Transport in Hamiltonian systems

MacKay, Robert S.; Meiss, James D.; Percival, I C

doi:10.1016/0167-2789(84)90270-7

Cited by 698 publications

(512 citation statements)

References 20 publications

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“…This structure is called a bubble of instability. 25) The real partω R = ε core −ε ph /2 is negative in a bubble withε core < 0 andε ph > 0.…”

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confidence: 99%

“…From the naive consideration given below, we can expect that the splitting instability should be less sensitive toR for R → ∞. According to the theory of Hamiltonian dynamical systems, 25) the dynamic instability can be induced by a mixing between positive and negative energy modes. Here, the negative energy mode corresponds to the core mode localized around the vortex core (r 1).…”

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confidence: 99%

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Is a Doubly Quantized Vortex Dynamically Unstable in Uniform Superfluids?

2018

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We revisit the fundamental problem of the splitting instability of a doubly quantized vortex in uniform singlecomponent superfluids at zero temperature. We analyze the system-size dependence of the excitation frequency of a doubly quantized vortex through large-scale simulations of the Bogoliubov-de Gennes equation, and find that the system remains dynamically unstable even in the infinite-system-size limit. Perturbation and semi-classical theories reveal that the splitting instability radiates a damped oscillatory phonon as an opposite counterpart of a quasi-normal mode.Introduction: Vortices appear in many branches of physics. In particular, the structure, stability, and dynamics of vortices in nonlinear fields share common features in many physical systems. 1) Quantized vortices are prototypes among those vortices, playing a key role in the fluid dynamics of superfluid helium and Bose-Einstein condensates (BECs). [2][3][4][5] In general, quantized vortices are characterized by the winding number of the phase of the superfluid order parameter around the vortex core. A vortex whose winding number l is more than unity is called an l-quantized or multiply quantized vortex (MQV). Since the energy of an l-quantized vortex is generally larger than the sum of energies of l singly quantized vortices (SQVs), an MQV is energetically unstable and splits into SQVs in uniform systems. 6) In fact, MQVs have never been observed in equilibrium. However, this argument does not eliminate the possibility that MQVs survive as a metastable state at very low temperatures when energy dissipation is negligible.To investigate the splitting instability precisely, we need to analyze the microscopic structure of the vortex core. It is difficult to demonstrate such an analysis in the strongly correlated superfluid 4 He. Experimentally, there is no established technique to prepare an MQV in helium superfluids as an initial state of the instability problem. The realization of MQV in the BECs of ultra-cold gases sheds light on this problem, and vortex splitting has been observed. [7][8][9] The MQV in trapped systems can be dynamically unstable, and split into vortices with smaller winding numbers according to the Bogoliubovde Gennes (BdG) analysis at zero temperature. [10][11][12][13][14][15][16][17] Dynamic instability may occur when the excitation modes have complex frequencies as a result of coupling or "mixing" between two modes with positive and negative excitation energies. The negative energy mode, called the core mode, is localized at the vortex core and decreases the angular momentum of the system by −l in the direction along the core. The positive energy mode is a collective mode of the condensate. The instability depends on the atomic interaction strength in a complicated

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“…This structure is called a bubble of instability. 25) The real partω R = ε core −ε ph /2 is negative in a bubble withε core < 0 andε ph > 0.…”

mentioning

confidence: 99%

mentioning

confidence: 99%

Is a Doubly Quantized Vortex Dynamically Unstable in Uniform Superfluids?

2018

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“…Following the highlights of the Kolmogorov-Arnol'd-Moser theorem, an effectual combination of mathematical skill and sophisticated computer performances has been revealing in more and more detail the process of destruction of invariant surfaces that takes place when a classical integrable system is subjected to a perturbation of increasing strength. It is now known that after the breakup of certain invariant surfaces (tori), some relics may be left in the form of invariant Cantor sets, which are named Cantori [2,3,4]. Now the necessity arises of understanding the quantal relevance of such intriguing classical results.…”

Section: Introductionmentioning

confidence: 99%

Destruction of classical cantori in the quantum Frenkel-Kontorova model

Borgonovi

Guarneri

Shepelyansky³

1990

Z. Physik B - Condensed Matter

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In this paper we investigate the properties of the quantized discrete Frenkel Kontorova model. The structure of the ground state is numerically analyzed by means of the Metropolis algorithm; special attention is given to the effects of quantization on the Cantori structure of the classical ground state. These quantum effects produce a new structure which can be approximately described by using a sawtooth map instead of the Standard map. The dependence of the quantum energy on temperature is also investigated and discussed in connection with these structural modifications.

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“…It has been argued that the algebraic decay is due to the hierarchical structure of phase space [1,2,3,4,5,6]. However, despite significant efforts, the mathematical understanding of the behavior described by Eq.…”

Section: Introductionmentioning

confidence: 99%

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Barra

Gilbert

2002

Journal of Statistical Physics

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We study the algebraic decay of the survival probability in open hierarchical graphs. We present a model of a persistent random walk on a hierarchical graph and study the spectral properties of the Frobenius-Perron operator. Using a perturbative scheme, we derive the exponent of the classical algebraic decay in terms of two parameters of the model. One parameter defines the geometrical relation between the length scales on the graph, and the other relates to the probabilities for the random walker to go from one level of the hierarchy to another. The scattering resonances of the corresponding hierarchical quantum graphs are also studied. The width distribution shows the scaling behavior P (Γ) ∼ 1/Γ.

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Transport in Hamiltonian systems

Cited by 698 publications

References 20 publications

Is a Doubly Quantized Vortex Dynamically Unstable in Uniform Superfluids?

Is a Doubly Quantized Vortex Dynamically Unstable in Uniform Superfluids?

Destruction of classical cantori in the quantum Frenkel-Kontorova model

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