Mittag-Leffler function

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

The Mittag-Leffler function, or combinations thereof, has been obtained from fractional rheological models, and it convincingly describes the behavior of a number of rubbery and nonrubbery polymeric substances [79, 85]. The numerical behavior of the Mittag-Leffler function is equivalent to asymptotic power-law patterns that are often used to fit experimental data, see the comparative discussion of data from early events in peptide folding in Ref. 86, where the asymptotic power-law was confronted with the stretched exponential fit function. 

The Mittag-Leffler function [44-46] can be viewed as a natural generalization of the exponential function. Within fractional dynamics, it replaces the traditional exponential relaxation patterns of moments, modes, or of the Kramers survival. It is an entire function that decays completely monotoni-cally for 0 < a < 1. It is the exact relaxation function for the underlying multiscale process, and it leads to the Cole-Cole behavior for the complex... 

The Mittag-Leffler function is defined through the inverse Laplace transform... 

We note in passing that the Mittag-Leffler function is the solution of the fractional relaxation equation [84]... 

The Mittag-Leffler function interpolates between the initial stretched exponential form... 

R. Gorenflo and F. Mainardi, Fractional oscillations and Mittag-Leffler functions, International Workshop on the Recent Advances in Applied Mathematics, State of Kuwait, May 4-7, 1996. Proceedings, Kuwait University, Department of Mathematics and Computer Science, 1996, p. 193. 

The Mittag-Leffler function has interesting properties in both the short-time and the long-time limits. In the short-time limit it yields the Kohlrausch-Williams-Watts Law from stress relaxation in rheology given by... 

Figure 12 displays the Mittag-Leffler function as well as the two asymptotes, the dashed curve being the stretched exponential and the dotted curve the inverse power law. What is apparent from this discussion is the long-time memory associated with the fractional relaxation process, being an inverse power law rather than the exponential of ordinary relaxation. It is apparent that the Mittag-Leffler function smoothly joins these two empirically determined asymptotic distributions. 

Figure 12. The solid curve is the Mittag-Leffler function, the solution to the fractional relaxation equation. The dashed curve is the stretched exponential (Kohlrausch-Williams-Watts Law), and the dotted curve is the inverse power law (Nutting Law).

Note the difference in the 5-dependence of the two coefficients of the right-hand side of Eq. (132). The inverse Laplace transform of the first term yields the Mittag-Leffler function as found in the homogeneous case above. The inverse Laplace transform of the second term is the convolution of the random force and a stationary kernel. The kernel is given by the series... 

In the case a = 1, the Mittag-Leffler function becomes the exponential, so that the solution to the fractional Langevin equation reduces to that for an Omstein-Uhlenbeck process... 

It is convenient to use Fox functions (generalized Mellin-Barnes integras) when solving equations with fractional derivatives because Laplace and Fourier transformations for Fox functions may be expressed via Fox functions with given parameters. The connection of Mittag-Leffler functions with Fox functions is as follows [216,217] ... 

We now express Mittag-Leffler function in terms of the Fox function [216] and using the formulae from Eq. (543) apply the inverse Fourier transformation... 

Now, on noting that the Laplace transform of the Mittag-Leffler function Ea[-(t/x)°] is [31]... 

Therefore, in the present context, the Kramers escape rate can be best understood as playing the role of a decay parameter in the Mittag-Leffler functions governing the highly nonexponential relaxation behavior of the system. 

We note that the characteristic times %int and xef do not exist in anomalous diffusion (a / 1). This is obvious from the properties of the Mittag-Leffler function [see Eqs. (62) and (63)]. However, we shall now demonstrate that the above time constants may also be used to characterize the dynamic susceptibility in anomalous relaxation. 

Equation (156) illustrates how anomalous relaxation influences the complex susceptibility arising from the slowest relaxation mode in the noninertial limit. Therefore, in anomalous relaxation the Kramers escape rate can be best understood as playing the role of a decay parameter in the Mittag-Leffler function governing the relaxation behavior of the system. We remark that the slowest decay mode, the parameters of which are determined by Eq. (156), will... 

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