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 L\'evy flight superdiffusion as a self-similar L\'evy process. The condition of self-similarity converts the infinitely divisible characteristic function of the L\'evy process into a stable characteristic function of the L\'evy motion. The L\'evy motion generalizes the Brownian motion on the base of the α-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\'evy flights. Some results concerning stationary probability distributions of L\'evy 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\'evy flights, recently obtained with different approaches.