Critical behavior of percolation process influenced by a random velocity field: One-loop approximation

Using the perturbation renormalization group, we investigate the influence of a random velocity field on the critical behavior of the directed-bond percolation process near its second-order phase transition between the absorbing and active phases. We use the Antonov-Kraichnan model with a finite correlation time to describe the advecting velocity field. To obtain information about the large-scale asymptotic behavior of the model, we use the field theory renormalization group approach. We analyze the model near its critical dimension via 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, and η is the deviation from the parabolic dispersion law for the velocity correlator. We find the fixed points with the corresponding stability regions in the leading order in the perturbation scheme.     

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.     

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.     

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.     

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.     

球面上等收敛算子的有界性

设Rⁿ 是n-维欧氏空间n≥3.用Ω_n表示Rⁿ 上的单位球面,对于函数f∈L（Ω_n）,E_N^δ（f）表示其Fourier-Laplace级数的δ阶Cesaro平均所决定的等收敛算子,其中,λ:=（n-2）/2,δ是熟知的临界指标.对于0<δ≤λ,令p₀:=(2λ)/（λ+δ）,本文主要证明了如下结果:     

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.     

Renormalization group and the <Emphasis Type="Italic">ɛ</Emphasis>-expansion: Representation of the <Emphasis Type="Italic">β</Emphasis>-function and anomalous dimensions by nonsingular integrals

In the framework of the renormalization group and the ɛ-expansion, we propose expressions for the β-function and anomalous dimensions in terms of renormalized one-irreducible functions. These expressions are convenient for numerical calculations. We choose the renormalization scheme in which the quantities calculated using R operations are represented by integrals that do not contain singularities in ɛ. We develop a completely automated calculation system starting from constructing diagrams, determining relevant subgraphs, combinatorial coefficients, etc., up to determining critical exponents. As an example, we calculate the critical exponents of the φ ³ model in the order ɛ ⁴ .     

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.     

Survival of contact processes on the hierarchical group

We consider contact processes on the hierarchical group, where sites infect other sites at a rate depending on their hierarchical distance, and sites become healthy with a constant recovery rate. If the infection rates decay too fast as a function of the hierarchical distance, then we show that the critical recovery rate is zero. On the other hand, we derive sufficient conditions on the speed of decay of the infection rates for the process to exhibit a nontrivial phase transition between extinction and survival. For our sufficient conditions, we use a coupling argument that compares contact processes on the hierarchical group with freedom two with contact processes on a renormalized lattice. An interesting novelty in this renormalization argument is the use of a result due to Rogers and Pitman on Markov functionals.     

Representation of Renormalization Group Functions By Nonsingular Integrals in a Model of the Critical Dynamics of Ferromagnets: The Fourth Order of The ε-Expansion

Using the representation for renormalization group functions in terms of nonsingular integrals, we calculate the dynamical critical exponents in the model of critical dynamics of ferromagnets in the fourth order of the ε-expansion. We calculate the Feynman diagrams using the sector decomposition technique generalized to critical dynamics problems.     

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.     

TheH model of critical dynamics: A proof of the multiplicative renormalizability of the simplifiedH 0 model

For the H₀ model of critical dynamics, which is obtained from the usual H model by omitting the term with the velocity derivative ∂_tv, renormalization multiplicity is proved, and its connection with the statistics is shown. In the proof, a universal general scheme is formulated that can be used to prove similar statements for any model of critical dynamics. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 122, No. 3, pp. 385–399, March, 2000     

On the Bifurcation and Stability of Rigidly Rotating Inviscid Liquid Bridges

Summary. We consider a mathematical model that describes the motion of an ideal fluid of finite volume that forms a bridge between two fixed parallel plates. Most importantly, this model includes capillarity effects at the plates and surface tension at the free surface of the liquid bridge. We point out that the liquid can stick to the plates due to the inner pressure even in the absence of adhesion forces. We use both the Hamiltonian structure and the symmetry group of this model to perform a bifurcation and stability analysis for relative equilibrium solutions. Starting from rigidly rotating, circularly cylindrical fluid bridges, which exist for arbitrary values of the angular velocity and vanishing adhesion forces, we find various symmetry-breaking bifurcations and prove corresponding stability results. Either the angular velocity or the angular momentum can be used as a bifurcation parameter. This analysis reduces to find critical points and corresponding definiteness properties of a potential function involving the respective bifurcation parameter. Received June 21, 1996; revision received October 2, 1997, and accepted for publication October 9, 1997     

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.     

Vapor-liquid phase transition: The Van Der Waals model

We develop the kinetic theory of critical phenomena in the Van der Waals model. Our approach is considerably different from the traditional phenomenological approach based on the scaling invariance hypothesis and the renormalization group method. From the analysis of the kinetic equation, we can calculate the dynamic and fluctuation characteristics and thus explain a number of experimental observations. The dynamic processes are investigated using self-consistent equations for the first and second moments of the distribution function. We use the corresponding Langevin equation to describe the fluctuation processes. The structure of the dissipative terms in the kinetic equation determines the source intensities. Analysis of experimental data for the temperature dependence of the heat capacity and for the molecular scattering spectra confirms the conclusions derived from the kinetic theory. This paper is published for discussion. Editorial Board. Translated from Teoreticheskaya i Matematicheskaya Fizika Vol. 115, No. 3, pp.437–458, June, 1998.