# Non-Gaussian Distributions to Random Walk in the Context of Memory Kernels

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Montroll-Weiss Formalism

#### 2.2. Generalised Diffusion Equation

## 3. Main Results

#### 3.1. Exponential Memory-Kernel and Non-Gaussian Solutions

#### 3.2. Mittag-Leffler Memory–Kernel and Non-Gaussian Solutions

#### 3.3. Diffusive Aspects of Non-Singular Diffusion Equations

#### 3.4. Random Walk Process with Stochastic Resetting and Memory

## 4. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Einstein, A. Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen. Ann. Phys.
**1905**, 322, 549–560. [Google Scholar] [CrossRef] [Green Version] - Von Smoluchowski, M. Zur kinetischen theorie der brownschen molekularbewegung und der suspensionen. Ann. Phys.
**1906**, 326, 756–780. [Google Scholar] [CrossRef] - Langevin, P. Sur la théorie du mouvement brownien. CR Acad. Sci. Paris
**1908**, 146, 530. [Google Scholar] - Perrin, J. L’agitation moléculaire et le mouvement brownien. Comptes Rendus Hebdomadaires des Séances de L’académie des Sciences
**1908**, 146, 967–970. [Google Scholar] - Perrin, J. Mouvement brownien et réalité moléculaire. Annales de Chimie et de Physique
**1909**, 18, 5–104. [Google Scholar] - Alves, L.G.; Winter, P.B.; Ferreira, L.N.; Brielmann, R.M.; Morimoto, R.I.; Amaral, L.A. Long-range correlations and fractal dynamics in C. elegans: Changes with aging and stress. Phys. Rev. E
**2017**, 96, 022417. [Google Scholar] [CrossRef] [PubMed] - Alves, L.G.; Scariot, D.B.; Guimarães, R.R.; Nakamura, C.V.; Mendes, R.S.; Ribeiro, H.V. Transient superdiffusion and long-range correlations in the motility patterns of trypanosomatid flagellate protozoa. PLoS ONE
**2016**, 11, e0152092. [Google Scholar] [CrossRef] [PubMed] - Reverey, J.F.; Jeon, J.H.; Bao, H.; Leippe, M.; Metzler, R.; Selhuber-Unkel, C. Superdiffusion dominates intracellular particle motion in the supercrowded cytoplasm of pathogenic Acanthamoeba castellanii. Sci. Rep.
**2015**, 5, 11690. [Google Scholar] [CrossRef] [PubMed] - Ribeiro, H.V.; Tateishi, A.A.; Alves, L.G.; Zola, R.S.; Lenzi, E.K. Investigating the interplay between mechanisms of anomalous diffusion via fractional Brownian walks on a comb-like structure. New J. Phys.
**2014**, 16, 093050. [Google Scholar] [CrossRef] [Green Version] - Metzler, R.; Jeon, J.H.; Cherstvy, A.G.; Barkai, E. Anomalous diffusion models and their properties: Non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking. Phys. Chem. Chem. Phys.
**2014**, 16, 24128–24164. [Google Scholar] [CrossRef] [PubMed] - Vahabi, M.; Schulz, J.H.; Shokri, B.; Metzler, R. Area coverage of radial Lévy flights with periodic boundary conditions. Phys. Rev. E
**2013**, 87, 042136. [Google Scholar] [CrossRef] [PubMed] - Sibatov, R.; Uchaikin, V. Dispersive transport of charge carriers in disordered nanostructured materials. J. Comput. Phys.
**2015**, 293, 409–426. [Google Scholar] [CrossRef] - Shikano, Y.; Wada, T.; Horikawa, J. Discrete-time quantum walk with feed-forward quantum coin. Sci. Rep.
**2014**, 4, 4427. [Google Scholar] [CrossRef] [PubMed] - Dos Santos Mendes, R.; Lenzi, E.K.; Malacarne, L.C.; Picoli, S.; Jauregui, M. Random Walks Associated with Nonlinear Fokker-Planck Equations. Entropy
**2017**, 19, 155. [Google Scholar] [CrossRef] - Plastino, A.; Curado, E.; Nobre, F.; Tsallis, C. From the nonlinear Fokker-Planck equation to the Vlasov description and back: Confined interacting particles with drag. Phys. Rev. E
**2018**, 97, 022120. [Google Scholar] [CrossRef] [PubMed] - Mendes, G.; Ribeiro, M.; Mendes, R.; Lenzi, E.; Nobre, F. Nonlinear Kramers equation associated with nonextensive statistical mechanics. Phys. Rev. E
**2015**, 91, 052106. [Google Scholar] [CrossRef] [PubMed] - Chen, W.B.; Wang, J.; Qiu, W.Y.; Ren, F.Y. Solutions for a time-fractional diffusion equation with absorption: influence of different diffusion coefficients and external forces. J. Phys. A
**2008**, 41, 045003. [Google Scholar] [CrossRef] - Podlubny, I. Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications; Elsevier: New York, NY, USA, 1998; Volume 198. [Google Scholar]
- Yang, X.J.; Machado, J.T.; Baleanu, D. Anomalous diffusion models with general fractional derivatives within the kernels of the extended Mittag-Leffler type functions. Rom. Rep. Phys.
**2017**, 69, 115. [Google Scholar] - Sabatier, J.; Agrawal, O.P.; Machado, J.T. Advances in Fractional Calculus; Springer: Berlin, Germany, 2007; Volume 4. [Google Scholar]
- Ha, S.Y.; Jung, J. Remarks on the slow relaxation for the fractional Kuramoto model for synchronization. J. Math. Phys.
**2018**, 59, 032702. [Google Scholar] [CrossRef] [Green Version] - Batool, F.; Akram, G. A novel approach for solitary wave solutions of the generalized fractional Zakharov-Kuznetsov equation. Indian J. Phys.
**2018**, 92, 111–119. [Google Scholar] [CrossRef] - Lenzi, E.K.; dos Santos, M.A.F.; Lenzi, M.K.; Neto, R.M. Solutions for a mass transfer process governed by fractional diffusion equations with reaction terms. Commun. Nonlinear Sci. Numer. Simul.
**2017**, 48, 307–317. [Google Scholar] [CrossRef] - Dos Santos, M.A.F.; Lenzi, M.K.; Lenzi, E.K. Anomalous Diffusion with an Irreversible Linear Reaction and Sorption-Desorption Process. Adv. Math. Phys.
**2017**, 2017. [Google Scholar] [CrossRef] - Lenzi, E.K.; Ribeiro, H.V.; dos Santos, M.A.F.; Rossato, R.; Mendes, R.S. Time dependent solutions for a fractional Schrödinger equation with delta potentials. J. Math. Phys.
**2013**, 54, 082107. [Google Scholar] [CrossRef] - Carpinteri, A.; Mainardi, F. Fractals and Fractional Calculus in Continuum Mechanics; Springer: Berlin, Germany, 2014; Volume 378. [Google Scholar]
- Liemert, A.; Sandev, T.; Kantz, H. Generalized Langevin equation with tempered memory kernel. Physica A
**2017**, 466, 356–369. [Google Scholar] [CrossRef] - Liemert, A.; Kienle, A. Fractional radiative transport in the diffusion approximation. J. Math. Chem.
**2018**, 56, 317–335. [Google Scholar] [CrossRef] - Liang, Y.; Chen, W. Continuous time random walk model with asymptotical probability density of waiting times via inverse Mittag-Leffler function. Commun. Nonlinear Sci. Numer. Simul.
**2018**, 57, 439–448. [Google Scholar] [CrossRef] - Le Vot, F.; Abad, E.; Yuste, S. Continuous-time random-walk model for anomalous diffusion in expanding media. Phys. Rev. E
**2017**, 96, 032117. [Google Scholar] [CrossRef] [PubMed] - Atangana, A.; Baleanu, D. New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model. Therm. Sci.
**2016**, 18. [Google Scholar] [CrossRef] - Caputo, M.; Fabrizio, M. A new definition of fractional derivative without singular kernel. Progr. Fract. Differ. Appl.
**2015**, 1, 1–13. [Google Scholar] - Baleanu, D.; Jajarmi, A.; Hajipour, M. On the nonlinear dynamical systems within the generalized fractional derivatives with Mittag-Leffler kernel. Nonlinear Dyn.
**2018**, 1–18, 1–18. [Google Scholar] [CrossRef] - Hristov, J. Steady-state heat conduction in a medium with spatial non-singular fading memory: Derivation of Caputo-Fabrizio space-fractional derivative with Jeffrey’s kernel and analytical solutions. Therm. Sci.
**2016**, 115. [Google Scholar] [CrossRef] - Hristov, J. Derivatives with non-singular kernels from the Caputo-Fabrizio definition and beyond: Appraising analysis with emphasis on diffusion models. Front. Fract. Calc.
**2017**, 1, 270–342. [Google Scholar] - Sandev, T.; Sokolov, I.M.; Metzler, R.; Chechkin, A. Beyond monofractional kinetics. Chaos Solitons Fractals
**2017**, 102, 210–217. [Google Scholar] [CrossRef] - Tateishi, A.A.; Ribeiro, H.V.; Lenzi, E.K. The role of fractional time-derivative operators on anomalous diffusion. Front. Phys.
**2017**, 5, 52. [Google Scholar] [CrossRef] - Sandev, T.; Chechkin, A.; Kantz, H.; Metzler, R. Diffusion and Fokker-Planck-Smoluchowski equations with generalized memory kernel. Fract. Calc. Appl. Anal.
**2015**, 18, 1006–1038. [Google Scholar] [CrossRef] - Montroll, E.W.; Weiss, G.H. Random walks on lattices. II. J. Math. Phys.
**1965**, 6, 167–181. [Google Scholar] [CrossRef] - Scher, H.; Montroll, E.W. Anomalous transit-time dispersion in amorphous solids. Phys. Rev. B
**1975**, 12, 2455. [Google Scholar] [CrossRef] - Shlesinger, M.F. Origins and applications of the Montroll-Weiss continuous time random walk. Eur. Phys. J. B
**2017**, 90, 93. [Google Scholar] [CrossRef] - Bouchaud, J.P.; Georges, A. Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications. Phys. Rep.
**1990**, 195, 127–293. [Google Scholar] [CrossRef] - Metzler, R.; Klafter, J. The random walk’s guide to anomalous diffusion: A fractional dynamics approach. Phys. Rep.
**2000**, 339, 1–77. [Google Scholar] [CrossRef] - Aghion, E.; Kessler, D.A.; Barkai, E. Asymptotic densities from the modified Montroll-Weiss equation for coupled CTRWs. Eur. Phys. J. B
**2018**, 91, 17. [Google Scholar] [CrossRef] - Yadav, A.; Fedotov, S.; Méndez, V.; Horsthemke, W. Propagating fronts in reaction-transport systems with memory. Phys. Lett. A
**2007**, 371, 374–378. [Google Scholar] [CrossRef] - Ben-Zvi, R.; Nissan, A.; Scher, H.; Berkowitz, B. A continuous time random walk (CTRW) integro-differential equation with chemical interaction. Eur. Phys. J. B
**2018**, 91, 15. [Google Scholar] [CrossRef] - Furnival, T.; Leary, R.K.; Tyo, E.C.; Vajda, S.; Ramasse, Q.M.; Thomas, J.M.; Bristowe, P.D.; Midgley, P.A. Anomalous diffusion of single metal atoms on a graphene oxide support. Chem. Phys. Lett.
**2017**, 683, 370–374. [Google Scholar] [CrossRef] - Montroll, E.W.; Scher, H. Random walks on lattices. IV. Continuous-time walks and influence of absorbing boundaries. J. Stat. Phys.
**1973**, 9, 101–135. [Google Scholar] [CrossRef] - Sandev, T.; Metzler, R.; Chechkin, A. From continuous time random walks to the generalized diffusion equation. Fract. Calc. Appl. Anal.
**2018**, 21, 10–28. [Google Scholar] [CrossRef] [Green Version] - Mainardi, F.; Pagnini, G.; Saxena, R. Fox H functions in fractional diffusion. J. Comput. Appl. Math.
**2005**, 178, 321–331. [Google Scholar] [CrossRef] - Barenblatt, G.; Zheltov, I.P.; Kochina, I. Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks [strata]. J. Appl. Math. Mech.
**1960**, 24, 1286–1303. [Google Scholar] [CrossRef] - Prudnikov, A.P.; Marichev, O.I. Integrals and Series. Vol. 4’ Laplace Transforms; Gordon and Breach Science Publishers: New York, NY, USA; London, UK; Tokyo, Japan, 1992; Volume 18. [Google Scholar]
- Evans, M.R.; Majumdar, S.N. Diffusion with stochastic resetting. Phys. Rev. Lett.
**2011**, 106, 160601. [Google Scholar] [CrossRef] [PubMed] - Chatterjee, A.; Christou, C.; Schadschneider, A. Diffusion with resetting inside a circle. Phys. Rev. E
**2018**, 97, 062106. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Pal, A.; Kundu, A.; Evans, M.R. Diffusion under time-dependent resetting. J. Phys. A Math. Theor.
**2016**, 49, 225001. [Google Scholar] [CrossRef] [Green Version] - Falcao, R.; Evans, M.R. Interacting Brownian motion with resetting. J. Stat. Mech. Theory Exp.
**2017**, 2017, 023204. [Google Scholar] [CrossRef] [Green Version] - Kuśmierz, Ł; Gudowska-Nowak, E. Optimal first-arrival times in Lévy flights with resetting. Phys. Rev. E
**2015**, 92, 052127. [Google Scholar] - Durang, X.; Henkel, M.; Park, H. The statistical mechanics of the coagulation–diffusion process with a stochastic reset. J. Phys. A Math. Theor.
**2014**, 47, 045002. [Google Scholar] [CrossRef] [Green Version] - Palyulin, V.V.; Mantsevich, V.N.; Klages, R.; Metzler, R.; Chechkin, A.V. Comparison of pure and combined search strategies for single and multiple targets. Eur. Phys. J. B
**2017**, 90, 170. [Google Scholar] [CrossRef]

**Figure 1.**This curves illustrate the changes of distribution $\rho (x,t)$ caused by the exponential kernel (Equation (3) in Equation (1)), and the usual random walk (Brownian motion) case $\mathcal{K}\left(t\right)=\delta \left(t\right)$ (or $\tau =0$ in Equation (3)), considering $\phi \left(x\right)=\delta \left(x\right)$, $\mathcal{D}{f}^{-1}=1$, $t=1$, and a family of characteristic times $\{{10}^{-2},{10}^{-1},0.5,1,2,4\}$ (in context Caputo-Fabrizio operator $\tau =(1-\alpha ){\alpha}^{-1}$ and $\mathcal{D}{f}^{-1}=1$).

**Figure 2.**This curves illustrate the changes of distribution $\rho (x,t)$ caused by the exponential kernel (Equation (3) in Equation (1)), considering $\phi \left(x\right)=\delta \left(x\right)$, $\mathcal{D}{f}^{-1}=1$ and $\tau =4$ to different times $\{0.1,1,2,4,6,8\}$ (in context of Caputo-Fabrizio operator $\alpha =0.2$ and $\mathcal{D}{f}^{-1}=1$).

**Figure 3.**This curves illustrate the changes of distribution $\rho (x,t)$ caused by the Mittag-Leffler kernel (Equation (4) in Equation (1)), and the usual (Brownian motion) case $\mathcal{K}\left(t\right)=\delta \left(t\right)$ (or $\alpha =1$ and $\tau \to 0$ in Equation (4)), considering $\phi \left(x\right)=\delta \left(x\right)$, $\mathcal{D}{b}^{-1}=1$, $t=1$, $\tau ={10}^{-2}$ and different values to index $\alpha $.

**Figure 4.**This curves illustrate the changes of distribution $\rho (x,t)$ caused by the Mittag-Leffler kernel (Equation (4) in Equation (1)), and the usual (Brownian motion) case $\mathcal{K}\left(t\right)=\delta \left(t\right)$ (or $\alpha =1$ and $\tau \to 0$ in Equation (4)). Considering ,$\phi \left(x\right)=\delta \left(x\right)$, $\mathcal{D}{b}^{-1}=1$, $t=1$, $\alpha =0.25$ and different values of parameter $\tau $.

**Figure 5.**These curves illustrate the changes of MSD caused by a set of fractional $\alpha $-index in Mittag-Leffler kernel, considering $\tau ={10}^{-4}$, $\mathcal{D}{b}^{-1}=0.5$ and $r=1$.

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Dos Santos, M.A.F.
Non-Gaussian Distributions to Random Walk in the Context of Memory Kernels. *Fractal Fract.* **2018**, *2*, 20.
https://doi.org/10.3390/fractalfract2030020

**AMA Style**

Dos Santos MAF.
Non-Gaussian Distributions to Random Walk in the Context of Memory Kernels. *Fractal and Fractional*. 2018; 2(3):20.
https://doi.org/10.3390/fractalfract2030020

**Chicago/Turabian Style**

Dos Santos, Maike A. F.
2018. "Non-Gaussian Distributions to Random Walk in the Context of Memory Kernels" *Fractal and Fractional* 2, no. 3: 20.
https://doi.org/10.3390/fractalfract2030020