# Fractional Criticality Theory and Its Application in Seismology

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Compound Fractional Poisson Process with Power-Law Distribution of Events Recurrence Frequencies

#### 2.1. Compound Fractional Poisson Process

#### 2.2. Distribution of Recurrence Frequencies of Events

## 3. Critical Indices and Process Instability

## 4. Calculation of the Distribution Parameters of the Events Recurrence Frequency Based on Seismic Data

#### 4.1. Calculation of b-Value

#### 4.2. Distributions of the First-Passage Times

#### 4.3. Approximation of the First-Passage Times Distributions

## 5. Results and Discussion

- The hereditarity parameter or the average of the exponent $\nu $ of the fractional derivative of the CFPP is calculated on the values in column 10 of the Table 2$$\frac{1}{k}\sum _{r=1}^{k}{\nu}_{r}=\nu =0.8675.$$According to the value of this parameter, we can conclude that the considered seismic process has «memory», so random events of deformation changes cannot be considered independent.Since $\nu <1$, the distributions (16) define the delayed relaxation of strains, which are associated with the hardening of the deformable medium and the accumulation of elastic energy, which may be the reason for the activation of the process.
- The fractional decay rate of CFPP states is determined by the parameter $\Lambda \phantom{\rule{0.166667em}{0ex}}\left[da{y}^{-\nu}\right]$ (13), (14), which is represented as follows:$$\Lambda =\sum _{r=1}^{k}{\lambda}_{r}=\sum _{r=1}^{k}{\left({\omega}_{r}\right)}^{\nu}.$$The $\Lambda $-value equal to the zero moment is calculated on the values in column 9 of Table 2 and in item 1,$$\Lambda =2.6401\phantom{\rule{4pt}{0ex}}da{y}^{-0.8675}={(3.0622/day)}^{0.8675}.$$
- The stability parameter of the CFPP takes the value$$(2b+1)\nu \approx 2.0641,$$
- The values of the critical indices (23) are equal to$${\nu}_{0}=0.4230,\phantom{\rule{4pt}{0ex}}{\nu}_{1}=0.8406,\phantom{\rule{4pt}{0ex}}{\nu}_{2}=1.2608.$$A comparison of the $\nu $-parameter (item 1) with the critical indices ${\nu}_{p}$ ($p=0,1,2$) shows that the seismic process is in a subcritical regime for the zero and first moments and in a supercritical regime for the second moment of distribution (13), which indicates the instability of deformation changes that can go into a non-stationary regime of the seismic process. The reason for this activation is indicated in item 1.This result means that the fractional decay rate $\Lambda ={(2b\Omega )}^{\nu}{S}_{\infty}$ of the seismic process, described by the parameters of the CFPP (14) and (18), and the average deformations (20), proportional to ${S}_{k,1}$ at $k=\infty $, are finite, and the divergence in the dispersion growth (21) caused by ${S}_{k,2}$ at $k=\infty $ leads to the instability of the process and its transition to a non-stationary regime considered in papers [13,14].The anomalous growth of fluctuations caused by the hereditarity of the seismic process is represented in $\mathbf{Var}\left(t\right)$ (21) by the second term, which is proportional to the square of the mean $\mathbf{E}\left(t\right)$ (20), which is different to the first term and proportional to the mean $\mathbf{E}\left(t\right)$. If the first term in $\mathbf{Var}\left(t\right)$ (21) describes an ordinary deformation, then the second term describes an anomalous one caused by the consolidation of scales. This is a collective or induced coherent effect, the analogue of which in quantum optics is superluminescence, and in phase transition physics is explosive boiling. In the absence of hereditarity, this effect disappears, because if we take $\nu =1$, then the second term of $\mathbf{Var}\left(t\right)$ in (21) will be zero based on the property of the gamma function $\mathrm{\Gamma}(z+1)=z\mathrm{\Gamma}\left(z\right)$.

## 6. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

CFPP | Compound Fractional Poisson Process |

LSM | Least Squares Method |

eCDF | Empirical Cumulative Distribution Function |

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**Figure 1.**(

**a**) Distribution of the earthquakes number depending on their energy class (magnitude) and (

**b**) its logarithmic form for classes $K\in [8.3,16.1]$. The dot graphs (black dots) are the distribution and its logarithmic form, where the approximation interval $K\in [9.2,12.9]$ is highlighted with red dots; the blue graph is an approximation of the linear part of logarithmic form of the distribution.

**Figure 2.**(

**a**) The step function eCDF of the first-passage times for class ${K}_{32}=12.3$. (

**b**) The eCDF approximation by the function (16) for class ${K}_{32}=12.3$, the dot graph (black dots in figure (

**b**)) is eCDF (the corresponding relative frequency is mapped to the middle of each interval), the blue graph is one-parameter approximation, the red graph is two-parameter approximation.

**Table 1.**Parameters of the Gutenberg–Richter law and statistical characteristics of approximating functions $F\left(X\right)$.

$\mathit{F}\left(\mathit{X}\right)$ | $[{\mathit{K}}_{1},\phantom{\rule{4pt}{0ex}}{\mathit{K}}_{2}]$^{1} | k | ${\mathit{N}}_{\mathit{total}}$ | a | b | R | F | $\tilde{\mathit{F}}$ | $\mathit{\epsilon},\%$ |
---|---|---|---|---|---|---|---|---|---|

${10}^{a-bX}$ $a-bX$ | $[9.2,\phantom{\rule{4pt}{0ex}}12.9]$ | 38 | 22,230 | $6.3815$ | $0.6897$ | $0.9857$ $0.8567$ | 1233 99 | $4.08$ | $1.658$ $1.687$ |

^{1}Approximation interval (for magnitudes respectively equal to $[{M}_{1},\phantom{\rule{4pt}{0ex}}{M}_{2}]=[2.93,\phantom{\rule{4pt}{0ex}}5.40]$).

Approximation by a Function (16) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

One-Parameter | Two-Parameter | |||||||||

${\mathit{K}}_{\mathit{r}}$ | ${\mathit{M}}_{\mathit{r}}$ | ${\mathit{n}}_{\mathit{r}}$ | $\mathit{RSS}$ | ${\mathit{\omega}}_{\mathit{r}},\phantom{\rule{0.166667em}{0ex}}{\mathit{day}}^{-\mathbf{1}}$ | ${\mathit{\nu}}_{\mathit{r}}$ | $\mathit{\epsilon},\%$ | $\mathit{RSS}$ | ${\mathit{\omega}}_{\mathit{r}},\phantom{\rule{0.166667em}{0ex}}{\mathit{day}}^{-\mathbf{1}}$ | ${\mathit{\nu}}_{\mathit{r}}$ | $\mathbf{\epsilon},\%$ |

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |

9.2 | 2.93 | 57 | 0.102 | 0.135 | 0.961 | 4.71 | 0.025 | 0.182 | 0.891 | 2.33 |

9.3 | 3.0 | 57 | 0.067 | 0.125 | 0.974 | 3.93 | 0.031 | 0.152 | 0.925 | 2.59 |

9.4 | 3.07 | 57 | 0.085 | 0.119 | 0.962 | 4.36 | 0.034 | 0.149 | 0.905 | 2.76 |

9.5 | 3.13 | 58 | 0.099 | 0.109 | 0.958 | 4.71 | 0.039 | 0.139 | 0.897 | 2.97 |

9.6 | 3.2 | 62 | 0.105 | 0.101 | 0.959 | 4.71 | 0.033 | 0.129 | 0.895 | 2.64 |

9.7 | 3.27 | 67 | 0.109 | 0.09 | 0.962 | 4.62 | 0.036 | 0.114 | 0.901 | 2.65 |

9.8 | 3.33 | 67 | 0.102 | 0.084 | 0.954 | 4.55 | 0.032 | 0.105 | 0.895 | 2.53 |

9.9 | 3.4 | 77 | 0.095 | 0.075 | 0.961 | 4.08 | 0.043 | 0.09 | 0.914 | 2.76 |

10.0 | 3.47 | 71 | 0.108 | 0.071 | 0.955 | 4.65 | 0.063 | 0.083 | 0.907 | 3.54 |

10.1 | 3.53 | 83 | 0.077 | 0.056 | 0.960 | 3.70 | 0.036 | 0.064 | 0.92 | 2.51 |

10.2 | 3.6 | 84 | 0.074 | 0.057 | 0.968 | 3.57 | 0.029 | 0.065 | 0.928 | 2.24 |

10.3 | 3.67 | 91 | 0.112 | 0.048 | 0.960 | 4.29 | 0.035 | 0.056 | 0.91 | 2.38 |

10.4 | 3.73 | 93 | 0.245 | 0.045 | 0.932 | 6.31 | 0.045 | 0.06 | 0.849 | 2.7 |

10.5 | 3.8 | 95 | 0.133 | 0.041 | 0.969 | 4.64 | 0.043 | 0.049 | 0.918 | 2.64 |

10.6 | 3.87 | 103 | 0.214 | 0.034 | 0.942 | 5.79 | 0.051 | 0.042 | 0.88 | 2.83 |

10.7 | 3.93 | 103 | 0.163 | 0.033 | 0.959 | 5.12 | 0.081 | 0.039 | 0.91 | 3.61 |

10.8 | 4.0 | 107 | 0.257 | 0.027 | 0.922 | 6.53 | 0.089 | 0.034 | 0.858 | 3.83 |

10.9 | 4.07 | 109 | 0.28 | 0.025 | 0.947 | 6.75 | 0.041 | 0.031 | 0.888 | 2.6 |

11.0 | 4.13 | 113 | 0.331 | 0.025 | 0.922 | 7.19 | 0.041 | 0.033 | 0.865 | 3.76 |

11.1 | 4.2 | 115 | 0.395 | 0.021 | 0.880 | 8.01 | 0.079 | 0.028 | 0.817 | 3.59 |

11.2 | 4.27 | 112 | 0.357 | 0.018 | 0.907 | 7.97 | 0.059 | 0.023 | 0.852 | 3.25 |

11.3 | 4.33 | 113 | 0.301 | 0.019 | 0.926 | 7.17 | 0.103 | 0.024 | 0.873 | 4.19 |

11.4 | 4.4 | 112 | 0.144 | 0.016 | 0.898 | 5.38 | 0.044 | 0.018 | 0.886 | 4.05 |

11.5 | 4.47 | 104 | 0.405 | 0.015 | 0.845 | 9.15 | 0.101 | 0.019 | 0.797 | 4.57 |

11.6 | 4.53 | 100 | 0.225 | 0.012 | 0.936 | 7.27 | 0.141 | 0.013 | 0.91 | 5.75 |

11.7 | 4.6 | 95 | 0.505 | 0.011 | 0.825 | 11.18 | 0.023 | 0.021 | 0.784 | 4.07 |

11.8 | 4.67 | 89 | 0.451 | 0.01 | 0.861 | 10.74 | 0.112 | 0.014 | 0.822 | 5.35 |

11.9 | 4.73 | 79 | 0.353 | 0.009 | 0.835 | 10.21 | 0.036 | 0.012 | 0.85 | 5.4 |

12.0 | 4.8 | 76 | 0.529 | 0.008 | 0.803 | 13.1 | 0.055 | 0.012 | 0.818 | 4.22 |

12.1 | 4.87 | 73 | 0.576 | 0.007 | 0.772 | 14.03 | 0.08 | 0.01 | 0.775 | 5.21 |

12.2 | 4.93 | 61 | 0.21 | 0.006 | 0.915 | 9.31 | 0.113 | 0.007 | 0.886 | 6.83 |

12.3 | 5.0 | 68 | 0.133 | 0.006 | 0.920 | 7.06 | 0.089 | 0.007 | 0.909 | 5.76 |

12.4 | 5.07 | 65 | 0.666 | 0.006 | 0.751 | 15.45 | 0.04 | 0.01 | 0.766 | 5.95 |

12.5 | 5.13 | 55 | 0.644 | 0.006 | 0.755 | 16.42 | 0.033 | 0.011 | 0.749 | 4.92 |

12.6 | 5.2 | 49 | 0.273 | 0.005 | 0.787 | 12.2 | 0.043 | 0.007 | 0.791 | 6.04 |

12.7 | 5.27 | 51 | 0.217 | 0.004 | 0.896 | 10.55 | 0.075 | 0.005 | 0.882 | 6.21 |

12.8 | 5.33 | 47 | 0.108 | 0.004 | 0.880 | 7.92 | 0.051 | 0.005 | 0.858 | 5.46 |

12.9 | 5.4 | 50 | 0.428 | 0.004 | 0.868 | 14.85 | 0.032 | 0.006 | 0.883 | 5.32 |

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**MDPI and ACS Style**

Shevtsov, B.; Sheremetyeva, O.
Fractional Criticality Theory and Its Application in Seismology. *Fractal Fract.* **2023**, *7*, 890.
https://doi.org/10.3390/fractalfract7120890

**AMA Style**

Shevtsov B, Sheremetyeva O.
Fractional Criticality Theory and Its Application in Seismology. *Fractal and Fractional*. 2023; 7(12):890.
https://doi.org/10.3390/fractalfract7120890

**Chicago/Turabian Style**

Shevtsov, Boris, and Olga Sheremetyeva.
2023. "Fractional Criticality Theory and Its Application in Seismology" *Fractal and Fractional* 7, no. 12: 890.
https://doi.org/10.3390/fractalfract7120890