Critical behaviour of two systems, subjected to the turbulent mixing, is studied by means of the field theoretic renormalization group. The first system, described by the equilibrium model A, corresponds to relaxational dynamics of a non-conserved order parameter. The second one is the strongly non-equilibrium reaction-diffusion system, known as Gribov process and equivalent to the Reggeon field theory. The turbulent mixing is modelled by the Kazantsev-Kraichnan "rapid-change" ensemble: time-decorrelated Gaussian velocity field with the power-like spectrum k^{-d-ξ}. Effects of compressibility of the fluid are studied. It is shown that, depending on the relation between the exponent ξand the spatial dimension d, the both systems exhibit four different types of critical behaviour, associated with four possible fixed points of the renormalization group equations. The most interesting point corresponds to a new type of critical behaviour, in which the nonlinearity and turbulent mixing are both relevant, and the critical exponents depend on d, ξand the degree of compressibility. For the both models, compressibility enhances the role of the nonlinear terms in the dynamical equations: the region in the d-ξplane, where the new nontrivial regime is stable, is getting much wider as the degree of compressibility increases. In its turn, turbulent transfer becomes more efficient due to combined effects of the mixing and the nonlinear terms.
Dynamic critical behavior in superfluid systems is considered in a presence of external stirring and advecting processes. The latter are generated by means of the Gaussian random velocity ensemble with white-noise character in time variable and self-similar spatial dependence. The main focus of this work is to analyze an effect of compressible modes on the critical behavior. The model is formulated through stochastic Langevin equations, which are then recast into Janssen-De Dominicis response formalism. Employing the field-theoretic perturbative renormalization group method we analyze large-sc…
Universal behavior is a typical emergent feature of critical systems. A paramount model of the non-equilibrium critical behavior is the directed bond percolation process that exhibits an active- to-absorbing state phase transition in the vicinity of a percolation threshold. Fluctuations of the ambient environment might affect or destroy the universality properties completely. In this work we assume that the random environment can be described by means of compressible velocity fluctu- ations. Using field-theoretic models and renormalization group methods we investigate large-scale and long-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 wi…
Motivated by the ubiquity of turbulent flows in realistic conditions, effects of turbulent advection on two models of classical non-linear systems are investigated. In particular, we analyze model A (according to the Hohenberg-Halperin classification [1]) of a non-conserved order parameter and a model of the direct bond percolation process. Having two paradigmatic representatives of distinct stochastic dynamics, our aim is to elucidate to what extent velocity fluctuations affect their scaling behavior. The main emphasis is put on an interplay between anisotropy and compressibility of the veloc…
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 equati…
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 possib…
The field theoretic renormalization group and the operator product expansion are applied to the stochastic model of a passively advected vector field. The advecting velocity field is generated by the stochastic Navier-Stokes equation with compressibility taken into account. The model is considered in the vicinity of space dimension $d=4$ and the perturbation theory is constructed within a double expansion scheme in $y$ and $ε=4-d$, where $y$ describes scaling behaviour of the random force that enters a stochastic equation for the velocity field. We show that the correlation functions of the pa…
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…
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 exis…
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 …
Phoresis, the drift of particles induced by scalar gradients in a flow, can result in an effective compressibility, bringing together or repelling particles from each other. Here, we ask whether this effect can affect the transport of particles in a turbulent flow. To this end, we study how the dispersion of a cloud of phoretic particles is modified when injected in the flow, together with a blob of scalar, whose effect is to transiently bring particles together, or push them away from the center of the blob. The resulting phoretic effect can be quantified by a single dimensionless number. Phe…
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 analyz…
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 relati…
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 Ave…
The analytic method for deriving the equation of state of a cell fluid model in the region above the critical temperature ($T \geqslant T_\text{c}$) is elaborated using the renormalization group transformation in the collective variables set. Mathematical description with allowance for non-Gaussian fluctuations of the order parameter is performed in the vicinity of the critical point on the basis of the $ρ^4$ model. The proposed method of calculation of the grand partition function allows one to obtain the equation for the critical temperature of the fluid model in addition to universal quanti…
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/…
The renormalization group method is applied in order to analyze models E and F of critical dynamics in the presence of velocity fluctuations generated by the stochastic Navier-Stokes equation. Results are given to the one-loop approximation for the anomalous dimension $γ_λ$ and fixed-points' structure. The dynamic exponent $z$ is calculated in the turbulent regime and stability of the fixed points for the standard model E is discussed.
We consider the model of a transverse vector (e.g. magnetic) field with the most general form of the nonlinearity, known as the ${\cal A}$ model, passively advected by a strongly compressible turbulent flow, governed by the randomly stirred Navier-Stokes equation. The full stochastic problem is equivalent to a certain renormalizable field theoretic model with an infrared attractive fixed point. Thus, the scaling behaviour for the large-scale, long-distance behaviour is established. However, the question whether the parameter ${\cal A}$ tends to a certain fixed-point value of the renormalizatio…
