Probability distribution fractal structures

The next step they have to generalize the mathematical formulation of the statistics of linear polymer to general fractal objects by assuming that the clusters formed by the filler particles can be described by a fractal shape. This assumption allows them to predict certain specific forms of the reinforcement, such as to work out the probability distribution for the filler clusters. By appropriate modelling of the filler structure they arrive at the generalization distribution to calculate the self-energy function which corresponds directly to the reinforcement factor. They then derive a new form of the Green function G which contains the effects of the filler particles. In this way, they are able to take into account of all the effects the shape of the filler particles, the spatial distribution of the particles, etc. 

The generation parameter defining the generation of ionizing trajectories in the self-similar structure in Fig. 10 is related to the number w of encounters of the two electrons at ri = T2 rather than to the ionization time. This interpretation is confirmed in Fig. 11 which shows the density n of trajectories starting with initial conditions uniformly distributed in the middle panel of Fig. 10 as function of the number w of encounters of the two electrons and of the ionization time T. The density n is proportional to minus the derivative of the survival probability with respect to the relevant variable (w or T). The logarithmic plot in Fig. 11a reveals an exponential decay of the density, n(w)ocexp(—0.27w), and hence also of the survival probability, as a function of the number of encounters of the two electrons, just as expected for a self-similar fractal set of trapped trajectories. The doubly logarithmic plot of the density of trajectories in Fig. 11b reveals a power-law decay of the density, (T) oc and hence... 

The multifractal spectrum reflects the distribution of probability measure of its corresponding fractal body (Sun, 2003). Therefore, it could better reflect the complexity and structural characteristics of the spatial distribution of the fault system. It is clear from Figure 5 that there are great differences in their shapes of the multifractal spectra of different regions. 

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