Levy distribution

Appendix A A Primer on Levy Distributions Appendix B The Ubiquitous Mittag-Leffler Function References... 

Systems that display strange kinetics no longer fall into the basin of attraction of the central limit theorem, as can be anticipated from the anomalous form (1) of the mean squared displacement. Instead, they are connected with the Levy-Gnedenko generalized central limit theorem, and consequently with Levy distributions [43], The latter feature asymptotic power-law behaviors, and thus the asymptotic power-law form of the waiting time pdf, w(r) AaT /r1+a, may belong to the family of completely asymmetric or one-sided Levy distributions L+, that is,... 

We choose the representation in terms of a Levy distribution for convenience because it includes the Brownian limit. Indeed, any waiting time pdf w(t) with the asymptotic power-law trend following Eq. (6) leads to the same results as obtained in the following for 0[Pg.229]

Let us examine the one-sided Levy distribution dF(X) = L+ (t/x) with the characteristic function... 

The essential difference between the two transition probability densities lies in the fact that for the gaussian distribution pw r, ) the different moments E[Xm], m = 1, 2,. . . , n, exist, while for the Cauchy distribution pc(j, x) they do not exist. The Levy distributions characterized by p(t, k) = exp -a k qT) with 0< <2U 127 128 play a prominent role in the theory of relaxation processes.129 133... 

The derivation of the Levy distribution from the proper extension of these arguments is straightforward. We assume that the second piece of information available to us, rather than being expressed in terms of the second-order derivative of Eq. (185), is given by... 

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. 

This is a property of remarkable interest for the settlement of the still open problem of defining the temperature associated to a given Levy distribution [80]. In fact, thanks to this truncation, the dynamic process of Eq. (195) is shown [87] to reach an equilibrium distribution characterized by a finite second moment. The explicit expression of this second moment is... 

Gaussian-like distribution of energy around the energy average. Other ensembles with non-Boltzmann distributions can enhance the sampling considerably for example, in the multi-canonical approach [97, 98], all the conformations are equiprobable in energy in Tsallis statistics [99], the distribution function includes Boltzmann, Lorentzian, and Levy distributions. 

Note that the integral term in Eq. (174) is the Reisz fractional derivative, first applied in this context by Seshadri and West [96] and whose solution is the symmetric Levy distribution. It is worth stressing that in the last few years the approaches based on fractional derivatives, of which Eq. (174) is an early example [95], have received an ever-increasing interest, as shown by the excellent review articles by Metzler and Klafter [97] and Sokolov et al. [78]. The symmetric Levy distribution that solves Eq. (174) is... 

The tails prevent [19] convergence to the Gaussian distribution for N -= oo, but not the existence of a limiting distribution. These distributions as we have seen are called stable distributions. If the concept of a Levy distribution is applied to an assembly of temporal random variables such as the x, of the present chapter, then w(x) is a long-tailed probability density function with long-time asymptotic behavior [7,37],... 

The divergence of all the global characteristic times for anomalous diffusion—as defined in their conventional sense (which is a natural consequence of the underlying Levy distribution), rendering them useless as a measure of the relaxation behavior—signifies the importance of characteristic times for such processes in terms of the frequency-domain representation of... 

Santra, S.B., B. Sapoval, and O. Haeberle. 1997. Levy distribution of collisions for Knudsen diffusion in fractal pores, p. 350-355. In J.L. Vehel et al. (ed.) Fractals in engineering. From theory to industrial applications. Springer-Verlag, New York. 

Width parameter C corresponds to the standard deviation of Gaussian distribution. Compared with Gaussian distribution, Levy distribution is characterized by long tails and infinite theoretical second (and higher) moments. We further define increments of log(K), where K refers to permeability, as... 

The Levy distribution and Levy process have many interesting properties, partially connected to random walks on fractals (see e.g. Lindenberg and West 1986). The right-hand-side of the evolution equation for a Levy process can not be constructed by,differential operators. The schematic form of the equation is ... 

---