Fractal time processes

T = J0°° w(t)tdt —> oo, manifesting the self-similar nature of this waiting process that has also prompted the coinage of fractal time processes [48]. Note that in the limit a —> 1, this waiting time pdf reduces to the singular form Lf (t/x) — 5(t - x) with finite T = % that leads back to the temporally local Markovian formulation of classical Brownian transport. In fact, for any waiting time pdf with a finite characteristic time T, one recovers the Brownian picture, such as for the Poissonian form w(t) = x xe t x. 

The essentially different nature of transport processes with y/ t) °c r should be stressed. Processes with this type of waiting time distribution function show an absence of scale. They exhibit very sporadic behavior. Long dormancies are followed by bursts of activity. They have been desaibed as fractal time processes (Schlesinger [1984]). Fractal space processes, in which the absence of scale is present in the spatial aspects of the transport, are considered later in this section. 

In Refs. 80 and 81 it is shown that the Mittag-Leffier function is the exact relaxation function for an underlying fractal time random walk process, and that this function directly leads to the Cole-Cole behavior [82] for the complex susceptibility, which is broadly used to describe experimental results. Furthermore, the Mittag-Leffier function can be decomposed into single Debye processes, the relaxation time distribution of which is given by a mod-... 

Conversely, the relationship (7.2) expresses a time-scale invariance (selfsimilarity or fractal scaling property) of the power-law function. Mathematically, it has the same structure as (1.7), defining the capacity dimension dc of a fractal object. Thus, a is the capacity dimension of the profiles following the power-law form that obeys the fundamental property of a fractal self-similarity. A fractal decay process is therefore one for which the rate of decay decreases by some exact proportion for some chosen proportional increase in time the self-similarity requirement is fulfilled whenever the exact proportion, a, remains unchanged, independent of the moment of the segment of the data set selected to measure the proportionality constant. 

In this section the notion of an allometric relation is generalized to include measures of time series. In this view, y is interpreted to be the variance and x the average value of the quantity being measured. The fact that these two central measures of a time series satisfy an allometric relation implies that the underlying time series is a fractal random process and therefore scales. It was first determined empirically that certain statistical data satisfy a power-law relation of the form given by Taylor [17] in Eq. (1), and this is where we begin our discussion of the allometric aggregation method of data analysis. 

Successive increments of mathematical fractal random processes are independent of the time step. Here D = 1.5 corresponds to a completely uncorrelated random process r = 0, such as Brownian motion, and D = 1.0 corresponds to a completely correlated process r= 1, such as a regular curve. Studies of various physiologic time series have shown the existence of strong long-time correlations in healthy subjects and demonstrated the breakdown of these correlations in disease see, for example, the review by West [56]. Complexity decreases with convergence of the Hurst exponent H to the value 0.5 or equivalently of the fractal dimension to the value 1.5. Conversely, system complexity increases as a single fractal dimension expands into a spectrum of dimensions. 

Both these Langevin equations are monofractal if the fluctuations are monofractal, which is to say, the time series given by the trajectory X(t) is a fractal random process if the random force is a fractal random process. 

The empirical evidence overwhelmingly supports the interpretation of the time series analysis that complex physiologic phenomena are described by fractal stochastic processes. Furthermore, the fractal nature of these time series is not constant in time but changes with the vagaries of the interaction of the system with its environment, and therefore these phenomena are multifractal. 

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 time behavior of the right-hand side of Eq. (90) indicates that the initial state Wo(x) decays slowly with a long-time tail unlike the exponential decay of normal diffusion, which is an indication of the fractal time character of the process. Equations (89) and (90) are fractional analogs of the conventional Fokker-Planck equation [Eq. (88)] giving rise to the Cole-Cole anomalous behavior. 

Fractals can also be inferred from morphogenesis (the development of form or structure), where a single time scale does not adequately address all time-dependent processes. The electrocardiogram seems to have fractal time properties, as well as electrical activity of a single neuron and beat-to-beat variability of the heart rate. There are also fractal (power law) variations in blood neutrophil counts. Further research will probably turn up other cases of self-similarity. 

Thus, from the said above it follows, that thermooxidative degradation process of PAr and PAASO melts proceeds in the fractal space with dimension A In such space degradation process can be presented schematically as devil s staircase [33]. Its horizontal sections correspond to temporal intervals, where the reaction does not proceed. In this case the degradation process is described with fractal time t using, which belongs to Cantor s setpoints [34]. If the reaction is considered in Euclidean space, then time belongs to real numbers sets. 

For the evolutionary processes with fractal time description the mathematical calculus of fractional differentiation and integration is used [34]. As it has been shown in Ref [35], in this case the fractional exponent... 

Devalues, obtained for PAr and PUAr poly condensation process, showed, that the indicated processes were realized by aggre tion cluster-cluster mechanism [49], i.e., by small macromolecular coils joining in larger ones [23], Thus, polycondensation process is a fractal object with dimension D. reaction. Such reaction can be presented schematically in a form of devil s staircase [80], Its horizontal parts correspond to temporal intervals, in which reaction is not realized. In this case polycondensation process is described with irsing fractal time t, which belongs to Kantor s set points [81], If polycondensation process is considered in Euclidean space, then time belongs to a real number set. 

The mathematical calculus of fractional differentiation and integration is used for the description of evolutionary processes with fractal time [81 ]. As it has been shown in paper [82], in this case the fractional exponent v coincides with fractal dimension of Kantor s set and indicates fraction of system states, maintaining during all evolution time t. Let us remind, that Kantor s set is considered in onedimensional Euchdean space (d = 1) and therefore its fractal dimension djfractal definition [52]. For fractal objects in Euclidean spaces with Mgher dimensions (d>l) dj,fractional part should be accepted as v or [83] ... 

Following the treatment [101], it is believed that the reference times of random influences on fractal aggregation process of macromolecules are much less than the reference times of the aggregation itself and, consequently, it is possible... 

M. F. Schlesinger [1984] Williams-Watts Dielectric Relaxation A Fractal Time Stochastic Process, J. Statist. Phys. 36, 639-648. 

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],... 

The comparison with the similar formula for polymers synthesis [7] shows their principal distinction the higher (in about 4 times) constant coefficient in the combustion case at the expense of process temperature enhancement. Let ns note, that the exponents a and P are interconnected according to the Eq. (2) combnstion front jumps ( Levy s flights ) intensity reinforcement resnlts in a growth and, hence, to fractal time x enhancement and vice versa. 

Hence, a solid-phase polymers deformation process is realized in fractal space with the dimension, which is equal to structure dimension d. In such space the deformation process can be presented schematically as the devil s staircase [39]. Its horizontal sections correspond to temporal intervals, where deformation is absent. In this case deformation process is described with using of fractal time t, which belongs to the points of Cantor s set [30]. If Euclidean object deformation is considered then time belongs to real numbers set. 

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