9 research outputs found
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
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 Renyi 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)].Comment: 5 page
Leading Pollicott-Ruelle Resonances and Transport in Area-Preserving Maps
The leading Pollicott-Ruelle resonance is calculated analytically for a
general class of two-dimensional area-preserving maps. Its wave number
dependence determines the normal transport coefficients. In particular, a
general exact formula for the diffusion coefficient D is derived without any
high stochasticity approximation and a new effect emerges: The angular
evolution can induce fast or slow modes of diffusion even in the high
stochasticity regime. The behavior of D is examined for three particular cases:
(i) the standard map, (ii) a sawtooth map, and (iii) a Harper map as an example
of a map with nonlinear rotation number. Numerical simulations support this
formula.Comment: 5 pages, 1 figur
Approach to equilibrium for a class of random quantum models of infinite range
We consider random generalizations of a quantum model of infinite range
introduced by Emch and Radin. The generalization allows a neat extension from
the class of absolutely summable lattice potentials to the optimal class
of square summable potentials first considered by Khanin and Sinai and
generalised by van Enter and van Hemmen. The approach to equilibrium in the
case of a Gaussian distribution is proved to be faster than for a Bernoulli
distribution for both short-range and long-range lattice potentials. While
exponential decay to equilibrium is excluded in the nonrandom case, it is
proved to occur for both short and long range potentials for Gaussian
distributions, and for potentials of class in the Bernoulli case. Open
problems are discussed.Comment: 10 pages, no figures. This last version, to appear in J. Stat. Phys.,
corrects some minor errors and includes additional references and comments on
the relation to experiment
Periodic Orbits and Escapes in Dynamical Systems
We study the periodic orbits and the escapes in two different dynamical
systems, namely (1) a classical system of two coupled oscillators, and (2) the
Manko-Novikov metric (1992) which is a perturbation of the Kerr metric (a
general relativistic system). We find their simple periodic orbits, their
characteristics and their stability. Then we find their ordered and chaotic
domains. As the energy goes beyond the escape energy, most chaotic orbits
escape. In the first case we consider escapes to infinity, while in the second
case we emphasize escapes to the central "bumpy" black hole. When the energy
reaches its escape value a particular family of periodic orbits reaches an
infinite period and then the family disappears (the orbit escapes). As this
family approaches termination it undergoes an infinity of equal period and
double period bifurcations at transitions from stability to instability and
vice versa. The bifurcating families continue to exist beyond the escape
energy. We study the forms of the phase space for various energies, and the
statistics of the chaotic and escaping orbits. The proportion of these orbits
increases abruptly as the energy goes beyond the escape energy.Comment: 28 pages, 23 figures, accepted in "Celestial Mechanics and Dynamical
Astronomy
Phase correlations in chaotic dynamics: a Shannon entropy measure
In the present work, we investigate phase correlations by recourse to the Shannon entropy. Using theoretical arguments, we show that the entropy provides an accurate measure of phase correlations in any dynamical system, in particular when dealing with a chaotic diffusion process. We apply this approach to different low-dimensional maps in order to show that indeed the entropy is very sensitive to the presence of correlations among the successive values of angular variables, even when it is weak. Later on, we apply this approach to unveil strong correlations in the time evolution of the phases involved in the Arnold’s Hamiltonian that lead to anomalous diffusion, particularly when the perturbation parameters are comparatively large. The obtained results allow us to discuss the validity of several approximations and assumptions usually introduced to derive a local diffusion coefficient in multidimensional near-integrable Hamiltonian systems, in particular the so-called reduced stochasticity approximation.Fil: Cincotta, Pablo Miguel. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - La Plata. Instituto de Astrofísica La Plata. Universidad Nacional de La Plata. Facultad de Ciencias Astronómicas y Geofísicas. Instituto de Astrofísica La Plata; ArgentinaFil: Giordano, Claudia Marcela. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - La Plata. Instituto de Astrofísica La Plata. Universidad Nacional de La Plata. Facultad de Ciencias Astronómicas y Geofísicas. Instituto de Astrofísica La Plata; Argentin