Scaling transformations, renormalization

Oono and Freed (1981b) applied the renormalization procedure of conformational space to describe the dynamics of chains. They performed the scaling transformation of the complete diffusion equation with the free energy of a chain system written in terms of the model of a continuous chain. Besides, the sialic quantities in Equation 13, the friction aHifficieiit of a chain fragment with its size less than a... 

The renormalization group transformation of the Wilson type for the macromolecular conformational space Ks is composed of the scaling transformation Ss and KsMlanoff s transformation /Cs -... 

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. 

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).

A truncation of the expansion (3.5) defines the zero- and first-order regular approximation (ZORA, FORA) (van Lenthe et al. 1993). A particular noteworthy feature of ZORA is that even in the zeroth order there is an efficient relativistic correction for the region close to the nucleus, where the main relativistic effects come from. Excellent agreement of orbital energies and other valence-shell properties with the results from the Dirac equation is obtained in this zero-order approximation, in particular in the scaled ZORA variant (van Lenthe et al. 1994), which takes the renormalization to the transformed large component approximately into account, using... 

Of course, each transformation from N to N + 1, is implemented by proper scale changes and by a re-numbering of the degrees of freedom. In this way, for an infinite system, it is possible to establish a one-to-one correspondence between the degrees of freedom of Nassociated with the same set renormalization transformation can be expressed formally as follows... 

However, the principles and the techniques of renormalization theory are not directly related to the existence of fields. They apply whenever one deals with a critical system, i.e. whenever one has to describe large-scale phenomena which depend only globally on the chemical microstructure. Thus, because an ensemble of long polymers in a solution constitutes a critical system, renormalization principles and renormalization techniques must be directly applicable to their study. Actually, this idea appeared quite naturally. It led to the decimation method which has been described previously and which lacks efficiency. However, the same idea can be applied in a much better way. This direct renormalization method (des Cloizeaux 1980)37,38 consists in adapting to polymers methods which had been successful in field theory.39 In other words, the aim is to bypass the Laplace de Gennes transformation (see Chapter 11). This method applies to semi-dilute solutions as well as to dilute solutions. 

The finite-size scaling approach to localization assumes that there is a unique function describing the length scaling of the renormalization transformation which is universal for systems of the same symmetry and dimension and that depends only on the Thouless number (22). This function is (3(g), where (for L a small multiple of L)... 

Figure I Scaling function for the renormalization transformation. The critical value, which is an unstable fixed point for the transformation, is indicated by gc. d is the number of dimensions. 3, the scaling function, and g, the Thouless number, are defined fully in the text.

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. 

Kenneth Geddes Wilson (born 1936), American theoretical physicist. The authorities of Cornell University worried by Wilson s low number of published papers. Pressed by his supervisors, he finally started to publish, and won in 1982 the Nobel prize for the renormalization theory. It is a theory of the mathematical transformations describing a system viewed at various scales (with variable resolution). The renormalization theory, as applied by Head-Gordon to the hydrocarbon molecule, first... 

The approach of Kiselev, based on the work of Sengers and co-workers and Kiselev and co-workers, " utilizes a renormalized Landau expansion that smoothly transforms the classical Helmholtz energy density into an equation that incorporates the fluctuation-induced singular scaling laws near the critical point, and reduces to the classical expression far from the critical point. The Helmholtz energy density is separated into ideal and residual terms, and the crossover function applied to the critical part of the Helmholtz energy Aa(AT, Av), where Aa(AT, Av) = a(T, v) — a, g(T, v) and the background contribution abg(T, v) is expressed as. 

The approximation we make in the transformation process is to neglect both the off-diagonal terms in the transformed Hamiltonian and the residual coupling in the wave function. At the same time as neglecting 8, we also renormalize rlf. This will introduce a scaling factor, which we will absorb into the definition of and O. With these approximations, the relations between the wave functions become... 

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)... 

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