Fractal time” process, fractional dynamics

We shall now almost exclusively concentrate on the fractal time random walk excluding inertial effects and the discrete orientation model of dielectric relaxation. We shall demonstrate how in the diffusion limit this walk will yield a fractional generalization of the Debye-Frohlich model. Just as in the conventional Debye relaxation, a fractional generalization of the Debye-Frohlich model may be derived from a number of very different models of the relaxation process (compare the approach of Refs. 22, 23, 28 and 34—36). The advantage of using an approach based on a kinetic equation such as the fractional Fokker-Planck equation (FFPE) however is that such a method may easily be extended to include the effects of the inertia of the dipoles, external potentials, and so on. Moreover, the FFPE (by use of a theorem of operational calculus generalized to fractional exponents and continued fraction methods) clearly indicates how many existing results of the classical theory of the Brownian motion may be extended to include fractional dynamics. 

The introduction of fractional derivative 5 /9r in the kinetic Equation (1) also allows taking into account random walks in fractal time (RWFT)—a temporal component of strange dynamic processes in turbulent mediums [5], The absence of some appreciable jumps in particles behavior serves as a distinctive feature of RWFT in addition mean-square displacement grows with t as r. Parameter a has the sense of fractal dimension of active time, in which real particles walks look as random process active time interval is proportional to r [5],... 

In this section we describe some of the essential features of fractal functions starting from the simple dynamical processes described by functions that are fractal (such as the Weierstrass function) and that are continuous everywhere but are nowhere differentiable. This idea of nondifferentiability leads to the introduction of the elementary definitions of fractional integrals and fractional derivatives starting from the limits of appropriately defined sums. We find that the relation between fractal functions and the fractional calculus is a deep one. For example, the fractional derivative of a regular function yields a fractal function of dimension determined by the order of the fractional derivative. Thus, the changes in time of phenomena that are best described by fractal functions are probably best described by fractional equations of motion, as well. In any event, this latter perspective is the one we developed elsewhere [52] and discuss herein. Others have also made inquiries along these lines [70] ... 

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