Stochastic fractal

Figure 10. A hierarchy of stochastic fractal pore development a. Straight cylindrical constant diameter, b. Straight variable diameter, c. Tourtous variable diameter, d. Tortous non-circular cross-section,...

D SPNs, and stochastic fractal geometry, would enable us to dispense with indirect laboratory measurements which require expensive instruments. 

T-H-M-C processes are significantly affected by subsurface heterogeneity, which results in scale-dependence of the related parameters. To handle this scale-dependent behavior, we need to characterize this heterogeneity and consider its effects at different scales. In this study, we demonstrate that the measured permeability data from Sellafield site, UK, are very well described by fractional Levy motion (fLm), a stochastic fractal. This finding has important implications for modeling large-scale coupled processes in heterogeneous fractured rocks. 

Data on the fractal forms of macromolecules, the existence of which is predetermined by thermodynamic nonequilibrium and by the presence of deterministic order, are considered. The limitations of the concept of polymer fractal (macromolecular coil), of the Vilgis concept and of the possibility of modelling in terms of the percolation theory and diffusion-limited irreversible aggregation are discussed. It is noted that not only macromolecular coils but also the segments of macromolecules between topological fixing points (crosslinks, entanglements) are stochastic fractals this is confirmed by the model of structure formation in a network polymer. 

Figure 11.6 shows the plots for the variation of versus r j for two amorphous polymers the plots correspond to relationship (11.28). In other words, a stable crack in polymer film samples is a stochastic fractal with the dimension 1.48. The linearity of plots shown in... 

In Figure 14.4 double logarithmic dependences 2 In 8 = /(In r) are presented for two amorphous polymers which have appeared to be linear and by virtue of it, correspond to Equation (14.7). Otherwise, the stable crack in film polymeric samples is a stochastic fractal with dimension 1.48. Linearity of the diagrams presented on Figure 14.4 reflects the self-similarity of a crack at different stages of its growth. Thus, at the macroscopic level polymers the fractal properties are also displayed. 

The examples considered do not reflect all the variety of fractalities in polymers. So, it is possible to show that stochastic fractals are not only macromolecular coils, but also sections of chains between clusters, crazes etc. These are all examples of the term multifractality with the reference to polymers. 

Perikinetic motion of small particles (known as colloids ) in a liquid is easily observed under the optical microscope or in a shaft of sunlight through a dusty room - the particles moving in a somewhat jerky and chaotic manner known as the random walk caused by particle bombardment by the fluid molecules reflecting their thermal energy. Einstein propounded the essential physics of perikinetic or Brownian motion (Furth, 1956). Brownian motion is stochastic in the sense that any earlier movements do not affect each successive displacement. This is thus a type of Markov process and the trajectory is an archetypal fractal object of dimension 2 (Mandlebroot, 1982). 

In the classical book [4], the distinct models dealing with ion channel kinetics are extensively discussed. One of the important results is the connection established between fractal scaling and stochastic modeling. Based on experimental data, Liebovitch et al. [394] assessed the dependence of the effective kinetic constant ka on the sufficient time scale for detection tG by a fractal scaling relationship ... 

The use of fractal geometry, both deterministic and non-deterministic i.e. (stochastic), to model natural processes has become an intensive research area in recent years. This has extended to include characterization and analysis of the configuration of void spaces within porous materials. Qualitative geometrical analysis have shown a wide variety of natural and synthetic materials ranging from rocks, trees and clouds to charcoal, quartz and aluminas, to posses fi-actal properties [11,12]. 

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. 

T. Gneiting and M. Schlather, Stochastic models that separate fractal dimension and the hurst effect. SIAM Rev. 46, 269-282 (2004). 

The regularities revealed in the theory of fractals and percolation have turned out to be generally true for heterogeneous stochastic media and, in particular, for composite materials. 

Use of the concept of the fractal set allows one to examine the dependence of physical properties on the behavior of hierarchical structures. Such structures appear in stochastic inhomogeneous medium. 

A. Stochastic Difference in Time Definition A Stochastic Model for a Trajectory Weights of Trajectories and Sampling Procedures Mean Field Approach, Fast Equilibration, and Molecular Labeling Stochastic Difference Equation in Length Fractal Refinement of Trajectories Parameterized by Length... 

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