Self-similar anomalous diffusion and Levy-stable laws

Учайкин, В. В.

doi:10.1070/pu2003v046n08abeh001324

Cited by 141 publications

(130 citation statements)

References 42 publications

Supporting

Mentioning

128

Contrasting

Order By: Relevance

“…Then the fractional Riemann-Liouville derivative of zero order is unity operator, implying that no fractional properties come into play for homogeneous spaces. Equation (28) when account is taken for the initial value problem can be rephrased [32] in terms of the Caputo fractional derivative [59,60] which shows a better behavior under transformations. Taking moments of the fractional diffusion equation (29) leads to the dispersion law in Eq.…”

Section: Non-markovian Diffusion Equationmentioning

confidence: 99%

“…[26,41,67,[70][71][72]; reviewed in Refs. [24,25,60,62]). Note that the fractional diffusion equation (46) is Markovian, in contrast to Eq.…”

Section: ∆N) (46)mentioning

confidence: 99%

“…(47) for 1 < 2µ ≤ 2, with Γ µ = −2 cos(πµ)Γ(2 − 2µ); and similarly for 0 < 2µ < 1 [26,60]. The two-sided improper integral is understood as the sum ∆n −∞ + +∞ ∆n .…”

Section: ∆N) (46)mentioning

confidence: 99%

“…(28) with long-time correlated dynamics near separatrix. The differintegration on the right-hand side has the analytical structure [59,60] of fractional time the so-called Riemann-Liouville fractional derivative,…”

Section: Non-markovian Diffusion Equationmentioning

confidence: 99%

See 3 more Smart Citations

Topological approximation of the nonlinear Anderson model

Milovanov

Iomin

2014

Phys. Rev. E

View full text Add to dashboard Cite

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.

show abstract

Section: Non-markovian Diffusion Equationmentioning

confidence: 99%

“…[26,41,67,[70][71][72]; reviewed in Refs. [24,25,60,62]). Note that the fractional diffusion equation (46) is Markovian, in contrast to Eq.…”

Section: ∆N) (46)mentioning

confidence: 99%

“…(47) for 1 < 2µ ≤ 2, with Γ µ = −2 cos(πµ)Γ(2 − 2µ); and similarly for 0 < 2µ < 1 [26,60]. The two-sided improper integral is understood as the sum ∆n −∞ + +∞ ∆n .…”

Section: ∆N) (46)mentioning

confidence: 99%

Section: Non-markovian Diffusion Equationmentioning

confidence: 99%

See 2 more Smart Citations

Topological approximation of the nonlinear Anderson model

Milovanov

Iomin

2014

Phys. Rev. E

View full text Add to dashboard Cite

show abstract

“…More and more situations has been reported in the literature supporting this [1,2,3,6,12]. The immediate causes for these anomalies could be attributed to fractal or multi-fractal character of phase space, Lèvy flights, dynamical traps, or long-range correlations which are present in many interesting applications [2,3,4,5]. The desolated once fractional order calculus revives again in order to discribe such problems.…”

Section: Introductionmentioning

confidence: 99%

Coupled oscillators with power-law interaction and their fractional dynamics analogues

Korabel

Zaslavsky

Tarasov

2007

Communications in Nonlinear Science and Numerical Simulation

View full text Add to dashboard Cite

The one-dimensional chain of coupled oscillators with long-range power-law interaction is considered. The equation of motion in the infrared limit are mapped onto the continuum equation with the Riesz fractional derivative of order α, when 0 < α < 2. The evolution of soliton-like and breather-like structures are obtained numerically and compared for both types of simulations: using the chain of oscillators and using the continuous medium equation with the fractional derivative.

show abstract

$$\sqrt{\epsilon }$$ Law: Centennial of the First Exact Result of Classical Radiative Transfer Theory

Ivanov

2021

Springer Series in Light Scattering

View full text Add to dashboard Cite

Self-similar anomalous diffusion and Levy-stable laws

Cited by 141 publications

References 42 publications

Topological approximation of the nonlinear Anderson model

Topological approximation of the nonlinear Anderson model

Coupled oscillators with power-law interaction and their fractional dynamics analogues

$$\sqrt{\epsilon }$$ Law: Centennial of the First Exact Result of Classical Radiative Transfer Theory

Contact Info

Product

Resources

About