Renormalization group, operator expansion, and anomalous scaling in a simple model of turbulent diffusion

Using the renormalization group method and the operator expansion in the Obukhov-Kraichnan model that describes the intermixing of a passive scalar admixture by a random Gaussian field of velocities with the correlator 〈v(t,x)v(t′,x)〉−〈v(t,x)v(t′,x′)〉∝δ(t−t′)|x−x′|^ε, we prove that the anomalous scaling in the inertial interval is caused by the presence of “dangerous” composite operators (powers of the local dissipation rate) whose negative critical dimensions determine the anomalous exponents. These exponents are calculated up to the second order of the ε expansion. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 120, No. 2, pp. 309–314, August, 1999.     

Renormalization group method and Julia sets

The renormalization group (RG) method has been used successfully in treating a variety of phase change and critical-point problems (Wilson KG, Kogut J. Phys Rev C 1974;12:75; Wilson KG. Rev Mod Phys 1975;773; Wilson KG. Phys Rev B 1971;3174). A relatively simple system is considered at the smallest scale; the problem is then renormalized in order to utilize the same system at next larger scale. The process is repeated at larger and larger scales. In the following we consider a model for the flow of a fluid through a porous-medium. The RG transformations for the flow of a fluid through a porous-medium in two and three dimensions are derived and generalized to the complex plane, and the types of the corresponding Julia sets are found and generated. Also, the RG transformation for Ising model on a square lattice is derived and the corresponding Julia set is found.     

Boundaries on spacetimes: causality,topology, and group actions

The future causal boundary on a spacetime serves to explicate the causal behavior of the spacetime at future infinity. The purely causal nature of this boundary has a categorically universal nature, the category being that of chronological sets. There is an associated topology with any chronological set, replicating the appropriate topology for a spacetime. Adding the future causal boundary (and using this topology) provides a quasi-compactification. The boundary for a product spacetime can be detailed in terms of the Riemannian factor M.      

Renormalization group in a fermionic hierarchical model

A study is made of a hierarchical model with spin values in a Grassmann algebra defined by a potential of general form. The action of the spin-block renormalization group in the space of Hamiltonians is reduced to a rational mapping of the space of coupling constants into itself. The methods of the theory of bifurcations are used to investigate the nontrivial fixed points of this mapping. A theorem establishing the existence of a thermodynamic limit of the model at these points in a certain neighborhood of a bifurcation value is proved.This work was done with financial support of the Russian Foundation for Fundamental Research (Grant 93-011-16099).State University, Kazan. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 101, No. 2, pp. 282–293, November, 1994.     

Renormalization group approach and short-distance expansion in theory of developed turbulence: Asymptotics of the triplex equal-time correlation function

Asymptotics of the triplex equal-time correlation function for the turbulence developed in incompressible fluids in the region of widely separated wave vector values is investigated using the renormalization group approach and short-distance expansion. The problem of the most essential composite operators contributing to these asymptotics is examined. For this purpose, the critical dimensions of a family of composite quadratic tensor operators in the velocity gradient are found. Considered in the one-loop approximation, the contribution of these operators turns out to be less substantial (although not significantly) than the contribution of the linear term. The derived asymptotics of the triplex correlator coincide in form with that predicted by the EDQNM approximation.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 105, No. 3, pp. 450–461, December, 1995.     

Renormalization group symmetries in problems of nonlinear geometrical optics

We construct renormalization group symmetries in the geometrical optics approximation for the boundary value problem of the system of equations describing the propagation of strong radiation in a nonlinear medium. Using the renormalization group symmetries, new exact and approximate analytic solutions to the equations of nonlinear geometrical optics are obtained. Explicit analytic expressions are presented that characterize the spatial evolution of a laser beam having an arbitrary dependence on intensity at the nonlinear medium boundary. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 111, No. 3, pp. 369–388, June, 1997.     

Renormalization group in a fermionic hierarchical model in projective coordinates

We study the renormalization group action in a fermionic hierarchical model in the space of coefficients determining the Grassmann-valued density of the free measure. This space is interpreted as the two-dimensional projective space. The renormalization group map is a homogeneous quadratic map and has a special geometric property that allows describing invariant sets and the global dynamics in the whole space.     

Renormalization group in turbulence theory: Exactly solvable Heisenberg model

An exactly solvable Heisenberg model describing the spectral balance conditions for the energy of a turbulent liquid is investigated in the renormalization group (RG) framework. The model has RG symmetry with the exact RG functions (the β-function and the anomalous dimension γ) found in two different renormalization schemes. The solution to the RG equations coincides with the known exact solution of the Heisenberg model and is compared with the results from the ε expansion, which is the only tool for describing more complex models of developed turbulence (the formal small parameter ε of the RG expansion is introduced by replacing a δ-function-like pumping function in the random force correlator by a powerlike function). The results, which are valid for asymptotically small ε, can be extrapolated to the actual value ε=2, and the few first terms of the ε expansion already yield a reasonable numerical estimate for the Kolmogorov constant in the turbulence energy spectrum. Translated from Teoreticheskaya i Matematicheskaya Fizika. Vol. 115, No. 2, pp. 245–262 May. 1998.     

Renormalization group and asymptotics of solutions of nonlinear parabolic equations

We present a general method for studying long-time asymptotics of nonlinear parabolic partial differential equations. The method does not rely on a priori estimates such as the maximum principle. It applies to systems of coupled equations, to boundary conditions at infinity creating a front, and to higher (possibly fractional) differential linear terms. We present in detail the analysis for nonlinear diffusion-type equations with initial data falling off at infinity and also for data interpolating between two different stationary solutions at infinity. In an accompanying paper, [5], the method is applied to systems of equations where some variables are “slaved,” such as the complex Ginzburg-Landau equation. © 1994 John Wiley & Sons, Inc.     

Renormalization group analysis for singularities in the wave beam self-focusing problem

A singular solution of the boundary value problem for the system of equations describing wave beam self-focusing is investigated by constructing renormalization group symmetries. New analytic expressions are found that characterize the spatial evolution of a beam with an arbitrary initial profile in a medium with cubic nonlinearity. The behavior of a Gaussian beam is thoroughly analyzed up to the moment the solution singularity is formed, and a hypothesis is proposed for describing the solution structure after the singularity occurs. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 119, No. 3, pp. 405–418, June, 1999.     

Renormalization group approach to function approximation and to improving subsequent approximations

We establish a relation between bijective functions and renormalization group transformations and find their renormalization group invariants. For these functions, taking into account that they are globally one-to-one, we propose several improved approximations (compared with the power series expansion) based on this relation. We propose using the obtained approximations to improve the subsequent approximations of physical quantities obtained, in particular, by one of the main calculation techniques in theoretical physics, i.e., by perturbation theory. We illustrate the effectiveness of the renormalization group approximation with several examples: renormalization group approximations of several analytic functions and calculation of the nonharmonic oscillator ground-state energy. We also generalize our approach to the case of set maps, both continuous and discrete.     

Trees, Renormalization and Differential Equations

The Butcher group and its underlying Hopf algebra of rooted trees were originally formulated to describe Runge–Kutta methods in numerical analysis. In the past few years, these concepts turned out to have far-reaching applications in several areas of mathematics and physics: they were rediscovered in noncommutative geometry, they describe the combinatorics of renormalization in quantum field theory. The concept of Hopf algebra is introduced using a familiar example and the Hopf algebra of rooted trees is defined. Its role in Runge–Kutta methods, renormalization theory and noncommutative geometry is described.