Levy-walk diffusion relation

Figure 11. DEA and SDA for (A) a fractal Gaussian intermittent noise with /(t) = exp[—t/y] with y = 25 and H—d — 0.75 the fractal Gaussian relation of equal exponents is satisfied. (B) A Levy-walk intermittent noise with /(t) oc and p = 2.5 note the bifurcation between H — 0.75 and 5 = 0.67 caused by the Levy-walk diffusion relation [66].

How general are our results From a stochastic point of view ergodicity breaking, Levy statistics, anomalous diffusion, aging, and fractional calculus, are all related. In particular ergodicity breaking is found in other models with power-law distributions, related to the underlying stochastic model (the Levy walk). For example, the well known continuous time random walk model also... 

Using DEA, we have established that there are statistical processes for which 8 = H and statistical processes for which 8 H, both of which scale. However, there is a third class of processes for which the scaling index is a function of the Hurst exponent, but the relation is not one of their being equal. This third class is the Levy random walk process (Levy diffusion) introduced by Shlesinger et al. [65] in their discussion of the application of Levy statistics to the understanding of turbulent fluid flow. 

A convenient way to formulate a dynamical equation for a Levy flight in an external potential is the space-fractional Fokker-Planck equation. Let us quickly review how this is established from the continuous time random walk. We will see below, how that equation also emerges from the alternative Langevin picture with Levy stable noise. Consider a homogeneous diffusion process, obeying relation (16). In the limit k — 0 and u > 0, we have X(k) 1 — CTa fe and /(w) 1 — uz, whence [52-55]... 

The main objective of this chapter is to establish the relation between the macroscopic equations like (3.1) and (3.5), the mesoscopic equations (3.2) and (3.3), etc., and the underlying microscopic movement of particles. We will show how to derive mesoscopic reaction-transport equations like (3.2) and (3.3) from microscopic random walk models. In particular, we will discuss the scaling procedures that lead to macroscopic reaction-transport equations. As an example, let us mention that the macroscopic reaction-diffusion equation (3.1) occurs as a result of the convergence of the random microscopic movement of particles to Brownian motion, while the macroscopic fractional equation (3.5) is closely related to the convergence of random walks with heavy-tailed jump PDFs to a-stable random processes or Levy flights. 

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