Levy walks with applications to turbulence and chaos

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.