Chaotic Billiards Research Articles

A Strong Pair Correlation Bound Implies the CLT for Sinai Billiards

We investigate the possibility of proving the Central Limit Theorem (CLT) for Dynamical Systems using only information on pair correlations. A strong bound on multiple correlations is known to imply the CLT (Chernov and Markarian in Chaotic Billiards, 2006). In Chernov’s paper (J. Stat. Phys. 122(6), 2006), such a bound is derived for dynamically Hölder continuous observables of dispersing Billiards. Here we weaken the regularity assumption and subsequently show that the bound on multiple correlations follows directly from the bound on pair correlations. Thus, a strong bound on pair correlations alone implies the CLT, for a wider class of observables. The result is extended to Anosov diffeomorphisms in any dimension. Some non-invertible maps are also considered.

Quantum chaotic scattering in microwave resonators

In a frequency range where a microwave resonator simulates a chaotic quantum billiard, we have measured moduli and phases of reflection and transmission amplitudes in the regimes of both isolated and of weakly overlapping resonances and for resonators with and without time-reversal invariance. Statistical measures for S -matrix fluctuations were determined from the data and compared with extant and/or newly derived theoretical results obtained from the random-matrix approach to quantum chaotic scattering. The latter contained a small number of fit parameters. The large data sets taken made it possible to test the theoretical expressions with unprecedented accuracy. The theory is confirmed by both a goodness-of-fit-test and the agreement of predicted values for those statistical measures that were not used for the fits, with the data.

Time-dependent focusing billiards and macroscopic realization of Maxwell's Demon

The Fermi acceleration is always inherent in completely chaotic time-dependent billiards. At the same time, the particle dynamics in nearly integrable billiard systems can be more complex. Using a simplified approach, we investigate time-dependent stadium-like billiards and show that at a certain particle velocity, Vr, a resonance between external periodical perturbations and the motion within stability islands of the unperturbed billiard can be observed. This resonance suppresses the Fermi acceleration of particles with velocities less than Vr. As a result, we observe a separation of billiard particles by their velocities. If V0 < Vr, the average particle velocity decreases, while the particles with V0 > Vr are on average accelerated. At the initial velocity V0 = Vr we observe a phase transition in the velocity distribution of particles: if V0 < Vr then the distribution approaches a stationary one, whereas for V0 > Vr the distribution is non-stationary and spreads toward the higher velocities. This phenomenon may be treated as a peculiar billiard Maxwell's Demon, when weak perturbations of a system lead the particle ensemble to separation. In other nearly integrable billiard systems similar resonances can lead to differences in the acceleration of particles with velocities smaller or larger than a resonance value.

Universality in level spacing fluctuations of a chaotic optical billiard

We study chaotic behavior of a classical optical stadium billiard model. We construct a matrix of time-of-travel along trajectories corresponding to a set of boundary points. We carry out a level spacing fluctuation analysis and compute the Dyson–Mehta spectral rigidity. The distribution of time-of-travel is approximately described by a Gaussian. The results for level spacing distribution and spectral rigidity show universal behavior.

Billiards in magnetic fields: A molecular dynamics approach

We present a computational scheme based on classical molecular dynamics to study chaotic billiards in static external magnetic fields. The method allows us to treat arbitrary geometries and several interacting particles. We test the scheme for rectangular single-particle billiards in magnetic fields and find a sequence of regularity islands at integer aspect ratios. In the case of two Coulomb-interacting particles the dynamics is dominated by chaotic behavior. However, signatures of quasiperiodicity can be identified at weak interactions, as well as regular trajectories at strong magnetic fields. Our scheme provides a promising tool to monitor the classical limit of many-electron semiconductor nanostructures and transport systems up to high magnetic fields.

Scattering Properties of Chaotic Microwave Billiards

Reflection and transmission measurements in microwave billiards with attached antennas allow the determination of all components of the scattering matrix including their phases. This is an extraordinary situation, since in usual scattering experiments in nuclear or atomic physics only reduced information such as the scattering cross-section can be obtained where the phase information is completely lost. This allows experimental tests of theoretical predictions of scattering theory inaccessible by any other method. As an example the distribution of reflection coefficients and of the line widths in a chaotic microwave billiard are discussed.

Focusing Components in Typical Chaotic Billiards Should be Absolutely Focusing

We demonstrate that the defocusing mechanism fails to work if not all focusing components of the boundary are absolutely focusing. More precisely, we construct billiard tables with arbitrary long free path away from a non-absolutely focusing component such that a nonlinearly stable periodic orbit exists. Therefore the only known standard procedure of constructing chaotic ergodic billiards works in general only if all focusing boundary components are absolutely focusing.

Avoided-level-crossing statistics in open chaotic billiards

We investigate a two-level model with a large number of open decay channels in order to describe avoided level crossing statistics in open chaotic billiards. This model allows us to describe the fundamental changes in the probability distribution of the avoided level crossings compared with the closed case. Explicit expressions are derived for systems with preserved and broken time-reversal symmetry. We find that the decay process induces a modification at small spacings of the probability distribution of the avoided level crossings due to an attraction of the resonances. The theoretical predictions are in complete agreement with the recent experimental results of Dietz [Phys. Rev. E 73, 035201 (2006)].

Non-Gaussian Curvature Spikes from Chaotic Billiards in Inflation Preheating

A new class of non-Gaussian curvature fluctuations zeta(pr)(x) identical with deltaN(chi(i)) arises from the postinflation preheating behavior of a noninflaton field chi(i). Its billiardlike chaotic dynamics imprints regular log-spaced narrow spikes in the number of preheating e-folds N(chi(i)). We perform highly accurate lattice simulations of supersymmetry-inspired quartic inflaton and coupling potentials in a separate-universe approximation to compute N(chi(i)) as a function of the (nearly homogeneous) initial condition chi(i). The superhorizon modes of chi(i)(x) result in positive spiky excursions in zeta(pr) and hence negative gravitational potential fluctuations added to the usual sign-independent inflaton-induced perturbations, observably manifested in large cosmic structures and as (polarized) temperature cosmic microwave background cold spots.

Semiclassical theory of nonlocal statistical measures: Residual Coulomb interactions

In a recent paper [Phys. Rev. Lett. 100, 164101 (2008)] and within the context of quantized chaotic billiards, random plane-wave and semiclassical theoretical approaches were applied to an example of a relatively new class of statistical measures, i.e., measures involving both complete spatial integration and energy summation as essential ingredients. A quintessential example comes from the desire to understand the short-range approximation to the first-order ground-state contribution of the residual Coulomb interaction. Billiards, fully chaotic or otherwise, provide an ideal class of systems on which to focus as they have proven to be successful in modeling the single-particle properties of a Landau-Fermi liquid in typical mesoscopic systems, i.e., closed or nearly closed quantum dots. It happens that both theoretical approaches give fully consistent results for measure averages, but that somewhat surprisingly for fully chaotic systems the semiclassical theory gives a much improved approximation for the fluctuations. Comparison of the theories highlights a couple of key shortcomings inherent in the random plane-wave approach. This paper contains a complete account of the theoretical approaches, elucidates the two shortcomings of the oft-relied-upon random plane-wave approach, and treats non-fully-chaotic systems as well.

Long-time saturation of the Loschmidt echo in quantum chaotic billiards

The Loschmidt echo (LE) (or fidelity) quantifies the sensitivity of the time evolution of a quantum system with respect to a perturbation of the Hamiltonian. In a typical chaotic system the LE has been previously argued to exhibit a long-time saturation at a value inversely proportional to the effective size of the Hilbert space of the system. However, until now no quantitative results have been known and, in particular, no explicit expression for the proportionality constant has been proposed. In this paper we perform a quantitative analysis of the phenomenon of the LE saturation and provide the analytical expression for its long-time saturation value for a semiclassical particle in a two-dimensional chaotic billiard. We further perform extensive (fully quantum mechanical) numerical calculations of the LE saturation value and find the numerical results to support the semiclassical theory.

Erratum: Quantum stress in chaotic billiards [Phys. Rev. E77, 066209 (2008)

Received 25 November 2008DOI:https://doi.org/10.1103/PhysRevE.79.019901©2009 American Physical Society

Book Review: Chaotic billiards

Book Review: Chaotic billiards

Scar-Like States in Dynamical Electron-Wavepackets in Chaotic Billiard

The time-evolution of the wavepacket inside chaotic and integrable two-dimensional (2D) nanostructures is numerically studied. We have found the enhancement around the classical periodic orbits during the time-evolution in the stadium billiard. It is similar to the scars in the standing wave of the chaotic billiards. The initial position and velocity, and the shape of the wavepacket are crucial for the enhancement, but we can observe that the remnant of the initial wavepacket travels along the unstable periodic orbit. Then the wavepacket gradually diffuses around the structure. This behavior has close relation to the dynamical properties of electrons in the structure, e.g., the conductivity, the magneto-resistance etc. The quantum fidelity, which can measure the robustness of dynamical states inside the nanostructures, is also discussed. [DOI: 10.1380/ejssnt.2009.721]

Directed transport in classical and quantum chaotic billiards

We construct an autonomous chaotic Hamiltonian ratchet as a channel billiard subdivided by equidistant walls attached perpendicularly to one side of the channel, leaving an opening on the opposite side. A static homogeneous magnetic field penetrating the billiard breaks time-reversal invariance and renders the classical motion partially chaotic. We show that the classical dynamics exhibits directed transport, owing to the asymmetric distribution of regular regions in phase space. The billiard is quantized by a numerical method based on a finite-element algorithm combined with the Landau gauge and the Bloch formalism for periodic potentials. We discuss features of the billiard eigenstates such as node lines and vortices in the probability flow. Evidence for directed quantum transport, inherited from the corresponding features of the classical dynamics, is presented in terms of level-velocity statistics.

Wavefunctional analysis of electron transfers in 2D nanostructures under external fields

AbstractThe time‐evolution of the wave packet inside chaotic and integrable two‐dimensional nanostructures is numerically studied. Recently the time‐evolution of quantum states is seriously considered to treat the dynamical property of nanostructures. By computer simulation, we have found the enhancement around the classical periodic orbits, which has been theoretically predicted. It is similar to the scars in the standing wave of the chaotic billiards and is also beyond the naive prediction of the random nodal patterns from the chaotic nanostructure. The initial position and velocity and the shape of the wave packet are crucial for the enhancement, however, we can observe that the remnant of the initial wave packet travels along the unstable periodic orbit. Then the wave packet gradually diffuses around the structure. This behavior has close relation to the dynamical properties of electrons in the structure, e.g., the conductivity, the magneto‐resistance etc. Copyright © 2008 John Wiley &amp; Sons, Ltd.

Heat Conductivity from Molecular Chaos Hypothesis in Locally Confined Billiard Systems

We study the transport properties of a large class of locally confined Hamiltonian systems, in which neighboring particles interact through hard-core elastic collisions. When these collisions become rare and the systems large, we derive a Boltzmann-like equation for the evolution of the probability densities. We solve this equation in the linear regime and compute the heat conductivity from a Green-Kubo formula. The validity of our approach is demonstrated by comparing our predictions with the results of numerical simulations performed on a new class of high-dimensional defocusing chaotic billiards.

Diamagnetic persistent currents for electrons in ballistic billiards subject to a point flux

We study the persistent current of noninteracting electrons subject to a pointlike magnetic flux in the simply connected chaotic Robnik-Berry quantum billiard and also in an annular analog thereof. For the simply connected billiard, we find a large diamagnetic contribution to the persistent current at small flux, which is independent of the flux and is proportional to the number of electrons (or equivalently the density since we keep the area fixed). The size of this diamagnetic contribution is much larger than mesoscopic fluctuations in the persistent current in the simply connected billiard. Moreover, it can ultimately be traced to the response of the angular-momentum $l=0$ levels (neglected in semiclassical expansions) on the unit disk to a pointlike flux at its center. The same behavior is observed for the annular billiard when the inner radius is much smaller than the outer one. The usual fluctuating persistent current and Anderson-like localization due to boundary scattering are seen when the annulus tends to a one-dimensional ring. We explore the conditions for the observability of this phenomenon.

Experimental Investigation of Energy Level Dynamics in Aluminium Blocks with Temperature Distribution

Using temperature distribution as an external parameter to change symmetry and measuring frequency spectrum of acoustic resonances in aluminium blocks, we investigate the statistics and dynamics of energy levels in the chaotic billiards. To extract the resonances accurately and eliminate the influence of noise, a filter-diagonalization method for harmonic inversion is used to overcome low resolution of conventional fast Fourier transform method for low quantity factor resonance systems. We present an improved and feasible simulation method to study chaotic characteristic of quantum systems experimentally.

Quantum stress in chaotic billiards

This paper reports on a joint theoretical and experimental study of the Pauli quantum-mechanical stress tensor T_{alphabeta}(x,y) for open two-dimensional chaotic billiards. In the case of a finite current flow through the system the interior wave function is expressed as psi=u+iv . With the assumption that u and v are Gaussian random fields we derive analytic expressions for the statistical distributions for the quantum stress tensor components T_{alphabeta} . The Gaussian random field model is tested for a Sinai billiard with two opposite leads by analyzing the scattering wave functions obtained numerically from the corresponding Schrödinger equation. Two-dimensional quantum billiards may be emulated from planar microwave analogs. Hence we report on microwave measurements for an open two-dimensional cavity and how the quantum stress tensor analog is extracted from the recorded electric field. The agreement with the theoretical predictions for the distributions for T_{alphabeta}(x,y) is quite satisfactory for small net currents. However, a distinct difference between experiments and theory is observed at higher net flow, which could be explained using a Gaussian random field, where the net current was taken into account by an additional plane wave with a preferential direction and amplitude.