Abstract: The Lorentz gas of Z²-periodic scatterers (or the so called Sinai billiards) can be used to model motion of electrons on a metal. We investigate the linear response for the system under various external forces (during both the flight and the collision). We give some characterizations under which the forced system is time-reversible, and derive an estimate of the electrical current generated by the forced system. Moreover, applying Pesin entropy formula and Young dimension formula, we get several characterizations of the non-equilibrium steady state of the forced system.

Abstract: Sinai-Ruelle-Bowen measures are the only physically observable invariant measures for billiard dynamical systems under small perturbations. These measures are singular, but as it was noted in our previous article [1], marginal distributions of spatial and angular coordinates are absolutely continuous. We generalize these facts and provide full mathematical proofs.

Abstract: Fitting quadratic curves (a.k.a. conic sections, or conics) to data points (digitized images) is a fundamental task in image processing and computer vision. This problem reduces to minimization of a certain function over the parameter space of conics. Here we undertake a thorough investigation of that space and the properties of the objective function on it. Our analysis leads us to many important conclusions relevant to the performance of minimization algorithms.

Abstract: We develop a new algorithm for fitting circles that does not have drawbacks commonly found in existing circle fits. Our fit achieves ultimate accuracy (to machine precision), avoids divergence, and is numerically stable even when fitting circles get arbitrary large. Lastly, our algorithm takes less than 10 iterations to converge, on average.

Abstract: We find an explicit expression for the long time evolution and stationary speed distribution of N point particles in 2D moving under the action of a weak external field E, and undergoing elastic collisions with either a fixed periodic array of convex scatterers, or with virtual random scatterers. The total kinetic energy of the N particles is kept fixed by a Gaussian thermostat which induces an interaction between the particles. We show analytically and numerically that for weak fields this distribution is universal, i.e. independent of the position or shape of the obstacles, \revision{as far as they form a dispersing billiard with finite horizon}, or the nature of the stochastic scattering. Our results are non-perturbative. They exploit the existence of two time scales; the velocity directions become uniformized in times of order unity while the speeds change only on a time scale of O(|E|^-2).

Abstract: We study the problem of projecting 2D points onto quadratic curves (ellipses, hyperbolas, parabolas). We investigate various projection algorithms focusing on those that are mathematically proven to produce (or converge to) correct results in all cases. Our tests demonstrate that those may be still unfit for practical use due to large computational errors. We present two new algorithms that are not only theoretically proven to converge, but achieve nearly perfect accuracy.

Abstract: In regression problems where covariates are subject to errors (albeit small) it often happens that maximum likelihood estimators (MLE) of relevant parameters have infinite moments. We study here circular and elliptic regression, i.e., the problem of fitting circles and ellipses to observed points whose both coordinates are measured with errors. We prove that several popular circle fits due to Pratt, Taubin, and others return estimates of the center and radius that have infinite moments. We also argue that estimators of the ellipse parameters (center and semiaxes) should have infinite moments, too.

Abstract: Dispersing billiards with cusps are deterministic dynamical systems with a mild degree of chaos, exhibiting "intermittent" behavior that alternates between regular and chaotic patterns. They are characterized by decay of correlations of order 1/n and a central limit theorem with a non-classical scaling factor of [n log n]^1/2. We show that the pth moments of the so normalized ergodic sums converge to the moments of the limit normal distribution only for p<2 and diverge for p>2. The critical second moments converge, but their limit is double the second moment of the normal distribution.

Abstract: Fitting straight lines and simple curved objects (circles, ellipses, etc.) to observed data points is a basic task in computer vision and modern statistics (errors-in-variables regression). We have investigated the problem of existence of the best fit in our previous paper (see Chernov et al. (2012)). Here we deal with the issue of uniqueness of the best fit.

Abstract: We study a gas of N hard disks in a box with semi-periodic boundary conditions. The unperturbed gas is hyperbolic and ergodic (these facts are proved for N=2 and expected to be true for all N>2). 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.

Abstract: The technique of “renormalization” for geometric estimation attracted much attention when it was proposed in early 1990s for having higher accuracy than any other then known methods. Later, it was replaced by minimization of the reprojection error. This paper points out that renormalization can be modified so that it outperforms reprojection error minimization. The key fact is that renormalization directly specifies equations to solve, just as the “estimation equation” approach in statistics, rather than minimizing some cost. Exploiting this fact, we modify the problem so that the solution has zero bias up to high order error terms; we call the resulting scheme hyper-renormalization. We apply it to ellipse fitting to demonstrate that it indeed surpasses reprojection error minimization. We conclude that it is the best method available today.

Abstract: Fitting geometric shapes to observed data points (images) is a popular task in computer vision and modern statistics (Errors-In-Variables regression). We investigate the problem of existence of the best fit using geometric and probabilistic approaches.

Abstract: We consider the two dimensional dissipative surface quasi-geostrophic equation on the unit square with mixed boundary conditions. Under some suitable assumptions on the initial stream function, we obtain existence and uniqueness of solutions in the form of a fast converging trigonometric series. We prove that the Fourier coefficients of solutions have a non-uniform decay: in one direction the decay is exponential and along the other direction it is only power like.

Abstract: We investigate analytically and numerically the spatial structure of the non-equilibrium stationary states (NESS) of a point particle moving in a two dimensional periodic Lorentz gas (Sinai Billiard). The particle is subject to a constant external electric field E as well as a Gaussian thermostat which keeps the speed |v| constant. We show that despite the singular nature of the SRB measure its projections on the space coordinates are absolutely continuous. We further show that these projections satisfy linear response laws for small E. Some of them are computed numerically. We compare these results with those obtained from simple models in which the collisions with the obstacles are replaced by random collisions. Similarities and differences are noted.

Abstract: We study the problem of fitting ellipses to observed points in the context of Errors-In-Variables regression analysis. The accuracy of fitting methods is characterized by their variances and biases. The variance has a theoretical lower bound (the KCR bound), and many practical fits attend it, so they are optimal in this sense. There is no lower bound on the bias, though, and in fact our higher order error analysis (developed just recently) shows that it can be eliminated, to the leading order. Kanatani and Rangarajan recently constructed an algebraic ellipse fit that has no bias, but its variance exceeds the KCR bound; so their method is optimal only relative to the bias. We present here a novel ellipse fit that enjoys both optimal features: the theoretically minimal variance and zero bias (both to the leading order). Our numerical tests confirm the superiority of the proposed fit over the existing fits.

Abstract: We study the cutting and packing (C&P) problems in two dimensions by using phi-functions. Our phi-functions describe the layout of given objects; they allow us to construct a mathematical model in which C\&P problems become constrained optimization problems. Here we define (for the first time) a complete class of basic phi-functions which allow us to derive phi-functions for all 2D objects that are formed by linear segments and circular arcs. Our phi-functions support translations and rotations of objects. In order to deal with restrictions on minimal or maximal distances between objects, we also propose adjusted phi-functions. Our phi-functions are expressed by simple linear and quadratic formulas without radicals. The use of radical-free phi-functions allows us to increase efficiency of optimization algorithms. We include several model examples.

Abstract: Dispersing billiards with cusps are deterministic dynamical systems with a mild degree of chaos, exhibiting ``intermittent'' behavior that alternates between regular and chaotic patterns. Their statistical properties are therefore weak and delicate. They are characterized by a slow (power-law) decay of correlations, and as a result the classical central limit theorem fails. We prove that a non-classical central limit theorem holds, with a scaling factor of [n log(n)]^1/2 replacing the standard n^1/2. We also derive the respective Weak Invariance Principle, and we identify the class of observables for which the classical CLT still holds.

Abstract: Regression models in which all variables are subject to errors are known as errors-in-variables (EIV) models. The respective parameter estimates have many unusual properties: their exact distributions are very hard to determine, and their absolute moments are often infinite (so that their mean and variance do not exist). In our paper Error analysis for circle fitting algorithms (see Electr. J. Stat. 3 (2009), pp. 886-911) we developed an unconventional statistical analysis that allowed us to effectively assess EIV parameter estimates and design new methods with superior characteristics. In this paper we validate our approach in a series of numerical tests.

Abstract: We study a nonlinear regression problem of fitting a circle (or a circular arc) to scattered data. We prove that under any standard assumptions on the statistical distribution of errors that are commonly adopted in the literature, the estimates of the circle center and radius have infinite moments. We also discuss methodological implications of this fact.

Abstract: In computer vision one often fits ellipses and other conics to observed points on a plane or ellipsoids/quadrics to spacial point clouds. The most accurate and robust fit is obtained by minimizing geometric (orthogonal) distances, but this problem has no closed form solution and most known algorithms are prohibitively slow. We revisit this issue based on recent advances by S. J. Ahn, D. Eberly, and our own. Ahn has sorted out various approaches and identified the most efficient one. Eberly has developed a fast method of projecting points onto ellipses/ellipsoids (and gave a proof of its convergence). We extend Eberly's projection algorithm to other conics, as well as quadrics in space. We also demonstrate that Eberly's projection method combined with Ahn's most efficient approach (and using Taubin's algebraic fit for initialization) makes a highly efficient fitting scheme working well for all quadratic curves and surfaces.

Abstract: Here we construct adaptive phi-functions for planar objects bounded by straight lines and circular arcs in the presence of upper and/or lower limits on mutual distances.

Abstract: In a planar periodic Lorentz gas, a point particle (electron) moves freely and collides with fixed round obstacles (ions). If a constant force (induced by an electric field) acts on the particle, the latter will accelerate, and its speed will approach infinity. To keep the kinetic energy bounded one can apply a Gaussian thermostat, which forces the particle's speed to be constant. Then an electric current sets in and one can prove Ohm's law and the Einstein relation. However, the Gaussian thermostat has been criticized as unrealistic, because it acts all the time, even during the free flights between collisions. We propose a new model, where during the free flights the electron accelerates, but at the collisions with ions its total energy is reset to a fixed level; thus our thermostat is restricted to the surface of the scatterers (the `walls'). We rederive all physically interesting facts proven for the Gaussian thermostat, including Ohm's law and the Einstein relation. In addition, we investigate the superconductivity phenomenon in the infinite horizon case.

Abstract: Local Ergodic Theorem (also known as `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 Local Ergodic Theorem for two dimensional billiards without using Ansatz.

Abstract: D. Dolgopyat is the winner of the second Brin Prize in Dynamical Systems (2009). This article overviews his remarkable achievements in a non-technical manner. It complements two other surveys of Dolgopyat's work, written by Y. Pesin and C. Liverani and published in this issue. This survey covers Dolgopyat's work on various physical models, including the Lorentz gas, Galton board, and some systems of hard disks.

This is a book focusing on an active research topic in modern statistics and image processing applications, e.g., nuclear physics, engineering, archaeology, medical studies, and other areas where round (circular) objects appear. The book gives a comprehensive coverage of this important and interesting topic, including its theoretical, statistical, computational, and historical aspects. Table of content: 0. Preface. 1. Introduction and historic overview. 2. Fitting lines. 3. Fitting circles: theory. 4. Geometric circle fits. 5. Algebraic circle fits. 6. General statistical analysis of curve fits. 7. Statistical analysis of circle fits. 8. Various "exotic" circle fits.

Abstract: The article is devoted to mathematical models and practical algorithms for solving the cutting and packing (C\&P) problem. We review and further enhance the main tool of our studies -- phi-functions. Those are constructed here for 2D and 3D objects (unlike other standard tools, such as no-fit polygons, which are restricted to the 2D geometry). We also demonstrate that in all realistic cases the phi-functions and their derivatives can be described by quite simple formulas without radicals and other complications. Lastly, a general solution strategy using the phi-functions is outlined and illustrated by several 2D and 3D examples.

Abstract: We prove that for all planar objects bounded by straight lines and circular arcs the phi-functions can be described by simple formulas without radicals.

Abstract: We study electrical current in two-dimensional periodic Lorentz gas in the presence of a weak homogeneous electric field. When the horizon is finite, i.e. the free flights between collisions are bounded, the resulting current J is proportional to the voltage difference E, i.e. J = ½ D^* E + o(||E||), where D^* is the diffusion matrix of the Lorentz particle moving freely without electrical field (see a mathematical proof in Chernov et al 1993). This formula agrees with classical Ohm's law and the Einstein relation. Here we investigate the more difficult model with infinite horizon. We find that infinite corridors between scatterers allow the particles (electrons) move faster resulting in an abnormal current (causing `superconductivity'). Precisely, the current is now given by J = ½ D E | log ||E|| | + O(||E||), where D is the `superdiffusion' matrix of the Lorentz particle moving freely without electrical field. This means that Ohm's law fails in this regime, but the Einstein relation (suitably interpreted) still holds. We also obtain new results for the infinite horizon Lorentz gas without external fields, complementing recent studies by Szasz and Varju (2007).

Abstract: In this paper we describe the results of numerical studies of solutions of the Navier-Stokes System (NSS) under the boundary conditions introduced recently in the paper by Dinaburg et al. First, we investigate the decay of Fourier modes, confirming the results and conjectures made in their paper. Second, we explore the growth of the total energy and enstrophy, which is possible under the adopted boundary conditions. We show that the solutions of the finite-dimensional Galerkin approximations to the NSS may diverge to infinity in finite time, i.e. their energy may blow up.

Abstract: We study a particle moving in R² under a constant (external) force and bouncing off a periodic array of convex domains (scatterers); the latter must satisfy a standard `finite horizon' condition to prevent `ballistic' (collision-free) motion. This model is known to physicists as Galton board (it is also identical to a periodic Lorentz gas). Previous heuristic and experimental studies have suggested that the particle's speed v(t) should grow as t^1/3 and its coordinate x(t) as t^2/3. We prove these conjectures rigorously; we also find limit distributions for the rescaled velocity t^--1/3v(t) and position t^--2/3x(t). In addition, quite unexpectedly, we discover that the particle's motion is recurrent. That means that a ball dropped on an idealized Galton board will roll down but from time to time it should bounce all the way back up (with probability one).

Abstract: We study hyperbolic systems with singularities and prove the coupling lemma and exponential decay of correlations under weaker assumptions than previously adopted in similar studies. Our new approach allows us to study the mixing rates of the reduced map for certain billiard models that could not be handled by the traditional techniques. These models include modified Bunimovich stadia, which are bounded by minor arcs, and flower-type regions that are bounded by major arcs.

Abstract: A classical model of Brownian motion consists of a heavy molecule submerged into a gas of light atoms in a closed container. In this work we study a 2D version of this model, where the molecule is a heavy disk of mass M >> 1 and the gas is represented by just one point particle of mass m=1, which interacts with the disk and the walls of the container via elastic collisions. Chaotic behavior of the particles is ensured by convex (scattering) walls of the container. We prove that the position and velocity of the disk, in an appropriate time scale, converge, as M → ∞, to a Brownian motion (possibly, inhomogeneous); the scaling regime and the structure of the limit process depend on the initial conditions. Our proofs are based on strong hyperbolicity of the underlying dynamics, fast decay of correlations in systems with elastic collisions (billiards), and methods of averaging theory.

Abstract: We present a detailed error analysis for all popular circle fitting methods -- geometric fit, Kasa fit, Pratt fit, and Taubin fit. Our error analysis goes deeper than the traditional expansion to the leading order. We also obtain higher order terms, which show exactly why and by how much circle fits differ from each other. Our analysis allows us to construct a new algebraic (non-iterative) circle fitting algorithm that outperforms all the existing methods, including the (previously regarded as unsurpassable) geometric fit.

Abstract: We study perturbations of Sinai billiards, where a small stationary force acts on the moving particle between its collisions with scatterers. In the previous work, we proved that the collision map preserved a unique Sinai-Ruelle-Bowen (SRB) measure that was Bernoulli and had exponential decay of correlations. Here we add several other statistical properties, including bounds on multiple correlations, the almost sure invariance principle (ASIP), the law of iterated logarithms, and a Kawasaki-type formula. We also show that the corresponding flow is Bernoulli and satisfies a central limit theorem.

Abstract: We consider several classes of chaotic billiards with slow (polynomial) mixing rates, which include Bunimovich's stadium and dispersing billiards with cusps. In recent papers by Markarian and the present authors, estimates on the decay of correlations were obtained that were sub-optimal (they contained a redundant logarithmic factor). We sharpen those estimates by removing that factor.

Abstract: We study a particle moving at unit speed in a channel made by connected self-similar billiard tables that grow in size by a factor r>1 from left to right (this model was recently introduced in physics literature). Let q(T) denote the position of the particle at time T. Our main result is the existence of an asymptotic distribution of q(T)/T as  T → ∞  and  {ln T / ln r} → p   for some  0 ≤ p < 1.

Abstract: We study the problem of fitting circles to scattered data. Unlike many other studies, we assume that the noise is (strongly) correlated; we adopt a particular model where correlations decay exponentially with the distance between data points. Our main results are formulas for the maximum likelihood estimates and their covariance matrix. Our study is motivated by (and applied to) arcs collected during archeological field work.

Abstract: We study a particle moving at unit speed in a self-similar Lorentz billiard channel; the latter consists of an infinite sequence of cells which are identical in shape but growing exponentially in size, from left to right. We present numerical computation of the drift term in this system and establish the logarithmic periodicity of the corrections to the average drift.

Abstract: We study a mechanical model known as Galton board - a particle rolling on a tilted plane under gravitation and bouncing off a periodic array of rigid pegs. Incidentally, this model is identical to a periodic Lorentz gas where an electron is driven by a uniform electric field. Previous heuristic and experimental studies have suggested that the particle's speed v(t) should grow as t^1/3 and its coordinate x(t) as t^2/3. We find exact limit distributions for the rescaled velocity t^--1/3v(t) and position t^--2/3x(t). In addition, we determine that the particle's motion is recurrent, i.e.\ the particle comes back to the top of the board with probability one.

Abstract: Stadia are popular models of chaotic billiards introduced by Bunimovich in 1974. They are analogous to dispersing billiards due to Sinai, but their fundamental technical characteristics are quite different. Recently many new results were obtained for various chaotic billiards, including sharp bounds on correlations and probabilistic limit theorems, and these results require new, more powerful technical apparatus. We present that apparatus here, in the context of stadia, and prove `regularity' properties.

Abstract: 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) Holder continuous functions F,G on the phase space of the flow the time correlation function is bounded by const e^-a|t|1/2, here t is the (continuous) time and a > 0.

Abstract: Dispersing billiards introduced by Sinai are uniformly hyperbolic and have strong statistical properties (exponential decay of correlations and various limit theorems). However, if the billiard table has cusps (corner points with zero interior angles), then its hyperbolicity is nonuniform and statistical properties deteriorate. Until now only heuristic and experiments results existed predicting the decay of correlations as O(1/n). We present a first rigorous analysis of correlations for dispersing billiards with cusps.

Abstract: This work results from our attempts to solve Boltzmann-Sinai's hypothesis about the ergodicity of hard ball gases. A crucial element in the studies of the dynamics of hard balls is the analysis of special hypersurfaces in the phase space consisting of degenerate trajectories (which lack complete hyperbolicity). We prove that if a flow-invariant hypersurface J in the phase space of a semi-dispersing billiard has a negative Lyapunov function, then the volume of the forward image of J grows at least linearly in time. Our proof is independent of the solution of the Boltzmann-Sinai hypothesis, and we provide a complete and self-contained argument here.

Abstract: We investigate numerical schemes for estimating parameters in computer vision problems (HEIV, FNS, renormalization method, and others). We prove mathematically that these algorithms converge rapidly, provided the noise is small. In fact, in just 1-2 iterations they achieve maximum possible statistical accuracy. Our results are supported by a numerical experiment. We also discuss the performance of these algorithms when the noise increases and/or outliers are present.

Abstract: Let X be a Poisson random variable with parameter λ. We prove that the variance of its square root, i.e. the variance of Y=X^1/2, approaches a constant (=0.25) as λ → ∞.

Information: This a 320 pages long monograph, it presents the fundamentals of the mathematical theory of chaotic billiards developed by Ya. Sinai and his school (L. Bunimovich et al.). The book covers all the basic facts, provides full proofs, intuitive explanations, and plenty of illustrations (about 130 figures). The reader can learn the theory and master its techniques by working on hundreds of exercises included in every chapter. The book starts with the most elementary examples and formal definitions, and then takes the reader step by step into the depth of Sinai's theory of hyperbolicity and ergodicity of chaotic billiards, as well as more recent achievements related to their statistical properties (decay of correlations and limit theorems). Appendices provide basic definitions and facts from ergodic theory, probability theory, and measure theory.

Abstract: Mathematical theory of billiards is a fascinating subject providing a fertile source of new problems as well as conjecture testing in dynamics, geometry, mathematical physics and spectral theory. This survey is devoted to planar hyperbolic billiards with emphasis on their applications in statistical physics, where they provide many physically interesting and mathematically tractable models.

Abstract: A new approach to statistical properties of hyperbolic dynamical systems emerged recently; it was introduced by L.-S. Young and modified by D. Dolgopyat. It is based on coupling method borrowed from probability theory. We apply it here to one of the most physically interesting models - Sinai billiards. It allows us to derive a series of new results, as well as make significant improvements in the existing results. First we establish sharp bounds on correlations (including multiple correlations). Then we use our correlation bounds to obtain the central limit theorem (CLT), the almost sure invariance principle (ASIP), the law of iterated logarithms, and integral tests.

Abstract: This work is devoted to 2D dispersing billiards with smooth boundary, i.e. periodic Lorentz gases (with and without horizon). We revisit several fundamental properties of these systems and make a number of improvements. The necessity of such improvements became obvious during our recent studies of gases of several particles. We prove here that local (stable and unstable) manifolds, as well as singularity curves, have uniformly bounded derivatives of all orders. We establish sharp estimates on the size of local manifolds, on distortion bounds, and on the Jacobian of the holonomy map.

Abstract: We describe a one-parameter family of dispersing (hence hyperbolic, ergodic and mixing) billiards where the correlation function of the collision map decays as 1/n^a (here n denotes the discrete time), in which the degree a>1 changes continuously with the parameter of the family, β. We also derive an explicit relation between the degree a and the family parameter β.

Abstract: 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.

Abstract: Fitting standard shapes or curves to incomplete data (which represent only a small part of the curve) is a notoriously difficult problem. Even if the curve is quite simple, such as an ellipse or a circle, it is hard to reconstruct it from noisy data sampled along a short arc. Here we study the least squares fit (LSF) of circular arcs to incomplete scattered data. We analyze theoretical aspects of the problem and reveal the cause of unstable behavior of conventional algorithms. We also find a remedy that allows us to build another algorithm that accurately fits circles to data sampled along arbitrarily short arcs.

Abstract: We consider a heavy piston in an infinite cylinder surrounded by ideal gases on both sides. The piston moves under elastic collisions with gas atoms. We assume here that the gases always exert equal pressures on the piston, hence the piston remains at the so called mechanical equilibrium. However, the temperatures and densities of the gases may differ across the piston. In that case some earlier studies by Gruber, Piasecki and others reveal a very slow motion (drift) of the piston in the direction of the hotter gas. At the same time the hotter gas slowly transfers its energy (heat) across the piston to the cooler gas. While the previous studies of this interesting phenomenon were only heuristic or experimental, we provide first rigorous proofs assuming that the velocity distribution of the ideal gas satisfies a certain `cutoff' condition.

Abstract: We analyze the stability of stationary solutions of a singular Vlasov type hydrodynamic equation (HE). This equation was derived (under suitable assumptions) as the hydrodynamical scaling limit of the Hamiltonian evolution of a system consisting of a massive piston immersed in an ideal gas of point particles in a box. We find explicit criteria for global stability as well as a class of solutions which are linearly unstable for a dense set of parameter values. We present evidence (but no proof) that when the mechanical system has initial conditions `close' to stationary stable solutions of the HE then it stays close to these solutions for a time which is long compared to that for which the equations have been derived. On the other hand if the initial state of the particle system is close to an unstable stationary solutions of the HE the mechanical motion follows for an extended time a perturbed solution of that equation: we find such approximate periodic solutions that are linearly stable.

Abstract: We study a popular algorithm for fitting polynomial curves to scattered data based on the least squares with gradient weights. We show that sometimes this algorithm admits a substantial reduction of complexity, and, furthermore, find precise conditions under which this is possible. It turns out that this is, indeed, possible when one fits circles but not ellipses or hyperbolas.

Abstract: We study the problem of fitting parameterized curves to noisy data. Under certain assumptions (known as Cartesian and radial functional models), we derive asymptotic expressions for the bias and the covariance matrix of the parameter estimates. We also extend Kanatani's version of the Cramer-Rao lower bound, which he proved for unbiased estimates only, to more general estimates that include many popular algorithms (most notably, the orthogonal least squares and algebraic fits). We then show that the gradient-weighted algebraic fit is statistically efficient and describe all other statistically efficient algebraic fits.

Abstract: Geometric properties of multi-dimensional dispersing billiards are studied in this paper. On the one hand, non-smooth behaviour in the singularity submanifolds of the system is discovered (this discovery applies to the more general class of semi-dispersing billiards as well). On the other hand, a self-contained geometric description for unstable manifolds is given, together with the proof of important regularity properties. All these issues are highly relevant to studying the ergodic and statistical behaviour of the dynamics.

Abstract: We investigate whether a search light, S, illuminating a tiny angle (`cone') with vertex A inside a bounded region Q with the mirror boundary ∂Q, will eventually illuminate the entire region Q. It is assumed that light rays hitting the corners of Q terminate. We prove that: (i) if Q is a circle or an ellipse, then either the entire Q or an annulus between two concentric circles/confocal ellipses (one of which is ∂Q) or the region between two confocal hyperbolas will be illuminated; (ii) if Q is a square or a dispersing (Sinai) or semidespirsing billiard, then the entire region Q is will be illuminated.

Abstract: We study a dynamical system consisting of a massive piston in a cubical container of large size L filled with an ideal gas. The piston has mass M ~ L² and undergoes elastic collisions with N ~ L³ non-interacting gas particles of mass m=1. We find that, under suitable initial conditions, there is, in the limit L → ∞, a scaling regime with time and space scaled by L, in which the motion of the piston and the one particle distribution of the gas satisfy autonomous coupled equations (hydrodynamical equations), so that the mechanical trajectory of the piston converges, in probability, to the solution of the hydrodynamical equations for a certain period of time. We also discuss heuristically the dynamics of the system on longer intervals of time.

Abstract: We continue the study of the time evolution of a system consisting of a piston in a cubical container of large size L filled with an ideal gas. The piston has mass M ~ L² and undergoes elastic collisions with N ~ L³ gas particles of mass m. In a previous paper, Lebowitz, Piasecki and Sinai considered a scaling regime, with time and space scaled by L, in which they argued heuristically that the motion of the piston and the one particle distribution of the gas satisfy autonomous coupled differential equations. Here we state exact results and sketch proofs for this behavior.

Abstract: We study numerically and theoretically (on a heuristic level) the time evolution of a gas confined to a cube of size L³ divided into two parts by a piston with mass M_L ~ L² which can only move in the x-direction. Starting with a uniform `double-peaked' (non Maxwellian) distribution of the gas and a stationary piston, we find that (a) after an initial quiescent period the system becomes unstable and the piston performs a damped oscillatory motion, and (b) there is a thermalization of the system leading to a Maxwellian distribution of the gas velocities. The time of the onset of the instability appears to grow like L log L while the relaxation time to the Maxwellian grows like L^7/2.

Abstract: The fundamental theorem (also called the local ergodic theorem) was introduced by Sinai and Chernov in 1987. It provides sufficient conditions on a phase point under which some neighborhood of that point belongs to one ergodic component. This theorem has been instrumental in many studies of ergodic properties of hyperbolic dynamical systems with singularities, both in 2-D and in higher dimensions. The existing proofs of this theorem implicitly use the assumption on the boundedness of the curvature of singularity manifolds. However, we found recently that, in general, this assumption fails in multidimensional billiards. Here the fundamental theorem is established under a weaker assumption on singularities, which we call Lipschitz decomposability. Then we show that whenever the scatterers of the billiard are defined by algebraic equations, the singularities are Lipschitz decomposable. Therefore, the fundamental theorem still applies to physically important models - among others to hard ball systems, Lorentz gases with spherical scatterers, and Bunimovich-Rehacek stadia.

Abstract: We study a class of open chaotic dynamical systems. Consider an expanding map of an interval from which a few small open subintervals are removed (thus creating `holes'). Almost every point of the original interval then eventually escapes through the holes, so there can be no absolutely continuous invariant measures. We construct a so called conditionally invariant measure that is equivalent to the Lebesgue measure. Our measure is unique and naturally generates an invariant measure, which is singular. These results generalize early work by Pianigiani, Yorke, Collet, Martinez and Schmidt, who studied similar maps under an additional Markov assumption. We do not assume any Markov property here and use `bounded variation' techniques rather than Markov coding. Our results supplement those of Keller, who studied analytic interval maps with holes by using different techniques.

Abstract: Consider a particle moving freely on the torus and colliding elastically with some fixed convex bodies. This model is called a periodic Lorentz gas, or a Sinai billiard. It is a Hamiltonian system with a smooth invariant measure, whose ergodic and statistical properties have been well investigated. Now let the particle be subjected to a small external force. This new system is not likely to have a smooth invariant measure. Then a Sinai-Ruelle-Bowen (SRB) measure describes the evolution of typical phase trajectories. We find general sufficient conditions on the external force under which the SRB measure for the collision map exists, is unique, and enjoys good ergodic and statistical properties, including Bernoulliness and an exponential decay of correlations.

Abstract: Let T: X → X be a deterministic dynamical system preserving a probability measure μ. A dynamical Borel-Cantelli lemma asserts that for certain sequences of subsets A_n⊂ X and μ-almost every point x ∈ X the inclusion T ⁿx ∈ A_n holds for infinitely many n's. We discuss here systems which are either symbolic (topological) Markov chain or Anosov diffeomorphisms preserving Gibbs measures. We find sufficient conditions on sequences of cylinders and rectangles, respectively, that ensure the dynamical Borel-Cantelli lemma.

Abstract: We discuss rigorous results and open problems on the decay of correlations for dynamical systems characterized by elastic collisions: hard balls, Lorentz gases, Sinai billiards and related models. Recently developed techniques for general dynamical systems with some hyperbolic behavior are discussed. These techniques give exponential decay of correlations for many classes of billiards and open the door to further investigations.

Abstract: We describe rigorous mathematical results on the Kolmogorov-Sinai entropy for Lorentz gases and hard ball systems (both finite and infinite). Exact formulas and asymptotic estimates of the entropy are discussed for various models.

Abstract: The linear super-Burnett coefficient gives corrections to the diffusion equation in the form of higher derivatives of the density. Like the diffusion coefficient, it can be expressed in terms of integrals of correlation functions, but involving four different times. The power law decay of correlations in real gases (with many moving particles) and the random Lorentz gas (with one moving particle and fixed scatterers) are expected to cause the super-Burnett coefficient to diverge in most cases. Here we show that the expression for the super-Burnett coefficient of the periodic Lorentz gas converges as a result of exponential decay of correlations and a nontrivial cancellation of divergent contributions.

Abstract: We study Anosov diffeomorphisms on surfaces with small holes. The points that are mapped into the holes disappear and never return. In our previous paper we proved the existence of a conditionally invariant measure μ₊. Here we show that the iterations of any initially smooth measure, after renormalization, converge to μ₊. We construct the related invariant measure on the repeller and prove that it is ergodic and K-mixing. We prove the escape rate formula, relating the escape rate to the positive Lyapunov exponent and the entropy.

Abstract: We give a rigorous proof of exponential decay of correlations for all major classes of planar dispersing billiards: periodic Lorentz gases with and without horizon and dispersing billiard tables with corner points.

Abstract: We study smooth hyperbolic systems with singularities and their SRB measures. Here we assume that the singularities are submanifolds, the hyperbolicity is uniform aside from the singularities, and one-sided derivatives exist on the singularities. We prove that the ergodic SRB measures exist, are finitely many, and mixing SRB measures enjoy exponential decay of correlations and a central limit theorem. These properties have been proved previously only for two-dimensional systems.

Abstract: We develop Markov approximations for very general suspension flows. Based on this, we obtain a stretched exponential bound on time correlation functions for 3-D Anosov flows that verify `uniform nonintegrability of foliations'. These include contact Anosov flows and geodesic flows on compact surfaces of variable negative curvature. Our bound on correlations is stable under small smooth perturbations.

Abstract: We study Anosov diffeomorphisms on surfaces in which some small `holes' are cut. The points that are mapped into those holes disappear and never return. We assume that the holes are arbitrary open domains with piecewise smooth boundary, and their sizes are small enough. The set of points whose trajectories stay away from holes in the past is a Cantor-like union of unstable fibers. We establish the existence and uniqueness of a conditionally invariant measure on this set, whose conditional distributions on unstable fibers are smooth. This generalizes previous works by Pianigiani, Yorke, and others.

Abstract: The billiard in a polygon is not always ergodic and never K-mixing or Bernoulli. Here we consider billiard tables by attaching disks to each vertex of an arbitrary simply connected, convex polygon. We show that the billiard on such a table is ergodic, K-mixing and Bernoulli.

Abstract: For a class of two-dimensional hyperbolic maps (which includes certain billiard systems) we construct finite generating partitions. Thus trajectories of the map can be labelled uniquely by doubly infinite symbol sequences, where the symbols correspond to the atoms of the partition. It is shown that the corresponding conditions are fulfilled in the case of the cardioid billiard, the stadium billiard (and other Bunimovich billiards), planar dispersing and semi-dispersing billiards.

Abstract: 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 the paper include periodic Lorentz gas, stadium and its modifications, and the gas of hard balls. Some open questions and numerical observations are discussed.

Abstract: We study Anosov diffeomorphisms on manifolds in which some `holes' are cut. The points that are mapped into those holes disappear and never return. The holes studied here are rectangles of a Markov partition. Such maps generalize Smale's horseshoes and certain open billiards. The set of nonwandering points of a map of this kind is a Cantor-like set called repeller. We construct invariant and conditionally invariant measures on the sets of nonwandering points. Then we establish ergodic, statistical, and fractal properties of those measures.

Abstract: We study Anosov diffeomorphisms on manifolds in which some `holes' are cut. The points that are mapped into our holes will disappear and never return. We study the case where the holes are rectangles of a Markov partition. Such maps with holes generalize Smale's horseshoes and certain open billiards. The set of nonwandering points of our map is a Cantor-like set we call a repeller. In our previous paper, we assumed that the map restricted to the remaining rectangles of the Markov partition is topologically mixing. Under this assumption we constructed invariant and conditionally invariant measures on the sets of nonwandering points. Here we relax the mixing assumption and extend our results to nonmixing and nonergodic cases.

Abstract: Let φ^t be a topologically mixing Anosov flow on a 3-D compact manifolds M. Every unstable fiber (horocycle) of such a flow is dense in M. Sinai proved in 1992 that the one-dimensional SBR measures on long segments of unstable fibers converge uniformly to the SBR measure of the flow. We establish an explicit bound on the rate of convergence in terms of integrals of Holder continuous functions on M.

Abstract: We investigate stationary nonequilibrium states of systems of particles moving according to Hamiltonian dynamics with specified potentials. The systems are driven away from equilibrium by Maxwell demon `reflection rules' at the walls. These deterministic rules conserve energy but not phase space volume, and the resulting global dynamics may or may not be time reversible (or even invertible). Using rules designed to simulate moving walls we can obtain a stationary shear flow. Assuming that for macroscopic systems this flow satisfies the Navier-Stokes equations, we compare the hydrodynamic entropy production with the average rate of phase space volume compression. We find that they are equal when the velocity distribution of particles incident on the walls is a local Maxwellian. An argument for a general equality of this kind, based on the assumption of local thermodynamic equilibrium, is given. Molecular dynamic simulations of hard disks in a channel produce a steady shear flow with the predicted behavior.

Abstract: We prove that those nonuniformly hyperbolic maps and flows (with singularities) that enjoy the K-property are also Bernoulli. In particular, many billiard systems, including those systems of hard balls and stadia that have the K-property, and hyperbolic billiards, such as the Lorentz gas in any dimension, are Bernoulli. We obtain the Bernoulli property for both the billiard flows and the associated maps on the boundary of the phase space.

Abstract: We investigate stationary nonequilibrium states of particles moving according to Hamiltonian dynamics with Maxwell demon `reflection rules' at the walls. These rules simulate, in an energy but not phase space volume conserving way, moving boundaries. The resulting dynamics may or may not be time reversible. In either case the average rates of phase space volume contraction and macroscopic entropy production are shown to be equal for stationary hydrodynamic shear flows, i.e. when the velocity distribution of particles incident on the walls is a local Maxwellian. Molecular dynamic simulations of hard discs in a channel produce a steady shear flow with the predicted behavior.

Abstract: We prove the central limit theorem and weak invariance principle for abstract dynamical systems based on bounds on their mixing coefficients. We also develop techniques of Markov approximations for dynamical systems. We apply our results to expanding interval maps, Axiom A diffeomorphisms, chaotic billiards and hyperbolic attractors.

Abstract: Recently Ya.B. Pesin introduced a large class of hyperbolic attractors, and for those attractors he established the Smale spectral decomposition. In this paper our main results are a stretched exponential bound on the decay of correlations and the central limit theorem. Also we will obtain conditions under which two well known attractors - those of Belykh and Lozi - are subject to our main results.

Abstract: We extend here the famous theory by Bunimovich and Sinai covering statistical properties of planar periodic Lorentz gas to the multidimensional version of that model. Namely, we establish a stretched exponential bound for the decay of correlations, prove the central limit theorem and Donsker's Invariance Principle for multidimensional periodic Lorentz gases with finite horizon. Our methods do not require Markov partitions, we use only a crude approximation to such partitions, which we call Markov sieves. Multidimensional Markov sieves are built-up and used here for the first time. Their construction is principally different from that of two-dimensional Markov sieves and requires more difficult and intricate considerations. We restrict ourselves primarily with heuristic, qualitative explanations. Full details of that construction are yet to be elaborated.

Abstract: We study nonequilibrium steady states in the Lorentz gas of periodic scatterers when an electric external field is applied and the particle kinetic energy is held fixed by a `thermostat' constructed according to Gauss' principle of least constraint ( a model problem previously studied numerically by Moran and Hoover). The resulting dynamics is reversible and deterministic, but does not preserve Liouville measure. For a sufficiently small field, we prove the following results: (1) existence of a unique stationary, ergodic measure obtained by forward evolution of initial absolutely continuous distributions, for which the Pesin entropy formula and Young's expression for the fractal dimension are valid; (2) exact identity of the steady-state thermodynamic entropy production, the asymptotic decay of the Gibbs entropy for the time-evolved distribution, and minus the sum of the Lyapunov exponents; (3) an explicit expression for the full nonlinear current response (Kawasaki formula); and (4) validity of linear response theory and Ohm's transport law, including the Einstein relation between conductivity and diffusion matrices. Results (2) and (4) yield also a direct relation between Lyapunov exponents and zero-field transport (=diffusion) coefficients. Although we restrict ourselves here to dimension d = 2, the results carry over to higher dimensions and to some other physical situations: e.g. with additional external magnetic fields. The proofs use a well-developed theory of small perturbations of hyperbolic dynamical systems and the method of Markov sieves, an approximation of Markov partitions.

Abstract: We study the Lorentz gas in small external electric and magnetic fields, E and B, with the particle kinetic energy held fixed by a Gaussian ``thermostat'' (a modification of a model of Moran and Hoover.) Here we prove rigorously that : (1) Starting from any smooth initial density, a unique stationary, ergodic measure (whose support is fractal) is approached for times t → ∞. (2) The steady-state electric current, J(B,E), is given by a Kawasaki formula and the entropy production JE/T, with T the `temperature', is equal to both the asymptotic decay rate of the Gibbs entropy and minus the sum of the Lyapunov exponents. (3) The Einstein relation and Kubo formulas hold, i.e. J(B,E) = σ(B)E + higher order terms, with the diffusion matrix D(B) at E = 0 given by k_BT times the symmetric part of the conductivity matrix.

Abstract: We consider hyperbolic dynamical systems with singularities such as billiards and similar Hamiltonian systems. For this class of systems we formulate the theorem on local ergodicity in a rather general form. Besides, the existing version of this theorem is strengthened here by discarding one of its assumptions.

Abstract: 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.