Renormalization-group theory

Fisher M 1983 Scaling, universality and renormalization group theory Critical Phenomena (Lecture Notes in Physics vol 186) (Berlin Springer)... 

Freed K F 1987 Renormalization Group Theory of Macromolecules (New York Wiley-Interscience)... 

Turbulence modeling capability (range of models). Eddy viscosity k-1, k-e, and Reynolds stress. k-e and Algebraic stress. Reynolds stress and renormalization group theory (RNG) V. 4.2 k-e. low Reynolds No.. Algebraic stress. Reynolds stress and Reynolds flux. k- Mixing length (user subroutine) and k-e. 

Freed KF (1987) Renormalization group theory of macromolecules. John Wiley, New York des Cloizeaux J, Jonnink G (1990) Polymers in solution Clarendon Oxford... 

In good solvents, the mean force is of the repulsive type when the two polymer segments come to a close distance and the excluded volume is positive this tends to swell the polymer coil which deviates from the ideal chain behavior described previously by Eq. (1). Once the excluded volume effect is introduced into the model of a real polymer chain, an exact calculation becomes impossible and various schemes of simplification have been proposed. The excluded volume effect, first discussed by Kuhn [25], was calculated by Flory [24] and further refined by many different authors over the years [27]. The rigorous treatment, however, was only recently achieved, with the application of renormalization group theory. The renormalization group techniques have been developed to solve many-body problems in physics and chemistry. De Gennes was the first to point out that the same approach could be used to calculate the MW dependence of global properties... 

K. F. Freed, Renormalization Group Theory of Macromolecules, John Wiley Sons, New York, 1987, p. 22. 

Freed KF (1987) Renormalization group theory of macromolecules. Wiley, New York... 

Fig. 14. Dependence of the interpenetration function R on the number of arms in star molecules. The full line represents the result of the renormalization group theory [90], the data points refer to measurements [77]. Reprinted with permission from [77]. Copyright [1983] American Society...

A much better agreement was obtained recently by renormalization group theory (RG), but this theory failed to describe the effect correctly when f>6. This is demonstrated by Fig. 14. 

The region around (S2) 2q s 1 is extremely sensitive to structure and, thus, cannot be described by some laws of universality similar to those of the scaling or renormalization group theory. Moreover, this region contains the most valuable information on any particular system. 

Despite arising from a diverse set of intermolecular or interatomic forces, as seen by our examples in the previous section, these phenomena give rise to a very similar set of characteristics near the phase transition. The observation of this kind of behavior began more than a century ago, but the explanation of why this should be so has occurred only within the past 25 years. The details associated with the explanation are developed in renormalization group theory whose treatment lies outside the scope of this text. We will present only a qualitative survey of the results of the theory. 

The correlation length is also found to scale in a power-law fashion, and it becomes very large at the transition temperature. One of the most significant results of renormalization group theory is to show that the behavior of the correlation length in the critical region is the basis of the power-law singularities observed in the other thermodynamic properties. 

The key aspects of the modern understanding of phase transitions and the development of renormalization group theory can be summarized as follows. First was the observation of power-law behavior and the realization that critical exponents were, to some extent, universal for all kinds of phase transitions. Then it became clear that theories that only treated the average value of the order parameter failed to account for the observed exponents. The recognition that power-law behavior could arise from functions that were homogeneous in the thermodynamic variables and the scale-invariant behavior of such functions... 

A. Carpinteri et al Cohesive crack model description of ductile to brittle size-scale transition dimensional analysis vs. renormalization group theory. Eng. Fract. Mech. 70(14) 1809-1839 (2003)... 

As an extension of this work viscometry was used to study the conformation of A3B stars (A=PI and B=PS) in the common good solvent toluene [65]. In this case the experimental data, together with the ones obtained from the previous work, were used to extract the dimensionless ratio gg. This ratio expresses quantitatively the effects of heterointeractions between unlike segments on the conformational properties of the copolymers. The oG values for the A3B case were higher than for the A2B case it seems that this is due to the increased segment density of A units in the vicinity of B units for A3B. The experimental values compared rather well with the ones obtained from renormalization group theory and Monte Carlo calculations taking into account the uncertainty in the asymmetry correction coefficient used in the calculations (see Table 2). 

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