Numerical experiments on quantum chaotic billiards

A recently proposed numerical technique for generation of high-quality unstructured meshes is combined with a finite-element method to solve the Helmholtz equation that describes the quantum mechanics of a particle confined in two-dimensional cavities. Different shapes are treated on equal footing, including Sinai, stadium, annular, threefold symmetric, mushroom, cardioid, triangle, and coupled billiards. The results are shown to be in excellent agreement with available measurements in flat microwave resonator counterparts with nonintegrable geometries.     

Numerical study of aD-dimensional periodic Lorentz gas with universal properties

We give the results of a numerical study of the motion of a point particle in ad-dimensional array of spherical scatterers (Sinai's billiard without horizon). We find a simple universal law for the Lyapounov exponent (as a function ofd) and a stretched exponential decay for the velocity autocorrelation as a function of the number of collisions. The diffusion seems to be anomalous in this problem. Ergodicity is used to predict the shape of the probability distribution of long free paths. Physical interpretations or clues are proposed.     

Recurrence of particles in static and time varying oval billiards

Dynamical properties are studied for escaping particles, injected through a hole in an oval billiard. The dynamics is considered for both static and periodically moving boundaries. For the static boundary, two different decays for the recurrence time distribution were observed after exponential decay for short times: A changeover to: (i) power law or; (ii) stretched exponential. Both slower decays are due to sticky orbits trapped near KAM islands, with the stretched exponential apparently associated with a single group of large islands. For time dependent case, survival probability leads to the conclusion that sticky orbits are less evident compared with the static case.     

Reaction matrix for Dirichlet billiards with attached waveguides

The reaction matrix of a cavity with attached waveguides connects scattering properties to properties of a corresponding closed billiard for which the waveguides are cut off by straight walls. On the one hand, this matrix is directly related to the S-matrix, on the other hand it can be expressed by a spectral sum over all eigenfunctions of the closed system. However, in the physically relevant situation where these eigenfunctions vanish on the impenetrable boundaries of the closed billiard, the spectral sum for the reaction matrix, as it was used before, fails to converge and does not reliably reproduce the scattering properties. We derive here a convergent representation of the reaction matrix in terms of eigenmodes satisfying Dirichlet boundary conditions and demonstrate its validity in the rectangular and the Sinai billiards.     

Noncoincidence of geodesic lengths and hearing elliptic quantum billiards

Assume that the planar region has aC ¹ boundary and is strictly convex in the sense that the tangent angle determines a point on the boundary. The lengths of invariant circles for the billiard ball map (or caustics) accumulate on ||. It follows from direct calculations and from relations between the lengths of invariant circles and the lengths of trajectories of the billiard ball map that under mild assumptions on the lengths of some geodesics the region satisfies the strong noncoincidence condition. This condition plays a role in recovering the lengths of closed geodesics from the spectrum of the Laplacian. Asymptotics for the lengths of invariant circles and an application to ellipses are discussed. In addition; some examples regarding strong non coincidence are given.     

Microwave experiments on chaotic billiards

We describe experiments using billiard-shaped microwave cavities, to test ideas in quantum chaos. The experimental method for observing cavity resonances to obtain the eigenvalues, and the advantages and limitations of the techniques, including the influence of absorption, are discussed. An experimental technique to obtain a 2D mapping of the wavefunction is described. Results are displayed for 36 of the low-lying wavefunctions of a Sinai billiard cavity consisting of a central disc in a rectangular enclosure. The wavefunctions demonstrate the influence of classical periodic orbits (PO), of which there are two types: non-isolated PO, which avoid the central disc, and isolated PO, which hit the central disc. Scarred states, including those associated with isolated PO, are directly observed.     

Integrability and ergodicity of classical billiards in a magnetic field

We consider classical billiards in plane, connected, but not necessarily bounded domains. The charged billiard ball is immersed in a homogeneous, stationary magnetic field perpendicular to the plane. The part of dynamics which is not trivially integrable can be described by a bouncing map. We compute a general expression for the Jacobian matrix of this map, which allows us to determine stability and bifurcation values of specific periodic orbits. In some cases, the bouncing map is a twist map and admits a generating function. We give a general form for this function which is useful to do perturbative calculations and to classify periodic orbits. We prove that billiards in convex domains with sufficiently smooth boundaries possess invariant tori corresponding to skipping trajectories. Moreover, in strong field we construct adiabatic invariants over exponentially large times. To some extent, these results remain true for a class of nonconvex billiards. On the other hand, we present evidence that the billiard in a square is ergodic for some large enough values of the magnetic field. A numerical study reveals that the scattering on two circles is essentially chaotic.     

Entropy, Lyapunov exponents, and mean free path for billiards

We review known results and derive some new ones about the mean free path, Kolmogorov-Sinai entropy, and Lyapunov exponents for billiard-type dynamical systems. We focus on exact and asymptotic formulas for these quantities. The dynamical systems covered in this paper include the priodic Lorentz gas, the stadium and its modifications, and the gas of hard balls. Some open questions and numerical observations are discussed.     

Integrability and pseudointegrability in billiards illustrated by the harmonic wedge

We discuss Liouville's theorem for nonsmooth integrable systems of the billiard type and give a scheme of calculation of angle-action variables for the flow. We also deal with the problem of pseudointegrability. We discuss the relationship between the continuous-time (flow) and the discrete-time (map) approaches. We treat all these aspects through a specific billiard—a wedge embedded in a two-dimensional isotropic harmonic potential. Varying the parameters provides two integrable and two pseudointegrable cases. It turns out that the dynamics of one of the latter is indeed integrable in a certain sense. We also address the problem of applying perturbation theory.     

A Stretched Exponential Bound on Time Correlations for Billiard Flows

We construct Markov approximations to the billiard flows and establish a stretched exponential bound on time-correlation functions for planar periodic Lorentz gases (also known as Sinai billiards). Precisely, we show that for any (generalized) Hölder continuous functions F,G on the phase space of the flow the time correlation function is bounded by const? \(e^{-a\sqrt{|t|}}\), here t ? ? is the (continuous) time and a > 0.     

Ergodic and statistical properties of piecewise linear hyperbolic automorphisms of the 2-torus

We study generic piecewise linear hyperbolic automorphisms of the 2-torus. We explain why the resulting dynamical system is ergodic and mixing and prove the exponential decay of correlations.     

A model of modulated diffusion. II. Numerical results on statistical properties

We investigate numerically the statistical properties of a model of modulated diffusion for which we have already computed analytically the diffusion coefficientD. Our model is constructed by adding a deterministic or random noise to the frequency of an integrable isochronous system. We consider in particular the central limit theorem and the invariance principle and we show that they follow wheneverD is positive and for any magnitude of the noise; we also investigate the asymptotic distribution in a case whenD=0.     

Statistical properties of dynamical systems with singularities

We prove statistical properties of two-dimensional hyperbolic dynamical systems with singularities. Bunimovich, Sinai, and Chernov proved a theorem on the subexponential decay of correlations and a central limit theorem for billiard systems. In this paper we use their techniques to prove the same results for abstract systems.     

On Decay of Correlations for Unbounded Spin Systems with Arbitrary Boundary Conditions

We propose a method based on cluster expansion to study the truncated correlations of unbounded spin systems uniformly in the boundary condition and in a possible external field. By this method we study the spin–spin truncated correlations of various systems, including the case of infinite range simply integrable interactions, and we show how suitable boundary conditions and/or external fields may improve the decay of the correlations.     

Decay of correlations in surface models

A convergent low-temperature expansion for a variety of models of twodimensional surfaces is presented. It yields existence of the thermodynamic limit for the pressure and correlation functions as well as analyticity inz =e ^– In addition, the estimates give exponential decay of truncated correlations, which proves the existence of a gap in the spectrum of the transfer matrix below the ground state eigenvalue. Two particular examples included in the general framework are the solid-on-solid and discrete Gaussian models.Supported in part by the National Science Foundation under grant No. PHY 79-16812.     

Canonical nonequilibrium ensembles and subdynamics

A method for constructing a canonical nonequilibrium ensemble for systems in which correlations decay exponentially has recently been proposed by Coveney and Penrose. In this paper, we show that the method is equivalent to the subdynamics formalism, developed by Prigogine and others, when the dimension of the subdynamic kinetic subspace is finite. The comparison between the two approaches helps to clarify the nature of the various operators used in the Brussels formalism. We discuss further the relationship between these two approaches, with particular reference to a simple discrete-time dynamical system, based on the baker's transformation, which we call the baker's urn.     

On the slow decay ofO(2) correlations in the absence of topological excitations: Remark on the Patrascioiu-Seiler model

For spin models withO(2)-invariant ferromagnetic interactions, the Patrascioiu-Seiler constraint is |arg(S(x))–arg(S(y))|₀ for all |x–y|=1. It is shown that in two-dimensional systems of two-component spins the imposition of such contraints with ₀ small enough indeed results in the suppression of exponential clustering. More explicitly, it is shown that in such systems on every scale the spin-spin correlation function obeys S(x)·S(y)1/(2|x–y|²) at any temperature, includingT=. The derivation is along the lines proposed by A. Patrascioiu and E. Seiler, with the yet unproven conjectures invoked there replaced by another geometric argument.Dedicated to Oliver Penrose on the occasion of his 65th birthday.     

Witten Laplacian Method for The Decay of Correlations

The aim of this paper is to apply direct methods to the study of integrals that appear naturally in Statistical Mechanics and Euclidean Field Theory. We provide weighted estimates leading to the exponential decay of the two-point correlation functions for certain classical convex unbounded models. The methods involve the study of the solutions of the Witten Laplacian equations associated with the Hamiltonian of the system.     

Non-Fermi-liquid behavior in UCu4+xAl8−x compounds

We report on experimental studies of the Kondo physics and the development of non-Fermi-liquid scaling in UCu_4+xAl_8−x family. We studied 7 different compounds with compositions between x=0 and 2. We measured electrical transport (down to 65 mK) and thermoelectric power (down to 1.8 K) as a function of temperature, hydrostatic pressure, and/or magnetic field.Compounds with Cu content below x=1.25 exhibit long-range antiferromagnetic order at low temperatures. Magnetic order is suppressed with increasing Cu content and our data indicate a possible quantum critical point at x_cr≈1.15. For compounds with higher Cu content, non-Fermi-liquid behavior is observed. Non-Fermi-liquid scaling is inferred from electrical resistivity results for the x=1.25 and 1.5 compounds. For compounds with even higher Cu content, a sharp kink occurs in the resistivity data at low temperatures, and this may be indicative of another quantum critical point that occurs at higher Cu compositions.For the magnetically ordered compounds, hydrostatic pressure is found to increase the Néel temperature, which can be understood in terms of the Kondo physics. For the non-magnetic compounds, application of a magnetic field promotes a tendency toward Fermi-liquid behavior. Thermoelectric power was analyzed using a two-band Lorentzian model, and the results indicate one fairly narrow band (10 meV and below) and a second broad band (around hundred meV). The results imply that there are two relevant energy scales that need to be considered for the physics in this family of compounds.