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Fractional diffusion Processes: Probability Distributions and Continuous Time Random Walk

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Processes with Long-Range Correlations

Part of the book series: Lecture Notes in Physics ((LNP,volume 621))

Abstract

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 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 ε (0, 2] and skewness θ (|θ| ≤ min {α, 2 - α}), and the first-order time derivative with a Caputo derivative of order β ε (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.

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Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Science, Free University of Berlin, Arnimallee 3, D-14195, Berlin, Germany

    Rudolf Gorenflo

  2. Dipartimento di Fisica, Università di Bologna, and INFN, Sezione di Bologna, Via Irnerio 46, I-40126, Bologna, Italy

    Francesco Mainardi

Authors
  1. Rudolf Gorenflo
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  2. Francesco Mainardi
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Editors and Affiliations

  1. Dept. of Mathematics, Indian Institute of Science, 560 012, Bangalore, India

    Govindan Rangarajan

  2. Center for Complex Systems and Brain Sciences, Florida Atlantic University, 33431, Boca Raton, FL, USA

    Mingzhou Ding

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Gorenflo, R., Mainardi, F. (2003). Fractional diffusion Processes: Probability Distributions and Continuous Time Random Walk. In: Rangarajan, G., Ding, M. (eds) Processes with Long-Range Correlations. Lecture Notes in Physics, vol 621. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44832-2_8

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  • DOI: https://doi.org/10.1007/3-540-44832-2_8

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