Roberto Venegeroles

Chaos around the superposition of a black-hole and a thin disk

Abstract Motivated by strong astronomical evidence supporting that huge black-holes might inhabit the center of many active galaxies, we have studied the integrability of oblique orbits of test particles around the exact superposition of a black-hole and a thin disk. We have considered the relativistic and the Newtonian limits. Exhaustive numerical analyses were performed, and bounded zones of chaotic behavior were found for both limits. An intrinsic relativistic gravitational effect is detected: the chaoticity of trajectories that do not cross the disk.

Lyapunov statistics and mixing rates for intermittent systems.

We consider here a recent conjecture stating that correlation functions and tail probabilities of finite time Lyapunov exponents would have the same power law decay in weakly chaotic systems. We demonstrate that this conjecture fails for a generic class of maps of the Pomeau-Manneville type. We show further that, typically, the decay properties of such tail probabilities do not provide significant information on key aspects of weakly chaotic dynamics such as ergodicity and instability regimes. Our approaches are firmly based on rigorous results, particularly the Aaronson-Darling-Kac theorem, and are also confirmed by exhaustive numerical simulations.

Pesin-type relation for subexponential instability

We address here the problem of extending the Pesin relation among positive Lyapunov exponents and the Kolmogorov–Sinai entropy to the case of dynamical systems exhibiting subexponential instabilities. By using a recent rigorous result due to Zweimuller, we show that the usual Pesin relation can be extended straightforwardly for weakly chaotic one-dimensional systems of the Pomeau–Manneville type, provided one introduces a convenient subexponential generalization of the Kolmogorov–Sinai entropy. We show, furthermore, that Zweimullers result provides an efficient prescription for the evaluation of the algorithm complexity for such systems. Our results are confirmed by exhaustive numerical simulations. We also point out and correct a misleading extension of the Pesin relation based on the Krengel entropy that has appeared recently in the literature.

Alternative numerical computation of one-sided Lévy and Mittag-Leffler distributions.

We consider here the recently proposed closed-form formula in terms of the Meijer G functions for the probability density functions g(α)(x) of one-sided Lévy stable distributions with rational index α=l/k, with 0<α<1. Since one-sided Lévy and Mittag-Leffler distributions are known to be related, this formula could also be useful for calculating the probability density functions ρ(α)(x) of the latter. We show, however, that the formula is computationally inviable for fractions with large denominators, being unpractical even for some modest values of l and k. We present a fast and accurate numerical scheme, based on an early integral representation due to Mikusinski, for the evaluation of g(α)(x) and ρ(α)(x), their cumulative distribution function, and their derivatives for any real index α∈(0,1). As an application, we explore some properties of these probability density functions. In particular, we determine the location and value of their maxima as functions of the index α. We show that α≈0.567 and 0.605 correspond, respectively, to the one-sided Lévy and Mittag-Leffler distributions with shortest maxima. We close by discussing how our results can elucidate some recently described dynamical behavior of intermittent systems.

Ergodic transitions in continuous-time random walks

We consider continuous-time random walk models described by arbitrary sojourn time probability density functions. We find a general expression for the distribution of time-averaged observables for such systems, generalizing some recent results presented in the literature. For the case where sojourn times are identically distributed independent random variables, our results shed some light on the recently proposed transitions between ergodic and weakly nonergodic regimes. On the other hand, for the case of nonidentical trapping time densities over the lattice points, the distribution of time-averaged observables reveals that such systems are typically nonergodic, in agreement with some recent experimental evidences on the statistics of blinking quantum dots. Some explicit examples are considered in detail. Our results are independent of the lattice topology and dimensionality.

Number of first-passage times as a measurement of information for weakly chaotic systems.

We consider a general class of maps of the interval having Lyapunov subexponential instability |δxt|∼|δx0|exp[Λt(x0)ζ(t)], where ζ(t) grows sublinearly as t→∞. We outline here a scheme [J. Stat. Phys. 154, 988 (2014)] whereby the choice of a characteristic function automatically defines the map equation and corresponding growth rate ζ(t). This matching approach is based on the infinite measure property of such systems. We show that the average information that is necessary to record without ambiguity a trajectory of the system tends to 〈Λ〉ζ(t), suitably extending the Kolmogorov-Sinai entropy and Pesins identity. For such systems, information behaves like a random variable for random initial conditions, its statistics obeying a universal Mittag-Leffler law. We show that, for individual trajectories, information can be accurately inferred by the number of first-passage times through a given turbulent phase-space cell. This enables us to calculate far more efficiently Lyapunov exponents for such systems. Lastly, we also show that the usual renewal description of jumps to the turbulent cell, usually employed in the literature, does not provide the real number of entrances there. Our results are supported by exhaustive numerical simulations.

Quantitative Universality for a Class of Weakly Chaotic Systems

We consider a general class of intermittent maps designed to be weakly chaotic, i.e., for which the separation of trajectories of nearby initial conditions is weaker than exponential. We show that all its spatio and temporal properties, hitherto regarded independently in the literature, can be represented by a single characteristic function ϕ. A universal criterion for the choice of ϕ is obtained within the Feigenbaum’s renormalization-group approach. We find a general expression for the dispersion rate ζ(t) of initially nearby trajectories and we show that the instability scenario for weakly chaotic systems is more general than that originally proposed by Gaspard and Wang (Proc. Natl. Acad. Sci. USA 85:4591, 1988). We also consider a spatially extended version of such class of maps, which leads to anomalous diffusion, and we show that the mean squared displacement satisfies σ2(t)∼ζ(t). To illustrate our results, some examples are discussed in detail.

Pesin’s Relation for Weakly Chaotic One-Dimensional Systems

We explore a recent rigorous result due to Zweimuller in order to propose an extension, for the case of weakly chaotic systems, of the usual Pesin’s relation between the Lyapunov exponent and the Kolmogorov-Sinai entropy for one-dimensional systems. We show, furthermore, that Zweimuller’s result does provide an efficient prescription for the evaluation of the algorithm complexity for systems exhibiting subexponential instabilities. Our results are confirmed by exhaustive numerical simulations. We also compare our proposal with a recent one base on the Krengel entropy.

Thermodynamic phase transitions for Pomeau-Manneville maps.

We study phase transitions in the thermodynamic description of Pomeau-Manneville intermittent maps from the point of view of infinite ergodic theory, which deals with diverging measure dynamical systems. For such systems, we use a distributional limit theorem to provide both a powerful tool for calculating thermodynamic potentials as also an understanding of the dynamic characteristics at each instability phase. In particular, topological pressure and Rényi entropy are calculated exactly for such systems. Finally, we show the connection of the distributional limit theorem with non-Gaussian fluctuations of the algorithmic complexity proposed by Gaspard and Wang [Proc. Natl. Acad. Sci. USA 85, 4591 (1988)].

Relativistic Weierstrass random walks.

The Weierstrass random walk is a paradigmatic Markov chain giving rise to a Lévy-type superdiffusive behavior. It is well known that special relativity prevents the arbitrarily high velocities necessary to establish a superdiffusive behavior in any process occurring in Minkowski spacetime, implying, in particular, that any relativistic Markov chain describing spacetime phenomena must be essentially Gaussian. Here, we introduce a simple relativistic extension of the Weierstrass random walk and show that there must exist a transition time t{c} delimiting two qualitative distinct dynamical regimes: the (nonrelativistic) superdiffusive Lévy flights, for tt{c} . Implications of this crossover between different diffusion regimes are discussed for some explicit examples. The study of such an explicit and simple Markov chain can shed some light on several results obtained in much more involved contexts.