The Kardar–Parisi–Zhang model of a random kinetic growth: effects of a randomly moving medium

Antonov, N. V.; Kakin, Polina I.; Lebedev, Nikita M.

doi:10.1088/1751-8121/ab5426

Cited by 10 publications

(17 citation statements)

References 111 publications

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“…We omit the one-loop calculations of the renormalization constants for brevity; see., e.g., [74] for similar calculations. In the one-loop approximation and minimal subtraction scheme (MS scheme) the renormalization constants Z ν and Z ν ⊥ read…”

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confidence: 79%

Effects of Turbulent Environment on Self-Organized Critical Behavior: Isotropy vs. Anisotropy

et al. 2020

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We study a self-organized critical system under the influence of turbulent motion of the environment. The system is described by the anisotropic continuous stochastic equation proposed by Hwa and Kardar [Phys. Rev. Lett.62: 1813 (1989)]. The motion of the environment is modelled by the isotropic Kazantsev–Kraichnan “rapid-change” ensemble for an incompressible fluid: it is Gaussian with vanishing correlation time and the pair correlation function of the form ∝δ(t−t′)/kd+ξ, where k is the wave number and ξ is an arbitrary exponent with the most realistic values ξ=4/3 (Kolmogorov turbulence) and ξ→2 (Batchelor’s limit). Using the field-theoretic renormalization group, we find infrared attractive fixed points of the renormalization group equation associated with universality classes, i.e., with regimes of critical behavior. The most realistic values of the spatial dimension d=2 and the exponent ξ=4/3 correspond to the universality class of pure turbulent advection where the nonlinearity of the Hwa–Kardar (HK) equation is irrelevant. Nevertheless, the universality class where both the (anisotropic) nonlinearity of the HK equation and the (isotropic) advecting velocity field are relevant also exists for some values of the parameters ε=4−d and ξ. Depending on what terms (anisotropic, isotropic, or both) are relevant in specific universality class, different types of scaling behavior (ordinary one or generalized) are established.

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confidence: 79%

Effects of Turbulent Environment on Self-Organized Critical Behavior: Isotropy vs. Anisotropy

et al. 2020

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“…Lastly, such coordinates do not preclude the existence of non-perturbative IR attractive fixed point that remains hidden in perturbative RG analysis. See a more detailed discussion of related issues in [68].…”

Section: Conclusion and Discussionmentioning

confidence: 99%

“…Recent attempts [67][68][69] revealed that turbulent or random environment (modelled either by stochastic Navier-Stokes equation or by a "rapid-change" Kazantsev-Kraichnan velocity ensemble) dominates scaling and "washes away" the kinetic roughening.…”

Section: Introductionmentioning

confidence: 99%

Stirred Kardar-Parisi-Zhang Equation with Quenched Random Noise: Emergence of Induced Nonlinearity

et al. 2022

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We study the stochastic Kardar-Parisi-Zhang equation for kinetic roughening where the time-independent (columnar or spatially quenched) Gaussian random noise f(t,x) is specified by the pair correlation function ⟨f(t,x)f(t′,x′)⟩∝δ(d)(x−x′), d 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 ε=4−d and the velocity exponent ξ (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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“…The anomalous dimensions calculated from the Feynman graphs have the following one-loop expressions (we omit details of calculations 17 for brevity):…”

Section: Renormalization and Fixed Points Of The Model With Turbulent...mentioning

confidence: 99%

Dimensional transmutation and nonconventional scaling behaviour in a model of self-organized criticality

Antonov,

Gulitskiy,

Kakin

et al. 2021

Preprint

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The paper addresses two unusual scaling regimes (types of critical behaviour) predicted by the field-theoretic renormalization group analysis for a self-organized critical system with turbulent motion of the environment. The system is modelled by the anisotropic stochastic equation for a "running sandpile" introduced by Hwa and Kardar in [Phys. Rev. Lett. 62: 1813(1989]. The turbulent motion is described by the isotropic Kazantsev-Kraichnan "rapid-change" velocity ensemble for an incompressible fluid. The original Hwa-Kardar equation allows for independent scaling of the spatial coordinates x (the coordinate along the preferred dimension) and x ⊥ (the coordinates in the orthogonal subspace to the preferred direction) that becomes impossible once the isotropic velocity ensemble is coupled to the equation. However, it is found that one of the regimes of the system's critical behaviour (the one where the isotropic turbulent motion is irrelevant) recovers the anisotropic scaling through "dimensional transmutation." The latter manifests as a dimensionless ratio acquiring nontrivial canonical dimension. The critical regime where both the velocity ensemble and the nonlinearity of the Hwa-Kardar equation are relevant simultaneously is also characterized by "atypical" scaling. While the ordinary scaling with fixed infra-red irrelevant parameters is impossible in this regime, the "restricted" scaling where the times, the coordinates, and the dimensionless ratio are scaled becomes possible. This result brings to mind scaling hypotheses modifications (Stell's weak scaling or Fisher's generalized scaling) for systems with significantly different characteristic scales.

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The Kardar–Parisi–Zhang model of a random kinetic growth: effects of a randomly moving medium

Cited by 10 publications

References 111 publications

Effects of Turbulent Environment on Self-Organized Critical Behavior: Isotropy vs. Anisotropy

Effects of Turbulent Environment on Self-Organized Critical Behavior: Isotropy vs. Anisotropy

Stirred Kardar-Parisi-Zhang Equation with Quenched Random Noise: Emergence of Induced Nonlinearity

Dimensional transmutation and nonconventional scaling behaviour in a model of self-organized criticality

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