Random Walk on a Rough Surface: Renormalization Group Analysis of a Simple Model
The field-theoretic renormalization group is applied to a simple model of a random walk on a rough fluctuating surface. We consider the Fokker–Planck equation for a particle in a uniform gravitational field. The surface is modeled by the generalized Edwards–Wilkinson linear stochastic equation for t...
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MDPI AG
2023-03-01
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| Online Access: | https://www.mdpi.com/2218-1997/9/3/139 |
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| author | Nikolay V. Antonov Nikolay M. Gulitskiy Polina I. Kakin Dmitriy A. Kerbitskiy |
| author_facet | Nikolay V. Antonov Nikolay M. Gulitskiy Polina I. Kakin Dmitriy A. Kerbitskiy |
| author_sort | Nikolay V. Antonov |
| collection | DOAJ |
| description | The field-theoretic renormalization group is applied to a simple model of a random walk on a rough fluctuating surface. We consider the Fokker–Planck equation for a particle in a uniform gravitational field. The surface is modeled by the generalized Edwards–Wilkinson linear stochastic equation for the height field. The full stochastic model is reformulated as a multiplicatively renormalizable field theory, which allows for the application of the standard renormalization theory. The renormalization group equations have several fixed points that correspond to possible scaling regimes in the infrared range (long times and large distances); all the critical dimensions are found exactly. As an example, the spreading law for the particle’s cloud is derived. It has the form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>R</mi><mn>2</mn></msup><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>≃</mo><msup><mi>t</mi><mrow><mn>2</mn><mo>/</mo><msub><mo>Δ</mo><mi>ω</mi></msub></mrow></msup></mrow></semantics></math></inline-formula> with the exactly known critical dimension of frequency <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mo>Δ</mo><mi>ω</mi></msub></semantics></math></inline-formula> and, in general, differs from the standard expression <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>R</mi><mn>2</mn></msup><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>≃</mo><mi>t</mi></mrow></semantics></math></inline-formula> for an ordinary random walk. |
| first_indexed | 2024-03-11T05:47:57Z |
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| id | doaj.art-44a028dd4fce40128a54be84f501e740 |
| institution | Directory Open Access Journal |
| issn | 2218-1997 |
| language | English |
| last_indexed | 2024-03-11T05:47:57Z |
| publishDate | 2023-03-01 |
| publisher | MDPI AG |
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| series | Universe |
| spelling | doaj.art-44a028dd4fce40128a54be84f501e7402023-11-17T14:15:59ZengMDPI AGUniverse2218-19972023-03-019313910.3390/universe9030139Random Walk on a Rough Surface: Renormalization Group Analysis of a Simple ModelNikolay V. Antonov0Nikolay M. Gulitskiy1Polina I. Kakin2Dmitriy A. Kerbitskiy3Department of Physics, Saint Petersburg State University, Universitetskaya nab. 7/9, St. Petersburg 199034, RussiaDepartment of Physics, Saint Petersburg State University, Universitetskaya nab. 7/9, St. Petersburg 199034, RussiaDepartment of Physics, Saint Petersburg State University, Universitetskaya nab. 7/9, St. Petersburg 199034, RussiaDepartment of Physics, Saint Petersburg State University, Universitetskaya nab. 7/9, St. Petersburg 199034, RussiaThe field-theoretic renormalization group is applied to a simple model of a random walk on a rough fluctuating surface. We consider the Fokker–Planck equation for a particle in a uniform gravitational field. The surface is modeled by the generalized Edwards–Wilkinson linear stochastic equation for the height field. The full stochastic model is reformulated as a multiplicatively renormalizable field theory, which allows for the application of the standard renormalization theory. The renormalization group equations have several fixed points that correspond to possible scaling regimes in the infrared range (long times and large distances); all the critical dimensions are found exactly. As an example, the spreading law for the particle’s cloud is derived. It has the form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>R</mi><mn>2</mn></msup><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>≃</mo><msup><mi>t</mi><mrow><mn>2</mn><mo>/</mo><msub><mo>Δ</mo><mi>ω</mi></msub></mrow></msup></mrow></semantics></math></inline-formula> with the exactly known critical dimension of frequency <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mo>Δ</mo><mi>ω</mi></msub></semantics></math></inline-formula> and, in general, differs from the standard expression <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>R</mi><mn>2</mn></msup><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>≃</mo><mi>t</mi></mrow></semantics></math></inline-formula> for an ordinary random walk.https://www.mdpi.com/2218-1997/9/3/139stochastic growthkinetic rougheningrandom walkrenormalization group |
| spellingShingle | Nikolay V. Antonov Nikolay M. Gulitskiy Polina I. Kakin Dmitriy A. Kerbitskiy Random Walk on a Rough Surface: Renormalization Group Analysis of a Simple Model Universe stochastic growth kinetic roughening random walk renormalization group |
| title | Random Walk on a Rough Surface: Renormalization Group Analysis of a Simple Model |
| title_full | Random Walk on a Rough Surface: Renormalization Group Analysis of a Simple Model |
| title_fullStr | Random Walk on a Rough Surface: Renormalization Group Analysis of a Simple Model |
| title_full_unstemmed | Random Walk on a Rough Surface: Renormalization Group Analysis of a Simple Model |
| title_short | Random Walk on a Rough Surface: Renormalization Group Analysis of a Simple Model |
| title_sort | random walk on a rough surface renormalization group analysis of a simple model |
| topic | stochastic growth kinetic roughening random walk renormalization group |
| url | https://www.mdpi.com/2218-1997/9/3/139 |
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