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On the structure of the singularity manifolds of dispersing billiards  [PDF]
Pavel Batchourine
Mathematics , 2005,
Abstract: In this paper we study the singularity manifolds of multidimensional strictly dispersing billiards and show that the proof of the Fundamental theorem for dispersing billiards remain valid for a dense set of finitely smooth scatterers.
An expansion estimate for dispersing planar billiards with corner points  [PDF]
Jacopo De Simoi,Imre Péter Tóth
Mathematics , 2012,
Abstract: It is known that the dynamics of planar billiards satisfies strong mixing properties (e.g. exponential decay of correlations) provided that some expansion condition on unstable curves is satisfied. This condition has been shown to always hold for smooth dispersing planar billiards, but it needed to be assumed separately in the case of dispersing planar billiards with corner points. We prove that this expansion condition holds for any dispersing planar billiard with corner points, no cusps and bounded horizon.
The Role of Diffraction in the Quantization of Dispersing Billiards  [PDF]
Harel Primack,Holger Schanz,Uzy Smilansky,Iddo Ussishkin
Physics , 1995, DOI: 10.1103/PhysRevLett.76.1615
Abstract: We study diffraction corrections to the semiclassical spectral density of dispersing (Sinai) billiards. They modify the contributions of periodic orbits (PO's), with at least one segment which is almost tangent to the concave part of the boundary. Given a wavenumber $k$, all the PO's with length up to the Heisenberg length $O(k)$ are required for quantization. We show that most of the contributions of PO's which are longer than a limit $O(k^{2/3})$ must be corrected for diffraction effects. For orbits which just miss tangency, the corrections are of the same magnitude as the semiclassical contributions themselves. Orbits which bounce at extreme forward angles give very small terms in the standard semiclassical theory. The diffraction corrections increase their amplitude substantially.
Scale Anomaly and Quantum Chaos in the Billiards with Pointlike Scatterers  [PDF]
Taksu Cheon,Takaomi Shigehara
Physics , 1995, DOI: 10.1103/PhysRevE.54.3300
Abstract: We argue that the random-matrix like energy spectra found in pseudointegrable billiards with pointlike scatterers are related to the quantum violation of scale invariance of classical analogue system. It is shown that the behavior of the running coupling constant explains the key characteristics of the level statistics of pseudointegrable billiards.
Lyapunov spectra of billiards with cylindrical scatterers: comparison with many-particle systems  [PDF]
Astrid S. de Wijn
Physics , 2005, DOI: 10.1103/PhysRevE.72.026216
Abstract: The dynamics of a system consisting of many spherical hard particles can be described as a single point particle moving in a high-dimensional space with fixed hypercylindrical scatterers with specific orientations and positions. In this paper, the similarities in the Lyapunov exponents are investigated between systems of many particles and high-dimensional billiards with cylindrical scatterers which have isotropically distributed orientations and homogeneously distributed positions. The dynamics of the isotropic billiard are calculated using a Monte-Carlo simulation, and a reorthogonalization process is used to find the Lyapunov exponents. The results are compared to numerical results for systems of many hard particles as well as the analytical results for the high-dimensional Lorentz gas. The smallest three-quarters of the positive exponents behave more like the exponents of hard-disk systems than the exponents of the Lorentz gas. This similarity shows that the hard-disk systems may be approximated by a spatially homogeneous and isotropic system of scatterers for a calculation of the smaller Lyapunov exponents, apart from the exponent associated with localization. The method of the partial stretching factor is used to calculate these exponents analytically, with results that compare well with simulation results of hard disks and hard spheres.
Proving The Ergodic Hypothesis for Billiards With Disjoint Cylindric Scatterers  [PDF]
Nandor Simanyi
Mathematics , 2002, DOI: 10.1088/0951-7715/17/1/001
Abstract: In this paper we study the ergodic properties of mathematical billiards describing the uniform motion of a point in a flat torus from which finitely many, pairwise disjoint, tubular neighborhoods of translated subtori (the so called cylindric scatterers) have been removed. We prove that every such system is ergodic (actually, a Bernoulli flow), unless a simple geometric obstacle for the ergodicity is present.
Semi-dispersing billiards with an infinite cusp  [PDF]
Marco Lenci
Physics , 2001, DOI: 10.1007/s00220-002-0710-7
Abstract: Let $f: [0, +\infty) \to (0, +\infty)$ be a sufficiently smooth convex function, vanishing at infinity. Consider the planar domain $Q$ delimited by the positive $x$-semiaxis, the positive $y$-semiaxis, and the graph of $f$. Under certain conditions on $f$, we prove that the billiard flow in $Q$ has a hyperbolic structure and, for some examples, that it is also ergodic. This is done using the cross section corresponding to collisions with the dispersing part of the boundary. The relevant invariant measure for this Poincar\'e section is infinite, whence the need to surpass the existing results, designed for finite-measure dynamical systems.
A Supersymmetry approach to billiards with randomly distributed scatterers  [PDF]
H. -J. Stoeckmann
Physics , 2002, DOI: 10.1088/0305-4470/35/25/302
Abstract: The density of states for a chaotic billiard with randomly distributed point-like scatterers is calculated, doubly averaged over the positions of the impurities and the shape of the billiard. Truncating the billiard Hamiltonian to a N x N matrix, an explicit analytic expression is obtained for the case of broken time-reversal symmetry, depending on rank N of the matrix, number L of scatterers, and strength of the scattering potential. In the strong coupling limit a discontinuous change is observed in the density of states as soon as L exceeds N.
Decay of correlations and invariance principles for dispersing billiards with cusps, and related planar billiard flows  [PDF]
Péter Bálint,Ian Melbourne
Statistics , 2008, DOI: 10.1007/s10955-008-9623-y
Abstract: Following recent work of Chernov, Markarian, and Zhang, it is known that the billiard map for dispersing billiards with zero angle cusps has slow decay of correlations with rate 1/n. Since the collisions inside a cusp occur in quick succession, it is reasonable to expect a much faster decay rate in continuous time. In this paper we prove that the flow is rapid mixing: correlations decay faster than any polynomial rate. A consequence is that the flow admits strong statistical properties such as the almost sure invariance principle, even though the billiard map does not. The techniques in this paper yield new results for other standard examples in planar billiards, including Bunimovich flowers and stadia.
Upgrading the Local Ergodic Theorem for planar semi-dispersing billiards  [PDF]
N. Chernov,N. Simanyi
Statistics , 2007, DOI: 10.1007/s10955-010-9927-6
Abstract: The Local Ergodic Theorem (also known as the `Fundamental Theorem') gives sufficient conditions under which a phase point has an open neighborhood that belongs (mod 0) to one ergodic component. This theorem is a key ingredient of many proofs of ergodicity for billiards and, more generally, for smooth hyperbolic maps with singularities. However the proof of that theorem relies upon a delicate assumption (Chernov-Sinai Ansatz), which is difficult to check for some physically relevant models, including gases of hard balls. Here we give a proof of the Local Ergodic Theorem for two dimensional billiards without using the Ansatz.
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