Anomalous Diffusion Due to Accelerator Modes in the Standard Map

Ishizaki, Ryuji; Horita, Takehiko; Kobayashi, Tatsuharu; Mori, Hazime

doi:10.1143/ptp.85.1013

Cited by 61 publications

(72 citation statements)

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“…It is worthwhile remarking that while accelerator modes give rise to anomalous diffusion (in momentum) in the standard map on the cylinder [31], the different scaling regimes discussed above hold whether the stable regions are accelerator modes or not. However, the suppression of the rate of the variance's approach to zero and the average Lyapunov or stability exponent to the ergodic average in the presence of small islands has similar origins.…”

Section: B the Variance As A Sensor For Small Elliptic Islandsmentioning

confidence: 96%

Fluctuations of finite-time stability exponents in the standard map and the detection of small islands

Tomsovic

Lakshminarayan

2007

Phys. Rev. E

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Some statistical properties of finite-time stability exponents in the standard map can be estimated analytically. The mean exponent averaged over the entire phase space behaves quite differently from all the other cumulants. Whereas the mean carries information about the strength of the interaction, and only indirect information about dynamical correlations, the higher cumulants carry information about dynamical correlations and essentially no information about the interaction strength. In particular, the variance and higher cumulants of the exponent are very sensitive to dynamical correlations and easily detect the presence of very small islands of regular motion via their anomalous time-scalings. The average of the stability matrix' inverse trace is even more sensitive to the presence of small islands and has a seemingly fractal behavior in the standard map parameter. The usual accelerator modes and the small islands created through double saddle node bifurcations, which come halfway between the positions in interaction strength of the usual accelerator modes, are clearly visible in the variance, whose time scaling is capable of detecting the presence of islands as small as 0.01% of the phase space. We study these quantities with a local approximation to the trace of the stability matrix which significantly simplifies the numerical calculations as well as allows for generalization of these methods to higher dimensions. We also discuss the nature of this local approximation in some detail. 05.45.Ac,45.05.+x

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Section: B the Variance As A Sensor For Small Elliptic Islandsmentioning

confidence: 96%

Fluctuations of finite-time stability exponents in the standard map and the detection of small islands

Tomsovic

Lakshminarayan

2007

Phys. Rev. E

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“…5 t o o k a d e f i n i t e j u m p . T h i s o c c u r r a n c e a t c l o s e t o π / 2 c a n n o t b e d i s c a r d e d and may be a kind of enhancement similar to the classical enhanced diffusion due to accelerator islands [16]). …”

Section: Resultsmentioning

confidence: 99%

Non Saturation of Energy in a Quantum Kicked Oscillator

Daly

Hefernan

1996

Acta Phys. Pol. A

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We present a quantum mapping for the kicked harmonic oscillator which relates the probability amplitudes of the undriven oscillator's eigenfunctions over successive kicks. We show how for various kick strengths the wave functions lave a linear energy increase up to the limit imposed b y the finite matrix size of the evolution matrix. We use this linear energy increase to define a quantum diffusion-like coefficient. We also show how this increase in energy causes the wave functions to spread out and become diifuse with fittle or no discernible structure. This model may serve as a paradigm for the study of quantum chaos.

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“…Low-speed momentum diffusion in nonlinear Hamiltonian systems has been studied [1][2][3][4][5][6][7][8][9][10][11][12] extensively using nonrelativistic, Newtonian mechanics. The statistical quantity that is typically used to study momentum diffusion is the mean square momentum displacement (MSMD) [1][2][3]7,10,11].…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…The statistical quantity that is typically used to study momentum diffusion is the mean square momentum displacement (MSMD) [1][2][3]7,10,11]. In previous studies [1][2][3] of momentum diffusion in the Newtonian standard map for the periodically delta-kicked particle, an initially non-localized semi-uniform ensemble of trajectories (where semi-uniform means that the initial positions are uniformly distributed but the initial momenta are all the same value) was typically used in the numerical calculation of the MSMD. These studies [2,3] of the Newtonian standard map have shown that, for parameter K where accelerator mode islands (these are stable regions in the chaotic sea [13] in which the particle accelerates continuously [13,14]) exist, the MSMD has a power-law dependence on the kick n: n α , where 1 < α < 2, i.e., the diffusion is anomalous.…”

Section: Introductionmentioning

confidence: 99%

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Accuracy of the non-relativistic approximation for momentum diffusion

Liang

Lan

2016

Eur. Phys. J. Plus

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Abstract. The accuracy of the non-relativistic approximation, which is calculated using the same parameter and the same initial ensemble of trajectories, to relativistic momentum diffusion at low speed is studied numerically for a prototypical nonlinear Hamiltonian system -the periodically delta-kicked particle. We find that if the initial ensemble is a non-localized semi-uniform ensemble, the non-relativistic approximation to the relativistic mean square momentum displacement is always accurate. However, if the initial ensemble is a localized Gaussian, the non-relativistic approximation may not always be accurate and the approximation can break down rapidly.

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Anomalous Diffusion Due to Accelerator Modes in the Standard Map

Cited by 61 publications

References 0 publications

Fluctuations of finite-time stability exponents in the standard map and the detection of small islands

Fluctuations of finite-time stability exponents in the standard map and the detection of small islands

Non Saturation of Energy in a Quantum Kicked Oscillator

Accuracy of the non-relativistic approximation for momentum diffusion

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