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Operator Lévy motion and multiscaling anomalous diffusion.

M M Meerschaert1, D A Benson, B Baeumer

  • 1Department of Mathematics, University of Nevada, Reno, Nevada 89557-0084, USA. mcubed@unr.edu

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|April 20, 2001
PubMed
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Anomalous diffusion exhibits directional scaling rates due to heavy-tailed particle jumps. Operator stable motions model this, showing different dispersion rates in principal directions, crucial for understanding tracer movement in aquifers.

Area of Science:

  • Physics
  • Mathematics
  • Geophysics

Background:

  • Anomalous diffusion is characterized by heavy-tailed (power-law) particle jumps.
  • Particle jump scaling rates can differ across directions.
  • Natural aquifer tracer movement shows varying scaling rates parallel and perpendicular to flow.

Purpose of the Study:

  • To introduce and analyze operator stable motions as a model for anomalous diffusion with directional scaling.
  • To describe the governing equations for these motions, incorporating generalized fractional operators.
  • To provide a mathematical framework for understanding anisotropic dispersion in natural systems.

Main Methods:

  • Operator stable motions are defined as limits of d-dimensional random jumps.
  • Scale invariance is characterized by c(H)Y(t)=Y(ct) with a dxd matrix H.

Related Experiment Videos

  • Eigenvalues of H determine directional diffusion rates proportional to t(1/alpha(j)).

    • Main Results:

    • Operator stable motions exhibit different Fickian or super-Fickian diffusion rates in principal directions.
    • The governing equation involves a spatial dispersion operator mixing fractional derivatives of varying orders.
    • Specific cases include fractional Laplacian with directional mixing and the Riesz potential.

    Conclusions:

    • Operator stable motions provide a framework for anomalous diffusion with anisotropic scaling.
    • The generalized fractional operator captures directional dispersion observed in natural aquifers.
    • This model enhances understanding of solute transport in heterogeneous media.