Superdiffusion and stable laws

Золотарев, В. М.; Учайкин, В. В.; Saenko, V. V.

doi:10.1134/1.558856

Cited by 37 publications

(43 citation statements)

References 9 publications

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“…We also note that the obtained t -dependence is the typical superdiffusion law within the framework of fractional kinetics, see Refs. [21], [22], [18], [23]. Basing on the numerical solution to the Langevin equations (2.1) we estimated numerically the moments M r (t; q, α) by averaging over 100 realizations, each consisting from 50000 time steps.…”

Section: Non-homogeneous Relaxation and Superdiffusionmentioning

confidence: 99%

Fractional kinetics for relaxation and superdiffusion in a magnetic field

2002

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We propose fractional Fokker-Planck equation for the kinetic description of relaxation and superdiffusion processes in constant magnetic and random electric fields. We assume that the random electric field acting on a test charged particle is isotropic and possesses non-Gaussian Levy stable statistics. These assumptions provide us with a straightforward possibility to consider formation of anomalous stationary states and superdiffusion processes, both properties are inherent to strongly non-equilibrium plasmas of solar systems and thermonuclear devices. We solve fractional kinetic equations, study the properties of the solution, and compare analytical results with those of numerical simulation based on the solution of the Langevin equations with the noise source having Levy stable probability density. We found, in particular, that the stationary states are essentially non-Maxwellian ones and, at the diffusion stage of relaxation, the characteristic displacement of a particle grows superdiffusively with time and is inversely proportional to the magnetic field.

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Section: Non-homogeneous Relaxation and Superdiffusionmentioning

confidence: 99%

Fractional kinetics for relaxation and superdiffusion in a magnetic field

2002

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“…Acting by the generator (18) on the Equation 4 and substituting the prolongation formulae (19) to (22), we obtain the determining equation…”

Section: Symmetries Of 2-dimensional Fractional Filtration Equation Wmentioning

confidence: 99%

“…Such types of anomalous diffusion equations are known and studied. [16][17][18][19][20][21][22] Nevertheless, symmetry properties of fractional equations with the Riesz potential are not yet investigated.…”

Section: Introductionmentioning

confidence: 99%

Lie group analysis of 2‐dimensional space‐fractional model for flow in porous media

Belevtsov

Lukashchuk

2018

Math Methods in App Sciences

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Using the Lie group analysis approach, we study the symmetry properties of 2‐dimensional space‐fractional filtration equation with the Riesz potential. This equation is derived by a fractional generalization of Darcy law and permits to describe the pressure evolution during 1‐phase fluid flow through naturally fractured porous medium. We construct the prolongation of the point transformation group on the Riesz potential and obtain a generalization of the Leibniz rule for the Riesz potential, which are necessary for investigating symmetry properties using the invariance principle. The Lie group of linearly autonomous point transformations is constructed for the considered equation. In a limiting case of zero‐order potential, the obtained symmetry group coincides with the group of point transformations admitted by the classical integer‐order 2‐dimensional linear heat equation. Also, the asymptotic behavior of a group invariant solution is investigated.

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“…Besides circuit models, this branch of mathematics has been found to be useful in modeling and analyzing phenomena, such as rheology [32][33][34][35][36], quantum physics [25,37,38], and diffusion [39][40][41]. Additionally, the non-integer order calculus has other advantages, such as the ability to model complex systems that contain long-range spatial interactions and memory effects [42].…”

Section: Introductionmentioning

confidence: 99%

Investigation of chaos and memory effects in the Bonhoeffer-van der Pol oscillator with a non-ideal capacitor

Brechtl

Xie

Liaw

2019

Communications in Nonlinear Science and Numerical Simulation

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In this paper, the voltage fluctuations of the Bonhoeffer van der pol oscillator system with a non-ideal capacitor were investigated. Here, the capacitor was modeled, using a fractional differential equation in which the order of the fractional derivative, α, is also a measure of the memory in the dielectric. The governing fractional differential equation was derived using two methods, namely a differential and integral approach. The former method utilized a hierarchical resistor-capacitor (RC) ladder model while the latter utilized the theory of the universal dielectricresponse. It was found that the dynamical behavior of the potential across the capacitor was affected by this parameter, and, therefore, the memory of the system. Additionally, findings indicate that an increase in the memory parameter was associated with an increase in the energy stored in the dielectric. Furthermore, the effects of the dynamical behavior of the voltage on the capacity of the dielectric to store energy was examined. It was found that oscillation death resulted in a higher amount of stored energy in the dielectric over time, as compared to behavior, which displayed relaxation oscillations or chaotic fluctuations. The relatively-lower stored energy resulting from the latter types of dynamical behavior appeared to be a consequence of the memory effect, where present accumulations of energy in the capacitor are affected by previous decreases in the potential. Hence, in this type of scenario, the dielectric material can be thought of as "remembering" the past behavior of the voltage, which leads to either a decrease, or an enhancement in the stored energy. Moreover, an increase in the fractional parameter α, under certain conditions, led to the earlier onset of the chaotic voltage oscillations across the capacitor. Furthermore, the corresponding phase portraits showed that the chaotic behavior was heightened, in general, with a decrease in α. The non-ideal capacitor was also found to have a transitory nature, where it becomes more like a resistor as α → 0, and conversely, more like a capacitor as α → 1.Here, a decrease in α was linked to an enhanced metallic character of the dielectric. Finally, a possible link between the complexity of the voltage noise fluctuations and the metallic character of the non-ideal capacitor will be discussed.

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Superdiffusion and stable laws

Cited by 37 publications

References 9 publications

Fractional kinetics for relaxation and superdiffusion in a magnetic field

Fractional kinetics for relaxation and superdiffusion in a magnetic field

Lie group analysis of 2‐dimensional space‐fractional model for flow in porous media

Investigation of chaos and memory effects in the Bonhoeffer-van der Pol oscillator with a non-ideal capacitor

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