Self-similarity and transport in the standard map

Benkadda, S.; Kassibrakis, Serge; White, R. B.; Zaslavsky, G. M.

doi:10.1103/physreve.55.4909

Cited by 76 publications

(42 citation statements)

References 25 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In the first case that we have considered the main structure of the phase space is a self-similar hierarchy of chains of satellite islands. The anomalous diffusion in the presence of such structures has been previously observed and described in a series of papers [4][5][6][33][34][35][36] . It has been shown that the transport exponent is determined by two parameters which describe the selfsimilarity of the islands-around-islands structure.…”

Section: Numerical Investigationsmentioning

confidence: 87%

See 1 more Smart Citation

Anomalous diffusion in two-dimensional potentials with hexagonal symmetry

Panoiu

2000

Chaos: An Interdisciplinary Journal of Nonlinear Science

View full text Add to dashboard Cite

The diffusion process in a Hamiltonian dynamical system describing the motion of a particle in a two-dimensional (2D) potential with hexagonal symmetry is studied. It is shown that, depending on the energy of the particle, various transport processes can exist: normal (Brownian) diffusion, anomalous diffusion, and ballistic transport. The relationship between these transport processes and the underlying structure of the phase space of the Hamiltonian dynamical system is investigated. The anomalous transport is studied in detail in two particular cases: in the first case, inside the chaotic sea there exist self-similar structures with fractal properties while in the second case the transport takes place in the presence of multilayered structures. It is demonstrated that structures of the second type can lead to a physical situation in which the transport becomes ballistic. Also, it is shown that for all cases in which the diffusive transport is anomalous the trajectories of the diffusing particles contain long segments of regular motion, the length of these segments being described by Levy probability density functions. Finally, the numerical values of the parameters which describe the diffusion processes are compared with those predicted by existing theoretical models. (c) 2000 American Institute of Physics.

show abstract

Section: Numerical Investigationsmentioning

confidence: 87%

“…Such unexpected behaviors have been observed for maps [1][2][3][4][5][6] as well as for incompressible two-dimensional fluid flows. [7][8][9] The main characteristic of an anomalous diffusion process is that the mean square displacement does not grow linearly in time…”

Section: Introductionmentioning

confidence: 92%

Anomalous diffusion in two-dimensional potentials with hexagonal symmetry

Panoiu

2000

Chaos: An Interdisciplinary Journal of Nonlinear Science

View full text Add to dashboard Cite

show abstract

“…curve breaks and unbounded diffusion in action space takes place. Nevertheless even for K >> 1 some small stability islands survive in the classical phase space, correspondent to accelerator modes [23][24][25]. In fig.7 the classical phase space of the map (8) for K = 2.21 with the stability islands of accelerator modes is shown.…”

Section: Kicked Rotator Modelmentioning

confidence: 99%

Effects of a nonlinear perturbation on dynamical tunneling in cold atoms

Artuso

Rebuzzini

2003

Phys. Rev. E

View full text Add to dashboard Cite

We perform a numerical analysis of the effects of a nonlinear perturbation on the quantum dynamics of two models describing non-interacting cold atoms in a standing wave of light with a periodical modulated amplitude A(t). One model is the driven pendulum, considered in ref. [1], and the other is a variant of the well-known Kicked Rotator Model. In absence of the nonlinear perturbation, the system is invariant under some discrete symmetries and quantum dynamical tunnelling between symmetric classical islands is found. The presence of nonlinearity destroys tunnelling, breaking the symmetries of the system. Finally, further consequences of nonlinearity in the kicked rotator case are considered.PACS numbers: 05.45.-a, 03.65.-w Tunneling is one of the most typical features of quantum mechanics, concerning oscillations between states that cannot be connected in the classical Hamiltonian dynamics by real trajectories. The original formulation of the problem involved states separated by potential barriers: the relevant features are semiclassically explained in terms of complex solutions of Hamilton equations for one-dimensional systems [2], while a proper treatment in higher dimensions is considerably subtler even when integrability is preserved [3]. Recently a novel kind of tunneling, involving transitions between classically separated regions in the presence of a non trivial structure of the phase space, has attracted considerable attention, both theoretically [4][5][6][7], and experimentally [8,1,9]. In particular a large fraction of these papers focus on physical settings realized with cold atoms in optical potentials. The recent widespread interest and experimental activity in Bose-Einstein condensation [10] suggest to check whether the presence of Gross-Pitaevskii nonlinearities [11] deeply influences the characteristic features of dynamical tunneling. Such a question was already raised and analyzed in the framework of the kicked oscillator [12], where it was observed how the nonlinear terms typically destroys quantum effects induced by symmetry [13,14]. The paper is organized as follows: we firstly give a few details and fix notations for the class of models we are going to consider, then analyze two cases: a driven pendulum and the kicked rotator; in the last section we briefly reconsider pioneering work made on the nonlinear kicked rotator [15,16] and supplement it with novel results for the quantum resonant case. I. GENERAL SETTING.We will consider models described by the following hamiltonianwhere p and ϑ are canonical momentum and position coordinates respectively, expressed in scaled dimensionless units, and A(t) is a periodic function of time. Under appropriate choices of the function A(t), the quantum version of this model describes an ensemble of noninteracting cold atoms in presence of a standing wave with a periodically modulated amplitude A(t). The connection between the scaled variables ϑ, p and the physical ones We consider two possible choices for the periodic function A(t):• The first model i...

show abstract

“…3 and 4 supporting the powerlaw distributions P(s) and P(n) are already significant for understanding the anomalous large scale plasma behavior, the analysis may be much more effective if we can obtain an equation governing the distribution of density fluctuations. Here, we propose to use phenomenological fractional generalization of the Fokker-Planck-Kolmogorov ͑FFPK͒ equation, 13,14 which was successfully applied to some chaotic maps 15,28,29 as well as to the running sand-pile model. 16 For this goal, we introduce a distribution function F s (s,t) that describes only a singular part of the full distribution function…”

Section: Description Of the Fractional Kineticsmentioning

confidence: 99%

Large-scale behavior of the tokamak density fluctuations

et al. 2000

Self Cite

View full text Add to dashboard Cite

An analysis of tokamak density fluctuations data permits the determination of two characteristic exponents. The exponents correspond to the powers of a power-law dependence of the distributions of the long-lasting monotonic change ͑''flight''͒ of the density and the time length of these changes. Speculation based on these results leads to construction of the fractional kinetic equation for the distribution function of the flights. The asymptotic transport properties of the particle density distribution function are directly connected with the exponents obtained from the density fluctuations data.

show abstract

Self-similarity and transport in the standard map

Cited by 76 publications

References 25 publications

Anomalous diffusion in two-dimensional potentials with hexagonal symmetry

Anomalous diffusion in two-dimensional potentials with hexagonal symmetry

Effects of a nonlinear perturbation on dynamical tunneling in cold atoms

Large-scale behavior of the tokamak density fluctuations

Contact Info

Product

Resources

About