Fractional Fokker-Planck dynamics: Numerical algorithm and simulations

Heinsalu, Els; Patriarca, Marco; Goychuk, Igor; Schmid, Gerhard; Hänggi, Peter

doi:10.1103/physreve.73.046133

Cited by 98 publications

(72 citation statements)

References 30 publications

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“…This represents a further element of the formal analogy between fractional and normal diffusion, besides, for example, the validity of the generalized Stratonovich formula (18) [19] and the fact that the stationary reduced probability density is the same for both cases [25]. Here, we present additional results which support and corroborate this formal analogy further.…”

Section: Probability Density: Anomalous Versus Normalsupporting

confidence: 79%

“…For a detailed description of the algorithm for the numerical simulations and of the employment of the Pareto or Mittag-Leffler distribution, we refer readers to the comprehensive work in [25].…”

Section: Set-up Of the Modelmentioning

confidence: 99%

“…The different small-time evolutions of the normal (left) and anomalous (right) reduced probability densitiesP(x, t) andP(x, t ) defined by equation (29), in the cosine potential (25). Curve labels (1)-(4) represent increasing values of re-scaled time t = 0.01, 0.02, 0.04, 0.11 for the normal case and of time t for the anomalous case, related to t through equation (28).…”

Section: Probability Density: Anomalous Versus Normalmentioning

confidence: 99%

“…As for the second moment, upon introducing the re-scaled time, Figure 4. The time evolutions of the probability densities characterizing the anomalous and normal diffusion processes in the periodic cosine potential (25). The re-scaled temperature is k B T / A = 0.5 and the fractional exponent is α = 0.5.…”

Section: Probability Density: Anomalous Versus Normalmentioning

confidence: 99%

See 3 more Smart Citations

Fractional diffusion in periodic potentials

Heinsalu¹,

Patriarca²,

Goychuk³

et al. 2007

J. Phys.: Condens. Matter

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Fractional, anomalous diffusion in space-periodic potentials is investigated. The analytical solution for the effective fractional diffusion coefficient in an arbitrary periodic potential is obtained in closed form in terms of two quadratures. This theoretical result is corroborated by numerical simulations for different shapes of the periodic potential. Normal and fractional spreading processes are contrasted via the time evolution of the corresponding probability densities in state space. While there are distinct differences occurring at small evolution times, a re-scaling of time yields a mutual matching between the long-time behaviours of normal and fractional diffusion.

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Section: Probability Density: Anomalous Versus Normalsupporting

confidence: 79%

Section: Set-up Of the Modelmentioning

confidence: 99%

Section: Probability Density: Anomalous Versus Normalmentioning

confidence: 99%

Section: Probability Density: Anomalous Versus Normalmentioning

confidence: 99%

See 2 more Smart Citations

Fractional diffusion in periodic potentials

Heinsalu¹,

Patriarca²,

Goychuk³

et al. 2007

J. Phys.: Condens. Matter

View full text Add to dashboard Cite

show abstract

“…Dispersion in complex plasmas (Ratynskaia et al, 2006), self-diffusion of surfactant molecules (Gambin et al, 2005), light in a cold atomic cloud (Labeyrie et al, 2003) and donor-acceptor electron pairs within a protein (Kou and Sunney Xie, 2004) are examples of the more recent experimental evidences. Several papers Heinsalu et al, 2006;Heinsalu et al, 2009) consider the fractional Fokker-Planck equation both in an analytical and a numerical approach. The fractional time approach is considered also in control theory, see, e.g., (Kaczorek, 2008;Guermah et al, 2008).…”

Section: Introductionmentioning

confidence: 99%

Numerical solutions to integral equations equivalent to differential equations with fractional time

Bandrowski¹,

Karczewska²,

Rozmej³

2010

International Journal of Applied Mathematics and Computer Science

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This paper presents an approximate method of solving the fractional (in the time variable) equation which describes the processes lying between heat and wave behavior. The approximation consists in the application of a finite subspace of an infinite basis in the time variable (Galerkin method) and discretization in space variables. In the final step, a large-scale system of linear equations with a non-symmetric matrix is solved with the use of the iterative GMRES method.

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Measurements and models of reactive transport in geological media

Berkowitz

Dror

Hansen

et al. 2016

Reviews of Geophysics

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Reactive chemical transport plays a key role in geological media across scales, from pore scale to aquifer scale. Systems can be altered by changes in solution chemistry and a wide variety of chemical transformations, including precipitation/dissolution reactions that cause feedbacks that directly affect the flow and transport regime. The combination of these processes with advective‐dispersive‐diffusive transport in heterogeneous media leads to a rich spectrum of complex dynamics. The principal challenge in modeling reactive transport is to account for the subtle effects of fluctuations in the flow field and species concentrations; spatial or temporal averaging generally suppresses these effects. Moreover, it is critical to ground model conceptualizations and test model outputs against laboratory experiments and field measurements. This review emphasizes the integration of these aspects, considering carefully designed and controlled experiments at both laboratory and field scales, in the context of development and solution of reactive transport models based on continuum‐scale and particle tracking approaches. We first discuss laboratory experiments and field measurements that define the scope of the phenomena and provide data for model comparison. We continue by surveying models involving advection‐dispersion‐reaction equation and continuous time random walk formulations. The integration of measurements and models is then examined, considering a series of case studies in different frameworks. We delineate the underlying assumptions, and strengths and weaknesses, of these analyses, and the role of probabilistic effects. We also show the key importance of quantifying the spreading and mixing of reactive species, recognizing the role of small‐scale physical and chemical fluctuations that control the initiation of reactions.

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Fractional Fokker-Planck dynamics: Numerical algorithm and simulations

Cited by 98 publications

References 30 publications

Fractional diffusion in periodic potentials

Fractional diffusion in periodic potentials

Numerical solutions to integral equations equivalent to differential equations with fractional time

Measurements and models of reactive transport in geological media

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