Introduction to the Theory of Lévy Flights

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

doi:10.1002/9783527622979.ch5

Cited by 135 publications

(110 citation statements)

References 94 publications

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“…These step lengths are taken from a Lévy distribution. A symmetric Lévy distribution [7][8][9] is given by L α,0 (x) = 1 2πˆ∞ −∞ dp e −|p| α e ipx ,…”

Section: Introductionmentioning

confidence: 99%

Lévy flights in the presence of a point sink of finite strength

Janakiraman

2017

Phys. Rev. E

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“…These step lengths are taken from a Lévy distribution. A symmetric Lévy distribution [7][8][9] is given by L α,0 (x) = 1 2πˆ∞ −∞ dp e −|p| α e ipx ,…”

Section: Introductionmentioning

confidence: 99%

Lévy flights in the presence of a point sink of finite strength

Janakiraman

2017

Phys. Rev. E

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“…If the variance of the jump length is infinite, the Central Limit Theorem does not apply [6,7,8,9,10], and the dynamics is anomalous. Lévy flights, which are dominated by rare but etremely large jumps, have proven quite suitable in modeling many physical systems, ranging from turbulent fluids to contaminant transport in fractured rocks, from chaotic dynamics to disordered quantum ensembles [3,5,11,12,13,14,15,16].…”

Section: Introductionmentioning

confidence: 99%

Fractional Laplacian in bounded domains

Zoia

Rosso

Kardar³

2007

Phys. Rev. E

232

290

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The fractional Laplacian operator, −(−△) α 2 , appears in a wide class of physical systems, including Lévy flights and stochastic interfaces. In this paper, we provide a discretized version of this operator which is well suited to deal with boundary conditions on a finite interval. The implementation of boundary conditions is justified by appealing to two physical models, namely hopping particles and elastic springs. The eigenvalues and eigenfunctions in a bounded domain are then obtained numerically for different boundary conditions. Some analytical results concerning the structure of the eigenvalues spectrum are also obtained.

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“…A generic property of such correlations is the contribution of slowly decaying tails of the underlying probability distribution to the averaged quantities. This leads to random walks, with a nonvanishing occurrence of long jumps [3][4][5] , as well as long-range ordering effects in organic and inorganic materials 2,[6][7][8][9] . Spin-1/2 chains with antiferromagnetic interaction, so called the spinPeierls systems 10,11 , as well as cross-coupled spin-1/2 chains, so called the spin-ladder systems 12,13 , are both prime examples of solids with long-range ordered phases.…”

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confidence: 99%

The random mass Dirac model and long-range correlations on an integrated optical platform

et al. 2013

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Long-range correlation-the non-local interdependence of distant events-is a crucial feature in many natural and artificial environments. In the context of solid state physics, impurity spins in doped spin chains and ladders with antiferromagnetic interaction are a prominent manifestation of this phenomenon, which is the physical origin of the unusual magnetic and thermodynamic properties of these materials. It turns out that such systems are described by a one-dimensional Dirac equation for a relativistic fermion with random mass. Here we present an optical configuration, which implements this one-dimensional random mass Dirac equation on a chip. On this platform, we provide a miniaturized optical test-bed for the physics of Dirac fermions with variable mass, as well as of antiferromagnetic spin systems. Moreover, our data suggest the occurence of long-range correlations in an integrated optical device, despite the exclusively short-ranged interactions between the constituting channels.

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Introduction to the Theory of Lévy Flights

Cited by 135 publications

References 94 publications

Lévy flights in the presence of a point sink of finite strength

Lévy flights in the presence of a point sink of finite strength

Fractional Laplacian in bounded domains

The random mass Dirac model and long-range correlations on an integrated optical platform

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