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|Title:||Renormalization group theory for systems far from equilibrium|
|Author(s):||Chen, Lin Yuan|
|Doctoral Committee Chair(s):||Goldenfeld, Nigel D.|
|Department / Program:||Physics|
|Degree Granting Institution:||University of Illinois at Urbana-Champaign|
|Abstract:||This thesis is largely concerned with the long-time or large-scale asymptotic behavior of a variety of non-equilibrium (physical) systems in the absence of noise, to which the renormalization group theory (RG) and the structural stability hypothesis are systematically applied.
The first type of problems I study are examples of systems possessing physical scale invariance, for which I focus on similarity solutions, where the exponents are anomalous, and traveling-wave problems, where there is an apparent non-uniqueness of solutions of front propagation. For anomalous nonlinear diffusion and propagation of fronts in porous media, I show that the presence of anomalous dimensions far from equilibrium can be explained systematically. The exponents and related scaling laws are calculated analytically by perturbative renormalization group (RG) theory. I study the propagation of superfluid turbulence, and find that preliminary experimental data are in qualitative agreement with the theoretical RG predictions: in particular, a sharp propagating front is observed.
For the velocity selection in pattern formation systems, it is proposed that only structurally stable fronts are reproducibly observable in practice. Combining the structural stability hypothesis with RG techniques enables one to predict the uniquely selected velocity. The results apply to both the linear-marginal-stability and nonlinear-marginal-stability cases under very general conditions. A variational principle is also implemented to identify the transition between different regimes. I also present a numerical implementation of the RG for the problems mentioned above, constructing similarity solutions and traveling waves. I show that the numerical RG method is computationally more efficient than direct numerical integration of equations, and yields more accurate results than previous analytical perturbative RG calculations, especially when (perturbation) parameters are not small.
The second type of problems I investigate are singular perturbation problems, classical or quantum, without apparent physical scale invariance. Conventionally, numerous (ad hoc) singular perturbation methods are made use of to solve them. A systematic renormalization group (RG) theory is developed as a unified framework for global singular perturbation methods. This is the first time that the direct correspondence between singular perturbation and RG is established. This study reveals several very general conclusions: (1) Singular perturbation methods are naturally understood as renormalized perturbation methods in physics, and (2) Amplitude equations are nothing but RG equations, and (3) in several aspects, the RG method is technically superior.
|Rights Information:||Copyright 1994 Chen, Lin Yuan|
|Date Available in IDEALS:||2011-05-07|
|Identifier in Online Catalog:||AAI9512327|