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Chaotic dynamics of super‐diffusion revisited
Author(s) -
Cushman John H.,
Park Moongyu,
O'Malley Daniel
Publication year - 2009
Publication title -
geophysical research letters
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.007
H-Index - 273
eISSN - 1944-8007
pISSN - 0094-8276
DOI - 10.1029/2009gl037399
Subject(s) - fractional brownian motion , brownian motion , statistical physics , hurst exponent , lyapunov exponent , diffusion , physics , mean squared displacement , fractal , diffusion process , exponent , flow (mathematics) , mathematics , mathematical analysis , mechanics , quantum mechanics , statistics , computer science , knowledge management , linguistics , innovation diffusion , philosophy , nonlinear system , molecular dynamics
Super‐diffusive mixing in geophysics occurs in atmospheric turbulence, near surface currents in the oceans, and fracture flow in the subsurface to name a few examples. Models of super‐diffusion have been around since L. F. Richardson's pioneering work in the 1920's. Here we construct a family of super‐diffusive stochastic processes X α ( t ), 0 < α , with independent, nonstationary increments, but a priori defined mean‐square displacement given by t α +1 . The case α = 2 corresponds to Richardson super‐diffusion. The family of processes is a nonstationary extension of Brownian motion and hence completely characterized by its first two moments. The Fokker‐Planck equation for X α ( t ) is classical diffusive with time dependent diffusion coefficient given by t α /2. In contrast to what is found in fractional Brownian motion, where the fractal dimension depends on the Hurst exponent, here the fractal dimension and hence complexity of the process is the same for all exponents α as that of classical Brownian motion. An analytical expression is developed for the finite‐size Lyapunov exponent and numerical examples presented.