Barrier Billiards Research Articles

Level Compressibility of Certain Random Unitary Matrices.

The value of spectral form factor at the origin, called level compressibility, is an important characteristic of random spectra. The paper is devoted to analytical calculations of this quantity for different random unitary matrices describing models with intermediate spectral statistics. The computations are based on the approach developed by G. Tanner for chaotic systems. The main ingredient of the method is the determination of eigenvalues of a transition matrix whose matrix elements equal the squared moduli of matrix elements of the initial unitary matrix. The principal result of the paper is the proof that the level compressibility of random unitary matrices derived from the exact quantisation of barrier billiards and consequently of barrier billiards themselves is equal to irrespective of the height and the position of the barrier.

Random matrices associated with general barrier billiards

The paper is devoted to the derivation of random unitary matrices whose spectral statistics is the same as statistics of quantum eigenvalues of certain deterministic two-dimensional barrier billiards. These random matrices are extracted from the exact billiard quantisation condition by applying a random phase approximation for high-excited states. An important ingredient of the method is the calculation of S-matrix for the scattering in the slab with a half-plane inside by the Wiener–Hopf method. It appears that these random matrices have the form similar to the one obtained by the author in (2022 J. Phys. A: Math. Theor. 55 024001) for a particular case of symmetric barrier billiards but with different choices of parameters. The local correlation functions of the resulting random matrices are well approximated by the semi-Poisson distribution which is a characteristic feature of various models with intermediate statistics. Consequently, local spectral statistics of the considered barrier billiards is (i) universal for almost all values of parameters and (ii) well described by the semi-Poisson statistics.

Barrier billiard and random matrices

The barrier billiard is the simplest example of pseudo-integrable models with interesting and intricate classical and quantum properties. Using the Wiener–Hopf method it is demonstrated that quantum mechanics of a rectangular billiard with a barrier in the centre can be reduced to the investigation of a certain unitary matrix. Under heuristic assumptions this matrix is substituted by a special low-complexity random unitary matrix of independent interest. The main results of the paper are (i) spectral statistics of such billiards is insensitive to the barrier height and (ii) it is well described by the semi-Poisson distributions.

Formation of superscar waves in plane polygonal billiards* *In memory of Charles Schmit.

Polygonal billiards constitute a special class of models. Though they have zero Lyapunov exponent their classical and quantum properties are involved due to scattering on singular vertices with angles with integer n. It is demonstrated that in the semiclassical limit the multiple singular scattering on such vertices, when optical boundaries of many scatters overlap, leads to the vanishing of quantum wave functions along straight lines built by these scatters. This phenomenon has an especially important consequence for polygonal billiards where periodic orbits (when they exist) form pencils of parallel rays restricted from the both sides by singular vertices. Due to the singular scattering on boundary vertices, waves propagated inside a periodic orbit pencil tend in the semiclassical limit to zero along pencil boundaries thus forming weakly interacting quasi-modes. Contrary to scars in chaotic systems, the discussed quasi-modes in polygonal billiards become almost exact for high-excited states and for brevity they are designated as superscars. Many pictures of eigenfunctions for a triangular billiard and a barrier billiard which have clear superscar structures are presented in the paper. Special attention is given to the development of quantitative methods of detecting and analysing such superscars. In particular, it is demonstrated that the overlap between superscar waves associated with a fixed periodic orbit and eigenfunctions of a barrier billiard is distributed according to the Breit-Wigner distribution typical for weakly interacting quasi-modes. For special sub-class of rational polygonal billiards called Veech polygons where all periodic orbits can be calculated analytically it is argued and checked numerically that their eigenfunctions are fractal in the Fourier space.

Transport in the barrier billiard.

We investigate transport properties of an ensemble of particles moving inside an infinite periodic horizontal planar barrier billiard. A particle moves among bars and elastically reflects on them. The motion is a uniform translation along the bars' axis. When the tangent of the incidence angle, α, is fixed and rational, the second moment of the displacement along the orthogonal axis at time n,〈S_{n}^{2}〉, is either bounded or asymptotic to Kn^{2}, when n→∞. For irrational α, the collision map is ergodic and has a family of weakly mixing observables, the transport is not ballistic, and autocorrelation functions decay only in time average, but may not decay for a family of irrational α's. An exhaustive numerical computation shows that the transport may be superdiffusive or subdiffusive with various rates or bounded strongly depending on the values of α. The variety of transport behaviors sounds reminiscent of well-known behavior of conservative systems. Considering then an ensemble of particles with nonfixed α, the system is nonergodic and certainly not mixing and has anomalous diffusion with self-similar space-time properties. However, we verified that such a system decomposes into ergodic subdynamics breaking self-similarity.

The non-smooth pitchfork bifurcation: a renormalization analysis

We give a renormalization group analysis of a system exhibiting a non-smooth pitchfork bifurcation to a strange non-chaotic attractor. For parameter choices satisfying two specified conditions, self-similar behaviour of the attractor on and near the bifurcation curve can be observed, which corresponds to a periodic orbit of an underlying renormalization operator. We examine the scaling properties for various parameter choices including the so-called pitchfork critical point. Finally, we study the autocorrelation function for the system and show that it is equivalent to that present in symmetric barrier billiards.

Renormalisation of correlations in a barrier billiard: Quadratic irrational trajectories

We present an analysis of autocorrelation functions in symmetric barrier billiards using a renormalisation approach for quadratic irrational trajectories. Depending on the nature of the barrier, this leads to either self-similar or chaotic behaviour. In the self-similar case we give an analysis of the half barrier and present a detailed calculation of the locations, asymptotic heights and signs of the main peaks in the autocorrelation function. Then we consider arbitrary barriers, illustrating that typically these give rise to chaotic correlations of the autocorrelation function which we further represent by showing the invariant sets associated with these correlations. Our main ingredient here is a functional recurrence which has been previously derived and used in work on the Harper equation, strange non-chaotic attractors and a quasi-periodically forced two-level system.

Chaotic correlations in barrier billiards with arbitrary barriers

We study autocorrelation functions in symmetric barrier billiards for golden mean trajectories with arbitrary barriers. Renormalization analysis reveals the presence of a chaotic invariant set and thus that, for a typical barrier, there are chaotic correlations. The chaotic renormalization set is the analogue of the so-called orchid that arises in a generalized Harper equation.

Properties of nodal domains in a pseudointegrable barrier billiard

We investigate experimentally the scaling properties of the number of nodal domains of the wave functions and the distribution of the nodal domain areas in a pseudointegrable barrier billiard. The number of nodal domains is smaller than expected for billiards whose dynamics is chaotic or regular. This reduction is explained by the appearance of superscars in the barrier billiard, i.e., regular structures, which are embedded into the underlying chaotic wave functions. Furthermore, the area distribution of nodal domains follows the predictions of a percolation model for chaotic systems.

Application of the trace formula in pseudointegrable systems

We apply periodic-orbit theory to calculate the integrated density of states N(k) of the quantum mechanical eigenvalues from the periodic orbits of pseudointegrable polygon and barrier billiards. We show that the results agree so well with the density of states obtained from numerical solutions of the Schrödinger equation that about the first 100 eigenvalues can be obtained directly from the periodic-orbit calculations with good accuracy.

Periodic Orbits and Semiclassical Form Factor in Barrier Billiards

Using heuristic arguments based on the trace formulas, we analytically calculate the semiclassical two-point correlation form factor for a family of rectangular billiards with a barrier of height irrational with respect to the side of the billiard and located at any rational position p/q from the side. To do this, we first obtain the asymptotic density of lengths for each family of periodic orbits by a Siegel-Veech formula. The result K(0)=1/2+1/q obtained for these pseudo-integrable, non-Veech billiards is different but not far from the value of 1/2 expected for semi-Poisson statistics and from values of K(0) obtained previously in the case of Veech billiards.

RENORMALIZATION IN QUASIPERIODICALLY FORCED SYSTEMS

We review our recent rigorous results on renormalization in a variety of quasiperiodically forced systems. Our results include a description of (i) self-similar fluctuations of localized states in the Harper equation, including the renormalization strange set (known as the orchid) in the generalized Harper equation; and (ii) self-similarities in the correlations of strange nonchaotic attractors, barrier billiards, and quantum two-level systems.

Self-similar correlations in a barrier billiard

We give a renormalization analysis of the self-similarity of autocorrelation functions in symmetric barrier billiards for golden mean trajectories. For the special case of a half-barrier we present a rigorous calculation of the asymptotic height of the main peaks in the autocorrelation function. Fundamental to this work is a detailed analysis of a functional recurrence equation which has previously been used in the analysis of fluctuations in the Harper equation and of correlations in strange non-chaotic attractors and in quantum two-level systems.

Spectral properties of quantized barrier billiards.

The properties of energy levels in a family of classically pseudointegrable systems, the barrier billiards, are investigated. An extensive numerical study of nearest-neighbor spacing distributions, next-to-nearest spacing distributions, number variances, spectral form factors, and the level dynamics is carried out. For a special member of the billiard family, the form factor is calculated analytically for small arguments in the diagonal approximation. All results together are consistent with the so-called semi-Poisson statistics.

Pseudointegrable Andreev billiard.

A circular Andreev billiard in a uniform magnetic field is studied. It is demonstrated that the classical dynamics is pseudointegrable in the same sense as for rational polygonal billiards. The relation to a specific polygon, the asymmetric barrier billiard, is discussed. Numerical evidence is presented indicating that the Poincaré map is typically weak mixing on the invariant sets. This link between these different classes of dynamical systems throws some light on the proximity effect in chaotic Andreev billiards.

Quantum-classical correspondence in polygonal billiards.

We show that wave functions in planar rational polygonal billiards (all angles rationally related to pi) can be expanded in a basis of quasistationary and spatially regular states. Unlike the energy eigenstates, these states are directly related to the classical invariant surfaces in the semiclassical limit. This is illustrated for the barrier billiard. We expect that these states are also present in integrable billiards with point scatterers or magnetic-flux lines.

Periodic orbits of an almost integrable system

The authors enumerate and classify all the periodic orbits of the ( pi /3)-rhombus billiard. The billiard flow for this system resides on a classically invariant surface of genus two. After transforming the rhombus billiard trajectories on an exactly equivalent set of trajectories on a barrier billiard (with barrier-to-gap ratio equal to 2), they present a criterion facilitating the complete enumeration of periodic billiard trajectories. Having enumerated the trajectories, they classify them distinctly and provide the underlying number-theoretic rationale for the same. Their analysis involves a 'polar construction' which facilitates the determination of lengths of periodic orbits and the areas of the bands in which they occur. It is clear from the analysis that there are no isolated periodic trajectories implying that all the periodic orbits close after an even number of reflections from the boundaries. These results are used to study the family-counting function F(x) (i.e. the number of families of periodic orbits of length less than x) as a function of x.