Levy statistics

Anomalous transport features have been reported for an increasing number of (complex) systems. Many of these systems underlie some sort of a generalized limit theorem that is connected with Levy statistics and thus with selfsimilar evolution patterns. This fact is mirrored in the long-time prevalence of power-law time behaviors of the related physical quantities. 

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... 

A. R. Bizzarri and S. Cannistraro, Levy statistics of vibrational mode fluctuations of single molecules from surface-enhanced Raman scattering. Phys. Rev. Lett. 94 068303 (2005). 

G. Margolin and E. Barkai, Nonergodicity of a time series obeying Levy statistics, cond-mat/ 0504454 (2005) and J. Stat. Phys. DOI 10.1007/sl0955-005-8076-9. 

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... 

At this stage, we are confident that a clear connection between Levy statistics and critical random events is established. We have also seen that non-Poisson renewal yields a class of GME with infinite memory, from within a perspective resting on trajectories with jumps that act as memory erasers. The non-Poisson and renewal character of these processes has two major effects. The former will be discussed in detail in Section XV, and the latter will be discussed in Section XVI. The first problem has to do with decoherence theory. As we shall see, decoherence theory denotes an approach avoiding the use of wave function collapses, with the supposedly equivalent adoption of quantum densities becoming diagonal in the pointer basis set. In Section XV we shall see that the decoherence theory is inadequate to derive non-Poisson renewal processes from quantum mechanics. In Section XVI we shall show that the non-Poisson renewal properties, revealed by the BQD experiments, rule out modulation as a possible approach to complexity. 

Notice that this information approach to Levy statistics is even more direct than the nonextensive thermodynamic approach. As shown in Ref. 52, the adoption of the method of entropy maximization, with the Shannon entropy replaced by the Tsallis entropy [53], does not yield directly the Levy distribution, but a probability density function n(x) whereby reiterated application of the convolution generates the stable Levy distribution. 

However, this derivation of Levy statistics does not seem to reproduce satisfactorily all the properties that emerged from the illustration of superdiffusion processes of Section VI. We think that this formal way of proceeding does not address directly what seems to us a still unsolved issue of modem physics, this being the connection between dynamics and thermodynamics. As pointed out by the authors of Ref. 80, a satisfactory connection between a noncanonical form of equilibrium and thermodynamics should satisfactorily address the problem of thermal contact between a system at equilibrium in a canonical state and a system at equilibrium in a noncanonical condition. In other words, we have to establish which is the temperature of Levy system, if we believe that this form of noncanonical equilibrium is compatible with thermodynamic equilibrium. 

This observation reinforces, rather than weakens, the importance of the result of Ref. 84 insofar as it shows that with merely dynamic arguments the authors of this paper did derive the appropriate form of Boltzmann principle. This result sets the challenge for the derivation of the thermodynamic properties of Levy statistics from the same dynamic approach as that used in Ref. 84 to derive canonical equilibrium. 

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. 

Thus, we observe that when the memory kernel in the fractional Langevin equation is random, the solution consists of the product of two random quantities giving rise to a multifractal process. This is Feller s subordination process. We apply this approach to the SRV time series data discussed in Section II and observe, for the statistics of the multiplicative exponent given by Levy statistics, the singularity spectrum as a function of the positive moments... 

Compte, A., and J. Camacho. 1997. Levy statistics in Taylor dispersion. Phys. Rev. E 56 5445-5449. 

Geva and Skinner [14] provided a theoretical interpretation of the static line shape properties in a glass (i.e., tunneling model and the Kubo-Anderson approach as means to quantify the line shape behavior (i.e., the time-dependent fluctuations of W are neglected). In [16], the distribution of static line shapes in a glass was found analytically and the relation of this problem to Levy statistics was demonstrated. 

These techniques fail for strong environmental fluctuations. An interesting case of strong fluctuations is that where the rate coefflcients obey Levy statistics. In the particular case of first-order processes, Levy fluctuations lead to... 

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