Billiard Dynamics Research Articles

Bipartite unitary gates and billiard dynamics in the Weyl chamber

Long time behavior of a unitary quantum gate $U$, acting sequentially on two subsystems of dimension $N$ each, is investigated. We derive an expression describing an arbitrary iteration of a two-qubit gate making use of a link to the dynamics of a free particle in a $3D$ billiard. Due to ergodicity of such a dynamics an average along a trajectory $V^t$ stemming from a generic two-qubit gate $V$ in the canonical form tends for a large $t$ to the average over an ensemble of random unitary gates distributed according to the flat measure in the Weyl chamber - the minimal $3D$ set containing points from all orbits of locally equivalent gates. Furthermore, we show that for a large dimension $N$ the mean entanglement entropy averaged along a generic trajectory coincides with the average over the ensemble of random unitary matrices distributed according to the Haar measure on $U(N^2)$.

Duality of caustics in Minkowski billiards

In this paper we study convex caustics in Minkowski billiards. We show that for the Euclidean billiard dynamics in a planar smooth, centrally symmetric, strictly convex body K, for every convex caustic which K possesses, the ‘dual’ billiard dynamics in which the table is the Euclidean unit ball and the geometry that governs the motion is induced by the body K, possesses a dual convex caustic. Such a pair of caustics are dual in a strong sense, and in particular they have the same perimeter, Lazutkin parameter (both measured with respect to the corresponding geometries), and rotation number. We show moreover that for general Minkowski billiards this phenomenon fails, and one can construct a smooth caustic in a Minkowski billiard table which possesses no dual convex caustic.

Splitting of separatrices, scattering maps, and energy growth for a billiard inside a time-dependent symmetric domain close to an ellipse

We study billiard dynamics inside an ellipse for which the axes lengths are changed periodically in time and an -small quartic polynomial deformation is added to the boundary. In this situation the energy of the particle in the billiard is no longer conserved. We show a Fermi acceleration in such system: there exists a billiard trajectory on which the energy tends to infinity. The construction is based on the analysis of dynamics in the phase space near a homoclinic intersection of the stable and unstable manifolds of the normally hyperbolic invariant cylinder Λ, parameterised by the energy and time, that corresponds to the motion along the major axis of the ellipse. The proof depends on the reduction of the billiard map near the homoclinic channel to an iterated function system comprised by the shifts along two Hamiltonian flows defined on Λ. The two flows approximate the so-called inner and scattering maps, which are basic tools that arise in the studies of the Arnold diffusion; the scattering maps defined by the projection along the strong stable and strong unstable foliations Wss,uu of the stable and unstable invariant manifolds at the homoclinic points. Melnikov type calculations imply that the behaviour of the scattering map in this problem is quite unusual: it is only defined on a small subset of Λ that shrinks, in the large energy limit, to a set of parallel lines t = const as .

On Local Integrability in Billiard Dynamics

ABSTRACTWe consider a billiard system in a convex domain symmetric in coordinate axes . Hence, there exists a periodic orbit γ of period 2 moving along the y-axis. We ask the following question. Is it possible to choose the domain such that the dynamics of the corresponding billiard map are locally (near γ) conjugated to the dynamics of the rigid rotation by the angle α? In [Treschev 13], numeric evidence for positive answer is given in the case when α/π is a Diophantine number. In this paper, we propose further numerical analysis of the problem. In particular , we show that the relative measure of the domain in the billiard phase space on which the dynamics is conjugated to the rigid rotation can reach 50%.

Equilibration of energy in slow–fast systems

Ergodicity is a fundamental requirement for a dynamical system to reach a state of statistical equilibrium. However, in systems with several characteristic timescales, the ergodicity of the fast subsystem impedes the equilibration of the whole system because of the presence of an adiabatic invariant. In this paper, we show that violation of ergodicity in the fast dynamics can drive the whole system to equilibrium. To show this principle, we investigate the dynamics of springy billiards, which are mechanical systems composed of a small particle bouncing elastically in a bounded domain, where one of the boundary walls has finite mass and is attached to a linear spring. Numerical simulations show that the springy billiard systems approach equilibrium at an exponential rate. However, in the limit of vanishing particle-to-wall mass ratio, the equilibration rates remain strictly positive only when the fast particle dynamics reveal two or more ergodic components for a range of wall positions. For this case, we show that the slow dynamics of the moving wall can be modeled by a random process. Numerical simulations of the corresponding springy billiards and their random models show equilibration with similar positive rates.

Quantum supersymmetric cosmological billiards and their hidden Kac-Moody structure

We study the quantum fermionic billiard defined by the dynamics of a quantized supersymmetric squashed three-sphere (Bianchi IX cosmological model within $D=4$ simple supergravity). The quantization of the homogeneous gravitino field leads to a 64-dimensional fermionic Hilbert space. We focus on the 15- and 20-dimensional subspaces (with fermion numbers ${N}_{F}=2$ and ${N}_{F}=3$) where there exist propagating solutions of the supersymmetry constraints that carry (in the small-wavelength limit) a chaotic spinorial dynamics generalizing the Belinskii-Khalatnikov-Lifshitz classical ``oscillatory'' dynamics. By exactly solving the supersymmetry constraints near each one of the three dominant potential walls underlying the latter chaotic billiard dynamics, we compute the three operators that describe the corresponding three potential-wall reflections of the spinorial state describing, in supergravity, the quantum evolution of the universe. It is remarkably found that the latter, purely dynamically-defined, reflection operators satisfy generalized Coxeter relations which define a type of spinorial extension of the Weyl group of the rank-3 hyperbolic Kac-Moody algebra $A{E}_{3}$.

Virtual billiards in pseudo–euclidean spaces: Discrete hamiltonian and contact integrability

The aim of the paper is to unify the efforts in the study of integrable billiards within quadrics in flat and curved spaces and to explore further the interplay of symplectic and contact integrability. As a starting point in this direction, we consider virtual billiard dynamics within quadrics in pseudo-Euclidean spaces. In contrast to the usual billiards, the incoming velocity and the velocity after the billiard reflection can be at opposite sides of the tangent plane at the reflection point. In the symmetric case we prove noncommutative integrability of the system and give a geometrical interpretation of integrals, an analog of the classical Chasles and Poncelet theorems and we show that the virtual billiard dynamics provides a natural framework in the study of billiards within quadrics in projective spaces, in particular of billiards within ellipsoids on the sphere ${\mathbb{S}^{n - 1}}$ and the Lobachevsky space $\mathbb H^{n-1}$.

Thermodynamics of a time-dependent and dissipative oval billiard: A heat transfer and billiard approach

We study some statistical properties for the behavior of the average squared velocity-hence the temperature-for an ensemble of classical particles moving in a billiard whose boundary is time dependent. We assume the collisions of the particles with the boundary of the billiard are inelastic, leading the average squared velocity to reach a steady-state dynamics for large enough time. The description of the stationary state is made by using two different approaches: (i) heat transfer motivated by the Fourier law and (ii) billiard dynamics using either numerical simulations and theoretical description.

The billiard inside an ellipse deformed by the curvature flow

The billiard dynamics inside an ellipse is integrable. It has zero topological entropy, four separatrices in the phase space, and a continuous family of convex caustics: the confocal ellipses. We prove that the curvature flow destroys the integrability, increases the topological entropy, splits the separatrices in a transverse way, and breaks all resonant convex caustics.

Dynamical properties of the soft-wall elliptical billiard.

Physical systems such as optical traps and microwave cavities are realistically modeled by billiards with soft walls. In order to investigate the influence of the wall softness on the billiard dynamics, we study numerically a smooth two-dimensional potential well that has the elliptical (hard-wall) billiard as a limiting case. Considering two parameters, the eccentricity of the elliptical equipotential curves and the wall hardness, which defines the steepness of the well, we show that (1) whereas the hard-wall limit is integrable and thus completely regular, the soft wall elliptical billiard exhibits chaos, (2) the chaotic fraction of the phase space depends nonmonotonically on the hardness of the wall, and (3) the effect of the hardness on the dynamics depends strongly on the eccentricity of the billiard. We further show that the limaçon billiard can exhibit enhanced chaos induced by wall softness, which suggests that our findings generalize to quasi-integrable systems.

Universal Hitting Time Statistics for Integrable Flows

The perceived randomness in the time evolution of “chaotic” dynamical systems can be characterized by universal probabilistic limit laws, which do not depend on the fine features of the individual system. One important example is the Poisson law for the times at which a particle with random initial data hits a small set. This was proved in various settings for dynamical systems with strong mixing properties. The key result of the present study is that, despite the absence of mixing, the hitting times of integrable flows also satisfy universal limit laws which are, however, not Poisson. We describe the limit distributions for “generic” integrable flows and a natural class of target sets, and illustrate our findings with two examples: the dynamics in central force fields and ellipse billiards. The convergence of the hitting time process follows from a new equidistribution theorem in the space of lattices, which is of independent interest. Its proof exploits Ratner’s measure classification theorem for unipotent flows, and extends earlier work of Elkies and McMullen.

Volumes of strata of moduli spaces of quadratic differentials: getting explicit values

The volumes of strata of Abelian or quadratic differentials play an important role in the study of dynamics on flat surfaces, related to dynamics in polygonal billiards. This article reviews all known ways to compute volumes in the quadratic case and provides explicit values of volumes of the strata of meromorphic quadratic differentials with at most simple poles in all low dimensions.

衝突，接触，滑り摩擦および有限回転運動を考慮した3Dビリヤード力学

衝突，接触，滑り摩擦および有限回転運動を考慮した3Dビリヤード力学

Violation of adiabaticity in magnetic billiards due to separatrix crossings.

We consider dynamics of magnetic billiards with curved boundaries and strong inhomogeneous magnetic field. We investigate a violation of adiabaticity of charged particle motion in this system. The destruction of the adiabatic invariance is due to the change of type of the particle trajectory: particles can drift along the boundary reflecting from it or rotate around the magnetic field at some distance from the boundary without collisions with it. Trajectories of these two types are demarcated in the phase space by a separatrix. Crossings of the separatrix result in jumps of the adiabatic invariant. We derive an asymptotic formula for such a jump and demonstrate that an accumulation of these jumps leads to the destruction of the adiabatic invariance.

The semiclassical theory of discontinuous systems and ray-splitting billiards

We analyze the semiclassical limit of spectral theory on manifolds whose metrics have jump-like discontinuities. Such systems are quite different from manifolds with smooth Riemannian metrics because the semiclassical limit does not relate to a classical flow but rather to branching (ray-splitting) billiard dynamics. In order to describe this system we introduce a dynamical system on the space of functions on phase space. To identify the quantum dynamics in the semiclassical limit we compute the principal symbols of the Fourier integral operators associated to reflected and refracted geodesic rays and identify the relation between classical and quantum dynamics. In particular we prove a quantum ergodicity theorem for discontinuous systems. In order to do this we introduce a new notion of ergodicity for the ray-splitting dynamics.

Chaotic billiard dynamics for herding

Chaotic billiard dynamics for herding

Computing Mather's $\beta$-function for Birkhoff billiards

This article is concerned with the study of Mather's $\beta$-function associated to Birkhoff billiards. This function corresponds to the minimal average action of orbits with a prescribed rotation number and, from a different perspective, it can be related to the maximal perimeter of periodic orbits with a given rotation number, the so-called Marked length spectrum . After having recalled its main properties and its relevance to the study of the billiard dynamics, we stress its connections to some intriguing open questions: Birkhoff conjecture and the isospectral rigidity of convex billiards. Both these problems, in fact, can be conveniently translated into questions on this function. This motivates our investigation aiming at understanding its main features and properties. In particular, we provide an explicit representation of the coefficients of its (formal) Taylor expansion at zero, only in terms of the curvature of the boundary. In the case of integrable billiards, this result provides a representation formula for the $\beta$-function near $0$. Moreover, we apply and check these results in the case of circular and elliptic billiards.

A boundary integral formalism for stochastic ray tracing in billiards.

Determining the flow of rays or non-interacting particles driven by a force or velocity field is fundamental to modelling many physical processes. These include particle flows arising in fluid mechanics and ray flows arising in the geometrical optics limit of linear wave equations. In many practical applications, the driving field is not known exactly and the dynamics are determined only up to a degree of uncertainty. This paper presents a boundary integral framework for propagating flows including uncertainties, which is shown to systematically interpolate between a deterministic and a completely random description of the trajectory propagation. A simple but efficient discretisation approach is applied to model uncertain billiard dynamics in an integrable rectangular domain.

Bicentennial of the Great Poncelet Theorem (1813–2013): Current advances

We present very recent results related to the Poncelet Theorem on the occasion of its bicentennial. We are telling the story of one of the most beautiful theorems of geometry, recalling for general mathematical audiences the dramatic historic circumstances which led to its discovery, a glimpse of its intrinsic appeal, and the importance of its relationship to dynamics of billiards within confocal conics. We focus on the three main issues: A) The case of pseudo-Euclidean spaces, for which we present a recent notion of relativistic quadrics and apply it to the description of periodic trajectories of billiards within quadrics. B) The relationship between so-called billiard algebra and the foundations of modern discrete differential geometry which leads to double-reflection nets. C) We present a new class of dynamical systems—pseudo-integrable billiards generated by a boundary composed of several arcs of confocal conics having nonconvex angles. The dynamics of such billiards have several extraordinary properties, which are related to interval exchange transformations and which generate families of flows that are minimal but not uniquely ergodic. This type of dynamics provides a novel type of Poncelet porisms—the local ones.

Ergodicity of the generalized lemon billiards

In this paper, we study a two-parameter family of convex billiard tables, by taking the intersection of two round disks (with different radii) in the plane. These tables give a generalization of the one-parameter family of lemon-shaped billiards. Initially, there is only one ergodic table among all lemon tables. In our generalized family, we observe numerically the prevalence of ergodicity among the some perturbations of that table. Moreover, numerical estimates of the mixing rate of the billiard dynamics on some ergodic tables are also provided.