Green function for a non-Markovian Fokker-Planck equation: comb-model and anomalous diffusion

Silva, L. R. da; Tateishi, A. A.; Lenzi, Marcelo Kaminski; Lenzi, E. K.; Silva, P. C. da

doi:10.1590/s0103-97332009000400025

Cited by 18 publications

(26 citation statements)

References 42 publications

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“…From relation (8), in the same way as in (17), for a finite integration in [0, L], one finds a result in the form of an incomplete γ function γ(a, x) =…”

Section: Fractal Structure Of Backbonesmentioning

confidence: 99%

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Fractional diffusion on a fractal grid comb

2015

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A grid comb model is a generalization of the well known comb model, and it consists of N backbones. For N = 1 the system reduces to the comb model where subdiffusion takes place with the transport exponent 1/2. We present an exact analytical evaluation of the transport exponent of anomalous diffusion for finite and infinite number of backbones. We show that for an arbitrarily large but finite number of backbones the transport exponent does not change. Contrary to that, for an infinite number of backbones, the transport exponent depends on the fractal dimension of the backbone structure.

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“…From relation (8), in the same way as in (17), for a finite integration in [0, L], one finds a result in the form of an incomplete γ function γ(a, x) =…”

Section: Fractal Structure Of Backbonesmentioning

confidence: 99%

“…They have been used for the understanding of continuous [6][7][8] and discrete [9] non-Markovian random walks. There are generalizations of this equation by introducing * sandev@pks.mpg.de † iomin@physics.technion.ac.il ‡ kantz@pks.mpg.de time fractional derivatives and integrals in (1) [10,11].…”

Section: Introductionmentioning

confidence: 99%

Fractional diffusion on a fractal grid comb

2015

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“…The fractional operator is also responsible for introducing a nonlinear time dependence in the mean square displacement of the system [17]. Thus, a large class of complex phenomena can be effectively described by extending the standard differential operator to a non-integer order [25][26][27][28][29][30][31][32][33][34]; indeed, as pointed out by West [35], the fractional calculus provides a suitable framework to deal with complex systems. Recently, researchers have made and promoted remarkable progress toward improving experimental techniques for investigating diffusive processes, mainly illustrated by the developments in the single-particle tracking technique [36][37][38][39].…”

Section: Introductionmentioning

confidence: 99%

The Role of Fractional Time-Derivative Operators on Anomalous Diffusion

2017

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The generalized diffusion equations with fractional order derivatives have shown be quite efficient to describe the diffusion in complex systems, with the advantage of producing exact expressions for the underlying diffusive properties. Recently, researchers have proposed different fractional-time operators (namely: the Caputo-Fabrizio and Atangana-Baleanu) which, differently from the well-known Riemann-Liouville operator, are defined by non-singular memory kernels. Here we proposed to use these new operators to generalize the usual diffusion equation. By analyzing the corresponding fractional diffusion equations within the continuous time random walk framework, we obtained waiting time distributions characterized by exponential, stretched exponential, and power-law functions, as well as a crossover between two behaviors. For the mean square displacement, we found crossovers between usual and confined diffusion, and between usual and sub-diffusion. We obtained the exact expressions for the probability distributions, where non-Gaussian and stationary distributions emerged. This former feature is remarkable because the fractional diffusion equation is solved without external forces and subjected to the free diffusion boundary conditions. We have further shown that these new fractional diffusion equations are related to diffusive processes with stochastic resetting, and to fractional diffusion equations with derivatives of distributed order. Thus, our results suggest that these new operators may be a simple and efficient way for incorporating different structural aspects into the system, opening new possibilities for modeling and investigating anomalous diffusive processes.

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“…The corresponding dynamic equation for the description of the continuous comb has been introduced in Ref. [19] and extensively studied [20][21][22][23][24][25]. In this approach the effective comb model equation reads…”

Section: Introductionmentioning

confidence: 99%

Comb Model with Slow and Ultraslow Diffusion

Sandev

Iomin

Kantz

et al. 2016

Math. Model. Nat. Phenom.

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We consider a generalised diffusion equation in two dimensions for modeling diffusion on a comblike structures. We analyse the probability distribution functions and we derive the mean squared displacement in x and y directions. Different forms of the memory kernels (Dirac delta, power-law, and distributed order) are considered. It is shown that anomalous diffusion may occur along both x and y directions. Ultraslow diffusion and some more general diffusive processes are observed as well. We give the corresponding continuous time random walk model for the considered two dimensional diffusion-like equation on a comb, and we derive the probability distribution functions which subordinate the process governed by this equation to the Wiener process.

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Green function for a non-Markovian Fokker-Planck equation: comb-model and anomalous diffusion

Cited by 18 publications

References 42 publications

Fractional diffusion on a fractal grid comb

Fractional diffusion on a fractal grid comb

The Role of Fractional Time-Derivative Operators on Anomalous Diffusion

Comb Model with Slow and Ultraslow Diffusion

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