Fractional Diffusion to a Cantor Set in 2D
A random walk on a two dimensional square in <inline-formula><math display="inline"><semantics><msup><mi>R</mi><mn>2</mn></msup></semantics></math></inline-formula> space with a hidden absorbing fractal set <inline-...
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| Format: | Article |
| Language: | English |
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MDPI AG
2020-11-01
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| Series: | Fractal and Fractional |
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| Online Access: | https://www.mdpi.com/2504-3110/4/4/52 |
| _version_ | 1797548615901642752 |
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| author | Alexander Iomin Trifce Sandev |
| author_facet | Alexander Iomin Trifce Sandev |
| author_sort | Alexander Iomin |
| collection | DOAJ |
| description | A random walk on a two dimensional square in <inline-formula><math display="inline"><semantics><msup><mi>R</mi><mn>2</mn></msup></semantics></math></inline-formula> space with a hidden absorbing fractal set <inline-formula><math display="inline"><semantics><msub><mi>F</mi><mi>μ</mi></msub></semantics></math></inline-formula> is considered. This search-like problem is treated in the framework of a diffusion–reaction equation, when an absorbing term is included inside a Fokker–Planck equation as a reaction term. This macroscopic approach for the 2D transport in the <inline-formula><math display="inline"><semantics><msup><mi>R</mi><mn>2</mn></msup></semantics></math></inline-formula> space corresponds to the comb geometry, when the random walk consists of 1D movements in the <i>x</i> and <i>y</i> directions, respectively, as a direct-Cartesian product of the 1D movements. The main value in task is the first arrival time distribution (FATD) to sink points of the fractal set, where travelling particles are absorbed. Analytical expression for the FATD is obtained in the subdiffusive regime for both the fractal set of sinks and for a single sink. |
| first_indexed | 2024-03-10T15:02:10Z |
| format | Article |
| id | doaj.art-746d57850fee4fd486700aadfedaa3f9 |
| institution | Directory Open Access Journal |
| issn | 2504-3110 |
| language | English |
| last_indexed | 2024-03-10T15:02:10Z |
| publishDate | 2020-11-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj.art-746d57850fee4fd486700aadfedaa3f92023-11-20T20:06:50ZengMDPI AGFractal and Fractional2504-31102020-11-01445210.3390/fractalfract4040052Fractional Diffusion to a Cantor Set in 2DAlexander Iomin0Trifce Sandev1Department of Physics, Technion, Haifa 32000, IsraelResearch Center for Computer Science and Information Technologies, Macedonian Academy of Sciences and Arts, Bul. Krste Misirkov 2, 1000 Skopje, MacedoniaA random walk on a two dimensional square in <inline-formula><math display="inline"><semantics><msup><mi>R</mi><mn>2</mn></msup></semantics></math></inline-formula> space with a hidden absorbing fractal set <inline-formula><math display="inline"><semantics><msub><mi>F</mi><mi>μ</mi></msub></semantics></math></inline-formula> is considered. This search-like problem is treated in the framework of a diffusion–reaction equation, when an absorbing term is included inside a Fokker–Planck equation as a reaction term. This macroscopic approach for the 2D transport in the <inline-formula><math display="inline"><semantics><msup><mi>R</mi><mn>2</mn></msup></semantics></math></inline-formula> space corresponds to the comb geometry, when the random walk consists of 1D movements in the <i>x</i> and <i>y</i> directions, respectively, as a direct-Cartesian product of the 1D movements. The main value in task is the first arrival time distribution (FATD) to sink points of the fractal set, where travelling particles are absorbed. Analytical expression for the FATD is obtained in the subdiffusive regime for both the fractal set of sinks and for a single sink.https://www.mdpi.com/2504-3110/4/4/52comb modelBrownian searchfractional Fokker–Planck equationsubdiffusionCantor set |
| spellingShingle | Alexander Iomin Trifce Sandev Fractional Diffusion to a Cantor Set in 2D Fractal and Fractional comb model Brownian search fractional Fokker–Planck equation subdiffusion Cantor set |
| title | Fractional Diffusion to a Cantor Set in 2D |
| title_full | Fractional Diffusion to a Cantor Set in 2D |
| title_fullStr | Fractional Diffusion to a Cantor Set in 2D |
| title_full_unstemmed | Fractional Diffusion to a Cantor Set in 2D |
| title_short | Fractional Diffusion to a Cantor Set in 2D |
| title_sort | fractional diffusion to a cantor set in 2d |
| topic | comb model Brownian search fractional Fokker–Planck equation subdiffusion Cantor set |
| url | https://www.mdpi.com/2504-3110/4/4/52 |
| work_keys_str_mv | AT alexanderiomin fractionaldiffusiontoacantorsetin2d AT trifcesandev fractionaldiffusiontoacantorsetin2d |