22 research outputs found
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 non-identical 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.Comment: 8 pages, final version to appear in PR
Alternative numerical computation of one-sided Levy 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 of
one-sided L\'evy stable distributions with rational index , with
. Since one-sided L\'evy and Mittag-Leffler distributions are known
to be related, this formula could also be useful for calculating the
probability density functions 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 and . We
present a fast and accurate numerical scheme, based on an early integral
representation due to Mikusinski, for the evaluation of and
, their cumulative distribution function and their derivatives
for any real index . 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 . We
show that and correspond,
respectively, to the one-sided L\'evy and Mittag-Leffler distributions with
shortest maxima. We close by discussing how our results can elucidate some
recently described dynamical behavior of intermittent systems.Comment: 6 pages, 5 figures. New references added, final version to appear in
PRE. Numerical code available at http://vigo.ime.unicamp.br/dist
Non-Gaussian features of chaotic Hamiltonian transport
Some non-Gaussian aspects of chaotic transport are investigated for a general
class of two-dimensional area-preserving maps. Kurtosis, in particular, is
calculated from the diffusion and the Burnett coefficients, which are obtained
analytically. A characteristic time scale delimiting the onset of the Markovian
regime for the master equation is established. Some explicit examples are
discussed.Comment: 19 pages, 6 Figures. v2: Grammatical corrections, new reference
Calculation of Superdiffusion for the Chirikov-Taylor Model
It is widely known that the paradigmatic Chirikov-Taylor model presents
enhanced diffusion for specific intervals of its stochasticity parameter due to
islands of stability, which are elliptic orbits surrounding accelerator mode
fixed points. In contrast with normal diffusion, its effect has never been
analytically calculated. Here, we introduce a differential form for the
Perron-Frobenius evolution operator in which normal diffusion and
superdiffusion are treated separately through phases formed by angular wave
numbers. The superdiffusion coefficient is then calculated analytically
resulting in a Schloemilch series with an exponent for the
divergences. Numerical simulations support our results.Comment: 4 pages, 2 figures (revised version
Leading Pollicott-Ruelle Resonances for Chaotic Area-Preserving Maps
Recent investigations in nonlinear sciences show that not only hyperbolic but
also mixed dynamical systems may exhibit exponential relaxation in the chaotic
regime. The relaxation rates, which lead the decay of probability distributions
and correlation functions, are related to the classical evolution resolvent
(Perron-Frobenius operator) pole logarithm, the so called Pollicott-Ruelle
resonances. In this Brief Report, the leading Pollicott-Ruelle resonances are
calculated analytically for a general class of area-preserving maps. Besides
the leading resonances related to the diffusive modes of momentum dynamics
(slow rate), we also calculate the leading faster rate, related to the angular
correlations. The analytical results are compared to the existing results in
the literature.Comment: 6 pages, 1 figure. See also: R. Venegeroles, Phys. Rev. Lett. 99,
014101 (2007) or arXiv:nlin/0608067v
Isochrone spacetimes
We introduce the relativistic version of the well-known Henon's isochrone
spherical models: static spherically symmetrical spacetimes in which all
bounded trajectories are isochrone in Henon's sense, i.e., their radial periods
do not depend on their angular momenta. Analogously to the Newtonian case,
these "isochrone spacetimes" have as particular cases the so-called Bertrand
spacetimes, in which all bounded trajectories are periodic. We propose a
procedure to generate isochrone spacetimes by means of an algebraic equation,
present explicitly several families of these spacetimes, and discuss briefly
their main properties. We identify, in particular, the family whose Newtonian
limit corresponds to the Henon's isochrone potentials and that could be
considered as the relativistic extension of the original Henon's proposal for
the study of globular clusters. Nevertheless, isochrone spacetimes generically
violate the weak energy condition and may exhibit naked singularities,
challenging their physical interpretation in the context of General Relativity.Comment: 14 pages, 2 figures. Final version accepted for publication in GR
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.8203111
Chaos around the superposition of a black-hole and a thin disk
Motivated by the strong astronomical evidences 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.Comment: Revtex, 13 pages, 3 figure
Relativistic Weierstrass random walks
The Weierstrass random walk is a paradigmatic Markov chain giving rise to a
L\'evy-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 delimiting two qualitative distinct dynamical
regimes: the (non-relativistic) superdiffusive L\'evy flights, for ,
and the usual (relativistic) Gaussian diffusion, for . 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.Comment: 5 pages, final version to appear in PR