Anomalous diffusion and fractional diffusion equation: anisotropic media and external forces

Andrade, Marcelo Freitas de; Lenzi, E. K.; Evangelista, L. R.; Mendes, R. S.; Malacarne, L. C.

doi:10.1016/j.physleta.2005.07.090

Cited by 23 publications

(6 citation statements)

References 39 publications

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“…In particular, Tsallis and others [4] have proposed that conventional Gibbs-Boltzmann statistics be generalized by extending the definition of entropy. On the other hand, it has been shown that an explanation using conventional statistics is possible if one uses a Fokker-Planck equation with either fractional derivatives [2] or time-dependent diffusion coefficients [7,8]. In addition, a study using path integral methods [9] showed that anomalous diffusion appears in quantum Brownian motion with colored noise.…”

Section: Introductionmentioning

confidence: 99%

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Anomalous diffusion in quantum Brownian motion with colored noise

Ford

O’Connell

2006

Phys. Rev. A

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Anomalous diffusion is discussed in the context of quantum Brownian motion with colored noise. It is shown that earlier results follow simply and directly from the fluctuation-dissipation theorem. The limits on the long-time dependence of anomalous diffusion are shown to be a consequence of the second law of thermodynamics. The special case of an electron interacting with the radiation field is discussed in detail. We apply our results to wave-packet spreading

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Section: Introductionmentioning

confidence: 99%

“…There has been a recurrence of interest in the phenomenon of anomalous diffusion. We refer to a sampling of the literature [1,2,3,4,5,6,7,8]. There have also appeared in the literature a variety of explanations for such behavior.…”

Section: Introductionmentioning

confidence: 99%

Anomalous diffusion in quantum Brownian motion with colored noise

Ford

O’Connell

2006

Phys. Rev. A

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“…Lévy processes in inhomogeneous media [10] and ergodicity breaking in heterogeneous diffusion processes [11] have been investigated, as well as the influence of external potentials on heterogeneous diffusion processes was recently considered in [12]. Time and power-law position dependent diffusion coefficient were also considered in the literature [13] in analysis of N-dimensional diffusion equation.…”

Section: Introductionmentioning

confidence: 99%

Heterogeneous diffusion in comb and fractal grid structures

Sandev

Schulz

Kantz

et al. 2018

Chaos, Solitons & Fractals

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We give an exact analytical results for diffusion with a power-law position dependent diffusion coefficient along the main channel (backbone) on a comb and grid comb structures. For the mean square displacement along the backbone of the comb we obtain behavior x 2 (t) ∼ t 1/(2−α) , where α is the powerlaw exponent of the position dependent diffusion coefficient D(x) ∼ |x| α . Depending on the value of α we observe different regimes, from anomalous subdiffusion, superdiffusion, and hyperdiffusion. For the case of the fractal grid we observe the mean square displacement, which depends on the fractal dimension of the structure of the backbones, i.e., x 2 (t) ∼ t (1+ν)/(2−α) , where 0 < ν < 1 is the fractal dimension of the backbones structure. The reduced probability distribution functions for both cases are obtained by help of the

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“…The use of fractional differential equations is becoming more popular in the field of hydrocarbon reservoirs, as it utilizes all the parameters previously stated. de Andrade et al (2005) discuss anomalous flow necessitating the use for fractional calculus, specificaly fractional diffusion equation. In this paper, the goal is to estimate the fractional order α of equation (1.1) which is difficult to isolate.…”

Section: Introductionmentioning

confidence: 99%

A Bayesian Estimation for the Fractional Order of the Differential Equation that Models Transport in Unconventional Hydrocarbon Reservoirs

Whitlinger,

Boone,

Ghanam

2017

Preprint

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The extraction of natural gas from the earth has been shown to be governed by differential equations concerning flow through a porous material. Recently, models such as fractional differential equations have been developed to model this phenomenon. One key issue with these models is estimating the fraction of the differential equation. Traditional methods such as maximum likelihood, least squares, and even method of moments are not available to estimate this parameter as traditional calculus methods do not apply. We develop a Bayesian approach to estimate the fraction of the order of the differential equation that models transport in unconventional hydrocarbon reservoirs. In this paper, we use this approach to adequately quantify the uncertainties associated with the error and predictions. A simulation study is presented as well to assess the utility of the modeling approach.

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Anomalous diffusion and fractional diffusion equation: anisotropic media and external forces

Cited by 23 publications

References 39 publications

Anomalous diffusion in quantum Brownian motion with colored noise

Anomalous diffusion in quantum Brownian motion with colored noise

Heterogeneous diffusion in comb and fractal grid structures

A Bayesian Estimation for the Fractional Order of the Differential Equation that Models Transport in Unconventional Hydrocarbon Reservoirs

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