Dynamical reduction of discrete systems based on the renormalization-group method

Kunihiro, Teiji; Matsukidaira, Junta

doi:10.1103/physreve.57.4817

Cited by 25 publications

(33 citation statements)

References 26 publications

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“…Possible relation between renormalizability and integrability of Hamilton systems was discussed by Yamaguchi and Nambu [39]. The method was proved to be applicable to discrete systems [40], too.…”

Section: Introductionmentioning

confidence: 99%

Renormalization-Group Method for Reduction of Evolution Equations; Invariant Manifolds and Envelopes

Ei¹,

Fujii²,

Kunihiro³

2000

Annals of Physics

145

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The renormalization group (RG) method as a powerful tool for reduction of evolution equations is formulated in terms of the notion of invariant manifolds. We start with derivation of an exact RG equation which is analogous to the Wilsonian RG equations in statistical physics and quantum field theory. It is clarified that the perturbative RG method constructs invariant manifolds successively as the initial value of evolution equations, thereby the meaning to set t 0 = t is naturally understood where t 0 is the arbitrary initial time. We show that the integral constants in the unperturbative solution constitutes natural coordinates of the invariant manifold when the linear operator A in the evolution equation is semi-simple, i.e., diagonalizable; when A is not semi-simple and has a Jordan cell, a slight modification is necessary because the dimension of the invariant manifold is increased by the perturbation. The RG equation determines the slow motion of the would-be integral constants in the unperturbative solution on the invariant manifold. We present the mechanical procedure to construct the perturbative solutions hence the initial values with which the RG equation gives meaningful results. The underlying structure of the reduction by the RG method as formulated in the present work turns out to completely fit to the universal one elucidated by Kuramoto some years ago. We indicate that the reduction procedure of evolution equations has a good correspondence with the renormalization procedure in quantum field theory; the counter part of the universal structure of reduction elucidated by Kuramoto may be the Polchinski's theorem for renormalizable field theories. We apply the method to interface dynamics such as kink-anti-kink and soliton-soliton interactions in the latter of which a linear operator having a Jordan-cell structure appears.

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Section: Introductionmentioning

confidence: 99%

Renormalization-Group Method for Reduction of Evolution Equations; Invariant Manifolds and Envelopes

Ei¹,

Fujii²,

Kunihiro³

2000

Annals of Physics

145

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“…Taking into account the expression 14) which relates the amplitude A to the renormalized amplitudeÃ(n), we obtain the renormalized resonant mapÃ 16) and represents C, D or G.…”

Section: Deutsche Physikalische Gesellschaftmentioning

confidence: 99%

“…The recently developed renormalization group (RG) method has been successfully applied to both continuous dynamical systems [12]- [14] and maps [15,16] that are of general interest in the physics of accelerators and beams. In a recent paper by Goto et al [17], a symplectic map chain was investigated by means of a new form of the regularized RG method previously introduced in [15].…”

Section: Introduction and Basic Equationsmentioning

confidence: 99%

Renormalization group reduction of the Hénon map and application to the transverse betatron motion in cyclic accelerators

Tzenov

Davidson

2003

New J. Phys.

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Abstract. The renormalization group (RG) method is applied to the study of discrete dynamical systems. As a particular example, the Hénon map is considered as being applied to describe the transverse betatron oscillations in a cyclic accelerator or storage ring possessing a FODO-cell structure with a single thin sextupole. A powerful RG method is developed that is valid correct to fourth order in the perturbation amplitude, and a technique for resolving the resonance structure of the Hénon map is also presented. This calculation represents an application of the RG method to the study of discrete dynamical systems in a unified manner capable of reducing the dynamics of the system both far from and close to resonances, thus preserving the symplectic symmetry of the original map. Contents

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“…We derive the Navier-Stokes equation explicitly from the Boltzmann equation for the first time; the microscopic expressions of the transport coefficients are given. (3) We will put an emphasis on the relation of the underlying mathematics of the RG method with the classical theory of envelopes in mathematical analysis [7,8,9,10,13].…”

Section: Introductionmentioning

confidence: 99%

“…The problems may be collectively called the reduction problem of dynamics. The RG method [6,7,8,9,10,11,12] might be a unified method for the reduction of dynamics as well as a powerful resummation method. Figure 1: The geometrical image of the reduction of the dynamics.…”

Section: Introductionmentioning

confidence: 99%

Application of the renormalization-group method to the reduction of transport equations

Kunihiro¹,

Tsumura²

2006

J. Phys. A: Math. Gen.

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We first give a comprehensive review of the renormalization group method for global and asymptotic analysis, putting an emphasis on the relevance to the classical theory of envelopes and on the importance of the existence of invariant manifolds of the dynamics under consideration. We clarify that an essential point of the method is to convert the problem from solving differential equations to obtaining suitable initial (or boundary) conditions: The RG equation determines the slow motion of the would-be integral constants in the unperturbative solution on the invariant manifold. The RG method is applied to derive the Navier-Stokes equation from the Boltzmann equation, as an example of the reduction of dynamics. We work out to obtain the transport coefficients in terms of the one-body distribution function.

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Dynamical reduction of discrete systems based on the renormalization-group method

Cited by 25 publications

References 26 publications

Renormalization-Group Method for Reduction of Evolution Equations; Invariant Manifolds and Envelopes

Renormalization-Group Method for Reduction of Evolution Equations; Invariant Manifolds and Envelopes

Renormalization group reduction of the Hénon map and application to the transverse betatron motion in cyclic accelerators

Application of the renormalization-group method to the reduction of transport equations

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