# Comb Model: Non-Markovian versus Markovian

^{1}

^{2}

^{3}

^{*}

*Fractal Fract’s*Editorial Board Members)

## Abstract

**:**

## 1. Introduction

#### 1.1. Preliminaries I: Subdiffusion versus Diffusion

#### 1.2. Preliminaries II: Fractional Fokker-Planck Equation

## 2. Non-Markovian Diffusion in $\mathit{m}+\mathit{n}$ Space

- (i)
- We start with $n=1$. Then Equation (25) yields$$\langle {x}^{2}\left(t\right)\rangle =\frac{1}{\sqrt{\pi D}}{\mathcal{L}}^{-1}\left[{s}^{-\frac{n}{2}}\right]=\frac{{t}^{\frac{1}{2}}}{\sqrt{\pi D}}.$$
- (ii)
- The case $n=2$ corresponds to ultra-slow diffusion [22]. To see this, we present Equation (25) in the explicit form,$$\langle {x}^{2}\left(t\right)\rangle =\frac{1}{2\pi D}{\int}_{-i\infty}^{+i\infty}\mathcal{L}\left[{t}^{-1}\right]\frac{{e}^{st}ds}{s}=\frac{1}{2\pi D}{\int}^{t}dt{\int}_{-i\infty}^{+i\infty}\mathcal{L}\left[{t}^{-1}\right]{e}^{st}ds.$$$$\langle {x}^{2}\left(t\right)\rangle =\frac{1}{2\pi D}ln\left(t\right)+\frac{C}{2\pi D\phantom{\rule{0.166667em}{0ex}}t}\approx \frac{1}{2\pi D}ln\left(t\right),\phantom{\rule{1.em}{0ex}}\mathrm{as}\phantom{\rule{1.em}{0ex}}t\to \infty ,$$
- (iii)
- For $n>2$, Equation (25) yields$$\begin{array}{c}\langle {x}^{2}\left(t\right)\rangle =\frac{2}{{\left[4\pi D\right]}^{\frac{n}{2}}}{\mathcal{L}}^{-1}\left\{\frac{1}{s}\mathcal{L}{\left[{t}^{-\frac{n}{2}}\right]}_{C}\right\}=\frac{2\Gamma \left(1-{\textstyle \frac{n}{2}}\right)}{{\left[4\pi D\right]}^{\frac{n}{2}}}{\mathcal{L}}^{-1}\left[{s}^{{\textstyle \frac{n}{2}}-2}\right]\hfill \\ \hspace{1em}\hspace{1em}=\frac{2\Gamma \left(1-{\textstyle \frac{n}{2}}\right)}{{\left[4\pi D\right]}^{\frac{n}{2}}}\left[{s}^{N-1}{s}^{-(N-{\textstyle \frac{n}{2}}+1)}\right]=\frac{2\Gamma \left(1-{\textstyle \frac{n}{2}}\right)}{{\left[4\pi D\right]}^{\frac{n}{2}}}{\left(\frac{d}{d\phantom{\rule{0.166667em}{0ex}}t}\right)}^{N-1}\Gamma \left(N-{\textstyle \frac{n}{2}}+1\right){t}^{N-{\textstyle \frac{n}{2}}}\\ \hfill =\frac{2}{{\left[4\pi D\right]}^{\frac{n}{2}}}\frac{\Gamma \left(N+1-{\textstyle \frac{n}{2}}\right)}{\Gamma \left(N+1-{\textstyle \frac{n}{2}}\right)}{t}^{1-{\textstyle \frac{n}{2}}}=\frac{2}{{\left[4\pi D\right]}^{\frac{n}{2}}}{t}^{1-{\textstyle \frac{n}{2}}}\to 0\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{1.em}{0ex}}\mathrm{as}\phantom{\rule{1.em}{0ex}}t\to \infty .\end{array}$$

## 3. Quantum Comb and Fractional Schrödinger Equation

#### Quantum Friction Due to Fingers

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. A Brief Survey on Fractional Integration

## Appendix B. Solution in the Form of the Fox H Function

## Appendix C. Asymptotic Forms of Green’s Function

## References

- White, S.R.; Barma, M. Field-induced drift and trapping in percolation networks. J. Phys. A Math. Gen.
**1984**, 17, 2995–3008. [Google Scholar] [CrossRef] - Weiss, G.H.; Havlin, S. Some properties of a random walk on a comb structure. Phys. A
**1986**, 134, 474–482. [Google Scholar] [CrossRef] - Arkhincheev, V.E.; Baskin, E.M. Anomalous diffusion and drift in the comb model of percolation clusters. Sov. Phys. JETP
**1991**, 73, 161–165. [Google Scholar] - Méndez, V.; Iomin, A. Comb-like models for transport along spiny dendrites. Chaos Solitons Fractals
**2013**, 53, 46–51. [Google Scholar] [CrossRef] [Green Version] - Iomin, A.; Méndez, V. Reaction-subdiffusion front propagation in a comblike model of spiny dendrites. Phys. Rev. E
**2013**, 88, 012706. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Sandev, T.; Iomin, A. Finite-velocity diffusion on a comb. Europhys. Lett.
**2018**, 124, 20005. [Google Scholar] [CrossRef] - Marsh, R.E.; Riauka, T.A.; McQuarrie, S.A. A review of basic principles of fractals and their application to pharmacokinetics. Q. J. Nucl. Med. Mol. Imaging
**2008**, 52, 278–288. [Google Scholar] - Sagi, Y.; Brook, M.; Almog, I.; Davidson, N. Observation of Anomalous Diffusion and Fractional Self-Similarity in One Dimension. Phys. Rev. Lett.
**2012**, 108, 093002. [Google Scholar] [CrossRef] - Iomin, A. Superdiffusive comb: Application to experimental observation of anomalous diffusion in one dimension. Phys. Rev. E
**2012**, 86, 032101. [Google Scholar] [CrossRef] [Green Version] - Agliari, E.; Blumen, A.; Cassi, D. Slow encounters of particle pairs in branched structures. Phys. Rev. E
**2014**, 89, 052147. [Google Scholar] [CrossRef] [Green Version] - Bénichou, O.; Illien, P.; Oshanin, G.; Sarracino, A.; Voituriez, R. Diffusion and Subdiffusion of Interacting Particles on Comblike Structures. Phys. Rev. Lett.
**2015**, 115, 220601. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Rebenshtok, A.; Barkai, E. Occupation times on a comb with ramified teeth. Phys. Rev. E
**2013**, 88, 052126. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Ribeiro, H.V.; Tateishi, A.A.; Alves, L.G.A.; 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] - Méndez, V.; Iomin, A.; Horsthemke, W.; Campos, D. Langevin dynamics for ramified structures. J. Stat. Mech. Theor. Exp.
**2017**, 2017, 063205. [Google Scholar] [CrossRef] [Green Version] - Forte, G.; Burioni, R.; Cecconi, F.; Vulpiani, A. Anomalous diffusion and response in branched systems: A simple analysis. J. Phys. Condens. Matter
**2013**, 25, 465106. [Google Scholar] [CrossRef] [Green Version] - Iomin, A.; Méndez, V.; Horsthemke, W. Fractional Dynamics in Comb-Like Structures; World Scientific: Singapore, 2018. [Google Scholar]
- 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] - Sandev, T.; Iomin, A.; Kantz, H.; Metzler, R.; Chechkin, A. Comb model with slow and ultraslow diffusion. Math. Model. Nat. Phenom.
**2016**, 11, 18–33. [Google Scholar] [CrossRef] - Sandev, T.; Iomin, A.; Méndez, V. Lévy processes on a generalized fractal comb. J. Phys. A Math. Theor.
**2016**, 49, 355001. [Google Scholar] [CrossRef] [Green Version] - Iomin, A.; Baskin, E. Negative superdiffusion due to inhomogeneous convection. Phys. Rev. E
**2005**, 71, 061101. [Google Scholar] [CrossRef] [Green Version] - Caputo, M. Linear Models of Dissipation whose Q is almost Frequency Independent–II. Geophys. J. R. Astron. Soc.
**1967**, 13, 529–539. [Google Scholar] [CrossRef] - Iomin, A.; Méndez, V. Does ultra-slow diffusion survive in a three dimensional cylindrical comb? Chaos Solitons Fractals
**2016**, 82, 142–147. [Google Scholar] [CrossRef] - Bateman, H.; Erdélyi, A. Tables of Integral Transforms; McGraw-Hill: New York, NY, USA, 1954; Volume I–II. [Google Scholar]
- Brychkov, Y.A.; Prudnikov, A.P. Integral Transformations of Generalised Functions; Nauka: Moscow, Russia, 1977. [Google Scholar]
- Laskin, N. Fractals and quantum mechanics. Chaos
**2000**, 10, 780–790. [Google Scholar] [CrossRef] [PubMed] - West, B.J. Quantum Lévy Propagators. J. Phys. Chem. B
**2000**, 104, 3830–3832. [Google Scholar] [CrossRef] - Naber, M. Time fractional Schrödinger equation. J. Math. Phys.
**2004**, 45, 3339–3352. [Google Scholar] [CrossRef] - Harrison, W.A. Solid State Theory; McGraw Hill: New York, NY, USA, 1970. [Google Scholar]
- Schulman, L. Techniques and Applications of Path Integration; Wiley: New York, NY, USA, 1981. [Google Scholar]
- Gaveau, B.; Schulman, L.S. Explicit time-dependent Schrödinger propagators. J. Phys. A Math. Gen.
**1986**, 19, 1833–1846. [Google Scholar] [CrossRef] - Iomin, A. Fractional evolution in quantum mechanics. Chaos Solitons Fractals X
**2019**, 1, 100001. [Google Scholar] [CrossRef] - Podlubny, I. Fractional Differential Equations; Academic Press: San Diego, CA, USA, 1999. [Google Scholar]
- Laskin, N. Time fractional quantum mechanics. Chaos Solitons Fractals
**2017**, 102, 16–28. [Google Scholar] [CrossRef] [Green Version] - Iomin, A. Fractional time quantum mechanics. In Handbook of Fractional Calculus with Applications. Applications in Physics, Part B; Tarasov, V., Ed.; De Gruyter: Berlin, Germany, 2019. [Google Scholar]
- Tarasov, V.E. Quantum Mechanics of Non-Hamiltonian and Dissipative Systems; Elsevier: Amsterdam, The Netherlands, 2008. [Google Scholar]
- Samko, S.G.; Kilbas, A.A.; Marichev, O.I. Fractional Integrals and Derivatives: Theory and Applications; Gordon and Breach: London, UK, 1993. [Google Scholar]
- Oldham, K.B.; Spanier, J. The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order; Academic Press: New York, NY, USA, 1974. [Google Scholar]
- Bateman, H.; Erdélyi, A. Higher Transcendental Functions; McGraw-Hill: New York, NY, USA, 1955; Volume I–III. [Google Scholar]
- Mathai, A.M.; Haubold, H.J. Special Functions for Applied Scientists; Springer: New York, NY, USA, 2008. [Google Scholar]
- Mathai, A.M.; Saxena, R.K.; Haubold, H.J. The H-Function: Theory and Applications; Springer: New York, NY, USA, 2010. [Google Scholar]

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Iomin, A.; Méndez, V.; Horsthemke, W.
Comb Model: Non-Markovian versus Markovian. *Fractal Fract.* **2019**, *3*, 54.
https://doi.org/10.3390/fractalfract3040054

**AMA Style**

Iomin A, Méndez V, Horsthemke W.
Comb Model: Non-Markovian versus Markovian. *Fractal and Fractional*. 2019; 3(4):54.
https://doi.org/10.3390/fractalfract3040054

**Chicago/Turabian Style**

Iomin, Alexander, Vicenç Méndez, and Werner Horsthemke.
2019. "Comb Model: Non-Markovian versus Markovian" *Fractal and Fractional* 3, no. 4: 54.
https://doi.org/10.3390/fractalfract3040054