Renormalization group

However, as given by group renormalization theory (45), the values of the universal exponents depend on the (thermodynamic) dimensionality of the system. For four dimensions (as required by the phase rule for the existence of tricritical points), the exponents have classical values. This means the values are multiples of 1/2. The dimensions of the volume of tietriangles are (31)... 

On the subject of stars and linear chains, the same authors have employed MC calculations based on Bishop and Clarke s pivot algorithm to study the validity of scaling and group renormalization theories of these interesting molecules. Dimensions and intrinsic viscosities were also calculated. - ... 

Our work addresses these two problems by adopting the solution theory of Hill (10, 11) and by adapting ideas from the Group Renormalization (12, 15) theory of polymer solutions to the prediction of the model parameters from the degree of polymerization of the phase forming polymers. 

In this section we "semi-empirically" adapt some scaling ideas from the Group Renormalization theory (12, 15) of polymer solutions to obtain expressions for the osmotic virial coefficients of Equations 6 and 7 in terms of the degree of polymerization. In the following discussion we will occasionally omit the indices on the osmotic virial coefficients for the sake of simplicity. 

From the Group Renormalization theory of polymer solutions (12) we know that "S" is proportional to "b" which depends on the nature of the polymer, the polymer concentration "c", and the degree of polymerization "N" raised to the power of an exponent "3u" as shown in Equation 12. 

With the use of the calculations performed by the methods of group renormalization of field theory [7] and by the Monte-Carlo method [8-10 it was shown, that the scaling index y of the polymeric star very nontrivi-ally depends on the number of the rays under the 5 increasing the index y firstly slowly is decreased to zero (at s 7), and after that under s>l it s sharply decreased taking the negative values up to J = -29 at 5 = 32 [5]. Such values are badly agreed with the physical interpretation of the sealing index. Probably, this caused by the absence of munerical estimations of z parameter and its possible dependence on s and N. 

It has to be specially noted that the set of S, for s real numbers is an Abelian group while that of >C, for s non-negative numbers is a commutative semigroup. S, and K, commute therefore, the sot of R, for s non-negative numbers is also a commutative semigroup called a renormalization transformation group (renormalization group). ... 

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

Goldenfeld L 1992 Lectures in Phase Transitions and Renormalization Group (New York Addison-Wesley) Goodstein D L 1974 States of Maffer(Englewood Cliffs, NJ Prentice-Hall and Dover)... 

WIson KG 1971 Renormalization group and critical phenomena. I. Renormalization group and the Kadanoff scaling picture Phys. Rev. B 4 3174-83... 

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. 

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. 

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... 

The scaling argument provides only the exponent but not the absolute numerical value for the constant. Therefore, for quantitative results, it should be completed by some more refined technique like the afore-mentioned renormalization group method [49],... 

In order to be useful in practice, the effective transport coefficients have to be determined for a porous medium of given morphology. For this purpose, a broad class of methods is available (for an overview, see [191]). A very straightforward approach is to assume a periodic structure of the porous medium and to compute numerically the flow, concentration or temperature field in a unit cell [117]. Two very general and powerful methods are the effective-medium approximation (EMA) and the position-space renormalization group method. 

As a second method to determine effective transport coefficients in porous media, the position-space renormalization group method will be briefly discussed. 

Figure 2.68 Grid model of a porous medium (left) and renormalization group transformation replacing a cluster of grid cells by a unit cell of larger scale (right).

Although in principle a powerful and elegant method, the position-space renormalization group method yields very complex expressions for the renormalized... 

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

The lattice gas model of Bell et al. [33] neither gave any detailed mechanism of the orientational ordering nor separated the contributions of the headgroup and the acyl chain. Lavis et al. [34] discussed Ref. 33 critically and concluded that the sharp kink point in the isotherm at transition was an artifact of the mean field approximation used. An improved correspondence to experimental data was claimed by the use of the real-space renormalization group method [35]. The same authors returned to the problem [35] and concluded that in addition to the orientation of the molecules, chain melting had to be included in a model which could interpret the phase transitions. 

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

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... 

Here it is our intention to show that for a system constituted by substrate phonons and laterally interacting low-frequency adsorbate vibrations which are harmonically coupled with the substrate, the states can be subclassified into independent groups by die wave vector K referring to the first Brillouin zone of the adsorbate lattice.138 As the phonon state density of a substrate many-fold exceeds the vibrational mode density of an adsorbate, for each adsorption mode there is a quasicontinuous phonon spectrum in every group of states determined by K (see Fig. 4.1). Consequently, we can regard the low-frequency collectivized mode of the adsorbate, t /(K), as a resonance vibration with the renormalized frequency and the reciprocal lifetime 7k-... 

---