Lattice fractal structures

In this context, Berry [277] studied the enzyme reaction using Monte Carlo simulations in 2-dimensional lattices with varying obstacle densities as models of biological membranes. That author found that the fractal characteristics of the kinetics are increasingly pronounced as obstacle density and initial concentration increase. In addition, the rate constant controlling the rate of the complex formation was found to be, in essence, a time-dependent coefficient since segregation effects arise due to the fractal structure of the reaction medium. In a similar vein, Fuite et al. [278] proposed that the fractal structure of the liver with attendant kinetic properties of drug elimination can explain the unusual... 

In the majority of numerical calculations of the anomalous frequency behavior of such composites (in particular, near the percolation threshold pc) under the action of an alternating current, lattice (discrete) models have been used, which were studied in terms of the transfer-matrix method [91,92] combined with the Frank-Lobb algorithm [93], Numerical calculations and the theoretical analysis of the properties of composites performed in Refs. 91-109 have allowed significant progress in the understanding of this phenomenon however, the dielectric properties of composites with fractal structures virtually have not been considered in the literature. 

An open problem in the theory is to ascertain whether eq. (16) also holds for infinitely ramified fractal structures. This problem can be tackled either by means of Green function renormalization or by considering lattice simulations. 

It should also be observed that, in the case of an infinitely ramified fractal structure fed from all the perimeter sites, an exponent 0 fairly close to 0 = 1/2 (i.e. the regular case) is obtained, 0.40 < 0 < 0.50. The small difference between the exponent 0 obtained for infinitely ramified fractals and in the regular Euclidean case makes it difficult to discriminate between the two scaling behaviors, especially if one makes use of numerical simulations on small lattices. 

In nature, ordered colloidal systems are unusual for two reasons. First, large scale lattices require monodispersity of the individual particle units within the lattice. Second, both diffusion limited aggregation and reaction limited aggregation in solution produce fractal structures rather... 

Up to now we considered pol5meric fiiactals behavior in Euclidean spaces only (for the most often realized in practice case fractals structure formation can occur in fractal spaces as well (fractal lattices in case of computer simulation), that influences essentially on polymeric fractals dimension value. This problem represents not only purely theoretical interest, but gives important practical applications. So, in case of polymer composites it has been shown [45] that particles (aggregates of particles) of filler form bulk network, having fractal dimension, changing within the wide enough limits. In its turn, this network defines composite polymer matrix structure, characterized by its fractal dimension polymer material properties. And on the contrary, the absence in particulate-filled polymer nanocomposites of such network results in polymer matrix structure invariability at nanofiller contents variation and its fractal dimension remains constant and equal to this parameter for matrix polymer [46]. 

This is another example of a stretched exponential behavior at long times. In principle, one can apply the same calculations to any network built from domains of arbitrary internal architecture, as long as the relaxation spectrum inside the domains obeys a power-law form, see Eq. 155. For instance, the networks may belong to any type of regular lattice topology (bcc, fee, tetrahedral, triangular, hexagonal) or even be fractal structures. 

We also note that a random walk on a fractal has S(t) t where d is the spectral dimension of the fractal lattice. So defect motion on fractal structures also leads to Eqn (1), with p independent of temperature. 

The approaches considered allow modeling of the primary texture of PS and the processes, limited by individual PBUs that mainly correspond to level III and partially to level IV in the hierarchical system of models (see Section 9.6.3). PBUs are identical in regular PSs, and simulation of numerous processes may be reduced to analysis of a process in a single PBU/C or PBU/P. An accurate modeling of the processes in irregular PSs requires the studies of the properties of structure and properties of the ensembles (clusters) of particles and pores (level IV of the system of models) and the lattices of such clusters (levels V to VII of the system of models). Let us consider the composition of clusters on the basis of fractal [127], and the lattices on the basis of percolation [8] theories. 

Simulation of structure formation on a lattice [7,100] demonstrated that randomly formed branched clusters also fulfill self-similarity conditions and gave fractal dimensions of [7,104,105] ... 

The behaviour of surface reaction is strongly influenced by structural variations of the surface on which the reaction takes place [23], Normally theoretical models and computer simulations for the study of surface reaction systems deal with perfect lattices such as the square or the triangular lattice. However, it has been shown that fractal-like structures give much better description of a real surface [24], In this Section we want to study the system (9.1.39) to (9.1.42). 

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