Fractal Modeling of Stochastic Processes and Development of Statistical Representations

Abstract

The effectiveness of using fractal models to consider complex multicomponent systems, especially those whose functioning is based on stochastic processes, is well known. In particular, they also include molecular systems, the study of which was considered the exclusive prerogative of the well-known methods of statistical physics. At first glance, it seems that the imposition of fractal modeling on a statistical problem is a kind of double simplification, which should narrow the area of applicability of such an approach. However, fractal modeling only tightened the requirements for a clearer and more precise formulation of statistical problems, refinement of basic concepts, ideas about distinguishing particles, etc. The paper shows for the first time that the statistics of molecular systems should be based on two statistical factors Gn and Fm. The basis for the factor Gn is the difference in the nature of the motion and interaction of particles, and the factor Fm is the difference in the ways of filling the phase cells. They together form physical statistics that are sensitive to changes in the nature of interactions in the system. At the same time, mathematical methods of statistics, insensitive to the nuances of interactions, describe the maximum chaos in the system, in fact, an ideal gas. One of the achievements of the study is the establishment of the fact that both quantum statistics are based only on the statistical factor Fm, based on the difference in the ways of filling the phase cells.