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

Abstract

scale-space (pure fractal), however, numerous works showed that fractal dimension was rather a multi-scale phenomena, thus a scale-and time-dependent quantity.In order to grasp all the complexity of multi-scale behaviours, a scale-entropy diffusion equation has been introduced [6]. Based on the scale-entropy diffusion equation and especially on the scale-entropy sink term, our aim is in the first part of the work, to explore several specific cases of the equation which would correspond to real multi-scale behaviours: (i) power law form for the scale-entropy sink term and (ii) polynomial form for the scale-entropy sink term. This leads to an interesting structure of what we called the 'fractal modes' of a multi-scale phenomena. In the second part of the paper, (iii) we numerically explore the intrinsic log-periodic structure of deterministic fractals, doing that, (iv) we develop a new measurement of fractal dimension based on a precise determination of transition scales, and (v) we introduce a new fractal diagram to show how scale, scale-dependent fractal dimension and scale-entropy sink are correlated. This gives us an original tool to interpret all the complexity of multi-scale behaviours. Finally, (vi) we build deterministic scale-dependent fractals with some specified properties, verified in the framework of the scale-entropy diffusion equation.The pure fractal geometry requires a strict self-similarity in the log-log scale analysis. Nevertheless, departure from linearity is widely popular in 'real' fractal geometries. Results from experiments on diverse phenomena (not thresholding or noise) [7,8] seems to behave as a parabolic fractal [9] and not really as a pure fractal. The parabolic fractal consists in an equipartition [10] of the scale-entropy sink through scale-space, but it can take any form leading to what we can call 'dissipative' fractals or 'cumulative' fractals. The idea of 'impure Von Koch' with altered fractal dimensions (possessing different internal similarities) was studied, for example, in [3], in our case, these similarities change at each scale of observation leading to scale-dependent geometry.

The effect of dragon-kings on the estimation of scaling law parameters

“…However, it is challenging to quantify reliably from data the value of the scaling exponents and their estimates usually come with caveats [6][7][8][9] . For example, Arcaute et al 10 raise the issue that, on the one hand, cities with extremely large populations such as London in the UK, tend to behave differently from the rest of cities within the urban system due to their socioeconomic role; therefore, they should possibly be analysed relative to other cities with extremely large populations and separately to the rest of cities within an urban system.…”

Fractal Geometric Model for Statistical Intermittency Phenomenon

“…Note that z p was different from p, unlike the homogeneous case. Some deviation from homogeneity occurred [50,56].…”

“…Note that 𝑧 𝑝 was different from p, unlike the homogeneous case. Some deviation from homogeneity occurred [50,56]. as functions as the scale r at scale range for (ln(10/4096) < ln(r/4096) < ln(205/4096)), where < ∆n i (r) p >∼ ∼ r z p .…”

“…Furthermore, at the geometric level, the concept of scale entropy S p (r) for each fluctuation level p in a scale r has been introduced to quantify the spatial extension of each skin in the intermittency hierarchy. It is defined as the logarithm of the space occupation rate w(r) [36,50]:…”

On the entropy production of the elemental construct of the constructal designed plate generating heat

“…The constructal design studies shapes from an elemental object until larger objects (opposite of the fractal reasoning), it is thus a method involving multiple scales. Certain shapes have been bounded to particular geometries like parabolic fractality related to scale dependence [25,26,27].…”