Tomáš Lučivjanský

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 a scaling regime [N. V. Antonov et al., Theor. Math. Phys. 110, 305 (1997)TMPHAH0040-577910.1007/BF02630456]. The aim of the present paper is to explore the existence of additional regimes, which 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 ɛ expansion, where y is the exponent associated with the random force and ɛ=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 ɛ. All calculations are performed in the leading one-loop approximation.

Field-theoretic technique for irreversible reaction processes

The single-species annihilation reaction A + A → O/ is studied in the presence of random advecting field. In order to determine possible infrared behaviour of the system all stable fixed points are presented to two-loop approximation in double (∈, Δ) expansion with the corresponding regions of stability. The main result of this paper is the calculation of all the renormalization constants and the decay exponent to the second-order precision as well as calculation of scaling function the mean particle number to the first order. Effects of random sources and sinks on reaction kinetics in the master-equation description have been investigated in the framework of a field-theoretic model, obtained by the “second quantization” a la Doi of the corresponding master equation. It has been demonstrated that random sources and sinks have a significant effect on the asymptotic behaviour of the model and two universality classes for their description have been identified by the scaling analysis. Results are compared with the Langevin-equation description of the same process.

Two-loop calculation of anomalous kinetics of the reaction A + A → Ø in randomly stirred fluid

The single-species annihilation reaction A + A → Ø is studied in the presence of a random velocity field generated by the stochastic Navier-Stokes equation. The renormalization group is used to analyze the combined influence of the density and velocity fluctuations on the long-time behavior of the system. The direct effect of velocity fluctuations on the reaction constant appears only from the two-loop order, therefore, all stable fixed points of the renormalization group and their regions of stability are calculated in the two-loop approximation in the two-parameter (ε, Δ) expansion. A renormalized integro-differential equation for the number density is put forward which takes into account the effect of density and velocity fluctuations at next-to-leading order. Solution of this equation in perturbation theory is calculated in a homogeneous system.

Superfluid phase transition with activated velocity fluctuations: Renormalization group approach.

A quantum field model that incorporates Bose-condensed systems near their phase transition into a superfluid phase and velocity fluctuations is proposed. The stochastic Navier-Stokes equation is used for a generation of the velocity fluctuations. As such this model generalizes model F of critical dynamics. The field-theoretic action is derived using the Martin-Siggia-Rose formalism and path integral approach. The regime of equilibrium fluctuations is analyzed within the perturbative renormalization group method. The double (ε,δ)-expansion scheme is employed, where ε is a deviation from space dimension 4 and δ describes scaling of velocity fluctuations. The renormalization procedure is performed to the leading order. The main corollary gained from the analysis of the thermal equilibrium regime suggests that one-loop calculations of the presented models are not sufficient to make a definite conclusion about the stability of fixed points. We also show that critical exponents are drastically changed as a result of the turbulent background and critical fluctuations are in fact destroyed by the developed turbulence fluctuations. The scaling exponent of effective viscosity is calculated and agrees with expected value 4/3.

Influence of hydrodynamic fluctuations on the phase transition in the E and F models of critical dynamics

We use the renormalization group method to study the E model of critical dynamics in the presence of velocity fluctuations arising in accordance with the stochastic Navier-Stokes equation. Using the Martin-Siggia-Rose theorem, we obtain a field theory model that allows a perturbative renormalization group analysis. By direct power counting and an analysis of ultraviolet divergences, we show that the model is multiplicatively renormalizable, and we use a two-parameter expansion in ∈ and δ to calculate the renormalization constants. Here, ∈ is the deviation from the critical dimension four, and δ is the deviation from the Kolmogorov regime. We present the results of the one-loop approximation and part of the fixedpoint structure. We briefly discuss the possible effect of velocity fluctuations on the arge-scale behavior of the model.

Numerical Solution of a Nonlinear Integro-Differential Equation

A discretization algorithm for the numerical solution of a nonlinear integrodifferential equation modeling the temporal variation of the mean number density a (t ) in the single-species annihilation reaction A + A → 0 is discussed. The proposed solution for the two-dimensional case (where the integral entering the equation is divergent) uses regularization and then finite differences for the approximation of the differential operator together with a piecewise linear approximation of a (t ) under the integral. The presented numerical results point to basic features of the behavior of the number density function a(t) and suggest further improvement of the proposed algorithm.

Effect of compressibility on the annihilation process

Using the renormalization group in the perturbation theory, we study the influence of a random velocity field on the kinetics of the single-species annihilation reaction at and below its critical dimension dc = 2. The advecting velocity field is modeled by a Gaussian variable self-similar in space with a finite-radius time correlation (the Antonov-Kraichnan model). We take the effect of the compressibility of the velocity field into account and analyze the model near its critical dimension using a three-parameter expansion in ∈, Δ, and η, where ∈ is the deviation from the Kolmogorov scaling, Δ is the deviation from the (critical) space dimension two, and η is the deviation from the parabolic dispersion law. Depending on the values of these exponents and the compressiblity parameter α, the studied model can exhibit various asymptotic (long-time) regimes corresponding to infrared fixed points of the renormalization group. We summarize the possible regimes and calculate the decay rates for the mean particle number in the leading order of the perturbation theory.