“FRACTALIZATION” OF THE SOLID-STATE PHYSICS

Abstract

It has been demonstrated that in modern solid-state physics, in accordance with the nonlinear and systems paradigms formulated by L. F. Chernogor in the late 1980s, many processes in open, nonlinear, dynamical systems are occurred to be very complex, nonlinear, short-time, ultra-wideband, or fractal.

Moreover, from the point of view of the fractal paradigm put forward in the early 2000s by V. V. Yanovsky, fractality is generally considered as one of the fundamental properties of the surrounding world. Therefore, the study of fractal characteristics, in particular, of natural physical processes and objects in the field of solid-state physics, is appeared to be relevant, interesting, useful and promising.

The main stages of the development of the fractal approach in general are briefly discussed. It is pointed that it was occurred to be too hard to formulate a strict and clear mathematical definition of a fractal. The definition of a fractal been formulated by. K. Falconer and been agreed by the most speciallists as the best for practical usage is considered in detail. Main numerical characteristics of a fractal as well as the modern classification of fractals are considered. The general principal differences existing between the mathematical fractals and physical (or natural) ones as well as between the mono-fractals and the multi-fractals are clearly explained. A review of the main methods for estimating the Hurst fractal dimension is provided.

The main existing directions of "fractalization" of modern solid-state physics are highlighted. Relevant examples are given.

It is noted that images with certain fractal properties play an important role in the "fractalization" of solid-state physics. The use of the two-dimensional Weierstrass function is proposed for modeling images with fractal properties. As an example, the modelling of the unidirectional twin structure observed in the YBa₂Cu₃O_7-δ crystal is considered. A comparison between the model image based on one-dimensional Weierstrass function with defined value of the Hurst fractal dimension and real experimental one is demonstrated.