Levy Flight Superdiffusion: An Introduction

Abstract

After a short excursion from discovery of Brownian motion to the Richardson "law of four thirds" in turbulent diffusion, the article introduces the Lévy flight superdiffusion as a self-similar Lévy process. The condition of self-similarity converts the infinitely divisible characteristic function of the Lévy process into a stable characteristic function of the Lévy motion. The Lévy motion generalizes the Brownian motion on the base of the $\alpha$-stable distributions theory and fractional order derivatives. The further development of the idea lies on the generalization of the Langevin equation with a non-Gaussian white noise source and the use of functional approach. This leads to the Kolmogorov's equation for arbitrary Markovian processes. As particular case we obtain the fractional Fokker-Planck equation for Lévy flights. Some results concerning stationary probability distributions of Lévy motion in symmetric smooth monostable potentials, and a general expression to calculate the nonlinear relaxation time in barrier crossing problems are derived. Finally we discuss results on the same characteristics and barrier crossing problems with Lévy flights, recently obtained with different approaches.

32 pages, 9 figures, to appear in Int. J. of Bifurcation and Chaos

References in corpus (7)

• The fundamental solution of the space-time fractional diffusion equation
• Escape driven by $\alpha$-stable white noises
• Cooling down Levy flights
• First exit times for Lévy-driven diffusions with exponentially light jumps
• Comment on "Mean First Passage Time for Anomalous Diffusion"
• Lifetime of metastable states and suppression of noise in Interdisciplinary Physical Models
• Mean number of visits to sites in Levy flights

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