Levy walks with applications to turbulence and chaos

Michael F. Shlesinger*, Joseph Klafter, Bruce J. West

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

63 Scopus citations

Abstract

Diffusion on fractal structures has been a popular topic of research in the last few years with much emphasis on the sublinear behavior in time of the mean square displacement of a random walker. Another type of diffusion is encountered in turbulent flows with the mean square displacement being superlinear in time. We introduce a novel stochastic process, called a Levy walk which generalizes fractal Brownian motion, to provide a statistical theory for motion in the fractal media which exists in a turbulent flow. The Levy walk describes random (but still correlated) motion in space and time in a scaling fashion and is able to account for the motion of particles in a hierarchy of coherent structures. We apply our model to the description of fluctuating fluid flow. When Kolmogorov's - 5 3 law for homogeneous turbulence is used to determine the memory of the Levy walk then Richardson's 4 3 law of turbulent diffusion follows in the Mandelbrot absolute curdling limit. If, as suggested by Mandelbrot, that turbulence is isotropic, but fractal, then intermittency corrections to the - 5 3 law follow in a natural fashion. The same process, with a different space-time scaling provides a description of chaos in a Josephson junction.

Original languageEnglish
Pages (from-to)212-218
Number of pages7
JournalPhysica A: Statistical Mechanics and its Applications
Volume140
Issue number1-2
DOIs
StatePublished - 1 Dec 1986
Externally publishedYes

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