Fractals complex systems

Synthesis processes in common case can be considered as a complex system of selforganization, developing during time, that results to formation of time-dependent fractal structures [3], In such reactions the important role is played by diffusive processes, which in the considered case have very specific nature. This specificity is due to the fact, that in chemical reactions not all reagents contacts occur with proper for reaction s product... 

More complex systems, porous electrodes, and fractals... 

The utility of the MD methods is apparent from the many influential investigations that have proceeded using the methods. The major drawback of the MD methods is the inability to consider more complex systems. For example, the influence of hydrodynamic forces between clusters of low fractal dimension remains to be quantified. It has been hypothesized that the influence would be much less than that expected between two impermeable spheres, and indeed experimental results seem to confirm this hypothesis [5]. 

Fractals, Diffusion, and Relaxation in Disordered Complex Systems A Special Volume of Advances in Chemical Physics, Volume 133, Part A, edited by William T. Coffey and Yuri P. Kalmykov. Series editor Stuart A Rice. 

FRACTALS, DIFFUSION, AND RELAXATION IN DISORDERED COMPLEX SYSTEMS... 

Strictly speaking, all naturally occurring power-laws in fractal or dynamic patterns are finite. Scale-free models nevertheless provide an efficient description of a wide variety of processes in complex systems [16,20,46,106]. This phenomenological fact is corroborated by the observation that the power-law properties of Levy processes persist strongly even in the presence of cutoffs [99]... 

Relaxation functions for fractal random walks are fundamental in the kinetics of complex systems such as liquid crystals, amorphous semiconductors and polymers, glass forming liquids, and so on [73]. Relaxation in these systems may deviate considerably from the exponential (Debye) pattern. An important task in dielectric relaxation of complex systems is to extend [74,75] the Debye theory of relaxation of polar molecules to fractional dynamics, so that empirical decay functions for example, the stretched exponential of Williams and Watts [76] may be justified in terms of continuous-time random walks. 

Ngai, K. L., Casalini, R., Capaccioli, S., Paluch, M., and Roland, C. M. (2006) Adv. Chem. Phys. in Chemical Physics Part B, Fractals, Diffusion and Relaxation in Disordered Complex Systems, 133B, 497-582... 

Adv. Chem. Phys. in Chemical Physics Part B, Fractals, Diffusion and Relaxation in Disordered Complex Systems, 133B, 497-582... 

The critical indices estimated from these relations fall into the admissible ranges of variation P = 0.39-0.40, V = 0.8-0.9, and t = 1.6-1.8, determined in terms of the percolation model for three-dimensional systems. The researchers [7] noted that not only numerical values but also the meanings of these values coincide. Thus the index P characterises the chain structure of a percolation cluster. The 1/p value, which serves as the index of the first subset of the fractal percolation cluster in the model considered [7], also determines the chain structure of the cluster. The index v is related to the cellular texture of the percolation cluster. The 2/df index of the second subset of the fractal percolation cluster is also associated with the cellular structure. By analogy, the index t defines the large-cellular skeleton of the fractal percolation cluster. The relationship between the critical percolation indices and the fractal dimension of the percolation cluster for three-dimensional systems and examples of determination of these values for filled polymers are considered in more detail in the book cited [7]. Thus, these critical indices are universal and significant for analysis of complex systems, the behaviour of which can be interpreted in terms of the percolation theory. 

We shall give some more examples below. However, even now we can conclude that the self-similarity and the fractal structure are not an excep>-tion but rather a rule in polymers and other complex systems. 

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