Functional self-similarity and renormalization group symmetry in mathematical physics

Kovalev, V. F.; Ширков, Дмитрий Васильевич

doi:10.1007/bf02557230

Cited by 21 publications

(12 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Both their dimension and the method of construction depend upon a model The use of infinitesimal approach results in constructing the RG-type symmetry with the help of regular methods of group analysis of DEs. Precisely, this regular algorithm naturally includes the RG=FS invariance condition in the general scheme of constructing and application of RGS generators (see also our recent review [49]). Within the infinitesimal approach this condition is formulated in terms of vanishing of canonical RG operator coordinates, which is especially important for Lie-Bäcklund RGS because finite transformations in this case are expressed as formal series.…”

Section: Overviewmentioning

confidence: 99%

The Bogoliubov renormalization group and solution symmetry in mathematical physics

Shirkov¹,

Kovalev²

2001

Physics Reports

View full text Add to dashboard Cite

Evolution of the concept known in the theoretical physics as the Renormalization Group (RG) is presented. The corresponding symmetry, that has been first introduced in QFT in mid-fifties, is a continuous symmetry of a solution with respect to transformation involving parameters (e.g., of boundary condition) specifying some particular solution.After short detour into Wilson's discrete semi-group, we follow the expansion of QFT RG and argue that the underlying transformation, being considered as a reparameterisation one, is closely related to the self-similarity property. It can be treated as its generalization, the Functional Self-similarity (FS).Then, we review the essential progress during the last decade of the FS concept in application to boundary value problem formulated in terms of differential equations. A summary of a regular approach recently devised for discovering the RG = FS symmetries with the help of the modern Lie group analysis and some of its applications are given.As a main physical illustration, we give application of new approach to solution for a problem of self-focusing laser beam in a non-linear medium.

show abstract

Section: Overviewmentioning

confidence: 99%

The Bogoliubov renormalization group and solution symmetry in mathematical physics

Shirkov¹,

Kovalev²

2001

Physics Reports

View full text Add to dashboard Cite

show abstract

“…Equating the right-hand sides of expressions (6) and (7) based on the equality of their left-hand sides, we obtain the functional equation…”

Section: The Rg Invariants Of Bijective Functionsmentioning

confidence: 99%

“…Functional equation (8) expresses the invariance of F under the RG transformations T (x 2 ) [5], [6], which are simultaneous one-parameter transformations of both arguments of F :…”

Section: The Rg Invariants Of Bijective Functionsmentioning

confidence: 99%

Renormalization group approach to function approximation and to improving subsequent approximations

Nikolaev

2010

Theor Math Phys

View full text Add to dashboard Cite

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 oneto-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.

show abstract

“…Here, we would like to outline the general principle behind the method. [20,21]. The RG symmetry is an exact symmetry of the solution and some boundary values.…”

Section: Applications Of Renormalization Group In Mathematical Physicsmentioning

confidence: 99%

Renormalization Group Analysis of Equilibrium and Non-equilibrium Charged Systems

Barkhudarov¹

2014

Springer Theses

View full text Add to dashboard Cite

In this thesis we investigate properties of equilibrium and non-equilibrium systems by means of renormalization group (RG) analysis. In the study of the d−dimensional Coulomb gas we have formulated a continuum model from the underlying hyper-cubic lattice and employed the irreducible differential formulation of the Wilson RG. We have identified a Thouless-Kosterletz transition in d = 2 and found no non-trivial fixed points for d > 2. As an example of a non-equilibrium system, we have investigated properties of quasi-neutral plasmas which are governed by stochastic magnetohydrodynamic (MHD) equations. The present method is based upon the Martin-Siggia-Rose field-theory formulation of stochastic dynamics. We develop a diagrammatic representation for the theory and carry out a momentum-shell RG of Wilson-Kadanoff type. An infinite set of diagrams is identified which are marginal in the RG sense. We have shown, in accordance with previous literature, that the same problem arises for the randomly-forced Navier-Stokes equation. The problem of marginal variables can be suppressed by working near equilibrium, where stochastic forcing represents thermal fluctuations.In a similar manner we have considered regimes when MHD equations are subject either to kinetic or magnetic forcing only. In such models the macroscopic limit can be taken such that all marginal terms are irrelevant and the dynamics is governed by linear equations. Furthermore, non-trivial fixed points are identified in such regimes and limiting values of either kinematic viscosity or magnetic diffusivity are derived. A consistent description of MHD dynamics far from equilibrium is still absent. We highlight some of the aspects of the functional integral formulation with regards to the symmetries of the system and propose possible ways in which the system can be studied non-pertubatively. 1

show abstract

Functional self-similarity and renormalization group symmetry in mathematical physics

Cited by 21 publications

References 18 publications

The Bogoliubov renormalization group and solution symmetry in mathematical physics

The Bogoliubov renormalization group and solution symmetry in mathematical physics

Renormalization group approach to function approximation and to improving subsequent approximations

Renormalization Group Analysis of Equilibrium and Non-equilibrium Charged Systems

Contact Info

Product

Resources

About