An Introduction to Fractional Diffusion (original) (raw)

Some recent advances in theory and simulation of fractional diffusion processes

Journal of Computational and Applied Mathematics, 2009

To offer an insight into the rapidly developing theory of fractional diffusion processes we describe in some detail three topics of current interest: (i) the well-scaled passage to the limit from continuous time random walk under power law assumptions to space-time fractional diffusion, (ii) the asymptotic universality of the Mittag-Leffler waiting time law in time-fractional processes, (iii) our method of parametric subordination for generating particle trajectories.

Fractional diffusion in inhomogeneous media

Journal of Physics A: Mathematical and General, 2005

Starting from the continuous time random walk (CTRW) scheme with the space-dependent waiting-time probability density function (PDF) we obtain the time-fractional diffusion equation with varying in space fractional order of time derivative. As an example, we study the evolution of a composite system consisting of two separate regions with different subdiffusion exponents and demonstrate the effects of non-trivial drift and subdiffusion whose laws are changed in the course of time.

LETTER TO THE EDITOR: Fractional diffusion in inhomogeneous media

J Phys a Math Gen, 2005

From power laws to fractional diffusion: the direct way

2007

Starting from the model of continuous time random walk that can also be considered as a compound renewal process we focus our interest on random walks in which the probability distributions of the waiting times and jumps have fat tails characterized by power laws with exponent between 0 and 1 for the waiting times, between 0 and 2 for the jumps. By stating the relevant lemmata (of Tauber type) for the distribution functions we need not distinguish between continuous and discrete space and time. We will see that by a well-scaled passage to the diffusion limit diffusion processes fractional in time as well as in space are obtained. The corresponding equation of evolution is a linear partial pseudo-differential equation with fractional derivatives in time and in space, the orders being equal to the above exponents. Such processes are enjoying increasing popularity in applications in physics, chemistry, finance and other fields, and their behaviour can be well approximated and visualized by simulation via various types of random walks. For their explicit solutions there are available integral representations that allow to investigate their detailed structure. For ease of presentation we restrict attention to the spatially one-dimensional symmetric situation.

Temporal Diffusion: From Microscopic Dynamics to Generalised Fokker–Planck and Fractional Equations

Journal of Statistical Physics

The temporal Fokker-Planck equation (Boon et al. in J Stat Phys 3/4: 527, 2003) or propagation-dispersion equation was derived to describe diffusive processes with temporal dispersion rather than spatial dispersion as in classical diffusion. We present two generalizations of the temporal Fokker-Planck equation for the first passage distribution function f j (r, t) of a particle moving on a substrate with time delays τ j. Both generalizations follow from the first visit recurrence relation. In the first case, the time delays depend on the local concentration, that is the time delay probability P j is a functional of the particle distribution function and we show that when the functional dependence is of the power law type, P j ∝ f ν−1 j , the generalized Fokker-Planck equation exhibits a structure similar to that of the nonlinear spatial diffusion equation where the roles of space and time are reversed. In the second case, we consider the situation where the time delays are distributed according to a power law, P j ∝ τ −1−α j (with 0 < α < 2), in which case we obtain a fractional propagationdispersion equation which is the temporal analog of the fractional spatial diffusion equation (with space and time interchanged). The analysis shows how certain microscopic mechanisms can lead to non-Gaussian distributions and non-classical scaling exponents.

Fractional diffusion Processes: Probability Distributions and Continuous Time Random Walk

Lecture Notes in Physics, 2003

A physical-mathematical approach to anomalous diffusion may be based on generalized diffusion equations (containing derivatives of fractional order in space or/and time) and related random walk models. By the spacetime fractional diffusion equation we mean an evolution equation obtained from the standard linear diffusion equation by replacing the second-order space derivative with a Riesz-Feller derivative of order α ∈ (0, 2] and skewness θ (|θ| ≤ min {α, 2 − α}), and the first-order time derivative with a Caputo derivative of order β ∈ (0, 1] . The fundamental solution (for the Cauchy problem) of the fractional diffusion equation can be interpreted as a probability density evolving in time of a peculiar self-similar stochastic process. We view it as a generalized diffusion process that we call fractional diffusion process, and present an integral representation of the fundamental solution. A more general approach to anomalous diffusion is however known to be provided by the master equation for a continuous time random walk (CTRW). We show how this equation reduces to our fractional diffusion equation by a properly scaled passage to the limit of compressed waiting times and jump widths. Finally, we describe a method of simulation and display (via graphics) results of a few numerical case studies.

Discrete random walk models for space–time fractional diffusion

Chemical Physics, 2002

A physical-mathematical approach to anomalous diffusion may be based on generalized diffusion equations (containing derivatives of fractional order in space or/and time) and related random walk models. By space-time fractional diffusion equation we mean an evolution equation obtained from the standard linear diffusion equation by replacing the second-order space derivative with a Riesz-Feller derivative of order α ∈ (0, 2] and skewness θ (|θ| ≤ min {α, 2 − α}), and the first-order time derivative with a Caputo derivative of order β ∈ (0, 1] . Such evolution equation implies for the flux a fractional Fick's law which accounts for spatial and temporal non-locality. The fundamental solution (for the Cauchy problem) of the fractional diffusion equation can be interpreted as a probability density evolving in time of a peculiar self-similar stochastic process that we view as a generalized diffusion process. By adopting appropriate finite-difference schemes of solution, we generate models of random walk discrete in space and time suitable for simulating random variables whose spatial probability density evolves in time according to this fractional diffusion equation.

Discrete and Continuous Random Walk Models for Space-Time Fractional Diffusion

Journal of Mathematical Sciences, 2006

A physical-mathematical approach to anomalous diffusion may be based on generalized diffusion equations (containing derivatives of fractional order in space or/and time) and related random walk models. By space-time fractional diffusion equation we mean an evolution equation obtained from the standard linear diffusion equation by replacing the second-order space derivative with a Riesz-Feller derivative of order α ∈ (0, 2] and skewness θ (|θ| ≤ min {α, 2 − α}), and the first-order time derivative with a Caputo derivative of order β ∈ (0, 1] . Such evolution equation implies for the flux a fractional Fick's law which accounts for spatial and temporal non-locality. The fundamental solution (for the Cauchy problem) of the fractional diffusion equation can be interpreted as a probability density evolving in time of a peculiar self-similar stochastic process that we view as a generalized diffusion process. By adopting appropriate finite-difference schemes of solution, we generate models of random walk discrete in space and time suitable for simulating random variables whose spatial probability density evolves in time according to this fractional diffusion equation.

An Introduction to Fractional Diffusion (original) (raw)

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