Fractional Diffusion to a Cantor Set in 2D
Abstract
:1. Introduction
Preliminaries: Subdiffusion and Boundary Conditions on a Comb
2. First Arrival Time Distribution for Subdiffusion
2.1. Fractal Set of Sinks
2.2. Single Sink
3. Discussion
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Appendix A. A Brief Survey on Fractional Integration
Appendix B. Fractional Fokker–Planck Equation
Solution in the Form of the Fox Function
References
- Palyulin, V.V.; Chechkin, A.V.; Metzler, R. Lévy flights do not always optimize random blind search for sparse targets. Proc. Natl. Acad. Sci. USA 2014, 111, 2931–2936. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Palyulin, V.V.; Chechkin, A.V.; Klages, R.; Metzler, R. Search reliability and search efficiency of combined Lévy– Brownian motion: Long relocations mingled with thorough local exploration. J. Phys. A Math. Theor. 2016, 49, 394002. [Google Scholar] [CrossRef]
- 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] [Green Version]
- Sandev, T.; Iomin, A.; Kocarev, L. Random search on comb. J. Phys. A Math. Theor. 2019, 52, 465001. [Google Scholar] [CrossRef]
- Bénichou, O.; Loverdo, C.; Moreau, M.; Voituriez, R. Two-dimensional intermittent search processes: An alternative to Lévy flight strategies. Phys. Rev. E 2006, 74, 020102. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Santos, M.C.; Viswanathan, G.M.; Raposo, E.P.; da Luz, M.G.E. Optimization of random searches on defective lattice networks. Phys. Rev. E 2008, 77, 041101. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Campos, D.; Bartumeus, F.; Raposo, E.P.; Méndez, V. First-passage times in multiscale random walks: The impact of movement scales on search efficiency. Phys. Rev. E 2015, 92, 052702. [Google Scholar] [CrossRef] [Green Version]
- Bressloff, P.C.; Newby, J.M. Quasi-steady-state analysis of two-dimensional random intermittent search processes. Phys. Rev. E 2011, 83, 061139. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Levernier, N.; Textor, J.; Bénichou, O.; Voituriez, R. Inverse square Lévy walks are not optimal search strategies for d≥2. Phys. Rev. Lett. 2020, 124, 080601. [Google Scholar] [CrossRef] [Green Version]
- Iomin, A.; Méndez, V.; Horsthemke, W. Fractional Dynamics in Comb-like Structures; World Scientific: Singapore, 2018. [Google Scholar]
- Newby, J.M.; Bressloff, P.C. Directed intermittent search for a hidden target on a dendritic tree. Phys. Rev. E 2009, 80, 021913. [Google Scholar] [CrossRef] [Green Version]
- Santamaria, F.; Wils, S.; De Schutter, E.; Augustine, G.J. Anomalous Diffusion in Purkinje Cell Dendrites Caused by Spines. Neuron 2006, 52, 635–648. [Google Scholar] [CrossRef] [Green Version]
- Santamaria, F.; Wils, S.; De Schutter, E.; Augustine, G.J. The diffusional properties of dendrites depend on the density of dendritic spines. Eur. J. Neurosci. 2011, 34, 561–568. [Google Scholar] [CrossRef] [Green Version]
- 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] [Green Version]
- Sandev, T.; Iomin, A. Finite-velocity diffusion on a comb. EPL (Europhys. Lett.) 2018, 124, 20005. [Google Scholar] [CrossRef] [Green Version]
- Iomin, A. Richardson diffusion in neurons. Phys. Rev. E 2019, 100, 010104. [Google Scholar] [CrossRef] [Green Version]
- Yuste, R. Dendritic Spines; MIT Press: Cambridge, MA, USA, 2010. [Google Scholar]
- Venneri, F.; Costanzo, S.; Borgia, A. Compact metamaterial absorber with fractal geometry. Electronics 2019, 8, 879. [Google Scholar] [CrossRef] [Green Version]
- 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]
- Redner, S. A Guide to First-Passage Processes; Cambridge University Press: Cambridge, MA, USA, 2001. [Google Scholar]
- Iomin, A.; Zaburdaev, V.; Pfohl, T. Reaction front propagation of actin polymerization in a comb-reaction system. Chaos Solitons Fractals 2016, 92, 115–122. [Google Scholar] [CrossRef]
- Chechkin, A.V.; Metzler, R.; Gonchar, V.Y.; Klafter, J.; Tanatarov, L.V. First passage and arrival time densities for Lévy flights and the failure of the method of images. J. Phys. A Math. Gen. 2003, 36, L537–L544. [Google Scholar] [CrossRef]
- Samko, S.G.; Kilbas, A.A.; Marichev, O.I. Fractional Integrals and Derivatives: Theory and Applications; Gordon and Breach: London, UK, 1993. [Google Scholar]
- Tarasov, V.E. Fractional generalization of Liouville equations. Chaos 2004, 14, 123–127. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Sandev, T.; Iomin, A.; Kantz, H. Fractional diffusion on a fractal grid comb. Phys. Rev. E 2015, 91, 032108. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Abramovitz, M.; Stegun, I.A. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables; Dover Publications: New York, NY, USA, 1972. [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]
- Uchaikin, V.; Sibatov, R. Fractional Kinetics in Solids; World Scientific: Singapore, 2013. [Google Scholar]
- Metzler, R.G.; Oshanin, S.R. First-Passage Phenomena, Their Applications; World Scientific: Singapore, 2014. [Google Scholar]
- Iomin, A. Anomalous diffusion in umbrella comb. arXiv 2020, arXiv:2009.14067. [Google Scholar]
- Podlubny, I. Fractional Differential Equations; Academic Press: San Diego, CA, USA, 1999. [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 [Volumes I-III]; McGraw-Hill: New York, NY, USA, 1953–1955. [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]
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Iomin, A.; Sandev, T. Fractional Diffusion to a Cantor Set in 2D. Fractal Fract. 2020, 4, 52. https://doi.org/10.3390/fractalfract4040052
Iomin A, Sandev T. Fractional Diffusion to a Cantor Set in 2D. Fractal and Fractional. 2020; 4(4):52. https://doi.org/10.3390/fractalfract4040052
Chicago/Turabian StyleIomin, Alexander, and Trifce Sandev. 2020. "Fractional Diffusion to a Cantor Set in 2D" Fractal and Fractional 4, no. 4: 52. https://doi.org/10.3390/fractalfract4040052