Fundamentals of Lévy Flight Processes

Chechkin, Aleksei V.; Gonchar, V.Yu.; Klafter, J.; Metzler, Ralf

doi:10.1002/0470037148.ch9

Cited by 76 publications

(121 citation statements)

References 76 publications

(146 reference statements)

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“…From Refs. [Chechkin et al, 2004[Chechkin et al, , 2006. The main tools to investigate the barrier crossing problem for Lévy flights are the first passage times, crossing times, arrival time and residence times.…”

Section: Barrier Crossingmentioning

confidence: 99%

Lévy Flight Superdiffusion: An Introduction

Dubkov

Spagnolo

Учайкин

2008

Int. J. Bifurcation Chaos

300

295

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After a short excursion from discovery of Brownian motion to the Richardson "law of four thirds" in turbulent diffusion, the article introduces the Lévy flight superdiffusion as a self-similar Lévy process. The condition of self-similarity converts the infinitely divisible characteristic function of the Lévy process into a stable characteristic function of the Lévy motion. The Lévy motion generalizes the Brownian motion on the base of the α-stable distributions theory and fractional order derivatives. The further development of the idea lies on the generalization of the Langevin equation with a non-Gaussian white noise source and the use of functional approach. This leads to the Kolmogorov's equation for arbitrary Markovian processes. As particular case we obtain the fractional Fokker-Planck equation for Lévy flights. Some results concerning stationary probability distributions of Lévy motion in symmetric smooth monostable potentials, and a general expression to calculate the nonlinear relaxation time in barrier crossing problems are derived. Finally we discuss results on the same characteristics and barrier crossing problems with Lévy flights, recently obtained with different approaches.

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Section: Barrier Crossingmentioning

confidence: 99%

Lévy Flight Superdiffusion: An Introduction

Dubkov

Spagnolo

Учайкин

2008

Int. J. Bifurcation Chaos

300

295

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“…The Lévy-type statistics, in fact, is observed in various scientific areas, where scale-invariance phenomena take place or can be suspected [45][46][47][48] (for a recent review on Lévy-flights see Ref. [49] and references there).…”

Section: Introductionmentioning

confidence: 99%

Non-Gaussian noise effects in the dynamics of a short overdamped Josephson junction

2010

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Abstract. The role of thermal and non-Gaussian noise on the dynamics of driven short overdamped Josephson junctions is studied. The mean escape time of the junction is investigated considering Gaussian, Cauchy-Lorentz and Lévy-Smirnov probability distributions of the noise signals. In these conditions we find resonant activation and the first evidence of noise enhanced stability in a metastable system in the presence of Lévy noise. For Cauchy-Lorentz noise source, trapping phenomena and power law dependence on the noise intensity are observed.

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“…The PDF of ΔΦ/ΔΦ0 for the fresh sample shows a skewed line shape with a fat tail, which we found could be fitted by a Levy stable distribution [28][29][30][31][32][33], as shown in Figure 3b by the solid red curve. Recently, the Levy stable distribution has found increasing interest in several applications in diverse fields for describing complex phenomena [28][29][30][31][32][33]. This kind of probability distribution does not have an analytical expression, but is represented by a characteristic function defined by four parameters: stability index α, skewness parameter β, scale parameter γ (width of the distribution) and location parameter δ.…”

Section: Mapping Of Axon Orientation Fluctuationsmentioning

confidence: 82%

Correlated Disorder in Myelinated Axons Orientational Geometry and Structure

et al. 2017

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While the ultrastructure of myelin is considered a quasi-crystalline stable system, nowadays its multiscale complex dynamics appear to play a key role in its functionality, degeneration and repair processes following neurological diseases and trauma. In this work, we investigated the fluctuation of the myelin supramolecular assembly by measuring the spatial distribution of orientation fluctuations of axons in a Xenopus Laevis sciatic nerve associated with nerve functionality. To this end, we used scanning micro X-ray diffraction (SµXRD), a non-invasive technique that has already been applied to other heterogeneous systems presenting complex geometries from microscale to nanoscale. We found that the orientation of the spatial fluctuations of fresh axons show a Levy flight distribution, which is a clear indication of correlated disorder. We found that the Levy flight distribution was missing in the aged nerve prepared in an unfresh state. This result shows that the spatial distribution of axon orientation fluctuations in unfresh nerve state loses the correlated disorder and assumes a random disorder behavior. This work provides a deeper understanding of the ultrastructure-function nerve relation and paves the way for the study of other materials and biomaterials using the SµXRD technique to detect fluctuations in their supramolecular structure.

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Fundamentals of Lévy Flight Processes

Cited by 76 publications

References 76 publications

Lévy Flight Superdiffusion: An Introduction

Lévy Flight Superdiffusion: An Introduction

Non-Gaussian noise effects in the dynamics of a short overdamped Josephson junction

Correlated Disorder in Myelinated Axons Orientational Geometry and Structure

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