Two-dimensional Billiards Research Articles

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.

Sinai billiard maps with Ruelle resonances

We construct families of two-dimensional Sinai billiards whose transfer operators have Ruelle resonances arbitrarily close to 1. Our method involves taking a large enough cover of an initial billiard table, and relating the transfer operator of the covering table to twisted transfer operators of the initial table. We also study the distribution of these resonances which are close to 1.

Integrable geodesic flows on orientable two-dimensional surfaces and topological billiards

The authors have recently introduced the class of topological billiards. Topological billiards are glued from elementary planar billiard sheets (bounded by arcs of confocal quadrics) along intervals of their boundaries. It turns out that the integrability of the elementary billiards implies that of the topological billiards. We show that all classical linearly and quadratically integrable geodesic flows on tori and spheres are Liouville equivalent to appropriate topological billiards. Moreover, the linear and quadratic integrals of the geodesic flows reduce to a single canonical linear integral and a single canonical quadratic integral on the billiard. These results are obtained within the framework of the Fomenko–Zieschang theory of the classification of integrable systems.

Asymptotic Eigenfunctions of the Operator ∇D(x)∇ Defined in a Two-Dimensional Domain and Degenerating on Its Boundary and Billiards with Semi-Rigid Walls

We propose a method for constructing asymptotic eigenfunctions of the operator L = ∇D(x1,x2)∇ defined in a domain Ω ? R 2 with coefficient D(x) degenerating on the boundary ∂Ω. Such operators arise, for example, in problems about long water waves trapped by coasts and islands. These eigenfunctions are associated with analogs of Liouville tori of integrable geodesic flows with the metric defined by the Hamiltonian system with Hamiltonian D(x)p2 and degenerating on ∂ Ω. The situation is unusual compared, say, with the case of integrable two-dimensional billiards, because the momentum components of trajectories on such “tori” are infinite over the boundary, where D(x) = 0, although their projections onto the plane R2 are compact sets, as a rule, diffeomorphic to annuli in R2. We refer to such systems as billiards with semi-rigid walls.

Few Islands Approximation of Hamiltonian System with divided Phase Space

It is well known that typical Hamiltonian systems have divided phase space consisting of regions with regular dynamics on KAM tori and region(s) with chaotic dynamics called chaotic sea(s). This complex structure makes rigorous analysis of such systems virtually impossible and significantly complicates numerical exploration of their dynamical properties. Hamiltonian systems with sharply divided phase space between regions of regular and chaotic dynamics are much easier to analyze, but there are only few cases or families of such systems known to date. In this article, we outline a new approach for a systematic construction, starting from a generic KAM Hamiltonian system, of a system with a sharply divided phase space with an arbitrary number of regular islands which are in one-to-one correspondence with islands of the initial KAM system. In this procedure a typical Hamiltonian system, for example a KAM billiard, is replaced by a sequence of Hamiltonian systems having an increasing (but finite) number of islands of regular motion. The islands in the substituting systems are sub-islands of the KAM islands in the initial system. We apply this idea to two-dimensional lemon-shaped billiards, where the substituting systems are obtained by replacing parts of the curved boundaries by chords, so that in the limit of infinite number of islands the boundary of the substituting system becomes arbitrary close to the original billiard’s boundary.

Integrable geodesic flows on orientable two-dimensional surfaces and topological billiards

Недавно авторами был введен класс топологических биллиардов. Топологические биллиарды получаются склейками элементарных плоских биллиардов-листов, ограниченных дугами софокусных квадрик, вдоль сегментов их границ. Интегрируемость элементарных биллиардов, оказывается, влечет интегрируемость топологических биллиардов. В статье доказывается, что классические линейно и квадратично интегрируемые геодезические потоки на торе и сфере лиувиллево эквиваленты подходящим топологическим биллиардам. При этом линейные и квадратичные интегралы геодезических потоков сводятся к одному каноническому линейному интегралу и к одному каноническому квадратичному интегралу на биллиарде. Эти результаты получены в рамках теории Фоменко-Цишанга классификации интегрируемых систем. Библиография: 64 наименования.

Exact solution to Lippmann-Schwinger equation for a circular billiard

We present an exact solution to the Lippmann-Schwinger equation for a two-dimensional circular billiard. After diagonalizing an integral operator whose kernel is a zeroth order Hankel function of first kind, we use its eigenfunctions and eigenvalues to obtain in a straightforward way the exact wavefunctions of the referred Lippmann-Schwinger equation.

Modeling Nondegenerate Bifurcations of Closures of Solutions for Integrable Systems with Two Degrees of Freedom by Integrable Topological Billiards

It is well known that surgeries of closures of solutions for integrable nondegenerate Hamiltonian systems with two degrees of freedom at a level of constant energy are classified by the so-called 3-atoms. These surgeries correspond to singular leaves of the Liouville foliation of three-dimensional isoenergetic surfaces. In this paper we prove the Fomenko conjecture that all such surgeries are modeled by integrable topological two-dimensional billiards (billiard books).

Extracting trajectory equations of classical periodic orbits from the quantum eigenmodes in two-dimensional integrable billiards.

The trajectory equations for classical periodic orbits in the equilateral-triangular and circular billiards are systematically extracted from quantum stationary coherent states. The relationship between the phase factors of quantum stationary coherent states and the initial positions of classical periodic orbits is analytically derived. In addition, the stationary coherent states with noncoprime parametric numbers are shown to correspond to the multiple periodic orbits, which cannot be explicable in the one-particle picture. The stationary coherent states are further verified to be linked to the resonant modes that are generally observed in the experimental wave system excited by a localized and unidirectional source. The excellent agreement between the resonant modes and the stationary coherent states not only manifests the importance of classical features in experimental systems but also paves the way to manipulate the mesoscopic wave functions localized on the periodic orbits for applications.

Entanglement Entropy in a Triangular Billiard

The Schrödinger equation for a quantum particle in a two-dimensional triangular billiard can be written as the Helmholtz equation with a Dirichlet boundary condition. We numerically explore the quantum entanglement of the eigenfunctions of the triangle billiard and its relation to the irrationality of the triangular geometry. We also study the entanglement dynamics of the coherent state with its center chosen at the centroid of the different triangle configuration. Using the von Neumann entropy of entanglement, we quantify the quantum entanglement appearing in the eigenfunction of the triangular domain. We see a clear correspondence between the irrationality of the triangle and the average entanglement of the eigenfunctions. The entanglement dynamics of the coherent state shows a dependence on the geometry of the triangle. The effect of quantum squeezing on the coherent state is analyzed and it can be utilize to enhance or decrease the entanglement entropy in a triangular billiard.

Quantum chaos for two-dimensional Sinai billiard

We study the classical and quantum correspondence for a two-dimensional Sinai billiard system. By using the Stationary state expansion method and Gutzwiller's periodic orbit theory, we analyze the quantum length spectrum obtained through the Fourier transformation of the quantum density of state for the Sinai billiard system, and by comparing the peak position with the length of the classical periodic orbit we find their excellent correspondence. We observe that some quantum states are localized near some short period orbits, forming the quantum scarred states or superscarred states. In this paper we also investigate the nearest-neighbor spacing distribution of levels for both concentric and nonconcentric Sinai billiard systems, and find that the concentric Sinai billiard system is nearintegrable, and for the nonconcentric Sinai billiard system with =3/8 its nearest-neighbor spacing distribution of levels transits from nearintegrable to the Wigner distribution as the distance between the two centers increases.

Competition between unlimited and limited energy growth in a two-dimensional time-dependent billiard

Abstract Some dynamical properties for a dissipative time-dependent Lorentz gas are studied. We assume that the size of the scatterers change periodically in time. We show that for some combination of the control parameters the particles come to a complete stop between the scatterers, but for some other cases, the average velocity grows unbounded. This is the first time that the unlimited energy growth is observed in a dissipative system. Finally, we study the behavior of the average velocity as a function of the number of collisions and we show that the system is scaling invariant with scaling exponents well defined.

Metrics with equatorial singularities on the sphere

Motivated by optimal control of affine systems stemming from mechanics, metrics on the two-sphere of revolution are considered; these metrics are Riemannian on each open hemisphere, whereas one term of the corresponding tensor becomes infinite on the equator. Length-minimizing curves are computed, and structure results on the cut and conjugate loci are given, extending those in Bonnard et al. (Ann Inst H Poincaré Anal Non Linéaire 26(4):1081–1098, 2009). These results rely on monotonicity and convexity properties of the quasi-period of the geodesics; such properties are studied on an example with elliptic transcendency. A suitable deformation of the round sphere allows to reinterpretate the equatorial singularity in terms of concentration of curvature and collapsing of the sphere onto a two-dimensional billiard.

Chaotic behaviors and fractal self-similar analysis of particles transport properties in RIKEN mesoscopic devices

The theoretical model of RIKEN mesoscopic device in our study is one kind of the two-dimensional Sinai billiards, which is an ideal model to investigate the chaotic and fractal behaviors in particle escape curves. In the analysis of the escape curves, we use two methods:qualitative comparison and quantitative calculation of the fractal dimensions. Then we obtain the influence of the distribution of chaotic areas caused by the opening width, cavity length, corner positions, arc radius, etc. In the results, we find the fractal self-similar structure of the escape curves in which the similarity is very good, and they display the chaotic property of the transmission in the RIKEN device. Moreover, we use the eye-style structure analysis and the comparation between similar ratios to investagate the fractal self-similar structure.

Coupling of bouncing-ball modes to the chaotic sea and their counting function

We study the coupling of bouncing-ball modes to chaotic modes in two-dimensional billiards with two parallel boundary segments. Analytically, we predict the corresponding decay rates using the fictitious integrable system approach. Agreement with numerically determined rates is found for the stadium and the cosine billiard. We use this result to predict the asymptotic behavior of the counting function N_{bb}(E)∼E^{δ}. For the stadium billiard we find agreement with the previous result δ=3/4. For the cosine billiard we derive δ=5/8, which is confirmed numerically and is well below the previously predicted upper bound δ=9/10.

Percolating level sets of the adjacency eigenvectors of d-regular graphs

One of the most surprising discoveries in quantum chaos was that nodal domains of eigenfunctions of quantum-chaotic billiards and maps in the semi-classical limit display critical percolation. Here we extend these studies to the level sets of the adjacency eigenvectors of d-regular graphs. Numerical computations show that the statistics of the largest level sets (the maximal connected components of the graph for which the eigenvector exceeds a prescribed value) depend critically on the level. The critical level is a function of the eigenvalue and the degree d. To explain the observed behaviour we study a random Gaussian waves ensemble over the d-regular tree. For this model, we prove the existence of a critical threshold. Using the local tree property of d-regular graphs, and assuming the (local) applicability of the random waves model, we can compute the critical percolation level and reproduce numerical simulations. These results support the random-waves model for random regular graphs, suggested in [1] and provide an extension to Bogomolny's percolation model [2] for two-dimensional chaotic billiards.

Numerical evaluation of the escape time of a classical point particle from an annular billiard

A two-dimensional annular billiard consisting of a region confined within two concentric circumferences, the outer of radius R and the inner of radius r, is considered. The escape time of a point particle projected at a given angle within this billiard is numerically evaluated in terms of the size of the opening in the billiard. The problem is solved by means of classical mechanics and can be of interest for advanced high school physics students or undergraduate college physics students

Boundary crisis and suppression of Fermi acceleration in a dissipative two-dimensional non-integrable time-dependent billiard

Some dynamical properties for a dissipative time-dependent oval-shaped billiard are studied. The system is described in terms of a four-dimensional nonlinear mapping. Dissipation is introduced via inelastic collisions of the particle with the boundary, thus implying that the particle has a fractional loss of energy upon collision. The dissipation causes profound modifications in the dynamics of the particle as well as in the phase space of the non-dissipative system. In particular, inelastic collisions can be assumed as an efficient mechanism to suppress Fermi acceleration of the particle. The dissipation also creates attractors in the system, including chaotic. We show that a slightly modification of the intensity of the damping coefficient yields a drastic and sudden destruction of the chaotic attractor, thus leading the system to experience a boundary crisis. We have characterized such a boundary crisis via a collision of the chaotic attractor with its own basin of attraction and confirmed that inelastic collisions do indeed suppress Fermi acceleration in two-dimensional time-dependent billiards.

Competition between suppression and production of Fermi acceleration

The behavior of the average velocity for a classical particle in the one-dimensional Fermi accelerator model under sawtooth external force is considered. For elastic collisions, it is known that the average velocity of the particle grows unlimitedly because of the discontinuities of the derivative of the moving wall's position with respect to time. However, and contrary to what was expected to be observed, the introduction of a friction force generated from a slip of a body against a rough surface leads to a boundary separating different regions of the phase space that yields the particle to either experience unlimited energy growth or suppression of Fermi acceleration. The Fermi acceleration is described by using scaling arguments. The formalism presented can be extended to two-dimensional time-dependent billiards as well as to higher-order mappings.

Chaotic properties of the truncated elliptical billiards

Chaotic properties of symmetrical two-dimensional stadium-like billiards with piecewise flat and elliptical segments are studied numerically and analytically. Their sensitivity to small variations of the shape parameters can be usefully applied for optimal construction of the dielectric and polymer optical microresonators. For the two-parameter truncated elliptical billiards (TEB) the existence and linear stability of several periodic orbits are investigated in the full parameter space. Poincaré plots are computed and used for evaluation of the chaotic fraction of the phase space by means of the box-counting method. A highly chaotic behavior prevails in the region of elongated elliptical arcs, where most of the existing orbits are either neutral or unstable. In the parameter space, the upper limit of the fully chaotic behavior is reached when the relation between the two shape parameters becomes δ = 1 - γ 2 , corresponding to truncated circles. Above this limit, for flattened elliptical arcs, mixed dynamics with numerous stable elliptic islands is found. In both regions parabolic orbits are present, many of them identical to orbits within an ellipse. These properties of the TEB differ remarkably from the behavior in the elliptical stadium billiards (ESB), where the chaotic region in the parameter space was strictly bounded from both sides. In order to follow the transition from TEB to ESB, a generalisation to a novel three-parameter family (GTEB) of stadium-like boundary shapes with elliptical arcs is proposed.