Levy Flight Superdiffusion: An Introduction

Details Levy Flight Superdiffusion: An Introduction Levy Flight Superdiffusion: An Introduction

2008-10-08

arxiv.org/abs/0810.1492v1

Intern. Journ. of Bifurcation and Chaos, Vol. 18, No. 9, 2649 - 2672 (2008)

Physics

Condensed Matter

Statistical Mechanics

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

Scientific paper

After a short excursion from discovery of Brownian motion to the Richardson "law of four thirds" in turbulent diffusion, the article introduces the L\'{e}vy flight superdiffusion as a self-similar L\'{e}vy process. The condition of self-similarity converts the infinitely divisible characteristic function of the L\'{e}vy process into a stable characteristic function of the L\'{e}vy motion. The L\'{e}vy 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 L\'{e}vy flights. Some results concerning stationary probability distributions of L\'{e}vy 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 L\'{e}vy flights, recently obtained with different approaches.

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