Statistical fractal

Family [9] considered the conformations of statistical branched fractals (which simulate branched polymers) formed in equilibrium processes in terms of the Flory theory. Using this approach, he found only three different states of statistical fractals, which were called uncoiled, compensated, and collapsed states. In particular, it was found that in thermally induced phase transitions, clusters occur in the compensated state and have nearly equal fractal dimensions ( 2.5). Recall that the value df = 2.5 in polymers corresponds to the gelation point this allows gelation to be classified as a critical phenomenon. 

The fractal dimension of purely statistical models, i.e., models without the effect of excluded volume, can be determined accurately [see Equation (11.9a)]. For linear polymers, this model corresponds to phantom random-walk. In the case of branched statistical fractals, the corresponding model is a statistical branched cluster, whose branching obeys the random-walk statistics. Since the root-mean-square distance between the random-walk ends is proportional to the number of walk steps N, then D = 2 irrespective of the space dimension. These types of structures have been studied [61, 75-77]. The value D = 4 irrespective of d was obtained for a branched fractal. Unlike ideal statistical models, models with excluded volume, i.e., those involving correlations, cannot be accurately solved in the general case. The Df values for these systems are usually found either using numerical methods such as the Monte Carlo method or taking into account the spatial position of a renormalisation group. 

What are the physical conditions under which a statistical fractal has the most branched structure It is clear that, when a large number of clusters are present in the system, they all occupy the available volume. The presence of other clusters restricts the degree of branching of each cluster a cluster is more branched in the isolated state than in a concentrated solution. For polymer coils, this is expressed as an increase in D from its value in a dilute solution in a good solvent to the value at the 0-point in concentrated solutions [73]. 

Yet another factor which also influences the shape of the cluster is whether or not attractive interactions are present. As the temperature increases, the attractive interactions diminish. For example, no interactions of this sort are found in isolated macromolecular coils in good solvents at high temperature [77]. Family [9] defined this state of a statistical fractal as an uncoiled state because in this case, it is characterised by the smallest fractal dimension. 

Then Family used the version of the Flory theory proposed by Isaacson and Lubensky [78] for determination of the fractal dimension of an isolated statistical cluster. The method is based on the determination of the most probable cluster conformation using the free energy of repulsion. Under the influence of elastic free energy, the radius of the real cluster tends to the radius of the ideal cluster that involves no repulsive interaction. 

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However, nonrandom fractals are not found in nature. Many objects existing in nature belong to random fractal, viz., statistical fractal, when the enlarged part is similar to the original object in a statistical sense. In practice, it is impossible to verily that all moments of the distributions are identical, and claims of statistical... 

Oleschko, K. 1998. Delesse principle and statistical fractal sets 1. Dimensional equivalents. Soil Till. Res. 49 255-264. 

Vlad, M. O. Cerofofini, G. F. Ross, J. Statistical fractal adsorption isotherms, linear energy relations, and power-law trapping-time distributions in porous media. Phys Rev. E 2000, 62, 837-844. 

Statistical fractals are generated by disordered (random) processes. An element of disorder is typical of most real physical phenomena and objects. The fact that disorder, i.e., the absence of any spatial correlation, is a sufficient condition for the formation of fractals was first noted by Mandelbrot [1]. A typical example of this type of fractal is the random-walk path. However, real physical systems are often inadequately described by purely statistical models. Among other reasons, this is due to the effect of excluded volume. The essence of this effect lies in the geometric restriction that forbids two different elements of a system to occupy the same volume in space. This restriction is to be taken into account in the corresponding modelling [10, 11]. The best-known examples of this type of models are self-avoiding random walk, lattice animals and statistical percolation. 

The state of a statistical fractal in which the excluded volume effect is compensated by the screening effect is referred to as the compensated state. For this state. Family has found that ... 

Thus, the fractal dimension D increases near the compensation point this is due either to the geometric screening of the excluded volume effect or to the enhancement of the attractive interaction. Big clusters near the critical point in thermally induced phase transitions and polymers at the 0-point are examples of statistical fractals in the... 

Study of the conformation of a statistical fractal in a system occurring below the critical point and in the isolated state at W2<0 showed that the fractal is very compact and has a globular conformation in both cases. This state of a fractal was called collapsed, by analogy with the collapsed state of polymers. 

For a purely statistical fractal for which D = 4, we obtain = 4. This means that this type of statistical cluster is compact up to d = 4. Therefore, Family [9] called this state of statistical fractals the collapsed state. 

In eontrastto deterministie liaetals, statistical fractals are constructed by random proeesses. The element of randomness makes them a more... 

FIGURE 1 Two examples of linear fractals Koch curve (a) and self-avoiding random walk (b), representing a deterministic fractal and statistical fractal, respectively. 

The second factor, which influences a cluster shape, is availability or absence of the attraction interactions. For example, the attraction interactions are less effective at high temperatures, than at low ones. Therefore the isolated statistical fractal at high temperatures will possess a more branched stmcture, than at low ones. The best examples of this type of fractals are isolated macromolecular coils in good solvents at high temperatures [7]. Family called such state of a statistical fractal extended, since in this case it has its least fractal dimension D. ... 

The calculation within Flory theory [8] allows determining Devalue for the extended state of branched statistical fractal according to the equation [6] ... 

There are no closed-form expressions (Muthukumar and Nickel 1987, des Cloizeaux and Jannink 1990) for other measures of polymer conformations of a swollen chain, such as the form factor and monomer density profiles. Nevertheless, the general laws discussed in Section 2.2 for statistical fractals are valid, with the approximate value of 5/3 for df. The monomer density decays with the radial distance r as... 

Statistical fractals are raised by imordered (random) processes. A disorder element is... 

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