Billiards with polynomial mixing rates

While many dynamical systems of mechanical origin, in particular billiards, are strongly chaotic -- enjoy exponential mixing, the rates of mixing in many other models are slow (algebraic, or polynomial). The dynamics in the latter are intermittent between regular and chaotic, which makes them particularly interesting in physical studies. However, mathematical methods for the analysis of systems with slow mixing rates were developed just recently and are still difficult to apply to realistic models. Here we reduce those methods to a practical scheme that allows us to obtain a nearly optimal bound on mixing rates. We demonstrate how the method works by applying it to several classes of chaotic billiards with slow mixing as well as discuss a few examples where the method, in its present form, fails.Comment: 39pages, 11 figue

Spectral and spatial observations of microwave spikes and zebra structure in the short radio burst of May 29, 2003

The unusual radio burst of May 29, 2003 connected with the M1.5 flare in AR 10368 has been analyzed. It was observed by the Solar Broadband Radio Spectrometer (SBRS/Huairou station, Beijing) in the 5.2-7.6 GHz range. It proved to be only the third case of a neat zebra structure appearing among all observations at such high frequencies. Despite the short duration of the burst (25 s), it provided a wealth of data for studying the superfine structure with millisecond resolution (5 ms). We localize the site of emission sources in the flare region, estimate plasma parameters in the generation sites, and suggest applicable mechanisms for interpretating spikes and zebra-structure generation. Positions of radio bursts were obtained by the Siberian Solar Radio Telescope (SSRT) (5.7 GHz) and Nobeyama radioheliograph (NoRH) (17 GHz). The sources in intensity gravitated to tops of short loops at 17 GHz, and to long loops at 5.7 GHz. Short pulses at 17 GHz (with a temporal resolution of 100 ms) are registered in the R-polarized source over the N-magnetic polarity (extraordinary mode). Dynamic spectra show that all the emission comprised millisecond pulses (spikes) of 5-10 ms duration in the instantaneous band of 70 to 100 MHz, forming the superfine structure of different bursts, essentially in the form of fast or slow-drift fibers and various zebra-structure stripes. Five scales of zebra structures have been singled out. As the main mechanism for generating spikes (as the initial emission) we suggest the coalescence of plasma waves with whistlers in the pulse regime of interaction between whistlers and ion-sound waves. In this case one can explain the appearance of fibers and sporadic zebra-structure stripes exhibiting the frequency splitting.Comment: 11 pages, 5 figures, in press; A&A 201

Beam propagation in a Randomly Inhomogeneous Medium

An integro-differential equation describing the angular distribution of beams is analyzed for a medium with random inhomogeneities. Beams are trapped because inhomogeneities give rise to wave localization at random locations and random times. The expressions obtained for the mean square deviation from the initial direction of beam propagation generalize the "3/2 law".Comment: 4 page

Deterministic Weak Localization in Periodic Structures

The weak localization is found for perfect periodic structures exhibiting deterministic classical diffusion. In particular, the velocity autocorrelation function develops a universal quantum power law decay at 4 times Ehrenfest time, following the classical stretched-exponential type decay. Such deterministic weak localization is robust against weak enough randomness (e.g., quantum impurities). In the 1D and 2D cases, we argue that at the quantum limit states localized in the Bravis cell are turned into Bloch states by quantum tunnelling.Comment: 5 pages, 2 figure

Deterministic Walks in Quenched Random Environments of Chaotic Maps

This paper concerns the propagation of particles through a quenched random medium. In the one- and two-dimensional models considered, the local dynamics is given by expanding circle maps and hyperbolic toral automorphisms, respectively. The particle motion in both models is chaotic and found to fluctuate about a linear drift. In the proper scaling limit, the cumulative distribution function of the fluctuations converges to a Gaussian one with system dependent variance while the density function shows no convergence to any function. We have verified our analytical results using extreme precision numerical computations.Comment: 18 pages, 9 figure

Adatom incorporation and step crossing at the edges of 2D nanoislands

Adatom incorporation into the ``faceted'' steps bordering the 2D nanoislands is analyzed. The step permeability and incorporation coefficients are derived for some typical growth situations. It is shown that the step consisting of equivalent straight segments can be permeable even in the case of fast egde migration if there exist factors delaying creation of new kinks. The step consisting of alternating rough and straight segments may be permeable if there is no adatom transport between neighboring segments through the corner diffusion.Comment: 3 pages, one figur

Upgrading the Local Ergodic Theorem for planar semi-dispersing billiards

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.Comment: 17 pages, 2 figure

Chaos and stability in a two-parameter family of convex billiard tables

We study, by numerical simulations and semi-rigorous arguments, a two-parameter family of convex, two-dimensional billiard tables, generalizing the one-parameter class of oval billiards of Benettin--Strelcyn [Phys. Rev. A 17, 773 (1978)]. We observe interesting dynamical phenomena when the billiard tables are continuously deformed from the integrable circular billiard to different versions of completely-chaotic stadia. In particular, we conjecture that a new class of ergodic billiard tables is obtained in certain regions of the two-dimensional parameter space, when the billiards are close to skewed stadia. We provide heuristic arguments supporting this conjecture, and give numerical confirmation using the powerful method of Lyapunov-weighted dynamics.Comment: 19 pages, 13 figures. Submitted for publication. Supplementary video available at http://sistemas.fciencias.unam.mx/~dsanders

Random billiards with wall temperature and associated Markov chains

By a random billiard we mean a billiard system in which the standard specular reflection rule is replaced with a Markov transition probabilities operator P that, at each collision of the billiard particle with the boundary of the billiard domain, gives the probability distribution of the post-collision velocity for a given pre-collision velocity. A random billiard with microstructure (RBM) is a random billiard for which P is derived from a choice of geometric/mechanical structure on the boundary of the billiard domain. RBMs provide simple and explicit mechanical models of particle-surface interaction that can incorporate thermal effects and permit a detailed study of thermostatic action from the perspective of the standard theory of Markov chains on general state spaces. We focus on the operator P itself and how it relates to the mechanical/geometric features of the microstructure, such as mass ratios, curvatures, and potentials. The main results are as follows: (1) we characterize the stationary probabilities (equilibrium states) of P and show how standard equilibrium distributions studied in classical statistical mechanics, such as the Maxwell-Boltzmann distribution and the Knudsen cosine law, arise naturally as generalized invariant billiard measures; (2) we obtain some basic functional theoretic properties of P. Under very general conditions, we show that P is a self-adjoint operator of norm 1 on an appropriate Hilbert space. In a simple but illustrative example, we show that P is a compact (Hilbert-Schmidt) operator. This leads to the issue of relating the spectrum of eigenvalues of P to the features of the microstructure;(3) we explore the latter issue both analytically and numerically in a few representative examples;(4) we present a general algorithm for simulating these Markov chains based on a geometric description of the invariant volumes of classical statistical mechanics