Renormalization group approximation

The renormalization group approximation of conformational space can be performed, in principle, with any desired accuracy, but at present, calculations can be carried out near AD space only. As a result, theoreticians use expansion in = 4 — d. This is obviously a temperature limitation of the method which is of rather general significance (Oono and Freed, 1981a). 

The calculational algorithm of the renormalization group approximation enables [77] to be calculated directly (without preaver2iging), as well as with the preaveraging of Ozeen s tensor. I he ratio of the [r/J s calculated in these two cases shows the effect of preaveraging... 

The possibility to calculate hydrodynamic quantities (eg, [i ]) both with preaveraging Ozeeii s tensor and directly (without any preaveraging) proves to be a serious advantage of the calculational procedure of the renormalization group approximation. This allowed researchers to estimate the error of determination of the hydrodynamic quantities caus[Pg.744]

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. 

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. 

The other difference refers to the way in which the different groups approximate A. Thus, while VCP focus the attention on a renormalization of the approximated 3-RDM which uses the complementary holes matrix, NY again inspire their procedure on the assumption of an analogy with the Dyson equation. Finally, Mazziotti uses a self-consistent algorithm which may be sumarized as The A-RDM is calculated by means of an algorithm which coincides with the one proposed by NY, expressed in a compact form as ... 

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. 

Fig. 9.3. The scaling function R (N,Cp)/R (A, Cj, — 0) (excluded volume limit, monodisperse) hh function of s = CpRg N,Cp = 0). The full line is the renormalization group result. (See Chap. 18.) The broken lijies give approximations as motivated bv the blob model...

---