Universality of Algebraic Decays in Hamiltonian Systems

Cristadoro, Giampaolo; Ketzmerick, Roland

doi:10.1103/physrevlett.100.184101

Cited by 82 publications

(133 citation statements)

References 18 publications

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“…(KAM) islands coexisting in a chaotic sea [21,25,42,43], and also in higher dimensions [49]. The power-law behavior is due to the nonhyperbolicity of the dynamics and it is present both for recurrences and escapes.…”

Section: Nonhyperbolic Hamiltonian Systems With Power-law Tailsmentioning

confidence: 99%

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Poincaré recurrences and transient chaos in systems with leaks

2009

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In order to simulate observational and experimental situations, we consider a leak in the phase space of a chaotic dynamical system. We obtain an expression for the escape rate of the survival probability applying the theory of transient chaos. This expression improves previous estimates based on the properties of the closed system and explains dependencies on the position and size of the leak and on the initial ensemble. With a subtle choice of the initial ensemble, we obtain an equivalence to the classical problem of Poincaré recurrences in closed systems, which is treated in the same framework. Finally, we show how our results apply to weakly chaotic systems and justify a split of the invariant saddle in hyperbolic and nonhyperbolic components, related, respectively, to the intermediate exponential and asymptotic power-law decays of the survival probability.

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Section: Nonhyperbolic Hamiltonian Systems With Power-law Tailsmentioning

confidence: 99%

“…[43] for the latest result that indicates that α ≈ 2.57). Here, for simplicity we write the asymptotic decay as n −α , but it is meant to describe the power-law like behavior usually observed.…”

Section: Nonhyperbolic Hamiltonian Systems With Power-law Tailsmentioning

confidence: 99%

Poincaré recurrences and transient chaos in systems with leaks

2009

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“…The processes of types I and II are paradigms of various physical situations and the difference found here may give hope to develop statistical theories for mixed systems as proposed by [8][9][10] and others, and to classify relevant averaging processes into few classes. We start the calculation from (12).…”

Section: Discussionmentioning

confidence: 99%

“…Recently an important contribution was made by Cristadoro and Ketzmerick who demonstrated the universality of the decay of correlations in the framework of the model of [8]. They examined an ensemble of such systems by using an arbitrary distribution of transition probabilities in phase space [9] . Guided by similar ideas, Ceder and Agam [10] used diagrammatic methods to calculate the exponent of the decay of correlations and its fluctuations and found that the fluctuations are large.…”

Section: Introductionmentioning

confidence: 99%

Diffusion for ensembles of standard maps

Alus¹,

Fishman²

2015

Phys. Rev. E

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Two types of random evolution processes are studied for ensembles of the standard map with driving parameter K that determines its degree of stochasticity. For one type of processes the parameter K is chosen at random from a Gaussian distribution and is then kept fixed, while for the other type it varies from step to step. In addition, noise that can be arbitrarily weak is added. The ensemble average and the average over noise of the diffusion coefficient is calculated for both types of processes. These two types of processes are relevant for two types of experimental situations as explained in the paper. Both types of processes destroy fine details of the dynamics, and the second process is found to be more effective in destroying the fine details. We hope that this work is a step in the efforts for developing a statistical theory for systems with mixed phase

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“…Several results observed in the literature also noticed this changeover. 28,31 A universality was proposed recently for the recurrence time 27 and power law decay was observed in different regions of the phase space for 2-D Hamiltonian systems [33][34][35][36] and for higher degrees of freedom Hamiltonian systems. 37 FIG .…”

Section: -7mentioning

confidence: 99%

Scaling investigation for the dynamics of charged particles in an electric field accelerator

Ladeira¹,

Leonel

2012

Chaos: An Interdisciplinary Journal of Nonlinear Science

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Some dynamical properties of an ensemble of trajectories of individual (non-interacting) classical particles of mass m and charge q interacting with a time-dependent electric field and suffering the action of a constant magnetic field are studied. Depending on both the amplitude of oscillation of the electric field and the intensity of the magnetic field, the phase space of the model can either exhibit: (i) regular behavior or (ii) a mixed structure, with periodic islands of regular motion, chaotic seas characterized by positive Lyapunov exponents, and invariant Kolmogorov-Arnold-Moser curves preventing the particle to reach unbounded energy. We define an escape window in the chaotic sea and study the transport properties for chaotic orbits along the phase space by the use of scaling formalism. Our results show that the escape distribution and the survival probability obey homogeneous functions characterized by critical exponents and present universal behavior under appropriate scaling transformations. We show the survival probability decays exponentially for small iterations changing to a slower power law decay for large time, therefore, characterizing clearly the effects of stickiness of the islands and invariant tori. For the range of parameters used, our results show that the crossover from fast to slow decay obeys a power law and the behavior of survival orbits is scaling invariant. The formalism of escape is used to study the dynamics and hence the transport of charged particles in an accelerator. The model consists of a periodically time dependent electric field limited to a certain region in space that furnishes or absorbs energy of a particle. After the particle leaves the electric field region, a constant magnetic field impels the particle to move in a circular trajectory. This magnetic field is responsible for bringing the particle back to the electric field region leaving the energy unchanged. The control parameters e and s represent the amplitude of the time-dependent electric field and the inverse of the magnetic field, respectively. The parameter e defines the nonlinear strength while ps defines the time that the particle spends in the magnetic region. The dynamics can be described by a two-dimensional nonlinear mapping. Depending on both e and s, the phase space presents either regular or mixed structure. For e ¼ 0, the nonlinear term vanishes and the particle's energy is constant in time. For s ¼ 1; 3; 5; …, there is a strong correlation between the frequency of oscillation of the electric field and the intensity of the magnetic field, and as a consequence, the motion of the particle is regular. For different control parameters than those discussed, the phase space presents mixed structure with islands of regular motion surrounded by chaotic seas and invariant KAM curves that prevent the particle from acquiring unbounded energy gain. We study the escape of particles through a hole in the phase space placed in the velocity axis by considering the position of the lowest energy KAM curve. An ensemb...

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Universality of Algebraic Decays in Hamiltonian Systems

Cited by 82 publications

References 18 publications

Poincaré recurrences and transient chaos in systems with leaks

Poincaré recurrences and transient chaos in systems with leaks

Diffusion for ensembles of standard maps

Scaling investigation for the dynamics of charged particles in an electric field accelerator

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