# Comb Model with Non-Static Stochastic Resetting and Anomalous Diffusion

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## Abstract

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## 1. Introduction

## 2. The Model

## 3. The Model with Memory Kernels Effects: Crossover between Anomalous Diffusion Regimes

#### 3.1. First Case: Exponential Memory in Diffusion Terms

#### 3.2. Second Case: Tempered Power-Law Memory in Diffusion Terms

## 4. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The different situation to walkers that moves on comb structure. Figure (

**a**) shows two walkers, one on x-axis ($y=0$) that admits the next step in both directions, the other walker out of the $x-axis$ ($y\ne 0$) can move only in y-axis to anyone $y\ne 0$. Figure (

**b**) shows only one walker on comb structure that is resetting to position ${x}_{\mathrm{resetting}}=vt$ on black point.

**Figure 2.**These figures represents time evolution behavior to marginal distributions to static stochastic resetting $v=0$. Figure (

**a**) represents the solution (17) and Figure (

**b**) represents the solution (19) to different time valuer. Both figures consider the parameter valuers as follow ${\mathcal{K}}_{x}={\mathcal{K}}_{y}=1$ and $\kappa =1$.

**Figure 3.**This figure represents time evolution behavior to marginal distribution in x-axis (Equation (15) to non-static stochastic resetting. In this figure, we consider the parameter valuers as follow ${\mathcal{K}}_{x}={\mathcal{K}}_{y}=1$, $v=1$ and $\kappa =1$.

**Figure 4.**These figures represent time evolution of MSD behaviors in Equations (23) and (24) to static stochastic resetting case, i.e., $v=0$. Figure (

**a**) represents the MSD on x-axis and Figure (

**b**) represents the MSD on y-axis. Both figures consider the parameter valuers as follow ${\mathcal{K}}_{x}=1$ and ${\mathcal{K}}_{y}=1$.

**Figure 5.**This figure represents time evolution of MSD behavior x axis (Equations (23) and (24)) to non-static stochastic resetting case. In this figure we consider the parameter valuers as follow ${\mathcal{K}}_{x}=1$, ${\mathcal{K}}_{y}=1$, $\kappa =0.01$ and different velocities to resetting point.

**Figure 6.**These figures represent time evolution to MSD behaviors to static stochastic resetting, i.e., $v=0$. Figure (

**a**) represents the MSD associated to Equation (31) and Figure (

**b**) represents the MSD associated to Equation (32) to different $\tau $ valuers. Both figures consider the parameter valuers as follows ${\mathcal{K}}_{x}={\mathcal{K}}_{y}=1$ and $\kappa =0.02$.

**Figure 7.**This figure represents time evolution of MSD behavior x axis (Equation (31)) to non-static stochastic resetting case. In this figure, we consider the parameter valuers as follow ${\mathcal{K}}_{x}=1$, ${\mathcal{K}}_{y}=1$, $\tau =0.001$, $\kappa =0.02$ and different velocities to resetting point.

**Figure 8.**Figure (

**a**) represents the MSD associated to Equation (35) for different $\alpha $ indexes with fixed $\beta =1$. Figure (

**b**) represents the MSD associated to Equation (36) for different $\beta $ valuers. Both figures consider the parameter valuers as follows ${\mathcal{K}}_{x}={\mathcal{K}}_{y}=1$, $\kappa =0.02$ and $\tau =0.01$.

**Figure 9.**This figure represents time evolution of MSD behavior x axis (Equation (35)) to non-static stochastic resetting case, considering ${\mathcal{K}}_{x}=1$, ${\mathcal{K}}_{y}=1$, $\tau =0.01$, $\kappa =0.02$, $\beta =1$, $\alpha =0.1$ and different velocities to resetting point.

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**MDPI and ACS Style**

Antonio Faustino dos Santos, M.
Comb Model with Non-Static Stochastic Resetting and Anomalous Diffusion. *Fractal Fract.* **2020**, *4*, 28.
https://doi.org/10.3390/fractalfract4020028

**AMA Style**

Antonio Faustino dos Santos M.
Comb Model with Non-Static Stochastic Resetting and Anomalous Diffusion. *Fractal and Fractional*. 2020; 4(2):28.
https://doi.org/10.3390/fractalfract4020028

**Chicago/Turabian Style**

Antonio Faustino dos Santos, Maike.
2020. "Comb Model with Non-Static Stochastic Resetting and Anomalous Diffusion" *Fractal and Fractional* 4, no. 2: 28.
https://doi.org/10.3390/fractalfract4020028