Renormalization Group Procedures

The role of undulation on the equilibrium of lipid bilayers was also examined by Lipowsky and Leibler,18 who used a nonlinear functional renormalization group approach, and by Sornette,14 who employed a linear functional renormalization approach. It was theoretically predicted that a critical unbinding transition (corresponding to a transition from a finite to an infinite swelling) can occur by varying either the temperature or the Hamaker constant. However, the renormalization group procedures do not offer quantitative information about the systems, when they are not in the close vicinity of this critical point. 

J.-P. MalrieuandN. Guihdry, Phys. Rev. B 63,5110,2001. These authors formulate a renormalization-group procedure where the renormalized Hamiltonian is defined as a Bloch effective Hamiltonian. This procedure is based on the real-space renormalization-group (RSRG) method (a) K. G. Wilson, Rev. Mod. Phys. 47, 773, 1975. (b) S.R. White and R.M. Noack, Phys. Rev. Lett. 68, 3487, 1992. 

Edwards continuous chain with its corresponding Hamiltonian IIq is obviously the most suitable, in every respect, model of a polymer chain. However, this model involves plenty of fine details of the conformational structure, which actually have no influence on the experimentally measured quantities, eg. the mean-srjuare end-to-end distance. The theoreticians (Freed, des Cloizeaux, Oono, Ohta, Duplanticr, Schaffer, et el.) have found such renormalization group procedures of the source Hamiltonian Ho to drive it to the fixed point Hamiltonian //, which allow access, by the conventional methods of statistical physics, to characteristic quantities close to their experimental values. 

Ideas which are not far from the renormalization group procedure can be found in Z. Alexandtowicz, 7. Chem. Phys. 49, 1599 (1968). 

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. 

Recently, the key problems associated with the failure of the old RG method have been identified and a different renormalization procedure based on the eigenvalues of the many-body density matrix of proper subsystems has been developed [64, 65]. This method has come to be known as the density matrix renormalization group (DMRG) method and has found dramatic success in dealing with... 

The density matrix renormalization group (DMRG) method is an efficient and accurate Hilbert space truncation procedure (White 1992 1993) that can be used to solve quantum mechanical models on very large systems. It is particularly suited for one-dimensional quantum lattice models, such as the 7r-electron models discussed in this book. This appendix contains a brief review of the DMRG method relevant for these models. A full discussion of the method and its various applications may be found in (Peschel et al. 1999), (Dukelsky and Pittel 2004), or (Schollwock 2005). 

Methods of revealing suid classifying divergences are related to a regularization procedure. The ways of eliminating divergences constitute the context of renormalization methods (eg. Wilson s renormalization group approach). 

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]

The chief feature is that the experimentally measured quantities become actually insensitive to the fine elements of the structure instead, they perceive just the scaled-enlarged pattern of the system s structure. Such a bridge between the theuretico-iiiatlieiiiatieal procedure of scaling the Hamiltonian (the renormalization group transformation) and an experimentally measured quantity offers considerable scope for studies on substances in their critical state. 

However, only in renormalization group methods did the ideas of step-by-step scaling transformations find their rigorous analytical and beautiful realization. One of such procedures was put forward by de (Jennes and described in his book. 

An alternative derivation of Eq. (B-9) is based on the decimation procedure. Rigour-ously, a detailed analysis in terms of renormalization group trajectories (cf. Ref. 22, Chap. 11 and Ref. 37) is required. This type of procedure is the theoretical basis behind the so-called blob model. The excluded volume effects are important at short-range scale, within a blob containing g monomers. At larger scale excluded volume interactions are screened. The mean-field approach will, therefore, be valid if the blob of size and volume is taken as the site. Then, in order to describe the thermodynamics, the following transformations must be carried out in Eq. (B-6)... 

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 ... 

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