that captures the essential dynamics required for describing the motion of a realistic billiard for arbitrary boundaries. Simulations of the model are applied to parabolic, wedge and hyperbolic billiards that are driven sinusoidally. Direct comparisons are made between the model?s predictions and previously published experimental data. It is shown that the data can be successfully modeled with a simple set of parameters without an assumption of exotic energy dependence.     

Fermi acceleration in time-dependent rectangular billiards due to multiple passages through resonances

We consider a slowly rotating rectangular billiard with moving boundaries and use canonical perturbation theory to describe the dynamics of a billiard particle. In the process of slow evolution, certain resonance conditions can be satisfied. Correspondingly, phenomena of scattering on a resonance and capture into a resonance happen in the system. These phenomena lead to destruction of adiabatic invariance and to unlimited acceleration of the particle.     

Escape and transport for an open bouncer: Stretched exponential decays

We consider time-dependence of dynamical transport, following a recent study of the stadium billiard in which classical transmission and reflection probabilities were shown to exhibit exponential or algebraic decays depending on the choice of the lead or “hole”, raising the question of whether this feature is due to special properties of the stadium. The system considered here is much more general, having a generic mixed phase space structure, time-dependence of the dynamics, and Fermi acceleration (trajectories with unbounded velocity). We propose an efficient numerical scheme for this model, observe escape and transport effects including similar asymmetry, and also clear stretched exponential decays.     

Power law decay of correlations in a billiard problem

A billiard problem with boundary arcs that meet tangentially is studied both analytically and numerically. It is shown that the presence of tangential vertices leads to velocity correlations which decay like 1/n wheren is the number of collisions. This result contrasts with related billiard and Lorentz models where velocity correlations decay exponentially.     

A Functional Analytic Approach to Perturbations of the Lorentz Gas

We present a functional analytic framework based on the spectrum of the transfer operator to study billiard maps associated with perturbations of the periodic Lorentz gas. We show that recently constructed Banach spaces for the billiard map of the classical Lorentz gas are flexible enough to admit a wide variety of perturbations, including: movements and deformations of scatterers; billiards subject to external forces; nonelastic reflections with kicks and slips at the boundaries of the scatterers; and random perturbations comprised of these and possibly other classes of maps. The spectra and spectral projections of the transfer operators are shown to vary continuously with such perturbations so that the spectral gap enjoyed by the classical billiard persists and important limit theorems follow.     

逃逸电子不稳定性

本文分析了逃逸电子的加速运动对微观不稳定性的影响,并指出,加速效应改变了波与粒子之间的动力学耦合关系,不稳定机制不能再归结为简单的共振粒子与波的耦合效应。本文在动力学方程和特征线方程中同时引入零级电场,以这种自洽的方式讨论不稳定性问题,并与不考虑加速效应的情形作了比较。结果表明,当共振速度远离逃逸区的边界时,加速效应不导致增长率的显著改变,但当共振速度落在逃逸区的边界附近时,加速效应可能对增长率发生重要影响。 关键词：     

Quantum Monodromy in Prolate Ellipsoidal Billiards

This is the third in a series of three papers on quantum billiards with elliptic and ellipsoidal boundaries. In the present paper we show that the integrable billiard inside a prolate ellipsoid has an isolated singular point in its bifurcation diagram and, therefore, exhibits classical and quantum monodromy. We derive the monodromy matrix from the requirement of smoothness for the action variables for zero angular momentum. The smoothing procedure is illustrated in terms of energy surfaces in action space including the corresponding smooth frequency map. The spectrum of the quantum billiard is computed numerically and the expected change in the basis of the lattice of quantum states is found. The monodromy is already present in the corresponding two-dimensional billiard map. However, the full three degrees of freedom billiard is considered as the system of greater relevance to physics. Therefore, the monodromy is discussed as a truly three-dimensional effect.     

Linearly stable orbits in 3 dimensional billiards

We construct linearly stable periodic orbits in a class of billiard systems in 3 dimensional domains with boundaries containing semispheres arbitrarily far apart. It shows that the results about planar billiard systems in domains with convex boundaries which have nonvanishing Lyapunov exponents cannot be easily extended to 3 dimensions.Supported in part by the NSF Grant DMS-8807077 and the Sloan Foundation     

The Husimi Distribution of Circular Billiard with?an?Applied?Uniform Magnetic Field

We present a theoretical computation of the Husimi distribution function in phase-space for studying the semiclassical dynamics of the circular electron billiard subjected to a constant magnetic field in the perpendicular direction. The results reveal that with the increase of the applied magnetic field the peaks of Husimi function tend to the billiard boundaries, along with the movements a periodic splitting-recombining (alternative single-double) peak structure is arisen. This fact implies the localization of the eigenstates and coincides to the classical trajectory distribution what we obtained by use of representation on the billiard boundary. It becomes possible to compare the local properties of the quantum and classical distributions. Our analysis provides a new perspective to understand the quantum-classical correspondence.     

Hyperacceleration in a stochastic Fermi-Ulam model

Fermi acceleration in a Fermi-Ulam model, consisting of an ensemble of particles bouncing between two, infinitely heavy, stochastically oscillating hard walls, is investigated. It is shown that the widely used approximation, neglecting the displacement of the walls (static wall approximation), leads to a systematic underestimation of particle acceleration. An improved approximative map is introduced, which takes into account the effect of the wall displacement, and in addition allows the analytical estimation of the long term behavior of the particle mean velocity as well as the corresponding probability distribution, in complete agreement with the numerical results of the exact dynamics. This effect accounting for the increased particle acceleration--Fermi hyperacceleration--is also present in higher-dimensional systems, such as the driven Lorentz gas.     

Observation of chaotic and regular dynamics in atom-optics billiards

We report on experimental observations of chaotic and regular motion of ultracold atoms confined by a billiard-shaped optical dipole potential induced by a rapidly scanning laser beam. To investigate the dynamics of the atoms confined by such an "atom-optics" billiard we measure the decay of the number of trapped atoms through a hole on the boundary. A fast and purely exponential decay, the clear signature of chaotic motion, is found for a stadium billiard, but not for a circular or an elliptical billiard, in agreement with theory. We also investigated the effects of decoherence, velocity spread, and gravity on regular and chaotic motion.     

Spectral Gap for a Class of Random Billiards

A random billiard is a random dynamical system similar to an ordinary billiard system except that the standard specular reflection law is replaced with a more general stochastic operator specifying the post-collision distribution of velocities for any given pre-collision velocity. We consider such collision operators for certain random billiards that we call billiards with microstructure. Collisions modeled by these operators can still be thought of as elastic and time reversible. The operators are canonically determined by a second (deterministic) billiard system that models “microscopic roughness” on the billiard table boundary. Our main purpose here is to develop some general tools for the analysis of the collision operator of such random billiards. Among the main results, we give geometric conditions for these operators to be Hilbert-Schmidt and relate their spectrum and speed of convergence to stationary Markov chains with geometric features of the microscopic billiard structure. The relationship between spectral gap and the shape of the microstructure is illustrated with several simple examples.     

Experiments on quantum chaos using microwave cavities: Results for the pseudo-integrable L-billiard

We describe microwave experiments used to study billiard geometries as model problems of non-integrability in quantum or wave mechanics. The experiments can study arbitrary 2-D geometries, including chaotic and even disordered billiards. Detailed results on an L-shaped pseudo-integrable billiard are discussed as an example. The eigenvalue statistics are well-described by empirical formulae incorporating the fraction of phase space that is non-integrable. The eigenfunctions are directly measured, and their statistical properties are shown to be influenced by non-isolated periodic orbits, similar to that for the chaotic Sinai billiard. These periodic orbits are directly observed in the Fourier transform of the eigenvalue spectrum.     

Time-Domain Mapping of Electromagnetic Ray Movement Inside 2D Anisotropic Region

The paper presents the extension of the time-domain mapping method applied to 2D billiard problem inside an anisotropic region bounded by ellipse [1]. In this paper, it has been considered the ray movement inside 2D anisotropic region bounded by arbitrary differentiable curve. It has been proved that the problem can be one-to-one mapped onto 2D mathemeatical billiard problem inside the region possessing isotropic properties by linear transformation of group velocity hodograph and boundary with the same coefficient, which is equal to anisotropy of the ray group velocity, simultaneously. The main features of the ray movement inside 2D anistropic region are discussed.     

Resonance features of the conductance of open billiards: Spin-orbit coupling effects

Transport properties of an open circular billiard with attached channels in the presence of the spin-orbit coupling are studied. It is shown that the inclusion of the spin-orbit coupling causes the occurrence of additional narrow Fano resonances on the energy dependence of the conductance. It is established that the resonance regions are associated with the energy levels in a corresponding closed billiard. The effect of small finite temperatures on the observability of the conductance resonances is estimated.     

Particle Dynamics in Time-Dependent Stadium-Like Billiards

Billiards in the form of a stadium with perturbed boundaries are considered. Investigations are primarily devoted to billiards having a near-rectangle form, but the results regarding the classical stadium with the boundary that consists of two semicircles and two parallel segments tangent to them, are also described. In the phase plane, areas corresponding to decrease and increase of the velocity of billiard particles are found. The average velocity of the particle ensemble as a function of the number of collisions with the boundary is obtained.