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 d _c = 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.     

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.     

Study of anomalous kinetics of the annihilation reaction <Emphasis Type="Italic">A</Emphasis> + <Emphasis Type="Italic">A</Emphasis> → Ø

Using the perturbative renormalization group, we study the influence of a random velocity field on the kinetics of the single-species annihilation reaction A + A → Ø at and below its critical dimension d_c = 2. We use the second-quantization formalism of Doi to bring the stochastic problem to a field theory form. We investigate the reaction in spaces of dimension d ～ 2 using a two-parameter expansion in ε and Δ, where ε is the deviation from the Kolmogorov scaling parameter and Δ is the deviation from the space dimension d = 2. We evaluate all the necessary quantities, including fixed points with their regions of stability, up to the second order of the perturbation theory.     

Anomalous scaling in statistical models of passively advected vector fields

We use the methods of the renormalization group and the operator product expansion to consider the problem of the stochastic advection of a passive vector field with the most general form of the nonlinear term allowed by the Galilean symmetry. The external velocity field satisfies the Navier-Stokes equation. We show that the correlation functions have anomalous scaling in the inertial range. The corresponding anomalous exponents are determined by the critical dimensions of tensor composite fields (operators) built from only the fields themselves. We calculate the anomalous dimensions in the leading order of the expansion in the exponent in the correlator of the external force in the Navier-Stokes equation (the oneloop approximation of the renormalization group). The anomalous exponents exhibit a hierarchy related to the anisotropy degree: the lower the rank of the tensor operator is, the lower its dimension. The leading asymptotic terms are determined by the scalar operators in both the isotropic and the anisotropic cases, which completely agrees with Kolmogorov’s hypothesis of local isotropy restoration.     

Anomalous scaling in the model of turbulent advection of a vector field

We consider the model of turbulent advection of a passive vector field ϕ by a two-dimensional random velocity field uncorrelated in time and having Gaussian statistics with a powerlike correlator. The renormalization group and operator product expansion methods show that the asymptotic form of the structure functions of the ϕ field in the inertial range is determined by the fluctuations of the energy dissipation rate. The dependence of the asymptotic form on the external turbulence scale is essential and has a powerlike form (anomalous scaling). The corresponding exponents are determined by the spectrum of the anomalous dimension matrices of operator families consisting of gradients of ϕ. We find a basis constructed from powers of the dissipation and enstrophy operators in which these matrices have a triangular form in all orders of the perturbation theory. In the two-loop approximation, we evaluate the anomalous-scaling exponents for the structure functions of an arbitrary order. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 3, pp. 467–487, March, 2006.     

The asymptotic characteristics of the solutions of the diffusion equation with a non-linear sink: A renormalization group approach

A non-linear generalization of the diffusion equation, which describes the mass or heat transfer accompanied with chemical reactions, is used to consider the spreading of an initially localized distribution. The use of a renormalization group method enabled the nature of the solution to be analysed for long times and two characteristics of its asymptotic behaviour to be distinguished. When the dimension of the space is greater than a certain critical value, a state of asymptotic freedom is attained for which the role of non-linearity is small and the evolution of the density distribution is governed by diffusion processes. When the dimension is less than the critical value, the non-linear term remains substantial for long periods of time and a state of incomplete self-similarity of the evolution of the density distribution is established. The exponent of the exponential dependence of the radius of the diffusion spot on time is calculated for this case. The relation between the renormalization group method and perturbation theory and difficulties in substantiating the method when applied to a given problem are discussed.     

Composite operators,operator expansion,and Galilean invariance in the theory of fully developed turbulence. Infrared corrections to Kolmogorov scaling

The quantum-field renormalization group and operator expansion are used to investigate the infrared asymptotic behavior of the velocity correlation function in the theory of fully developed turbulence. The scaling dimensions of all composite operators constructed from the velocity field and its time derivatives are calculated, and their contributions to the operator expansion are determined. It is shown that the asymptotic behavior of the equal-time correlation function is determined by Galilean-invariant composite operators. The corrections to the Kolmogorov spectrum associated with the operators of canonical dimension 6 are found. The consequences of Galilean invariance for the renormalized composite operators are considered.State University, St. Petersburg. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 100, No. 3, pp. 382–401, September, 1994.     

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

Applying the standard field theory renormalization group to the model of landscape erosion introduced by Pastor-Satorras and Rothman yields 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 β-functions). We show that the one-loop counterterm can nevertheless be expressed in terms of a known function V (h) in the original stochastic equation and its derivatives with respect to the height field h. Its Taylor expansion yields the full infinite set of the one-loop renormalization constants, β-functions, and anomalous dimensions. Instead of a set of fixed points, there arises a two-dimensional surface of fixed points that quite probably contains infrared attractive regions. If that is the case, then the model exhibits scaling behavior in the infrared range. The corresponding critical exponents turn out to be nonuniversal because they depend on the coordinates of the fixed point on the surface, but they satisfy certain universal exact relations.     

Epsilon-expansion in the N-component ϕ 4 model

The formalism of projection Hamiltonians is applied to the N-component O(N)-invariant ϕ⁴ model in the Euclidean and p-adic spaces. We use two versions of the ε-expansion (with ε = 4 − d and with ε = α − 3d/2, where α is the renormalization group parameter) and evaluate the critical indices ν and η up to the second order of the perturbation theory. The results for the (4− d)-expansion then coincide with the known results obtained via the quantum-field renormalization-group methods. Our calculations give evidence that in dimension three, both expansions describe the same non-Gaussian fixed point of the renormalization group. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 3, pp. 365–384, March, 2006.     

Effects of turbulent transfer on critical behavior

Using the field theory renormalization group, we study the critical behavior of two systems subjected to turbulent mixing. The first system, described by the equilibrium model A, corresponds to the relaxational dynamics of a nonconserved order parameter. The second system is the strongly nonequilibrium reaction-diffusion system, known as the Gribov process or directed percolation process. The turbulent mixing is modeled by the stochastic Navier-Stokes equation with a random stirring force with the correlator ∞ δ(t − t′)p ^4−d−y, where p is the wave number, d is the space dimension, and y is an arbitrary exponent. We show that the systems exhibit various types of critical behavior depending on the relation between y and d. In addition to known regimes (original systems without mixing and a passively advected scalar field), we establish the existence of new strongly nonequilibrium universality classes and calculate the corresponding critical dimensions to the first order of the double expansion in y and ɛ = 4 − d (one-loop approximation).     

Directed-bond percolation subjected to synthetic compressible velocity fluctuations: Renormalization group approach

We study the directed-bond percolation process (sometimes called the Gribov process because it formally resembles Reggeon field theory) in the presence of irrotational velocity fluctuations with long-range correlations. We use the renormalization group method to investigate the phase transition between an active and an absorbing state. All calculations are in the one-loop approximation. We calculate stable fixed points of the renormalization group and their regions of stability in the form of expansions in three parameters (ε, y, η). We consider different regimes corresponding to the Kraichnan rapid-change model and a frozen velocity field.     

Tree Quantum Field Theory

We propose a new formalism for quantum field theory (QFT) which is neither based on functional integrals, nor on Feynman graphs, but on marked trees. This formalism is constructive, i.e., it computes correlation functions through convergent rather than divergent expansions. It applies both to Fermionic and Bosonic theories. It is compatible with the renormalization group, and it allows to define non-perturbatively differential renormalization group equations. It accommodates any general stable polynomial Lagrangian. It can equally well treat noncommutative models or matrix models such as the Grosse–Wulkenhaar model. Perhaps most importantly it removes the space-time background from its central place in QFT, paving the way for a non-perturbative definition of field theory in non-integer dimension.     

Zone Structure of the Renormalization Group Flow in a Fermionic Hierarchical Model

The Gaussian part of the Hamiltonian of the four-component fermion model on a hierarchical lattice is invariant under the block-spin transformation of the renormalization group with a given degree of normalization (the renormalization group parameter). We describe the renormalization group transformation in the space of coefficients defining the Grassmann-valued density of a free measure as a homogeneous quadratic map. We interpret this space as a two-dimensional projective space and visualize it as a disk. If the renormalization group parameter is greater than the lattice dimension, then the unique attractive fixed point of the renormalization group is given by the density of the Grassmann delta function. This fixed point has two different (left and right) invariant neighborhoods. Based on this, we classify the points of the projective plane according to how they tend to the attracting point (on the left or right) under iterations of the map. We discuss the zone structure of the obtained regions and show that the global flow of the renormalization group is described simply in terms of this zone structure.     

Transfer of a Passive Vector Admixture by a Two-Dimensional Turbulent Flow

We consider a model of a passive vector field transfer by a random two-dimensional transverse velocity field that is uncorrelated in time and has Gaussian spatial statistics given by a powerlike correlator. We use the renormalization group and the operator product expansion techniques to show that the asymptotic approximation of the structure functions of a vector field in the inertial range is determined by the energy dissipation fluctuations. The dependence of the asymptotic approximation on the external scale of turbulence is essential and has a powerlike form (the case of an anomalous scaling). The corresponding exponents are calculated in the one-loop approximation for structure functions of an arbitrary order.     

Temperature Green’s functions in Fermi systems: The superconducting phase transition

We consider the model of an equilibrium Fermi system of arbitrary-spin particles with the density-densitytype interaction. Based on the microscopic Hamiltonian in the formalism of temperature Green’s functions, we find critical modes and construct an effective action describing a neighborhood of the phase transition point. A renormalization group analysis of the obtained model leads to the standard critical behavior indices for spin-1/2 fermions but shows that in the system of higher-spin fermions, a first-order phase transition occurs whose temperature exceeds the standard estimates for the temperature of a second-order phase transition.     

Polymer stars in three dimensions. Three-loop results

We study the scaling properties of self-avoiding polymer stars and networks of arbitrarily given but fixed topologies. We use the massive field theory renormalization group framework to calculate the critical exponents governing the universal properties (star exponents). Calculations are performed directly in three dimensions; renormalization group functions are obtained in the three-loop approximation. Resulting asymptotic series for the star exponents are resummed with the help of the Padé-Borel and conformal mapping transformations.Republished from Teoreticheskaya i Matematicheskaya Fizika, Vol. 108, No. 1, pp. 34–50, October, 1996.     

Bose condensation: The viscosity critical dimension and developed turbulence

We propose a model for studying the mutual influence of critical fluctuations in the vicinity of the critical point of phase transition to a superfluid state and the velocity fluctuations in the developed turbulence regime. We demonstrate the presence of two different regimes: the turbulence regime and the equilibrium regime. We show that the standard critical behavior can break in the turbulence regime. The viscosity becomes an infrared-irrelevant parameter in the equilibrium regime. We justify the assumption that the viscosity critical dimension in this regime is determined by critical indices of the critical behavior statistical model, which are currently known with sufficient accuracy.