The renormalization group

An appropriate model of the Reynolds stress tensor is vital for an accurate prediction of the fluid flow in cyclones, and this also affects the particle flow simulations. This is because the highly rotating fluid flow produces a. strong nonisotropy in the turbulent structure that causes some of the most popular turbulence models, such as the standard k-e turbulence model, to produce inaccurate predictions of the fluid flow. The Reynolds stress models (RSMs) perform much better, but one of the major drawbacks of these methods is their very complex formulation, which often makes it difficult to both implement the method and obtain convergence. The renormalization group (RNG) turbulence model has been employed by some researchers for the fluid flow in cyclones, and some reasonably good predictions have been obtained for the fluid flow. 

D. J. Amit. Field Theory, the Renormalization Group and Critical Phenomena. Singapore World Scientific Publishing, 1984. 

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... 

CoLDENFELD, N., Ixctures on Phase Transitions and the Renormalization Group, Addison Wesley, Boston (1992). 

Our group has made extensive use of the RNG k-e model (Nijemeisland and Dixon, 2004), which is derived from the instantaneous Navier-Stokes equations using the Renormalization Group method (Yakhot and Orszag, 1986) as opposed to the standard k-e model, which is based on Reynolds averaging. The... 

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...

Wilson, K.G. The renormalization group critical phenomena and the kondo problem. Rev. Mod. 

Two methods appear to be very powerful for the study of critical phenomena field theory as a description of many-body systems, and cell methods grouping together sets of neighboring sites and describing them by an effective Hamiltonian. Both methods are based on the old idea that the relevant scale of critical phenomena is much larger than the interatomic distance and this leads to the notion of scale invariance and to the statistical applications of the renormalization group technique.93... 

I am now at the end of my series of flashes on the Solvay Conferences in Physics. I hope that, in spite of its shortness and incompleteness, it may help in stimulating two kinds of considerations. Those of the first kind regard the extraordinary develoment undergone during the last 70 years by our views on the physical world, many parts of which in present days appear to be dominated by a few general concepts, such as those of exact and approximate symmetry, and to be treatable by mathematical procedures such as the application of the renormalization group. The other kind of considerations concerns the role that the Solvay Conferences in Physics have played in the development of physics during the last 70 years, and the unique value they will maintain, even in the future, as sources of information for the historians of science. 

See, for example, P. Pfeuty and G. Tolouse, Introduction to the Renormalization Group and to Critical Phenomena, Wiley, London, 1977. 

The associated crossover will be calked concentration crossover . The neighborhood of the 0-liuiit will be addressed as 0-region1, etc. The renormalization group will be found to suggest the use of some modified variables, which however does not change the essential contents of the limits. 

Finally, for completeness in Appendix A 7.1 we consider the formal relation of the continuous chain model to a field theoretic Hamiltonian, used to describe critical phenomena in ferrornagnets. It was this relation discovered by de Genries [dG72] and extended by Des Cloizeaux [Clo75, which initiated the application of the renormalization group to polymer solutions and led to the embedding into the larger realm of critical phenomena. 

The relation (A 7.5) first was derived by de Gennes, initiating the renormalization group or scaling approach to polymer solutions. Again these expressions need some explanation. Equation (A 7.6) defines G/ (r, r 7 q) as path integral , summing over all continuous paths r(s), 0 < s < Rh r(0) = r r(Rg) = r. It is properly defined as the continuous chain limit of the discrete... 

We discuss here the basic ideas of the renormalization group, using the discrete chain model. This is not the most elegant or powerful approach, and in Part Til of this book we will present a much more efficient scheme. However, the present approach is conceptually the simplest, and it allows us to explain all the relevant features dilatation symmetry and scaling, fixed points and universality, crossover. Furthermore, technical aspects like the e-expansion also come up. We are then prepared to discuss the Qualitative concept of scaling in its general form and to work out some consequences. 

In Chap. 6 we learned that in the excluded volume limit ftc > 0,n —> oo, the cluster expansion breaks down, simply because it orders according to powers of z = j3enef2 —> oo. To proceed, we need a new idea, going beyond perturbation theory. The new concept is known as the Renormalization Group (RG), which postulates, proves, and exploits the fascinating scale invariance property of the theory. 

We can now state clearly what in the context of the renormalization group we mean by the excluded volume limit... 

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