# Structural Properties of Vicsek-like Deterministic Multifractals

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## Abstract

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## 1. Introduction

## 2. Theoretical Background

#### 2.1. Multifractals

#### 2.2. Small-Angle Scattering

- $F\left(\mathit{q}\right)\to F\left(\beta \mathit{q}\right)$ if the particle’s length is scaled as $L\to \beta L$,
- $F\left(\mathit{q}\right)\to F\left(\mathit{q}\right){e}^{\u2013i\mathit{q}\xb7\mathit{a}}$ if the particle is translated by the vector $\mathit{r}\to \mathit{r}+\mathit{a}$,
- $F\left(\mathit{q}\right)=\left(\right)open="["\; close="]">{V}_{I}{F}_{I}\left(\mathit{q}\right)+{V}_{II}{F}_{II}\left(\mathit{q}\right)$, if the particle can be decomposed as a union of two non-overlapping subsets I and $II$.

## 3. Results and Discussions

#### 3.1. Construction of the Multifractal Model

#### 3.2. Dimension Spectra

#### 3.3. Pair Distance Distribution Function

#### 3.4. Small-Angle Scattering Form Factor

## 4. Conclusions

- If ${\beta}_{\mathrm{s}1}<<{\beta}_{\mathrm{s}2}$, the system is highly heterogeneous and structural parameters are more clearly visible in pddf (see Figure 3a), since the mass fractal region of the scattering intensity is very short Figure 4a). The scaling factor ${\beta}_{\mathrm{s}1}$ is extracted from the periodicity of large groups of distances, while ${\beta}_{\mathrm{s}2}$ can be extracted in a relatively good approximation, from the periodicity of smaller groups found inside larger ones. The number of fractal iterations coincide with the number of large distinct groups in pddf.
- If ${\beta}_{\mathrm{s}1}\lesssim {\beta}_{\mathrm{s}2}$, separation of pddf in distinct groups of distance is not very clear since the values of distances arising from each of the scaling factors begin to mix with each other (see Figure 3b,c), and thus extracting exact values of the scaling factors can become a very difficult task. However, in the reciprocal space, the corresponding mass fractal region of scattering intensity is characterized by a succession of maxima and minima on a power-law decay (generalized power-law decay) and the value of the largest scaling factor can be clearly estimated from the periodicity of minima. In addition, the fractal dimension can be obtained from the scattering exponent of this power-law decay while the fractal iteration number can be obtained from the number of the minima.
- If ${\beta}_{\mathrm{s}1}={\beta}_{\mathrm{s}2}$, the system reduces to a single scale fractal. Structural properties of such systems have been studied elsewhere (see Reference [30]).

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**(Color online) First three iterations of the two-scale multifractal models. Upper row: ${\beta}_{{\mathrm{s}}_{1}}=0.1$ and ${\beta}_{{\mathrm{s}}_{2}}=0.8$ (Model M1). Note that for $m=3$ the disks of radii ${l}_{0}{\beta}_{{\mathrm{s}}_{1}}^{3}/2=0.0005{l}_{0}$ are too small to be seen in the figure (at the given size). Middle row: ${\beta}_{{\mathrm{s}}_{1}}=0.2$ and ${\beta}_{{\mathrm{s}}_{2}}=0.6$ (Model M2). Lower row: ${\beta}_{{\mathrm{s}}_{1}}=0.3$ and ${\beta}_{{\mathrm{s}}_{2}}=0.4$ (Model M3). Black, orange and green colors denote the disks generated at iterations $m=1$, $m=2$, and respectively at $m=3$.

**Figure 2.**(Color online) Dimension spectra ${D}_{\mathrm{s}}$ for the three multifractal models: M1 (black), M2 (red), M3 (green). The intersection of the vertical line with each horizontal (dashed) line gives the box-counting dimension ${D}_{0}$.

**Figure 3.**(Color online) The coefficients ${C}_{p}$ (orange dots) in Equation (13) for the pair distribution function of the considered multifractal models at $m=4$. (

**a**) Model M1; (

**b**) Model M2; (

**c**) Model M3. For a better visualization of pddf grouping the vertical line (blue) for each distance is shown.

**Figure 4.**(Color online) Scattering form factor (Equation (21)) for monodisperse (black) and polydisperse (red) multifractal models at $m=4$. (

**a**) Model M1; (

**b**) Model M2; (

**c**) Model M3. Vertical lines indicate the lower and upper edges of mass fractal region.

**Figure 5.**(Color online) The quantity $I\left(q\right){q}^{{\mathrm{D}}_{0}}$, where ${D}_{0}$ is the box-counting fractal dimension for monodisperse (black) and polydisperse (red) multifractal models at $m=4$. (

**a**) Model M1; (

**b**) Model M2; (

**c**) Model M3. Vertical lines indicate the lower and upper edges of the mass fractal region.

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**MDPI and ACS Style**

Anitas, E.M.; Marcelli, G.; Szakacs, Z.; Todoran, R.; Todoran, D.
Structural Properties of Vicsek-like Deterministic Multifractals. *Symmetry* **2019**, *11*, 806.
https://doi.org/10.3390/sym11060806

**AMA Style**

Anitas EM, Marcelli G, Szakacs Z, Todoran R, Todoran D.
Structural Properties of Vicsek-like Deterministic Multifractals. *Symmetry*. 2019; 11(6):806.
https://doi.org/10.3390/sym11060806

**Chicago/Turabian Style**

Anitas, Eugen Mircea, Giorgia Marcelli, Zsolt Szakacs, Radu Todoran, and Daniela Todoran.
2019. "Structural Properties of Vicsek-like Deterministic Multifractals" *Symmetry* 11, no. 6: 806.
https://doi.org/10.3390/sym11060806