Volume 22, Issue 2, June 2012

The dynamics in weakly chaotic Hamiltonian systems strongly depends on initial conditions (ICs) and little can be affirmed about generic behaviors. Using two distinct Hamiltonian systems, namely one particle in an open rectangular billiard and four particles globally coupled on a discrete lattice, we show that in these models, the transition from integrable motion to weak chaos emerges via chaotic stripes as the nonlinear parameter is increased. The stripes represent intervals of initial conditions which generate chaotic trajectories and increase with the nonlinear parameter of the system. In the billiard case, the initial conditions are the injection angles. For higherdimensional systems and small nonlinearities, the chaotic stripes are the initial condition inside which Arnold diffusion occurs.
 FOCUS ISSUE: STATISTICAL MECHANICS AND BILLIARDTYPE DYNAMICAL SYSTEMS


Introduction to Focus Issue: Statistical mechanics and billiardtype dynamical systems
View Description Hide DescriptionDynamical systems of the billiard type are of fundamental importance for the description of numerous phenomena observed in many different fields of research, including statistical mechanics, Hamiltonian dynamics, nonlinear physics, and many others. This Focus Issue presents the recent progress in this area with contributions from the mathematical as well as physical stand point.

Billiards: A singular perturbation limit of smooth Hamiltonian flows
View Description Hide DescriptionNonlinear multidimensional Hamiltonian systems that are not near integrable typically have mixed phase space and a plethora of instabilities. Hence, it is difficult to analyze them, to visualize them, or even to interpret their numerical simulations. We survey an emerging methodology for analyzing a class of such systems: Hamiltonians with steep potentials that limit to billiards.

Many faces of stickiness in Hamiltonian systems
View Description Hide DescriptionWe discuss the phenomenon of stickiness in Hamiltonian systems. By visual examples of billiards, it is demonstrated that one must make a difference between internal (within chaotic sea(s)) and external (in vicinity of KAM tori) stickiness. Besides, there exist two types of KAMislands, elliptic and parabolic ones, which demonstrate different abilities of stickiness.

Statistical properties of the system of two falling balls
View Description Hide DescriptionWe consider the motion of two point masses along a vertical halfline that are subject to constant gravitational force and collide elastically with each other and the floor. This model was introduced by Wojtkowski who established hyperbolicity and ergodicity in case the lower ball is heavier. Here, we investigate the dynamics in discrete time and prove that, for an open set of the external parameter (the relative mass of the lower ball), the system mixes polynomially—modulo logarithmic factors, correlations decay as —and satisfies the Central Limit Theorem.

Stable regimes for hard disks in a channel with twisting walls
View Description Hide DescriptionWe study a gas of N hard disks in a box with semiperiodic boundary conditions. The unperturbed gas is hyperbolic and ergodic (these facts are proved for N = 2 and expected to be true for all ). We study various perturbations by twisting the outgoing velocity at collisions with the walls. We show that the dynamics tends to collapse to various stable regimes, however we define the perturbations, and however small they are.

Chaos in the square billiard with a modified reflection law
View Description Hide DescriptionThe purpose of this paper is to study the dynamics of a square billiard with a nonstandard reflection law such that the angle of reflection of the particle is a linear contraction of the angle of incidence. We present numerical and analytical arguments that the nonwandering set of this billiard decomposes into three invariant sets, a parabolic attractor, a chaotic attractor, and a set consisting of several horseshoes. This scenario implies the positivity of the topological entropy of the billiard, a property that is in sharp contrast with the integrability of the square billiard with the standard reflection law.

Structure and evolution of strange attractors in nonelastic triangular billiards
View Description Hide DescriptionWe study nonelastic billiard dynamics in an equilateral triangular table. In such dynamics, collisions with the walls of the table are not elastic, as in standard billiards; rather, the outgoing angle of the trajectory with the normal vector to the boundary at the point of collision is a uniform factor λ < 1 smaller than the incoming angle. This leads to contraction in phase space for the discretetime dynamics between consecutive collisions, and hence to attractors of zero Lebesgue measure, which are almost always fractal strange attractors with chaotic dynamics, due to the presence of an expansion mechanism. We study the structure of these strange attractors and their evolution as the contraction parameter λ is varied. For , we prove rigorously that the attractor has the structure of a Cantor set times an interval, whereas for larger values of λ gaps arise in the Cantor structure. For λ close to 1, the attractor splits into three transitive components, whose basins of attraction have fractal boundaries.

Resonances within chaos
View Description Hide DescriptionA chaotic system under periodic forcing can develop a periodically visited strange attractor. We discuss simple models in which the phenomenon, quite easy to see in numerical simulations, can be completely studied analytically.

Billiards with a given number of (k,n)orbits
View Description Hide DescriptionWe consider billiard dynamics inside a smooth strictly convex curve. For each pair of integers (k,n), we focus our attention on the billiard trajectory that traces a closed polygon with n sides and makes k turns inside the billiard table, called a (k,n)orbit. Birkhoff proved that a strictly convex billiard always has at least two (k,n)orbits for any relatively prime integers k and n such that . In this paper, we show that Birkhoff’s lower bound is optimal by presenting examples of strictly convex billiards with exactly two (k,n)orbits. We generalize the result to billiards with given even numbers of orbits for a finite number of periods.

Classification of symmetric periodic trajectories in ellipsoidal billiards
View Description Hide DescriptionWe classify nonsingular symmetric periodic trajectories (SPTs) of billiards inside ellipsoids of without any symmetry of revolution. SPTs are defined as periodic trajectories passing through some symmetry set. We prove that there are exactly classes of such trajectories. We have implemented an algorithm to find minimal SPTs of each of the 12 classes in the 2D case ( ) and each of the 112 classes in the 3D case ( ). They have periods 3, 4, or 6 in the 2D case and 4, 5, 6, 8, or 10 in the 3D case. We display a selection of 3D minimal SPTs. Some of them have properties that cannot take place in the 2D case.

Three unequal masses on a ring and soft triangular billiards
View Description Hide DescriptionThe dynamics of three soft interacting particles on a ring is shown to correspond to the motion of one particle inside a soft triangular billiard. The dynamics inside the soft billiard depends only on the masses ratio between particles and softness ratio of the particles interaction. The transition from soft to hard interactions can be appropriately explored using potentials for which the corresponding equations of motion are well defined in the hard wall limit. Numerical examples are shown for the soft Todalike interaction and the error function.

Chaotic and Arnold stripes in weakly chaotic Hamiltonian systems
View Description Hide DescriptionThe dynamics in weakly chaotic Hamiltonian systems strongly depends on initial conditions (ICs) and little can be affirmed about generic behaviors. Using two distinct Hamiltonian systems, namely one particle in an open rectangular billiard and four particles globally coupled on a discrete lattice, we show that in these models, the transition from integrable motion to weak chaos emerges via chaotic stripes as the nonlinear parameter is increased. The stripes represent intervals of initial conditions which generate chaotic trajectories and increase with the nonlinear parameter of the system. In the billiard case, the initial conditions are the injection angles. For higherdimensional systems and small nonlinearities, the chaotic stripes are the initial condition inside which Arnold diffusion occurs.

Quantifying intermittency in the open drivebelt billiard
View Description Hide DescriptionA “drivebelt” stadium billiard with boundary consisting of circular arcs of differing radius connected by their common tangents shares many properties with the conventional “straight” stadium, including hyperbolicity and mixing, as well as intermittency due to marginally unstable periodic orbits (MUPOs). Interestingly, the roles of the straight and curved sides are reversed. Here, we discuss intermittent properties of the chaotic trajectories from the point of view of escape through a hole in the billiard, giving the exact leading order coefficient of the survival probability P(t) which is algebraic for fixed hole size. However, in the natural scaling limit of small hole size inversely proportional to time, the decay remains exponential. The big distinction between the straight and drivebelt stadia is that in the drivebelt case, there are multiple families of MUPOs leading to qualitatively new effects. A further difference is that most marginal periodic orbits in this system are oblique to the boundary, thus permitting applications that utilise total internal reflection such as microlasers.

Effect of noise in open chaotic billiards
View Description Hide DescriptionWe investigate the effect of whitenoise perturbations on chaotic trajectories in open billiards. We focus on the temporal decay of the survival probability for generic mixedphasespace billiards. The survival probability has a total of five different decay regimes that prevail for different intermediate times. We combine new calculations and recent results on noise perturbed Hamiltonian systems to characterize the origin of these regimes and to compute how the parameters scale with noise intensity and billiard openness. Numerical simulations in the annular billiard support and illustrate our results.

Lorentz process with shrinking holes in a wall
View Description Hide DescriptionWe ascertain the diffusively scaled limit of a periodic Lorentz process in a strip with an almost reflecting wall at the origin. Here, almost reflecting means that the wall contains a small hole waning in time. The limiting process is a quasireflected Brownian motion, which is Markovian, but not strong Markovian. Local time results for the periodic Lorentz process, having independent interest, are also found and used.

Billiard dynamics: An updated survey with the emphasis on open problems
View Description Hide DescriptionThis is an updated and expanded version of our earlier survey article [E. Gutkin, “Billiard dynamics: a survey with the emphasis on open problems,” Regular Chaotic Dyn. 8, 1–13 (2003)]. Section I introduces the subject matter. Sections II–IV expose the basic material following the paradigm of elliptic, hyperbolic, and parabolic billiard dynamics. In Sec. V, we report on the recent work pertaining to the problems and conjectures exposed in the survey [E. Gutkin, “Billiard dynamics: a survey with the emphasis on open problems,” Regular Chaotic Dyn. 8, 1–13 (2003)]. Besides, in Sec. V we formulate a few additional problems and conjectures. The bibliography has been updated and considerably expanded.

A twostage approach to relaxation in billiard systems of locally confined hard spheres
View Description Hide DescriptionWe consider the threedimensional dynamics of systems of many interacting hard spheres, each individually confined to a dispersive environment, and show that the macroscopic limit of such systems is characterized by a coefficient of heat conduction whose value reduces to a dimensional formula in the limit of vanishingly small rate of interaction. It is argued that this limit arises from an effective loss of memory. Similarities with the diffusion of a tagged particle in binary mixtures are emphasized.

Spreading of energy in the DingDong model
View Description Hide DescriptionWe study the properties of energy spreading in a lattice of elastically colliding harmonic oscillators (DingDong model). We demonstrate that in the regular lattice the spreading from a localized initial state is mediated by compactons and chaotic breathers. In a disordered lattice, the compactons do not exist, and the spreading eventually stops, resulting in a finite configuration with a few chaotic spots.

Fermi acceleration in timedependent rectangular billiards due to multiple passages through resonances
View Description Hide DescriptionWe consider a slowly rotating rectangular billiard with moving boundaries and use canonical perturbation theory to describe the dynamics of a billiard particle. In the process of slow evolution, certain resonance conditions can be satisfied. Correspondingly, phenomena of scattering on a resonance and capture into a resonance happen in the system. These phenomena lead to destruction of adiabatic invariance and to unlimited acceleration of the particle.

A consistent approach for the treatment of Fermi acceleration in timedependent billiards
View Description Hide DescriptionThe standard description of Fermi acceleration, developing in a class of timedependent billiards, is given in terms of a diffusion process taking place in momentum space. Within this framework, the evolution of the probability density function (PDF) of the magnitude of particle velocities as a function of the number of collisions n is determined by the FokkerPlanck equation (FPE). In the literature, the FPE is constructed by identifying the transport coefficients with the ensemble averages of the change of the magnitude of particle velocity and its square in the course of one collision. Although this treatment leads to the correct solution after a sufficiently large number of collisions have been reached, the transient part of the evolution of the PDF is not described. Moreover, in the case of the FermiUlam model (FUM), if a standard simplification is employed, the solution of the FPE is even inconsistent with the values of the transport coefficients used for its derivation. The goal of our work is to provide a selfconsistent methodology for the treatment of Fermi acceleration in timedependent billiards. The proposed approach obviates any assumptions for the continuity of the random process and the existence of the limits formally defining the transport coefficients of the FPE. Specifically, we suggest, instead of the calculation of ensemble averages, the derivation of the onestep transition probability function and the use of the ChapmanKolmogorov forward equation. This approach is generic and can be applied to any timedependent billiard for the treatment of Fermiacceleration. As a first step, we apply this methodology to the FUM, being the archetype of timedependent billiards to exhibit Fermi acceleration.
