Fractional Fokker-Planck equation for ultraslow kinetics

By Aleksei V. Chechkin, Joseph Klafter, Igor M. Sokolov. 2003

155 citations recorded by the source corpus.

Several classes of physical systems exhibit ultraslow diffusion for which the mean-squared displacement at long times grows as a power of the logarithm of time (strong anomaly) and share the interesting property that the probability distribution of particle's position at long times is a double-sided exponential. We show that such behaviors can be adequately described by a distributed-order fractional Fokker-Planck equations with a power law weighting function. We discuss the equations and the properties of their solutions, and connect this description with a scheme based on continuous-time random walks.

Subfields: Modeling and Simulation, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/332694320297047ae374543661bdbfd3158c14c9

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Citations in the corpus

Finite difference approximations for fractional advection-dispersion flow equations (2004, background): “However, many practical problems require a model with variable coe4cients [3,11].”
The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics (2004, methodology): “In contrast, for LFs, we will report that the method of images breaks down, and the details of the process (especially the Lévy index) enter only marginally into the first passage time density (FPTD).”
Finite difference methods for two-dimensional fractional dispersion equation (2006, methodology): “Keywords: Two-dimensional fractional partial differential equation; Implicit Euler method; Multi-dimensional fractional PDE; Numerical fractional PDE; Alternating direction implicit methods for fractional problems”
Numerical solution for diffusion equations with distributed order in time using a Chebyshev collocation method (2017, background): “[10] where they use an extra constant that represents a characteristic time of the problem.”
L\'evy walks (2014, methodology): “This line of research led to the formulation of the fractional Fokker-Planck (Barkai, 2001; Barkai et al., 2000b; Chechkin et al., 2003; Heinsalu et al., 2007; Metzler et al., 1999) and Klein-Kramers equations (Barkai and Silbey, 2000; Dieterich et al., 2008; Eule et al., 2007; Friedrich 20 et al.,…”
Finite di$erence approximations for fractional advection-dispersion &ow equations (2004, methodology): “Keywords: Finite di$erence approximation; Stability; Backward Euler method; Implicit Euler method; Radial dispersion; Radial advection; Fractional di$usion; Fractional derivative; Fractional advection–dispersion; Numerical fractional ADE”
Distributed order calculus and equations of ultraslow diffusion (2007, background): “A number of recent publications by physicists (see [3, 4, 5, 25, 38] and references there) is devoted to the case where the mean square displacement has a logarithmic growth.”
VECTOR GRÜNWALD FORMULA FOR FRACTIONAL DERIVATIVES (2004, background): “Introduction Fractional derivatives have been around for centuries [22, 26] but recently they have found new applications in physics [2, 6, 7, 9, 15, 18, 19, 29], hydrology [1, 4, 5, 10, 14, 28], and finance [24, 25, 27].”