Sinai Billiard Research Articles

Intersections of energy surfaces in quantized Sinai billiards

Degeneracies in the spectra of quantized Sinai billiards are numerically investigated. Two effects recently predicted are observed, namely, (i) a geometric phase of nπ during a circuit surrounding n conical intersections and (ii) neighboring glancing and conical intersections. Experiments in isomorphic flat microwave resonators exhibit excellent agreement with computed eigenvalues and eigenfunctions.

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.

Ergodic billiards that are not quantum unique ergodic

Partially rectangular domains are compact two-dimensional Riemannian manifolds $X$, either closed or with boundary, that contain a flat rectangle or cylinder. In this paper we are interested in partially rectangular domains with ergodic billiard flow; examples are the Bunimovich stadium, the Sinai billiard or Donnelly surfaces. We consider a one-parameter family $X_t$ of such domains parametrized by the aspect ratio $t$ of their rectangular part. There is convincing theoretical and numerical evidence that the Laplacian on $X_t$ with Dirichlet or Neumann boundary conditions is not quantum unique ergodic (QUE). We prove that this is true for all $t \in [1,2]$ excluding, possibly, a set of Lebesgue measure zero. This yields the first examples of ergodic billiard systems proven to be non-QUE.

Real cone contractions and analyticity properties of the characteristic exponents

In this paper, we use the Hilbert projective metric and Birkhoff's theorem to prove a generalization of a result of Ruelle. We consider a random product of operators preserving a family of proper closed convex cones. We show that, without any topological assumption, the characteristic exponent depends real analytically on the data of the problem. As an example, we apply this result to cocycles in Sinai billiards: the characteristic exponent behaves real analytically under small perturbations of the cocycle in an appropriate Banach space.

Universality of level spacing distributions in classical chaos

We suggest that random matrix theory applied to a matrix of lengths of classical trajectories can be used in classical billiards to distinguish chaotic from non-chaotic behavior. We consider in 2D the integrable circular and rectangular billiard, the chaotic cardioid, Sinai and stadium billiard as well as mixed billiards from the Limaçon/Robnik family. From the spectrum of the length matrix we compute the level spacing distribution, the spectral auto-correlation and spectral rigidity. We observe non-generic (Dirac comb) behavior in the integrable case and Wignerian behavior in the chaotic case. For the Robnik billiard close to the circle the distribution approaches a Poissonian distribution. The length matrix elements of chaotic billiards display approximate GOE behavior. Our findings provide evidence for universality of level fluctuations—known from quantum chaos—to hold also in classical physics.

Stability in High Dimensional Steep Repelling Potentials

The appearance of elliptic periodic orbits in families of n-dimensional smooth repelling billiard-like potentials that are arbitrarily steep and limit to Sinai billiards is established for any finite n. For typical potentials, the stability regions in the parameter space scale as a power-law in 1/n and in the steepness parameter. Thus, it is shown that even though these systems have a uniformly hyperbolic (albeit singular) limit, the ergodicity of this limit system is destroyed in the more realistic smooth setting. The considered example is highly symmetric and is not directly linked to the smooth many particle problem. Nonetheless, the possibility of explicitly constructing stable motion in smooth n degrees of freedom systems limiting to strictly dispersing billiards is now established.

Sinai Billiards under Small External Forces II

Consider a particle moving freely on the torus and colliding elastically with some fixed convex bodies. This model is called a periodic Lorentz gas, or a Sinai billiard. It is a Hamiltonian system with a smooth invariant measure, whose ergodic and statistical properties have been well investigated. Now let the particle be subjected to a small external force. This new system is not likely to have a smooth invariant measure. Then a Sinai-Ruelle-Bowen (SRB) measure describes the evolution of typical phase trajectories. We find general sufficient conditions on the external force under which the SRB measure for the collision map exists, is unique, and enjoys good ergodic and statistical properties, including Bernoulliness and an exponential decay of correlations.

A Stretched Exponential Bound on Time Correlations for Billiard Flows

We construct Markov approximations to the billiard flows and establish a stretched exponential bound on time-correlation functions for planar periodic Lorentz gases (also known as Sinai billiards). Precisely, we show that for any (generalized) Holder continuous functions F,G on the phase space of the flow the time correlation function is bounded by const⋅ \(e^{-a\sqrt{|t|}}\), here t ∊ ℝ is the (continuous) time and a > 0.

The Distribution of the Free Path Lengths in the Periodic Two-Dimensional Lorentz Gas in the Small-Scatterer Limit

We study the free path length and the geometric free path length in the model of the periodic two-dimensional Lorentz gas (Sinai billiard). We give a complete and rigorous proof for the existence of their distributions in the small-scatterer limit and explicitly compute them. As a corollary one gets a complete proof for the existence of the constant term $$c=2-3\ln 2+\frac{27\zeta(3)}{2\pi^2}$$ in the asymptotic formula $$h(T)=-2 \ln \epsilon +c+o(1)$$ of the KS entropy of the billiard map in this model, as conjectured by P. Dahlqvist.

Nonergodicity of the motion in three-dimensional steep repelling dispersing potentials

It is demonstrated numerically that smooth three degrees of freedom Hamiltonian systems that are arbitrarily close to three-dimensional strictly dispersing billiards (Sinai billiards) have islands of effective stability, and hence are nonergodic. The mechanism for creating the islands is corners of the billiards domain.

Eigenfunctions for Partially Rectangular Billiards

In this note, we further develop the methods of Burq and Zworski (2005) to study eigenfunctions for billiards which have rectangular components: these include the Bunimovich billiard, the Sinai billiard, and the recently popular pseudointegrable billiards (Bogomolny et al., 1999). The results are an application of a “black-box” point of view as presented in Burq and Zworski (2004).

Power spectrum analysis of experimental Sinai quantum billiards

We perform a power spectrum analysis of energy level fluctuations in experimental quantum Sinai billiards. In spite of experimental limitations we find the 1 / f behavior expected for quantum chaotic systems. Moreover, the deviations observed at low frequencies are easily explained within the semiclassical periodic orbit theory. In conclusion, it is shown that the power spectrum analysis of energy level fluctuations is a robust signature of quantum chaos.

Advanced Statistical Properties of Dispersing Billiards

A new approach to statistical properties of hyperbolic dynamical systems emerged recently; it was introduced by L.-S. Young and modified by D. Dolgopyat. It is based on coupling method borrowed from probability theory. We apply it here to one of the most physically interesting models—Sinai billiards. It allows us to derive a series of new results, as well as make significant improvements in the existing results. First we establish sharp bounds on correlations (including multiple correlations). Then we use our correlation bounds to obtain the central limit theorem (CLT), the almost sure invariance principle (ASIP), the law of iterated logarithms, and integral tests.

Rate of convergence in the multidimensional central limit theorem for stationary processes. Application to the Knudsen gas and to the Sinai billiard

We show how Rio's method [Probab. Theory Related Fields 104 (1996) 255--282] can be adapted to establish a rate of convergence in ${\frac{1}{\sqrt{n}}}$ in the multidimensional central limit theorem for some stationary processes in the sense of the Kantorovich metric. We give two applications of this general result: in the case of the Knudsen gas and in the case of the Sinai billiard.

Physical insight into superdiffusive dynamics of Sinai billiard through collision statistics

We report on distinct steady-motion dynamic regimes in chaotic Sinai billiard (SB). A numerical study on elastic reflections from the SB boundary (square wall of length L and circle obstacle of radius R) is carried out for different R / L . The research is based on the exploration of the generalized diffusion equation and on the analysis of wall-collision and the circle-collision distributions observed at late times. The asymptotes for the diffusion coefficient D R and the corresponding diffusion exponent z R are established for all geometries. The universal ( R-independent) diffusion with D 1 ∽ t 1 / 3 and z 1 = 1.5 replaces the ballistic motion regime ( z 0 = 1 ) attributed to square billiard ( R = 0 ). Geometrically, this superdiffusive regime is bounded by small radii 0 < R < R 1 ( R 1 / L = 2 4 ), when both diagonal and non-diagonal Bleher's corridors are open in the correspondent square lattice of Lorentz gas (LG) model. The relaxation dynamics observed is ensured by the universal diffusive propagation of the regular-type and the bouncing-ball-type orbits. Within the random walk scheme, this superdiffusive regime is due to the Lévy flights between the long-distant scatterers. With the increase in circle radius ( R 1 ⩽ R < R 2 , R 2 = L / 2 ), the diagonal corridor closes, but the arc-touching effects continue to bring the long-living bouncing-ball orbits back to the non-diagonal infinite corridors. This transient non-universal dynamics with 1.5 < z R < 2 , also associated with the trapping of regular orbits, seems to be characteristic of non-fully hyperbolic billiard systems. In SB with finite horizon ( R ⩾ R 2 ), all the principal corridors are closed and the interplay between square and circle boundaries generates the known chaotic dynamics, attributed to the fully hyperbolic systems. This is also observed through the normal Brownian diffusion ( z 2 = 2 ) and the Gaussian statistics, proved for both kinds of collisions.

Current statistics for wave transmission through an open Sinai billiard: effects of net currents.

Transport through quantum and microwave cavities is studied by analytic and numerical techniques. In particular, we consider the statistics for a finite net probability current (Poynting vector) <j> flowing through an open ballistic Sinai billiard to which two opposite leads/wave guides are attached. We show that if the net probability current is small, the scattering wave function inside the billiard is well approximated by a Gaussian random complex field. In this case, the current statistics are universal and obey simple analytic forms. For larger net currents, these forms still apply over several orders of magnitudes. However, small characteristic deviations appear in the tail regions. Although the focus is on electron and microwave billiards, the analysis is relevant also to other classical wave cavities as, for example, open planar acoustic reverberation rooms, elastic membranes, and water surface waves in irregularly shaped vessels.

Reaction matrix for Dirichlet billiards with attached waveguides

The reaction matrix of a cavity with attached waveguides connects scattering properties to properties of a corresponding closed billiard for which the waveguides are cut off by straight walls. On the one hand, this matrix is directly related to the S-matrix, on the other hand it can be expressed by a spectral sum over all eigenfunctions of the closed system. However, in the physically relevant situation where these eigenfunctions vanish on the impenetrable boundaries of the closed billiard, the spectral sum for the reaction matrix, as it was used before, fails to converge and does not reliably reproduce the scattering properties. We derive here a convergent representation of the reaction matrix in terms of eigenmodes satisfying Dirichlet boundary conditions and demonstrate its validity in the rectangular and the Sinai billiards.

Thermopower of a multiprobe ballistic conductor

The thermopower of a multiprobe ballistic conductor in the form of caterpillarlike Sinai billiard is experimentally investigated. The magnetic-field dependence of both longitudinal thermopower and NernstEttingshausen effect exhibits commensurability oscillations, which are more pronounced than the corresponding oscillations in the magnetoresistance. Results of computer calculations based on the generalized Landauer

Quantum transport in lattices of coupled electronic billiards

A new system with dynamic chaos—2D lattice of single Sinai billiards coupled through quantum dots—is studied experimentally. Localization in such a system was found to be substantially suppressed, because the characteristic size of the billiard for g≤1 (g is conductance measured in e2/h units) is the localization length rather than the de Broglie wavelength of an electron, as in the usual 2D electron system. Lattice ballistic effects (commensurate peaks in the magnetoresistance) for g≪1, as well as extremely large magnetoresistance caused by the interference in chaotic electron trajectories, were found. Thus, this system is shown to be characterized by simultaneous existence of effects that are inherent in order (commensurate peaks of magnetoresistance), disorder (percolation charge transport), and chaos (weak localization in chaotic electron trajectories).

Crossover from regular to irregular behavior in current flow through open billiards.

We discuss signatures of quantum chaos in terms of distributions of nodal points, saddle points, and streamlines for coherent electron transport through two-dimensional billiards, which are either nominally integrable or chaotic. As typical examples of the two cases we select rectangular and Sinai billiards. We have numerically evaluated distribution functions for nearest distances between nodal points and found that there is a generic form for open chaotic billiards through which a net current is passed. We have also evaluated the distribution functions for nodal points with specific vorticity (winding number) as well as for saddle points. The distributions may be used as signatures of quantum chaos in open systems. All distributions are well reproduced using random complex linear combinations of nearly monochromatic states in nominally closed billiards. In the case of rectangular billiards with simple sharp-cornered leads the distributions have characteristic features related to order among the nodal points. A flaring or rounding of the contact regions may, however, induce a crossover to nodal point distributions and current flow typical for quantum chaos. For an irregular arrangement of nodal points, as for example in the Sinai billiard, the quantum flow lines become very complex and volatile, recalling chaos among classical trajectories. Similarities with percolation are pointed out.