A mixed SOC-turbulence model for nonlocal transport and Lévy-fractional Fokker–Planck equation

Rasmussen

2018

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We formulate the problem of confined Lévy flight on a comb. The comb represents a sawtoothlike potential field V(x), with the asymmetric teeth favoring net transport in a preferred direction. The shape effect is modeled as a power-law dependence V(x)∝|Δx|^{n} within the sawtooth period, followed by an abrupt drop-off to zero, after which the initial power-law dependence is reset. It is found that the Lévy flights will be confined in the sense of generalized central limit theorem if (i) the spacing between the teeth is sufficiently broad, and (ii) n>4-μ, where μ is the fractal dimension of the flights. In particular, for the Cauchy flights (μ=1), n>3. The study is motivated by recent observations of localization-delocalization of transport avalanches in banded flows in the Tore Supra tokamak and is intended to devise a theory basis to explain the observed phenomenology.

Section: A Basic Equations Nonlocality and The Cauchy Limitmentioning

confidence: 99%

Section: A Basic Equations Nonlocality and The Cauchy Limitmentioning

confidence: 99%

Section: The Non-homogeneity Issuementioning

confidence: 99%

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Lévy flights on a comb and the plasma staircase

Rasmussen

2018

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“…That means that the dynamical order parameter acts as input control parameter for the nonlocal order [8,40]; thus giving rise to a statistics of the Lévy type through self-organization [41]. Then the nonlinear wave can generate via a back-reaction on the medium a channel along which it propagates to large distances on Lévy flights [40].…”

Section: Introductionmentioning

confidence: 99%

Topological approximation of the nonlinear Anderson model

Iomin

2014

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We study the phenomena of Anderson localization in the presence of nonlinear interaction on a lattice. A class of nonlinear Schrödinger models with arbitrary power nonlinearity is analyzed. We conceive the various regimes of behavior, depending on the topology of resonance-overlap in phase space, ranging from a fully developed chaos involving Lévy flights to pseudochaotic dynamics at the onset of delocalization. It is demonstrated that the quadratic nonlinearity plays a dynamically very distinguished role in that it is the only type of power nonlinearity permitting an abrupt localizationdelocalization transition with unlimited spreading already at the delocalization border. We describe this localization-delocalization transition as a percolation transition on the infinite Cayley tree (Bethe lattice). It is found in vicinity of the criticality that the spreading of the wave field is subdiffusive in the limit t → +∞. The second moment of the associated probability distribution grows with time as a powerlaw ∝ t α , with the exponent α = 1/3 exactly. Also we find for superquadratic nonlinearity that the analog pseudochaotic regime at the edge of chaos is self-controlling in that it has feedback on the topology of the structure on which the transport processes concentrate. Then the system automatically (without tuning of parameters) develops its percolation point. We classify this type of behavior in terms of self-organized criticality (SOC) dynamics in Hilbert space. For subquadratic nonlinearities, the behavior is shown to be sensitive to details of definition of the nonlinear term. A transport model is proposed based on modified nonlinearity, using the idea of "stripes" propagating the wave process to large distances. Theoretical investigations, presented here, are the basis for consistency analysis of the different localization-delocalization patterns in systems with many coupled degrees of freedom in association with the asymptotic properties of the transport.

“…A derivation using transition probabilities in reciprocal space has been obtained in Refs. [50,51]. As is well known, the Riesz operator in Eq.…”

Section: A Continuous Time Random Walks In Wave Number Spacementioning

confidence: 96%

Subdiffusive Lévy flights in quantum nonlinear Schrödinger lattices with algebraic power nonlinearity

Iomin

2019

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We report a new result concerning the dynamics of an initially localized wave packet in quantum nonlinear Schrödinger lattices with a disordered potential. A class of nonlinear lattices with subquadratic power nonlinearity is considered. We show that there exists a parameter range for which an initially localized wave packet can spread along the lattice to unlimited distances, but the phenomenon is purely quantum, and is hindered in the corresponding classical lattices. The mechanism for this spreading is moreover very peculiar and assumes that the components of the wave field may form coupled states by tunneling under the topological barriers caused by multiple discontinuities in the operator space. Then these coupled states thought of as quasiparticle states can propagate to long distances on Lévy flights with a distribution of waiting times. The overall process is subdiffusive and occurs as a competition between long-distance jumps of the quasiparticle states, on the one hand, and long-time trapping phenomena mediated by clustering of unstable modes in wave number space, on the other hand. The kinetic description of the transport, discussed in this work, is based on fractional-derivative equations allowing for both (i) non-Markovianity of the spreading process as a result of attractive interaction among the unstable modes; this interaction is then described in terms of the familiar Lennard-Jones potential; and (ii) the effect of long-range correlations in wave number space tending to introduce fast channels for the transport, the so-called "stripes." We argue that the notion of stripes is key to understand the topological constraints behind the quantum spreading, and we involve the idea of stripy ordering to obtain self-consistently the parameters of the associated waiting-time and jump-length distributions. Finally, we predict the asymptotic laws for quantum transport and show that the relevant parameter determining these laws is the exponent of the power-law defining the type of the nonlinearity. The results, presented here, shed light on the origin of Lévy flights in quantum nonlinear lattices with disorder. arXiv:1905.05467v1 [cond-mat.stat-mech]