Stirred Kardar-Parisi-Zhang Equation with Quenched Random Noise: Emergence of Induced Nonlinearity
We study the stochastic Kardar-Parisi-Zhang equation for kinetic roughening where the time-independent (columnar or spatially quenched) Gaussian random noise <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow>...
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2022-01-01
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| author | Polina I. Kakin Mikhail A. Reiter Maria M. Tumakova Nikolay M. Gulitskiy Nikolay V. Antonov |
| author_facet | Polina I. Kakin Mikhail A. Reiter Maria M. Tumakova Nikolay M. Gulitskiy Nikolay V. Antonov |
| author_sort | Polina I. Kakin |
| collection | DOAJ |
| description | We study the stochastic Kardar-Parisi-Zhang equation for kinetic roughening where the time-independent (columnar or spatially quenched) Gaussian random noise <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi mathvariant="bold">x</mi><mo>)</mo></mrow></semantics></math></inline-formula> is specified by the pair correlation function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>⟨</mo><mi>f</mi><mrow><mo>(</mo><mi>t</mi><mo>,</mo><mi mathvariant="bold">x</mi><mo>)</mo></mrow><mi>f</mi><mrow><mo>(</mo><msup><mi>t</mi><mo>′</mo></msup><mo>,</mo><msup><mi mathvariant="bold">x</mi><mo>′</mo></msup><mo>)</mo></mrow><mo>⟩</mo></mrow><mo>∝</mo><msup><mi>δ</mi><mrow><mo>(</mo><mi>d</mi><mo>)</mo></mrow></msup><mrow><mo>(</mo><mrow><mi mathvariant="bold">x</mi><mo>−</mo><msup><mi mathvariant="bold">x</mi><mo>′</mo></msup></mrow><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, <i>d</i> being the dimension of space. The field-theoretic renormalization group analysis shows that the effect of turbulent motion of the environment (modelled by the coupling with the velocity field described by the Kazantsev-Kraichnan statistical ensemble for an incompressible fluid) gives rise to a new nonlinear term, quadratic in the velocity field. It turns out that this “induced” nonlinearity strongly affects the scaling behaviour in several universality classes (types of long-time, large-scale asymptotic regimes) even when the turbulent advection appears irrelevant in itself. Practical calculation of the critical exponents (that determine the universality classes) is performed to the first order of the double expansion in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ε</mi><mo>=</mo><mn>4</mn><mo>−</mo><mi>d</mi></mrow></semantics></math></inline-formula> and the velocity exponent <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ξ</mi></semantics></math></inline-formula> (one-loop approximation). As is the case with most “descendants” of the Kardar-Parisi-Zhang model, some relevant fixed points of the renormalization group equations lie in “forbidden zones”, i.e., in those corresponding to negative kinetic coefficients or complex couplings. This persistent phenomenon in stochastic non-equilibrium models requires careful and inventive physical interpretation. |
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| series | Universe |
| spelling | doaj.art-78a27853b4bd457cb35918c060083f3f2023-11-23T22:22:52ZengMDPI AGUniverse2218-19972022-01-01827210.3390/universe8020072Stirred Kardar-Parisi-Zhang Equation with Quenched Random Noise: Emergence of Induced NonlinearityPolina I. Kakin0Mikhail A. Reiter1Maria M. Tumakova2Nikolay M. Gulitskiy3Nikolay V. Antonov4Department of Physics, Saint Petersburg State University, Universitetskaya nab. 7/9, 199034 St. Petersburg, RussiaDepartment of Physics, Saint Petersburg State University, Universitetskaya nab. 7/9, 199034 St. Petersburg, RussiaL.D. Landau Institute for Theoretical Physics, Ak. Semenova 1-A, 142432 Chernogolovka, RussiaDepartment of Physics, Saint Petersburg State University, Universitetskaya nab. 7/9, 199034 St. Petersburg, RussiaDepartment of Physics, Saint Petersburg State University, Universitetskaya nab. 7/9, 199034 St. Petersburg, RussiaWe study the stochastic Kardar-Parisi-Zhang equation for kinetic roughening where the time-independent (columnar or spatially quenched) Gaussian random noise <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi mathvariant="bold">x</mi><mo>)</mo></mrow></semantics></math></inline-formula> is specified by the pair correlation function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>⟨</mo><mi>f</mi><mrow><mo>(</mo><mi>t</mi><mo>,</mo><mi mathvariant="bold">x</mi><mo>)</mo></mrow><mi>f</mi><mrow><mo>(</mo><msup><mi>t</mi><mo>′</mo></msup><mo>,</mo><msup><mi mathvariant="bold">x</mi><mo>′</mo></msup><mo>)</mo></mrow><mo>⟩</mo></mrow><mo>∝</mo><msup><mi>δ</mi><mrow><mo>(</mo><mi>d</mi><mo>)</mo></mrow></msup><mrow><mo>(</mo><mrow><mi mathvariant="bold">x</mi><mo>−</mo><msup><mi mathvariant="bold">x</mi><mo>′</mo></msup></mrow><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, <i>d</i> being the dimension of space. The field-theoretic renormalization group analysis shows that the effect of turbulent motion of the environment (modelled by the coupling with the velocity field described by the Kazantsev-Kraichnan statistical ensemble for an incompressible fluid) gives rise to a new nonlinear term, quadratic in the velocity field. It turns out that this “induced” nonlinearity strongly affects the scaling behaviour in several universality classes (types of long-time, large-scale asymptotic regimes) even when the turbulent advection appears irrelevant in itself. Practical calculation of the critical exponents (that determine the universality classes) is performed to the first order of the double expansion in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ε</mi><mo>=</mo><mn>4</mn><mo>−</mo><mi>d</mi></mrow></semantics></math></inline-formula> and the velocity exponent <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ξ</mi></semantics></math></inline-formula> (one-loop approximation). As is the case with most “descendants” of the Kardar-Parisi-Zhang model, some relevant fixed points of the renormalization group equations lie in “forbidden zones”, i.e., in those corresponding to negative kinetic coefficients or complex couplings. This persistent phenomenon in stochastic non-equilibrium models requires careful and inventive physical interpretation.https://www.mdpi.com/2218-1997/8/2/72kinetic rougheningcritical behaviourturbulencerenormalization group |
| spellingShingle | Polina I. Kakin Mikhail A. Reiter Maria M. Tumakova Nikolay M. Gulitskiy Nikolay V. Antonov Stirred Kardar-Parisi-Zhang Equation with Quenched Random Noise: Emergence of Induced Nonlinearity Universe kinetic roughening critical behaviour turbulence renormalization group |
| title | Stirred Kardar-Parisi-Zhang Equation with Quenched Random Noise: Emergence of Induced Nonlinearity |
| title_full | Stirred Kardar-Parisi-Zhang Equation with Quenched Random Noise: Emergence of Induced Nonlinearity |
| title_fullStr | Stirred Kardar-Parisi-Zhang Equation with Quenched Random Noise: Emergence of Induced Nonlinearity |
| title_full_unstemmed | Stirred Kardar-Parisi-Zhang Equation with Quenched Random Noise: Emergence of Induced Nonlinearity |
| title_short | Stirred Kardar-Parisi-Zhang Equation with Quenched Random Noise: Emergence of Induced Nonlinearity |
| title_sort | stirred kardar parisi zhang equation with quenched random noise emergence of induced nonlinearity |
| topic | kinetic roughening critical behaviour turbulence renormalization group |
| url | https://www.mdpi.com/2218-1997/8/2/72 |
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