A scale-entropy diffusion equation to explore scale-dependent fractality

In the last three decades, fractal geometry became a mathematical tool widely used in physics. Nevertheless, it has been observed that real multi-scale phenomena display a departure to fractality that implies an impossibility to define the multi-scale features with an unique fractal dimension, leading to variations in the scale-space. The scale-entropy diffusion equation theorizes the organization of the scale dynamics involving scale-dependent fractals. A study of the theory is possible through the scale-entropy sink term in the equation and corresponds to precise behaviours in scale-space. In the first part of the paper, we study the scale space features when the scale-entropy sink term is modified. The second part is a numerical investigation and analysis of several solutions of the scale-entropy diffusion equation. By a precise measurement of the transition scales tested on truncated deterministic fractals, we developed a new simple method to estimate fractal dimension which appears to be much better than a classical method. Furthermore, we show that deterministic fractals display intrinsic log-periodic oscillations of the fractal dimension. In order to represent this complex behaviour, we introduce a departure to fractal diagram linking scale-space, scale-dependent fractal dimension and scale-entropy sink. Finally, we construct deterministic scale-dependent fractals and verify the results predicted by the scale-entropy diffusion equation.