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N. V. Antonov

Publications and source records attributed to N. V. Antonov.

At least 37 records · Page 2Linked to original sources
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.

cond-mat.stat-mech↗
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.

cond-mat.stat-mech↗
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.

cond-mat.stat-mech↗
arXiv2016

Scaling in erosion of landscapes: Renormalization group analysis of a model with turbulent mixing

The model of landscape erosion, introduced in [{\it Phys. Rev. Lett.} {\bf 80}: 4349 (1998); {\it J. Stat. Phys.} {\bf 93}: 477 (1998)] and modified in [{\it Theor. Math. Phys.} - in press; arXiv:1602.00432], is advected by anisotropic velocity field. The field is Gaussian with vanishing correlation time and 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 a certain preferred direction -- the $d$-dimensional generalization of the ensemble introduced by Avellaneda and Majda [{\it Commun. Math. Phys.} {\bf 131}: 381 (1990)]. Analogous to the case without advection, the model is multiplicatively renormalizable and has infinitely many coupling constants. The one-loop counterterm is derived in a closed form in terms of the certain function $V(h)$, entering the original stochastic equation, and its derivatives with respect to the height field $h(t,{\bf x})$. The full infinite set of the one-loop renormalization constants, $β$-functions and anomalous dimensions is obtained from the Taylor expansion of the counter-term. Instead of a two-dimensional surface of fixed points there is two such surfaces; they are likely to contain infrared attractive region(s). If that is the case, the model exhibits scaling behaviour in the infrared range. The corresponding critical exponents are nonuniversal because they depend on the coordinates of the fixed points on the surface; they also satisfy certain universal exact relation.

cond-mat.stat-mech↗
arXiv2016

Gribov process advected by the synthetic compressible velocity ensemble: Renormalization Group Approach

The direct bond percolation process (Gribov process) is studied in the presence of irrotational velocity fluctuations with long-range correlations. The perturbative renormalization group is employed in order to analyze the effects of finite correlation time on the long-time behavior of the phase transition between an active and an ab- sorbing state. The calculation is performed to the one-loop order. Stable fixed points of the renormalization group and their regions of stability are obtained within the three-parameter $(ε,y,η)$- expansion. Different regimes corresponding to the rapid- change limit and frozen velocity field are discussed.

cond-mat.stat-mech↗
arXiv2016

Scaling in erosion of landscapes: Renormalization group analysis of a model with infinitely many couplings

Standard field theoretic renormalization group is applied to the model of landscape erosion introduced by R. Pastor-Satorras and D. H. Rothman [Phys. Rev. Lett. 80: 4349 (1998); J. Stat. Phys. 93: 477 (1998)] yielding unexpected results: the model is multiplicatively renormalizable only if it involves infinitely many coupling constants ( i.e., the corresponding renormalization group equations involve infinitely many beta-functions). Despite this fact, the one-loop counterterm can be derived albeit in a closed form in terms of the certain function $V(h)$, entering the original stochastic equation, and its derivatives with respect to the height field $h$. Its Taylor expansion gives rise to the full infinite set of the one-loop renormalization constants, beta-functions and anomalous dimensions. Instead of a set of fixed points, there is a two-dimensional surface of fixed points that is likely to contain infrared attractive region(s). If that is the case, the model exhibits scaling behaviour in the infrared range. The corresponding critical exponents are nonuniversal through the dependence on the coordinates of the fixed point on the surface, but satisfy certain universal exact relations.

cond-mat.stat-mech↗
arXiv2016

Directed percolation process in the presence of velocity fluctuations: Effect of compressibility and finite correlation time

The direct bond percolation process (Gribov process) is studied in the presence of random velocity fluctuations generated by the Gaussian self-similar ensemble with finite correlation time. We employ the renormalization group in order to analyze a combined effect of the compressibility and finite correlation time on the long-time behavior of the phase transition between an active and an absorbing state. The renormalization procedure is performed to the one-loop order. Stable fixed points of the renormalization group and their regions of stability are calculated in the one-loop approximation within the three-parameter $(\eps,y,η)$-expansion. Different regimes corresponding to the rapid-change limit and frozen velocity field are discussed, and their fixed points' structure is determined in numerical fashion.

cond-mat.stat-mech↗
arXiv2015

Directed Percolation Process Advected by Compressible Velocity Field

The directed percolation process in the vicinity of non-equilibrium phase transition is studied by the means of field theoretic methods. It will be assumed that percolation takes place in a compressible environment, which will be generated by the modified Kraichnan model. We will discuss differ- ences with the incompressible case and describe how the given model can be renormalized. The renormalization constants will be given to the one-loop order.

nlin.CD↗
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 $L$.

cond-mat.stat-mech↗
arXiv2015

Effects of random environment on a self-organized critical system: Renormalization group analysis of a continuous model

We study effects of random fluid motion on a system in a self-organized critical state. The latter is described by the continuous stochastic model, proposed by Hwa and Kardar [{\it Phys. Rev. Lett.} {\bf 62}: 1813 (1989)]. 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 a certain preferred direction -- the $d$-dimensional generalization of the ensemble introduced by Avellaneda and Majda [{\it Commun. Math. Phys.} {\bf 131}: 381 (1990)]. Using the field theoretic renormalization group we show that, depending on the relation between the exponent $ξ$ and the spatial dimension $d$, the system reveals different types of large-scale, long-time scaling behaviour, associated with the three possible fixed points of the renormalization group equations. They correspond to ordinary diffusion, to passively advected scalar field (the nonlinearity of the Hwa--Kardar model is irrelevant) and to the "pure" Hwa--Kardar model (the advection is irrelevant). For the special choice $ξ=2(4-d)/3$ both the nonlinearity and the advection are important. The corresponding critical exponents are found exactly for all these cases.

cond-mat.stat-mech↗
arXiv2015

Anomalous scaling in magnetohydrodynamic turbulence: Effects of anisotropy and compressibility in the kinematic approximation

The field theoretic renormalization group and the operator product expansion are applied to the model of passive vector (magnetic) field advected by a random turbulent velocity field. The latter is governed by the Navier--Stokes equation for compressible fluid, subject to external random force with the covariance $\propto δ(t-t') k^{4-d-y}$, where $d$ is the dimension of space and $y$ is an arbitrary exponent. From physics viewpoints, the model describes magnetohydrodynamic turbulence in the so-called kinematic approximation, where the effects of the magnetic field on the dynamics of the fluid are neglected. The original stochastic problem is reformulated as a multiplicatively renormalizable field theoretic model; the corresponding renormalization group equations possess an infrared attractive fixed point. It is shown that various correlation functions of the magnetic field and its powers demonstrate anomalous scaling behavior in the inertial-convective range already for small values of~$y$. The corresponding anomalous exponents, identified with scaling (critical) dimensions of certain composite fields ("operators" in the quantum-field terminology), can be systematically calculated as series in $y$. The practical calculation is performed in the leading one-loop approximation, including exponents in anisotropic contributions. It should be emphasized that, in contrast to Gaussian ensembles with finite correlation time, the model and the perturbation theory presented here are manifestly Galilean covariant.

cond-mat.stat-mech↗
arXiv2015

Random interface growth in random environment: Renormalization group analysis of a simple model

We study effects of turbulent mixing on the random growth of an interface in the problem of the deposition of a substance on a substrate. The growth is modelled by the well-known Kardar--Parisi--Zhang model. The turbulent advecting velocity field is modelled by the Kraichnan's rapid-change ensemble: Gaussian statistics with the correlation function $\langle vv\rangle \propto δ(t-t') \, k^{-d-ξ}$, where $k$ is the wave number and $0<ξ<2$ is a free parameter. Effects of compressibility of the fluid are studied. Using the field theoretic renormalization group we show that, depending on the relation between the exponent $ξ$ and the spatial dimension $d$, the system reveals different types of large-scale, long-time asymptotic behaviour, associated with four possible fixed points of the renormalization group equations. In addition to known regimes (ordinary diffusion, ordinary growth process, and passively advected scalar field), existence of a new nonequilibrium universality class is established. Practical calculations of the fixed point coordinates, their regions of stability and critical dimensions are calculated to the first order of the double expansion in $ξ$ and $\varepsilon=2-d$ (one-loop approximation). It turns out that for incompressible fluid, the most realistic values $ξ=4/3$ or 2 and $d=1$ or 2 correspond to the case of passive scalar field, when the nonlinearity of the KPZ model is irrelevant and the interface growth is completely determined by the turbulent transfer. If the compressibility becomes strong enough, the crossover in the critical behaviour occurs, and these values of $d$ and $ξ$ fall into the region of stability of the new regime, where the advection and the nonlinearity are both important.

cond-mat.stat-mech↗
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.

cond-mat.stat-mech↗
arXiv2014

Anomalous scaling of passive scalar fields advected by the Navier-Stokes velocity ensemble: Effects of strong compressibility and large-scale anisotropy

The field theoretic renormalization group and the operator product expansion are applied to two models of passive scalar quantities (the density and the tracer fields) advected by a random turbulent velocity field. The latter is governed by the Navier--Stokes equation for compressible fluid, subject to external random force with the covariance $\propto δ(t-t') k^{4-d-y}$, where $d$ is the dimension of space and $y$ is an arbitrary exponent. The original stochastic problems are reformulated as multiplicatively renormalizable field theoretic models; the corresponding renormalization group equations possess infrared attractive fixed points. It is shown that various correlation functions of the scalar field, its powers and gradients, demonstrate anomalous scaling behavior in the inertial-convective range already for small values of $y$. The corresponding anomalous exponents, identified with scaling (critical) dimensions of certain composite fields ("operators" in the quantum-field terminology), can be systematically calculated as series in $y$. The practical calculation is performed in the leading one-loop approximation, including exponents in anisotropic contributions. It should be emphasized that, in contrast to Gaussian ensembles with finite correlation time, the model and the perturbation theory presented here are manifestly Galilean covariant. The validity of the one-loop approximation and comparison with Gaussian models are briefly discussed.

cond-mat.stat-mech↗
arXiv2013

Critical behaviour of the O(n)-$ϕ^{4}$ model with an antisymmetric order parameter

Critical behaviour of the O(n)-symmetric $ϕ^{4}$-model with an antisymmetric tensor order parameter is studied by means of the field-theoretic renormalization group (RG) in the leading order of the $\varepsilon=4-d$-expansion (one-loop approximation). For $n=2$ and 3 the model is equivalent to the scalar and the O(3)-symmetric vector models, for $n\ge4$ it involves two independent interaction terms and two coupling constants. It is shown that for $n>4$ the RG equations have no infrared (IR) attractive fixed points and their solutions (RG flows) leave the stability region of the model. This means that fluctuations of the order parameter change the nature of the phase transition from the second-order type (suggested by the mean-field theory) to the first-order one. For $n=4$, the IR attractive fixed point exists and the IR behaviour is non-universal: if the coupling constants belong to the basin of attraction for the IR point, the phase transition is of the second order and the IR critical scaling regime realizes. The corresponding critical exponents $ν$ and $η$ are presented in the order $\varepsilon$ and $\varepsilon^{2}$, respectively. Otherwise the RG flows pass outside the stability region and the first-order transition takes place.

cond-mat.stat-mech↗
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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