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 in the theory of two-dimensional turbulence: Instability of the fixed point with respect to weak anisotropy

A statistical model of fully developed turbulence in two-dimensional space is considered by means of the renormalization group method in the weak anisotropy approximation. It is shown that the corresponding fixed point of the renormalization group equations is not infrared stable. Hence, the weak, anisotropy approximation is not valid for describing two-dimensional turbulence. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 112, No. 3, pp. 417–427, September, 1997.     

Renormalization group method applied to the primitive equations

In this article we study the limit, as the Rossby number ε goes to zero, of the primitive equations of the atmosphere and the ocean. From the mathematical viewpoint we study the averaging of a penalization problem displaying oscillations generated by an antisymmetric operator and by the presence of two time scales.     

Convergence of solutions for a network approximation of the two-dimensional Laplace equation in a domain with a system of absolutely conducting disks

A boundary value problem for the Laplace equation describing the (electric, thermal, etc.) field of a system of ideally conducting disks of radius R is considered. The solution to the problem is analyzed under the condition that the characteristic distance δ between the disks is small. It was previously proved that the original continuous problem can be approximated as δ → 0 by a finite-dimensional network problem in the sense that the effective conductivities (energies) of the continuous problem are close to those of its network model. It is shown that the potentials of the ideally conducting disks determined from the continuous problem and the network model are also close to each other as δ → 0, and the difference between the potentials is O(ε^1/4), where ε = δ/R is the characteristic relative distance between the disks.     

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 the problem of fully developed turbulence of a compressible fluid

A model of fully developed turbulence of a compressible liquid (gas), based on the stochastic Navier-Stokes equation, is considered by means of the renormalization group. It is proved that the model is multiplicatively renormalized in terms of the “velocity-logarithm of density” variables. The scaling dimensions of the fields and parameters are calculated in the one-loop approximation. Dependence of the effective sound velocity and the Mach number on the integral turbulence scale L is studied. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 110, No. 3, pp. 385–398, March, 1997.     

On the Brauer group of a two-dimensional local field

The two-dimensional local field K = F _q((u))((t)), char K = p, and its Brauer group Br(K) are considered. It is proved that, if L = K(x) is the field extension for which we have x ^p ? x = ut ^?p =: h, then the condition that (y, f | h]_K = 0 for any y ε K is equivalent to the condition f ε Im(Nm(L*)).     

Computation of entropy increase for Lorentz gas and hard disks

Entropy functionals are computed for non-stationary distributions of particles of Lorentz gas and hard disks. The distributions consisting of beams of particles are found to have the largest amount of entropy and entropy increase. The computations show exponentially monotonic increase during initial time of rapid approach to equilibrium. The rate of entropy increase is bounded by sums of positive Lyapounov exponents.     

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.     

Two interacting hard disks within a circular cavity: towards a quasi-equilibrium quantal equation of states

Two interacting hard disks confined in a circular cavity are investigated. Each disk shows a free motion except when bouncing elastically with its partner and with the boundary wall. According to the analysis of Lyapunov exponents, this system is classically nonintegrable and almost chaotic because of the (short-range) interaction between the disks. The system can be quantized by incorporating the excluded volume effect for the wave function. Eigenvalues and eigenfunctions are obtained by tuning the relative size between the disks and the billiard. The pressure P is defined as the derivative of each eigenvalue with respect to the cavity volume V. Since the energy spectra of eigenvalues versus the disk size show a multitude of level repulsions, P–V characteristics shows the anomalous pressure fluctuations accompanied by many van der Waals-like peaks in each of excited eigenstates taken as a quasi-equilibrium. For each eigenstate, we calculate the expectation values of the square distance between two disks, and point out their relationship with the pressure fluctuations. Role of Bose and Fermi statistics is also investigated.     

The two-dimensional uncoupled problem of magnetothermoelasticity for a magnetically hard layer

We propose a method for approximate solution of the problem of finding the thermally stressed state of a magnetically hard layer that is subject to the action of an external steady magnetic field. We obtain the general solution of the temperature fields and displacements for the two-dimensional problem. Bibliography: 5 titles.Translated fromMatematicheskie Melody i Fiziko-Mekhanicheskie Polya, Issue 30, 1989, pp. 63–69.     

Diffeomorphism classification of finite group actions on disks

In this paper we discuss the diffeomorphism classification of finite group actions on disks. We answer the question when an action on a space M can be extended to an action on a disk such that the action is free away from M. Let the singular set consist of the points with nontrivial isotropy group. We show (under some dimension assumptions) that disks with diffeomorphic neighborhoods of the singular set can be imbedded into each other. As a consequence we find a classification of group actions on disks in terms of the neighborhood of the singular set and an element in the Whitehead group of G.     

Order of a function on the Bruschlinsky group of a two-dimensional polyhedron

Homotopy classes of mappings of a compact polyhedron X to the circle T form an Abelian group B(X), which is called the Bruschlinsky group and is cananically isomorphic to H ¹ (X; ℤ), Let L be an Abelian group, and let f: B(X) → L be a function. One says that the order of f does not exceed r if for each mapping a: X → T the value f([a]) is ℤ-linearly expressed via the characteristic function I _r (a): (X × T) ^r → ℤ of (Γ _a ) ^r , where Γ _a ⊂ X × T is the graph of a. The (algebraic) degree of f is not greater than r if the finite differences of f of order r + 1 vanish. Conjecturally, the order of f is equal to the algebraic degree of f. The conjecture is proved in the case where dim X ≤ 2. Bibliography: 1 title.     

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 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.