TY - JOUR
T1 - Levy walks with applications to turbulence and chaos
AU - Shlesinger, Michael F.
AU - Klafter, Joseph
AU - J. West, Bruce
PY - 1986/12/1
Y1 - 1986/12/1
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=0009515051&partnerID=8YFLogxK
U2 - 10.1016/0378-4371(86)90224-4
DO - 10.1016/0378-4371(86)90224-4
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AN - SCOPUS:0009515051
SN - 0378-4371
VL - 140
SP - 212
EP - 218
JO - Physica A: Statistical Mechanics and its Applications
JF - Physica A: Statistical Mechanics and its Applications
IS - 1-2
ER -