Levy walk statistics

Kimmich and coworkers have studied the magnetic relaxation dispersion of liquids adsorbed on or contained in microporous inorganic materials such as glasses and packed silica (34-43) and analyze the relaxation dispersion data using Levy walk statistics for surface diffusion to build... 

Stapf S, Kimmich R and Seitter R.-O. (1995) Proton and deuteron field-cycHng NMR relaxometry of liquids in porous glasses Evidence for Levy-Walk statistics. Physical Review Letters 75 2855. 

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... 

As a step toward the study of thermodynamic equilibrium in the case of anomalous statistical physics, in Section VII we study how the generators of anomalous diffusion respond to external perturbation. The ordinary linear response theory is violated and, in some conditions, is replaced by a different kind of linear response. In Section VIII we review the results of an ambitious attempt at deriving thermodynamics from dynamics for the main purpose of exploring a dynamic approach to the still unsettled issue of the thermodynamics of Levy statistics. The Levy walk perspective seems to be the only possible way to establish a satisfactory connection between dynamics and thermodynamics in... 

The adoption of Eq. (195) yields an equilibrium distribution that is similar to the WS statistics, with the main difference, however, that the inverse power law is truncated by two peaks, at = W/y and = —W/y. Note that the Levy walk noise i (f) is generated according to the renewal prescriptions of Section VI that is, we use the waiting time distribution /(f) of Eq. (92) and, according to the... 

Figure 3. Comparison of the trajectories of a Gaussian (left) and a Levy (right) process, the latter with index a = 1.5. While both trajectories are statistically self-similar, the Levy walk trajectory possesses a fractal dimension, characterizing the island structure of clusters of smaller steps, connected by a long step. Both walks are drawn for the same number of steps ( 7000).

Anomalous diffusion is often caused by memory effects and Levy-type statistics [185, 53], Specifically, superdiffusion is observed for random walks with heavytailed jump length distributions and subdiffusion for heavy-tailed waiting time distributions, see Sect. 3.4. The latter type of distribution can be caused by traps that have an infinite mean waiting time [185]. For reviews of anomalous diffusion see, e.g., [298,299, 229]. 

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. 

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