arXiv subjects

N. M. Gulitskiy

Publications and source records attributed to N. M. Gulitskiy.

At least 19 recordsLinked to original sources

arXiv2024

Random Walk on a Random Surface: Implications of Non-perturbative Concepts and Dynamical Emergence of Galilean Symmetry

We study a model of random walk on a fluctuating rough surface using the field-theoretic renormalization group (RG). The surface is modelled by the well-known Kardar--Parisi--Zhang (KPZ) stochastic equation while the random walk is described by the standard diffusion equation for a particle in a uniform gravitational field. In the RG approach, possible types of infrared (IR) asymptotic (long-time, large-distance) behaviour are determined by IR attractive fixed points. Within the one-loop RG calculation (the leading order in $\varepsilon=2-d$, $d$ being the spatial dimension), we found six possible fixed points or curves of points. Two of them can be IR attractive: the Gaussian point (free theory) and the nontrivial point where the KPZ surface is rough but its interaction with the random walk is irrelevant. For those perturbative fixed points, the spreading law for a particles' cloud coincides with that for ordinary random walk, $R(t)\sim t^{1/2}$. We also explored consequences of the presumed existence in the KPZ model of a non-perturbative strong-coupling fixed point. We found that it gives rise to an IR attractive fixed point in the full-scale model with the nontrivial spreading law $R(t)\sim t^{1/z}$, where the exponent $z<2$ can be inferred from the non-perturbative analysis of the KPZ model. Thus, the spreading becomes faster on a rough fluctuating surface in comparison to a smooth one. What is more, the Galilean-type symmetry inherent for the pure KPZ model extends dynamically to the IR asymptotic behaviour of the Green's functions of the full model.

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arXiv2024

Field Theoretic Renormalization Group in an Infinite-Dimensional Model of Random Surface Growth in Random Environment

The influence of a random environment on the dynamics of a fluctuating rough surface is investigated using a field theoretic renormalization group. The environment motion is modelled by the stochastic Navier--Stokes equation, which includes both a fluid in thermal equilibrium and a turbulent fluid. The surface is described by the generalized Pavlik's stochastic equation. As a result of fulfilling the renormalizability requirement, the model necessarily involves an infinite number of coupling constants. The one-loop counterterm is derived in an explicit closed form. The corresponding renormalization group equations demonstrate the existence of three two-dimensional surfaces of fixed points in the infinite-dimensional parameter space. If the surfaces contain IR attractive regions, the problem allows for the large-scale, long-time scaling behaviour. For the first surface (advection is irrelevant) the critical dimensions of the height field $Δ_{h}$, the response field $Δ_{h'}$ and the frequency $Δ_ω$ are non-universal through the dependence on the effective couplings. For the other two surfaces (advection is relevant) the dimensions are universal and they are found exactly.

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arXiv2023

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 random walk on a rough fluctuating surface. We consider the Fokker--Planck equation for a particle in a uniform gravitational field. The surface is modelled 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 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, large distances); all the critical dimensions are found exactly. As an example, the spreading law for particle's cloud is derived. It has the form $R^2(t)\simeq t^{2/Δ_ω}$ with the exactly known critical dimension of frequency $Δ_ω$ and, in general, differs from the standard expression $R^2(t)\simeq t$ for ordinary random walk.

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arXiv2022

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

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 [{\it Phys. Rev. Lett.} {\bf 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 $x_{\parallel}$ (the coordinate along the preferred dimension) and ${\bf x_{\bot}}$ (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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arXiv2021

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 $f(t,{\bf x})$ is specified by the pair correlation function $\langle f(t,{\bf x})f(t',{\bf x'}) \rangle \propto δ^{(d)} ({\bf x-x'})$, $d$ 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 $\varepsilon=4-d$ 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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arXiv2021

Effects of Turbulent Environment and Random Noise on Self-Organized Critical Behavior: Universality vs Nonuniversality

Self-organized criticality in the Hwa-Kardar model of "running sandpile" [Phys. Rev. A 45, 7002 (1992)] with a turbulent motion of the environment taken into account is studied with the field theoretic renormalization group (RG). The turbulent flow is modelled by the synthetic $d$-dimensional anisotropic velocity ensemble introduced by Avellaneda and Majda [Commun. Math. Phys.131: 381 (1990)] with finite correlation time. The Hwa-Kardar model with time-independent (spatially quenched) random noise is considered alongside the original model with the white noise. The aim of the present paper is to explore fixed points of RG equations which determine the possible types of universality classes (regimes of critical behavior of the system) and critical dimensions of the measurable quantities. Our calculations demonstrate that influence of the type of the random noise is extremely large: in contrast to the case of the white noise where the system possess three fixed points, the case of the spatially quenched noise involves four fixed points with overlapping stability regions. This means that in the latter case the critical behavior of the system depends not only on the global parameters of the system which is the usual case, but also on the initial values of the charges (coupling constants) of the system. These initial conditions determine the specific fixed point which will be reached by RG flow. Since now the critical properties of the system are not defined strictly by its parameters the situation may be interpreted as universality violation. Such systems are not forbidden but they are very rare. It is especially interesting that the same model without turbulent motion of the environment does not predict this nonuniversal behavior and demonstrates the "usual" one with prescribed universality classes instead [J.~Stat. Phys. 178, 392 (2020)].

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arXiv2020

Effects of turbulent environment on self-organized critical behavior: Isotropy vs Anisotropy

We study a self-organized critical system under influence of turbulent motion of the environment. The system is described by the anisotropic continuous stochastic equation proposed by Hwa and Kardar [{\it Phys. Rev. Lett.} {\bf 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 $\proptoδ(t-t') / k^{d+ξ}$, where $k$ is the wave number and $ξ$ is an arbitrary exponent with the most realistic values $ξ= 4/3$ (Kolmogorov turbulence) and $ξ\to 2$ (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 $\varepsilon=4-d$ 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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arXiv2020

Effects of turbulent environment on the surface roughening: The Kardar-Parisi-Zhang model coupled to the stochastic Navier-Stokes equation

The Kardar-Parisi-Zhang model of non-equilibrium critical behaviour (kinetic surface roughening) with turbulent motion of the environment taken into account is studied by the field theoretic renormalization group approach. The turbulent motion is described by the stochastic Navier-Stokes equation with the random stirring force whose correlation function includes two terms that allow one to account both for a turbulent fluid and for a fluid in thermal equilibrium. The renormalization group analysis performed in the leading order of perturbation theory (one-loop approximation) reveals six possible types of scaling behaviour (universality classes). The most interesting values of the spatial dimension $d=2$ and~$3$ correspond to the universality class of a pure turbulent advection where the nonlinearity of the Kardar--Parisi--Zhang model is irrelevant.

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arXiv2019

Advection by Compressible Turbulent Flows: Renormalization Group Study of Vector and Tracer Admixture

Advection-diffusion problems of magnetic field and tracer field are analyzed using the field theoretic perturbative renormalization group. Both advected fields are considered to be passive, i.e., without any influence on the turbulent environment, and advecting velocity field is generated by compressible version of stochastic Navier-Stokes equation. The model is considered in the vicinity of space dimension $d=4$ and is a continuation of previous work [N.V. Antonov et al., Phys. Rev. E 95, 033120 (2017)]. The perturbation theory near the special dimension $d=4$ is constructed within a double expansion scheme in $y$ (which describes scaling behavior of the random force that enters a stochastic equation for the velocity field) and $ε=4-d$. We show that up to one-loop approximation both types of advected fields exhibit similar universal scaling behavior. In particular, we demonstrate this statement on the inertial range asymptotic behavior of the correlation functions of advected fields. The critical dimensions of tensor composite operators are calculated in the leading order of $(y,ε)$ expansion.

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arXiv2018

Stochastic Navier-Stokes equation for a compressible fluid: two-loop approximation

A model of fully developed turbulence of a compressible fluid is briefly reviewed. It is assumed that fluid dynamics is governed by a stochastic version of Navier-Stokes equation. We show how corresponding field theoretic-model can be obtained and further analyzed by means of the perturbative renormalization group. Two fixed points of the RG equations are found. The perturbation theory is constructed within formal expansion scheme in parameter $y$, which describes scaling behavior of random force fluctuations. Actual calculations for fixed points' coordinates are performed to two-loop order.

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arXiv2018

Statistical symmetry restoration in fully developed turbulence: Renormalization group analysis of two models

In this paper we consider the model of incompressible fluid described by the stochastic Navier-Stokes equation with finite correlation time of a random force. Inertial-range asymptotic behavior of fully developed turbulence is studied by means of the field theoretic renormalization group within the one-loop approximation. It is corroborated that regardless of the values of model parameters and initial data, the inertial-range behavior of the model is described by limiting case of vanishing correlation time. It indicates that the Galilean symmetry of the model violated by the "colored" random force is restored in the inertial range. This regime corresponds to the only nontrivial fixed point of the renormalization group equation. The stability of this point depends on the relation between the exponents in the energy spectrum ${\cal E} \propto k^{1-y}$ and the dispersion law $ω\propto k^{2-η}$. The second analyzed problem is the passive advection of a scalar field by this velocity ensemble. Correlation functions of the scalar field exhibit anomalous scaling behavior in the inertial-convective range. We demonstrate that in accordance with Kolmogorov's hypothesis of the local symmetry restoration, the main contribution to the operator product expansion is given by the isotropic operator, while anisotropic terms should be considered only as corrections.

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arXiv2017

Stochastic Navier-Stokes equation and advection of a tracer field: One-loop renormalization near $d=4$

The renormalization group approach and the operator product expansion technique are applied to the model of a tracer field advected by the Navier-Stokes velocity ensemble for a compressible fluid. The model is considered in the vicinity of the specific space dimension $d=4$. The properties of the equal-time structure functions are investigated. The multifractal behaviour of various correlation functions is established. All calculations are performed in the leading one-loop approximation.

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arXiv2017

Turbulent compressible fluid: renormalization group analysis, scaling regimes, and anomalous scaling of advected scalar fields

We study a model of fully developed turbulence of a compressible fluid, based on the stochastic Navier-Stokes equation, by means of the field theoretic renormalization group. In this approach, scaling properties are related to the fixed points of the renormalization group equations. Previous analysis of this model near the real-world space dimension 3 identified some scaling regime [Theor. Math. Phys., 110, 3 (1997)]. The aim of the present paper is to explore the existence of additional regimes, that could not be found using the direct perturbative approach of the previous work, and to analyze the crossover between different regimes. It seems possible to determine them near the special value of space dimension $4$ in the framework of double $y$ and $\varepsilon$ expansion, where $y$ is the exponent associated with the random force and $\varepsilon=4-d$ is the deviation from the space dimension $4$. Our calculations show that there exists an additional fixed point that governs scaling behavior. Turbulent advection of a passive scalar (density) field by this velocity ensemble is considered as well. We demonstrate that various correlation functions of the scalar field exhibit anomalous scaling behavior in the inertial-convective range. The corresponding anomalous exponents, identified as scaling dimensions of certain composite fields, can be systematically calculated as a series in $y$ and $\varepsilon$. All calculations are performed in the leading one-loop approximation.

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arXiv2016

Advection of a passive scalar field by turbulent compressible fluid: renormalization group analysis near $d = 4$

The field theoretic renormalization group (RG) and the operator product expansion (OPE) are applied to the model of a density field advected by a random turbulent velocity field. The latter is governed by the stochastic Navier-Stokes equation for a compressible fluid. The model is considered near the special space dimension $d = 4$. It is shown that various correlation functions of the scalar field exhibit anomalous scaling behaviour in the inertial-convective range. The scaling properties in the RG+OPE approach are related to fixed points of the renormalization group equations. In comparison with physically interesting case $d = 3$, at $d = 4$ additional Green function has divergences which affect the existence and stability of fixed points. From calculations it follows that a new regime arises there and then by continuity moves into $d = 3$. The corresponding anomalous exponents are identified with scaling dimensions of certain composite fields and can be systematically calculated as series in $y$ (the exponent, connected with random force) and $ε=4-d$. All calculations are performed in the leading one-loop approximation.

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arXiv2016

Renormalization group analysis of a turbulent compressible fluid near $d = 4$ : Crossover between local and non-local scaling regimes

We study scaling properties of the model of fully developed turbulence for a compressible fluid, based on the stochastic Navier-Stokes equation, by means of the field theoretic renormalization group (RG). The scaling properties in this approach are related to fixed points of the RG equation. Here we study a possible existence of other scaling regimes and an opportunity of a crossover between them. This may take place in some other space dimensions, particularly at $d = 4$. A new regime may there arise and then by continuity moves into $d = 3$. Our calculations have shown that there really exists an additional fixed point, that may govern scaling behaviour.

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arXiv2015

Passive advection of a vector field: Anisotropy, finite correlation time, exact solution and logarithmic corrections to ordinary scaling

In this work we study the generalization of the problem, considered in [{\it Phys. Rev. E} {\bf 91}, 013002 (2015)], to the case of {\it finite} correlation time of the environment (velocity) field. The model describes a vector (e.g., magnetic) field, passively advected by a strongly anisotropic turbulent flow. Inertial-range asymptotic behavior is studied by means of the field theoretic renormalization group and the operator product expansion. The advecting velocity field is Gaussian, with finite correlation time and preassigned pair correlation function. Due to the presence of distinguished direction ${\bf n}$, all the multiloop diagrams in this model are vanish, so that the results obtained are exact. The inertial-range behavior of the model is described by two regimes (the limits of vanishing or infinite correlation time) that correspond to the two nontrivial fixed points of the RG equations. Their stability depends on the relation between the exponents in the energy spectrum ${\cal E} \propto k_{\bot}^{1-ξ}$ and the dispersion law $ω\propto k_{\bot}^{2-η}$. In contrast to the well known isotropic Kraichnan's model, where various correlation functions exhibit anomalous scaling behavior with infinite sets of anomalous exponents, here the corrections to ordinary scaling are polynomials of logarithms of the integral turbulence scale $L$.

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arXiv2015

Logarithmic violation of scaling in strongly anisotropic turbulent transfer of a passive vector field

Inertial-range asymptotic behavior of a vector (e.g., magnetic) field, passively advected by a strongly anisotropic turbulent flow, is studied by means of the field theoretic renormalization group and the operator product expansion. The advecting velocity field is Gaussian, not correlated in time, with the pair correlation function of the form $\propto δ(t-t') / k_{\bot}^{d-1+ξ}$, where $k_{\bot}=|{\bf k}_{\bot}|$ and ${\bf k}_{\bot}$ is the component of the wave vector, perpendicular to the distinguished direction (`direction of the flow') -- the $d$-dimensional generalization of the ensemble introduced by Avellaneda and Majda [{\it Commun. Math. Phys.} {\bf 131}: 381 (1990)]. The stochastic advection-diffusion equation for the transverse (divergence-free) vector field includes, as special cases, the kinematic dynamo model for magnetohydrodynamic turbulence and the linearized Navier--Stokes equation. In contrast to the well known isotropic Kraichnan's model, where various correlation functions exhibit anomalous scaling behavior with infinite sets of anomalous exponents, here the dependence on the integral turbulence scale $L$ has a logarithmic behavior: instead of power-like corrections to ordinary scaling, determined by naive (canonical) dimensions, the anomalies manifest themselves as polynomials of logarithms of $L$. The key point is that the matrices of scaling dimensions of the relevant families of composite operators appear nilpotent and cannot be diagonalized. The detailed proof of this fact is given for correlation functions of arbitrary order.

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arXiv2013

Anomalous scaling and large-scale anisotropy in magnetohydrodynamic turbulence: Two-loop renormalization-group analysis of the Kazantsev--Kraichnan kinematic model

The field theoretic renormalization group and operator product expansion are applied to the Kazantsev--Kraichnan kinematic model for the magnetohydrodynamic turbulence. The anomalous scaling emerges as a consequence of the existence of certain composite fields ("operators") with negative dimensions. The anomalous exponents for the correlation functions of arbitrary order are calculated in the two-loop approximation (second order of the renormalization-group expansion), including the anisotropic sectors. The anomalous scaling and the hierarchy of anisotropic contributions become stronger due to those second-order contributions.

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