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

@article{Kunihiro2006ApplicationOT,
  title={Application of the renormalization-group method to the reduction of transport equations},
  author={T. Kunihiro and K. Tsumura},
  journal={Journal of Physics A},
  year={2006},
  volume={39},
  pages={8089-8104}
}
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… Expand

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References

SHOWING 1-10 OF 56 REFERENCES
Renormalization Group Method Applied to Kinetic Equations: Roles of Initial Values and Time
The so-called renormalization group (RG) method is applied to derive kinetic and transport equations from the respective microscopic equations. The derived equations include the Boltzmann equation inExpand
The Renormalization-Group Method Applied to Asymptotic Analysis,of Vector Fields
The renormalization group method of Goldenfeld, Oono and their collaborators is applied to asymptotic analysis of vector fields. The method is formulated on the basis of the theory of envelopes, asExpand
A geometrical formulation of the renormalization group method for global analysis II: Partial differential equations
It is shown that the renormalization group (RG) method for global analysis can be formulated in the context of the classical theory of envelopes: Several examples from partial differential equationsExpand
Renormalization-Group Method for Reduction of Evolution Equations; Invariant Manifolds and Envelopes
Abstract 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 anExpand
Dynamical Reduction of Discrete Systems Based on the Renormalization Group Method
The renormalization group (RG) method is extended for global asymptotic analysis of discrete systems. We show that the RG equation in the discretized form leads to difference equations correspondingExpand
Renormalization group and singular perturbations: Multiple scales, boundary layers, and reductive perturbation theory.
  • Chen, Goldenfeld, Oono
  • Mathematics, Medicine
  • Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics
  • 1996
TLDR
It is shown that the renormalization group equation may be interpreted as an amplitude equation, and from this point of view develop reductive perturbation theory for partial differential equations describing spatially extended systems near bifurcation points, deriving both amplitude equations and the center manifold. Expand
Renormalization group theory for global asymptotic analysis.
We show with several examples that renormalization group (RG) theory can be used to understand singular and reductive perturbation methods in a unified fashion. Amplitude equations describing slowExpand
Renormalization Group and Strong Interactions
The renormalization-group method of Gell-Mann and Low is applied to field theories of strong interactions. It is assumed that renormalization-group equations exist for strong interactions whichExpand
Renormalization group equation for critical phenomena
An exact renormalization equation is derived by making an infinitesimal change in the cutoff in momentum space. From this equation the expansion for critical exponents around dimensionality 4 and theExpand
On linearized hydrodynamic modes in statistical physics
We formulate the linearized generalized Boltzmann equation as an (asymmetric) eigenvalue problem. This problem has five eigenvalues which tend to zero when the uniformity parameter tends to zero: toExpand
...
1
2
3
4
5
...