Chaotic Billiards Research Articles

Classical and quantum elliptical billiards: Mixed phase space and short-range correlations in singlets and doublets

Billiards are flat cavities where a particle is free to move between elastic collisions with the boundary. In chaos theory, these systems are simple prototypes. The conservative dynamics of a billiard may vary from regular to chaotic, depending only on the shape of the boundary. This work aims to shed light into the quantization of classically chaotic systems. We present numerical results on classical and quantum properties in two bi-parametric families of billiards, namely Elliptical Stadium Billiard (ESB) and Elliptical-C3 Billiards (E-C3B). Both families entail elliptical perturbations of chaotic billiards with originally circular sectors on their borders. Our numerical calculations provide evidence that these elliptical families may exhibit a mixed classical dynamics, identified by the chaotic fraction of the phase space, the parameter χc<1. We use this quantity to guide our analysis of quantum spectra. We explore the short-range correlations through nearest neighbor spacing distribution p(s), revealing that in the mixed region of the classical phase space, p(s) is well described by the Berry–Robnik–Brody (BRB) distributions for the ESB. In agreement with the expectation from the so-called ergodic parameter α=tH/tT, the ratio between the Heisenberg time and the classical diffusive-like transport time. Our findings indicate the possibility of quantum dynamical localization when α<1. For the E-C3B family, the eigenstates can be split into singlets and doublets. BRB successfully describes p(s) for singlets as the previous family in the mixed region. However, for doublets, new distributions recently introduced in the literature come into play, providing descriptions for p(s) with a focus on cases where χc=1. We observed that as χc decreases, the p(s) distributions simultaneously deviate from the Gaussian Orthogonal Ensemble (GOE) for singlets, and Gaussian Unitary Ensemble (GUE) for doublets.

Random-hopping approach to fluctuation phenomena in quantum dots with chiral symmetry.

We propose a numerical approach to study mesoscopic fluctuations in quantum dots with chiral symmetry. Our method involves applying the random-hopping model to a tight-binding Hamiltonian, allowing us to calculate the conductance and shot-noise power distributions for systems belonging to the three chiral symmetry classes of random matrix theory. Furthermore, we demonstrate that the spectral fluctuations of quantum dots belonging to the Wigner-Dyson symmetry classes of random matrix theory can be obtained by applying the random-hopping model to a scattering region that was originally integrable, thus bypassing the need to use the boundaries of chaotic billiards.

Systematically Constructing Mesoscopic Quantum States Relevant to Periodic Orbits in Integrable Billiards from Directionally Resolved Level Distributions

Two-dimensional quantum billiards are one of the most important paradigms for exploring the connection between quantum and classical worlds. Researchers are mainly focused on nonintegrable and irregular shapes to understand the quantum characteristics of chaotic billiards. The emergence of the scarred modes relevant to unstable periodic orbits (POs) is one intriguing finding in nonintegrable quantum billiards. On the other hand, stable POs are abundant in integrable billiards. The quantum wavefunctions associated with stable POs have been shown to play a key role in ballistic transport. A variety of physical systems, such as microwave cavities, optical fibers, optical resonators, vibrating plates, acoustic waves, and liquid surface waves, are used to analogously simulate the wave properties of quantum billiards. This article gives a comprehensive review for the subtle connection between the quantum level clustering and the classical POs for three integrable billiards including square, equilateral triangle, and circular billiards.

Chaos Detection by Fast Dynamic Indicators in Reflecting Billiards.

The propagation of electromagnetic waves in a closed domain with a reflecting boundary amounts, in the eikonal approximation, to the propagation of rays in a billiard. If the inner medium is uniform, then the symplectic reflection map provides the polygonal rays' paths. The linear response theory is used to analyze the stability of any trajectory. The Lyapunov and reversibility error invariant indicators provide an estimate of the sensitivity to a small initial random deviation and to a small random deviation at any reflection, respectively. A family of chaotic billiards is considered to test the chaos detection effectiveness of the above indicators.

Thermodynamic uncertainty relations in mesoscopic devices.

We investigate the thermodynamic uncertainty relations (TURs) in mesoscopic devices for all universal symmetry classes of Wigner-Dyson and Dirac (chiral). The observables of interest include the TUR (MS), which is defined in terms of the ratio between the mean noise and mean conductance, as well as a new TUR (R) proposed in this article, which is based on the ensemble mean of the noise-to-conductance ratio. A detailed study is made on the quantum interference corrections associated with the TURs. We also analyze the influence of orbital and sublattice/chiral degrees of freedom for the validity of the observables in these chaotic mesoscopic billiards. Our investigation is based on the concatenation between the Landauer-Büttiker theory, the Mahaux-Wendeinmüller theory, and the TURs. We simulate the universal mesoscopic chaotic quantum dots using the random-matrix theory and compare our numerical results with the pertinent experimental data. The results were obtained for a different number of channels and tunneling rates that vary from the opaque to the ideal regime and, in all cases, demonstrate a clear phenomenological distinction between the TURs. In particular, the opaque regime engenders remarkable differences between the observables, even in the semiclassical regime, which characterizes a clear violation of the central limit theorem. Furthermore, we show that the phenomenology of the quantum interference corrections is strikingly robust, surprisingly exhibiting an order of magnitude greater than the supposedly leading semiclassical term for the TUR (R).

Marked Length Spectral determination of analytic chaotic billiards with axial symmetries

Marked Length Spectral determination of analytic chaotic billiards with axial symmetries

Ray Engineering from Chaos to Order in 2D Optical Cavities

AbstractChaos, namely exponential sensitivity to initial conditions, is generally considered a nuisance, inasmuch as it prevents long‐term predictions in physical systems. Here, an easily accessible approach to undo deterministic chaos and tailor ray trajectories in arbitrary 2D optical billiards by introducing spatially varying refractive index therein is presented. A new refractive index landscape is obtained by a conformal mapping, which makes the trajectories of the chaotic billiard fully predictable and the billiard fully integrable. Moreover, trajectory rectification can be pushed a step further by relating chaotic billiards with non‐Euclidean geometries. Two examples are illustrated by projecting billiards built on a sphere as well as the deformed spacetime outside a Schwarzschild black hole, which respectively lead to all periodic orbits and spiraling trajectories remaining away from the boundaries of the transformed 2D billiards/cavities. An implementation of this method is proposed, which enables real‐time control of chaos and can further contribute to a wealth of potential applications in the domain of optical microcavities.

Solar Hydrogen Variable Speed Control of Induction Motor Based on Chaotic Billiards Optimization Technique

This paper introduces a brand-new, inspired optimization algorithm (the chaotic billiards optimization (C-BO) approach) to effectively develop the optimal parameters for fuzzy PID techniques to enhance the dynamic response of the solar–hydrogen drive of an induction motor. This study compares fuzzy-PID-based C-BO regulators to fuzzy PID regulators based on particle swarm optimization (PSO) and PI-based PSO regulators to provide speed control in solar–hydrogen, induction-motor drive systems. The model is implemented to simulate the production and storage of hydrogen while powering an induction-motor drive which provides a great solution for the renewable energy storage problem in the case of solar pumping systems. MATLAB/Simulink 2021a is used to simulate and analyze the entire operation. The laboratory prototype is implemented in real time using a DSP-DS1104 board. Based on the simulation and experimental results, the proposed fuzzy-PID-based C-BO has reduced speed peak overshoot by 45.3% when compared to a fuzzy PID based PSO speed regulator and by 68.13% when compared to a PI-based PSO speed controller in the case of a large-scale motor. Additionally, the proposed speed regulator has a 6.1% faster speed rising time than a fuzzy-PID-based PSO and a 9.5% faster speed rising time than a PI–PSO speed controller. It has an excellent dynamic responsiveness value when compared to the other speed regulators.

Bound on energy dependence of chaos

We conjecture a chaos energy bound, an upper bound on the energy dependence of the Lyapunov exponent for any classical/quantum Hamiltonian mechanics and field theories. The conjecture states that the Lyapunov exponent $\ensuremath{\lambda}(E)$ grows no faster than linearly in the total energy $E$ in the high energy limit. In other words, the exponent $c$ in $\ensuremath{\lambda}(E)\ensuremath{\propto}{E}^{c}(E\ensuremath{\rightarrow}\ensuremath{\infty})$ satisfies $c\ensuremath{\le}1$. This chaos energy bound stems from thermodynamic consistency of out-of-time-order correlators and applies to any classical/quantum system with finite $N/\mathrm{large}$ $N$ ($N$ is the number of degrees of freedom) under plausible physical conditions on the Hamiltonians. To the best of our knowledge the chaos energy bound is satisfied by any classically chaotic Hamiltonian system known, and is consistent with the cerebrated chaos bound by Maldacena, Shenker, and Stanford, which is for quantum cases at large $N$. We provide arguments supporting the conjecture for generic classically chaotic billiards and multiparticle systems. The existence of the chaos energy bound may put a fundamental constraint on physical systems and the Universe.

Microbial narrow-escape is facilitated by wall interactions

Cells have evolved efficient strategies to probe their surroundings and navigate through complex environments. From metastatic spread in the body to swimming cells in porous materials, escape through narrow constrictions—a key component of any structured environment connecting isolated microdomains—is one ubiquitous and crucial aspect of cell exploration. Here, using the model microalgae Chlamydomonas reinhardtii, we combine experiments and simulations to achieve a tractable realization of the classical Brownian narrow-escape problem in the context of active confined matter. Our results differ from those expected for Brownian particles or leaking chaotic billiards and demonstrate that cell-wall interactions substantially modify escape rates and, under generic conditions, expedite spread dynamics.Received 4 October 2021Accepted 15 February 2022DOI:https://doi.org/10.1103/PhysRevResearch.4.L022029Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasBiological fluid dynamicsBiological movementMarine ecologySwimmingPhysical SystemsMicroswimmersBiological PhysicsInterdisciplinary PhysicsNonlinear Dynamics

Energy diffusion and prethermalization in chaotic billiards under rapid periodic driving.

We study the energy dynamics of a particle in a billiard subject to a rapid periodic drive. In the regime of large driving frequencies ω, we find that the particle's energy evolves diffusively, which suggests that the particle's energy distribution η(E,t) satisfies a Fokker-Planck equation. We calculate the rates of energy absorption and diffusion associated with this equation, finding that these rates are proportional to ω^{-2} for large ω. Our analysis suggests three phases of energy evolution: Prethermalization on short timescales, then slow energy absorption in accordance with the Fokker-Planck equation, and finally a breakdown of the rapid driving assumption for large energies and high particle speeds. We also present numerical simulations of the evolution of a rapidly driven billiard particle, which corroborate our theoretical results.

Multifractal magnetoconductance fluctuations in mesoscopic systems.

We perform a multifractal detrended fluctuation analysis of the magnetoconductance data of two standard types of mesoscopic systems: a disordered nanowire and a ballistic chaotic billiard, with two different lattice structures. We observe in all cases that multifractality is generally present and that it becomes stronger in the quantum regime of conduction, i.e., when the number of open scattering channels is small. We argue that this behavior originates from correlations induced by the magnetic field, which can be characterized through the distribution of conductance increments in the corresponding "stochastic time series," with the magnetic field playing the role of a fictitious time. More specifically, we show that the distributions of conductance increments are well fitted by q Gaussians and that the value of the parameter q is a useful quantitative measure of multifractality in magnetoconductance fluctuations.

Two particles on a chain with disordered interaction: Localization and dissociation of bound states and mapping to chaotic billiards

We consider two particles hopping on a chain with a contact interaction between them. At strong interaction, there is a molecular bound state separated by a direct gap from a continuous band of atomic states. Introducing weak disorder in the interaction, the molecular state becomes Anderson localized. At stronger disorder, part of the molecular band delocalizes and dissociates due to its hybridization to the atomic band. We characterize these different regimes by computing the density of states, the inverse participation ratio, the level-spacing statistics and the survival probability of an initially localized state. The atomic band is best described as that of a rough billiard for a single particle on a square lattice that shows signatures of quantum chaos. In addition to typical "chaotic states", we find states that are localized along only one direction. These "separatrix states" are more localized than chaotic states, and similar in this respect to scarred states, but their existence is due to the separatrix iso-energy line in the interaction-free dispersion relation, rather than to unstable periodic orbits.

Convergence to a-stable Lévy motion for chaotic billiards with several cusps at flat points

We consider billiards with several possibly non-isometric and asymmetric cusps at flat points; the case of a single symmetric cusp was studied previously in Zhang (2017 Dynamical Systems, Ergodic Theory, and Probability: in Memory of Kolya Chernov (Contemporary Mathematics vol 698) (Providence, RI: American Mathematical Society) pp 287–316) and Jung and Zhang (2018 Ann. Henri Poincaré 19 3815–53). In particular, we show that properly normalized Birkhoff sums of Hölder observables, with respect to the billiard map, converge in Skorokhod’s M1-topology to an -stable Lévy motion, where depends on the ‘curvature’ of the flattest points and the skewness parameter depends on the values of the observable at those same points. Previously, Jung and Zhang (2018 Ann. Henri Poincaré 19 3815–53) proved convergence of the one-point marginals to totally skewed -stable distributions for a symmetric cusp. The limits we prove here are stronger, since they are in the functional sense, but also allow for more varied behaviour due to the presence of multiple cusps. In particular, the general limits we obtain allow for any skewness parameter, as opposed to just the totally skewed cases. We also show that convergence in the stronger J1-topology is not possible.

A novel image encryption algorithm based on chaotic billiards

In this paper, a novel symmetric image encryption algorithm is proposed on the basis of the chaotic billiard (Sinai billiard). These chaotic systems, characterized by the highest degree of chaos, have been recently introduced in cryptography. The design of the algorithm relies on the random walk of three-point particles that move freely in the billiard. It includes several rounds of the confusion and diffusion processes. In each ciphering round, the permutation of the image pixels is performed according to the random numbers obtained from the output of a particle system, by using a new calculation technique. Thus, the pixel values are hidden by the pseudo-random sequences generated by the systems of the other two particles. The simulation results demonstrate that the algorithm is simple to implement and is highly secure. It has successfully passed all security tests and has shown the desirable properties in a good secure cryptosystem.

Partial Time-Reversal Invariance Violation in a Flat, Superconducting Microwave Cavity with the Shape of a Chaotic Africa Billiard.

We report on the experimental realization of a flat, superconducting microwave resonator, a microwave billiard, with partially violated time-reversal (T) invariance, induced by inserting a ferrite into the cavity and magnetizing it with an external magnetic field perpendicular to the resonator plane. In order to prevent its expulsion caused by the Meissner-Ochsenfeld effect, we used a cavity of which the top and bottom plate were made from niobium, a superconductor of type II, and cooled it down to liquid-helium temperature T_{LHe}≃4 K. The cavity had the shape of a chaotic Africa billiard. Superconductivity rendered possible the accurate determination of complete sequences of the resonance frequencies and of the widths and strengths of the resonances, an indispensable prerequisite for the unambiguous detection of T invariance violation, especially when it is only partially violated. This allows for the first time the precise specification of the size of T invariance violation from the fluctuation properties of the resonance frequencies and from the strength distribution, which actually depends sensitively on it and thus provides a most suitable measure. For this purpose we derived an analytical expression for the latter which is valid for isolated resonances in the range from no T invariance violation to complete violation.

Effect of chaos in a one-channel time-reversal acoustic mirror.

Time-reversal of propagating waves has been intensely studied during the last years and successfully implemented in different experimental contexts. It has been argued that ergodic or chaotic ray dynamics improve the refocusing resolution. In this work we consider this fundamental aspect by studying the reversion of sound waves in two-dimensional reflecting cavities numerically. The boundary of the enclosure is deformed from a rectangle with regular ray dynamics to a completely chaotic hyperbolic billiard. We observed that both the regular and chaotic cases display a prominent refocusing peak, and also that in the first scenario many secondary maxima appear. We developed measures of the spatial and temporal contrasts of the reconstructed signal in order to gain insight on these phenomena and to distinguish between cases. The results obtained point to the necessity for a reconsideration of what is usually understood by successful time-reversal processes.

Klein paradox in chaotic Dirac billiards

We investigate the Schrödinger (non-relativistic) and the Dirac (“relativistic”) billiards in the universal regime. The study is based on a non-ideal quantum resonant scattering numerical simulation. We show universal results that reveal anomalous behaviour on the conductance, on the shot-noise power and on the respective eigenvalues distributions. In particular, we demonstrate the Klein’s paradox in the graphene and tunable suppression/amplification transitions on the typical observables of the quantum dots.

Experimental Microwave Scattering in Polygonal Billiards

Fluctuations in the one-port scattering and normalized impedance matrices in three polygonal and one chaotic time-reversal invariant microwave billiards are experimentally investigated, in several levels of coupling and absorption, at room temperature and at 77 K. The observed distributions of reflection coefficient, phase of the scattering matrix, resistance and reactance exhibit no fingerprint of a given geometry. At low frequencies, the results are consistent with earlier theoretical models by López, Mello and Seligman and by Zheng, Antonsen and Ott, who independently predicted that the scattering fluctuations might be the same for the Wigner and Poisson level spacing distributions in the lossless cavity. The uniqueness of the observed scattering statistics at higher absorption levels is discussed with respect to inherent limitations posed by the experimental technique.

Ordered level spacing probability densities

Spectral statistics of quantum systems have been studied in detail using the nearest neighbour level spacings, which for generic chaotic systems follows random matrix theory predictions. In this work, the probability density of the and spacings from a given level are introduced. Analytical predictions are derived using a matrix model. The density is generalized to the spacing density, which allows for investigating long-range correlations. For larger the probability density of spacings is well described by a Gaussian. Using these spacings we propose the ratio of the to the as an alternative to the ratio of successive spacings. For a Poissonian spectrum the density of the ratio is flat, whereas for the three Gaussian ensembles repulsion at small values is found. The ordered spacing statistics and their ratio are numerically studied for the integrable circle billiard, the chaotic cardioid billiard, the standard map and the zeroes of the Riemann zeta function. Very good agreement with the predictions is found.